TOWARDS AN INTEGRATED APPROACH TO MEASUREMENT, ANALYSIS AND MODELING OF CORTICAL NETWORKS

EDITED BY: A. Ravishankar Rao, Guillermo A. Cecchi and Ehud Kaplan PUBLISHED IN: Frontiers in Neural Circuits

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ISSN 1664-8714 ISBN 978-2-88919-762-0 DOI 10.3389/978-2-88919-762-0

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## **TOWARDS AN INTEGRATED APPROACH TO MEASUREMENT, ANALYSIS AND MODELING OF CORTICAL NETWORKS**

Topic Editors:

**A. Ravishankar Rao,** Fairleigh Dickinson University, USA **Guillermo A. Cecchi,** IBM Thomas J. Watson Research Center, USA **Ehud Kaplan,** Icahn School of Medicine at Mount Sinai, USA

Figure taken from: Riera JJ, Goto T and Kawashima R (2014) A methodology for fast assessments to the electrical activity of barrel fields in vivo: from population inputs to single unit outputs. Front. Neural Circuits 8:4. doi: 10.3389/fncir.2014.00004.

The amount of data being produced by neuroscientists is increasing rapidly, driven by advances in neuroimaging and recording techniques spanning multiple scales of resolution. The availability of such data poses significant challenges for their processing and interpretation.

To gain a deeper understanding of the surrounding issues, the Editors of this e-Book reached out to an interdisciplinary community, and formed the Cor-tical Networks Working Group. The genesis of this e-Book thus began with this Working Group through support from the National Institute for Mathematical and Biological Synthesis in the USA. The Group consisted of scientists from neuroscience, physics, psychology and computer science, and

meetings were held in person. (A detailed list of the group members is presented in the Editorial that follows.)

At the time we started, in 2010, the term "big data" was hardly in existence, though the volume of data we were handling would certainly have qualified. Furthermore, there was significant interest in harnessing the power of supercomputers to perform large scale neuronal simulations, and in creating specialized hardware to mimic neural function.

We realized that the various disciplines represented in our Group could and should work together to accelerate progress in Neuroscience. We searched for common threads that could define the foundation for an integrated approach to solve important problems in the field.

We adopted a network-centric perspective to address these challenges, as the data are derived from structures that are themselves network-like. We proposed three inter-twined threads, consisting of measurement of neural activity, analysis of network structures deduced from this activity, and modeling of network function, leading to theoretical insights. This approach formed the foundation of our initial call for papers.

When we issued the call for papers, we were not sure how many papers would fall into each of these threads. We were pleased that we found significant interest in each thread, and the number of submissions exceeded our expectations. This is an indication that the field of neuroscience is ripe for the type of integration and interchange that we had anticipated.

We first published a special topics issue after we received a sufficient number of submissions. This is now being converted to an e-book to strengthen the coherence of its contributions. One of the strong themes emerging in this e-book is that network-based measures capture better the dynamics of brain processes, and provide features with greater discriminative power than point-based measures. Another theme is the importance of network oscillations and synchrony. Current research is shedding light on the principles that govern the establishment and maintenance of network oscillation states. These principles could explain why there is impaired synchronization between different brain areas in schizophrenics and Parkinson's patients. Such research could ultimately provide the foundation for an understanding of other psychiatric and neurodegenerative conditions.

The chapters in this book cover these three main threads related to cortical networks. Some authors have combined two or more threads within a single chapter. We expect the availability of related work appearing in a single e-book to help our readers see the connection between different research efforts, and spur further insights and research.

**Citation:** Rao, A. R., Cecchi, G. A., Kaplan, E., eds. (2016). Towards an Integrated Approach to Measurement, Analysis and Modeling of Cortical Networks. Lausanne: Frontiers Media. doi: 10.3389/978-2-88919-762-0

# Table of Contents


Jorge J. Riera, Takakuni Goto and Ryuta Kawashima

*160 Modulatory effects of inhibition on persistent activity in a cortical microcircuit model*

Xanthippi Konstantoudaki, Athanasia Papoutsi, Kleanthi Chalkiadaki, Panayiota Poirazi and Kyriaki Sidiropoulou


Carmen C. Canavier, Shuoguo Wang and Lakshmi Chandrasekaran

*238 Sudden synchrony leaps accompanied by frequency multiplications in neuronal activity*

Roni Vardi, Amir Goldental, Shoshana Guberman, Alexander Kalmanovich, Hagar Marmari and Ido Kanter

*247 Transient dynamics and rhythm coordination of inferior olive spatio-temporal patterns*

Roberto Latorre, Carlos Aguirre, Mikhail I. Rabinovich and Pablo Varona

# Editorial: Towards an integrated approach to measurement, analysis and modeling of cortical networks

A. Ravishankar Rao<sup>1</sup> \*, Guillermo A. Cecchi <sup>2</sup> and Ehud Kaplan<sup>3</sup>

*<sup>1</sup> Gildart Haase School of Computer Sciences and Engineering, Fairleigh Dickinson University, Teaneck, NJ, USA, <sup>2</sup> IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, <sup>3</sup> Icahn School of Medicine at Mount Sinai, New York, NY, USA*

Keywords: cortical networks, neural dynamics, graph measures, emergent properties

## 1. INTRODUCTION

Recent technological advances have led to an unprecedented increase in the volume and detail of neuroscientific data, creating significant challenges for their processing and interpretation. We approach this challenge through a network-centric perspective, as we believe that brain function is fundamentally determined by patterns of connectivity between neurons, and the resulting dynamics. This is in contrast to traditional computational neuroscience techniques that focus on models of individual neurons and compartments. Progress, in consequence, is essential on (at least) three major fronts: measurement of neural activity, analysis of network structures deduced from this activity, and modeling of network function, leading to theoretical insights.

The measurement front spans the range from multi-electrode recordings to whole-brain measurements using imaging. Several basic scientific questions arise: What do we need to measure in brain networks? Are there theoretical constraints that would dictate this? How do we design our experiments to generate the most meaningful data? How do we record from awake/behaving animals, or even from multiple animals interacting socially?

The analysis front consists of creating network models from the measurements. Some promising techniques explore the estimation of networks using causality. However, several open questions remain: How do we define the fundamental units within the network? Are these units fixed or do they evolve dynamically? How do we infer connectivity between network elements? How do we identify functional clustering, based on the individual neuronal features? How do we quantify and interpret the activity of multiple neurons via multi-unit recordings, especially when there is no stimulus-response paradigm?

The modeling front can proceed in several directions. From the extracted network we can identify topological regularities, such as motifs and cycles. An interesting research direction is to analyze the relationship between the structure of the network, as represented by its motifs, and its function. A growing body of work is examining the relationship between network structure and phenomena such as stability and synchrony. For instance, neurons in the hippocampus could be modeled as a network wherein hubs consisting of hub neurons promote synchrony, while cycles in this network may cause instability. The theme of synchrony as an important network phenomenon emerges in several articles in this research topic (Canavier et al., 2013; Latorre et al., 2013; Tibau et al., 2013; Vardi et al., 2013; Cavallari et al., 2014; Chary and Kaplan, 2014; Konstantoudaki et al., 2014; Ratnadurai-Giridharan et al., 2014).

We emphasize that the three fronts consisting of measurement, analysis and modeling are interdependent, but must evolve synergistically. The model and theoretical understanding need to be grounded in constraints produced by the measurement process. Insights derived from modeling can be used to drive novel experiments and measurement techniques. An emerging trend deploys

#### Edited by:

*Manuel S. Malmierca, University of Salamanca, Spain*

Reviewed by: *Heather Read, University of Connecticut, USA*

> \*Correspondence: *A. Ravishankar Rao dr\_ravirao@hotmail.com*

Received: *29 May 2015* Accepted: *28 September 2015* Published: *20 October 2015*

#### Citation:

*Rao AR, Cecchi GA and Kaplan E (2015) Editorial: Towards an integrated approach to measurement, analysis and modeling of cortical networks. Front. Neural Circuits 9:61. doi: 10.3389/fncir.2015.00061* active probing and network manipulation through viral vectors and optogenetic methods.

We expect that by aligning existing and future research along these fronts, we will be able to answer questions at the system level. We can view this development as a generalization of the Hubel-Wiesel approach which characterizes feed-forward sensory coding to approaches that characterize dynamic network-level interactions with the input signals. We can derive value from our understanding of network function by applying it to brain-related disorders, such as schizophrenia, drug addiction, or autism. For instance, differences between default mode networks of ASD (autism spectrum disorder) subjects and normals have been reported, among other psychiatric and neurodegenerative conditions. Cortical network properties ultimately determine how different network oscillation states are established and maintained and defining these principles could explain why there is impaired synchronization between different brain areas in schizophrenics and Parkinson's patients. Overall, network-based measures capture better the dynamics of brain processes, and provide features with greater discriminative power than point-based measures.

The articles in this research topic cover these different aspects of cortical networks. To guide the reader, we provide below a brief summary of each article, and relate it to the overall theme of the research topic.

## 2. MECHANISTIC MODELS OF NEURONAL DYNAMICS

Rothganger et al. (2014) present a model design platform, N2A, which has the potential to speed up the process of designing and validating biologically realistic models. By utilizing a hierarchical representation of neural information, N2A allows models from different users to be combined. N2A natively implements standard computations in sensitivity analysis and uncertainty quantification, which allows users to validate models easily. They demonstrate the versatility of N2A through several examples.

Ratnadurai-Giridharan et al. (2014) develop a biophysically relevant network model of the CA1 subfield, and investigate the relationship between network properties and the susceptibility of CA1 to exhibit interictal spikes (IIS). They investigate the conditions under which synchronization of paroxysmal depolarization shift (PDS) events evoked in CA1 pyramidal (Py) cells can trigger an IIS. Like other papers in this research topic, they explore the conditions necessary for and consequences of synchrony, and find that spontaneous IISs closely depend on the degree of the network's intrinsic excitability.

Bhattacharya et al. (2014) present a study of a thalamocortico-thalamic (TCT) implementation on SpiNNaker (Spiking Neural Network architecture), a hardware platform inspired by the processing parallelism, and energy efficiency of biological neural networks. Their system presents similar dynamic and spectral features to EEG in the sleep-wake transition, and could lead to much larger TCT models.

## 3. DESCRIPTIVE AND MODEL-BASED MEASUREMENTS OF EXPERIMENTAL DATA

Dey et al. (2014) use Resting State fMRI functional connectivity and a combination of topological and neuroanatomical features to implement predictive modeling on a dataset of Attention Deficit Hyperactive Disorder (ADHD) and control subjects, and obtain a high predictive accuracy, over 70 for 50% chance. The use of graph-theoretic and anatomical features emphasizes the notion that different brain functions (and dysfunctions) are an emergent property of the interaction between specific brain areas.

Alonso et al. (2014) test a specific hypothesis derived from theorizing the brain as a system determined by emergent properties, namely dynamical criticality. Studying ECoG recordings of anesthesia induction in humans, they show that depth of anesthesia is concomitant with increased dynamical stability, as estimated by the eigenvalues of fitted movingwindow auto-regressive models. They further demonstrate that this stabilization effect cannot be explained by the spectral changes associated with anesthesia, which are typically used to characterize the transition to unconsciousness.

Almeida-Filho et al. (2014) study multi-electrode recordings in the hippocampus and early visual and sensory cortices of rats during and after novel object exploration, as well as during the sleep cycle. They identified cell assemblies as a linear combination of the units' activity, and determined phase relationships between these assemblies. They computed a graph whose nodes correspond to assemblies, and edges correspond to phases. They use graph-theoretic features to perform high accuracy predictive modeling with a simple classifier (Naive Bayes).

Vardi et al. (2013) propose a mechanism that allows timelags among populations of spiking neurons to drop from several tens of milliseconds to nearly zero-lag synchrony. The mechanism allows sudden leaps out of synchrony, hence creating short epochs of synchrony. They obtained results by enforcing conditioned stimulations on neurons embedded within a large cortical network in vitro. Their simulations support the proposed underlying biological mechanisms: the increase of neuronal response latency to ongoing stimulations and temporal or spatial summation required to generate evoked spikes.

Tibau et al. (2013) monitored the development of neuronal cultures, and recorded their activity using calcium fluorescence imaging. They demonstrate that the power spectrum can be used as a signature of the state of the network, for instance, when inhibition is active or silent, as well as a measure of the network's connectivity strength. The power spectrum identifies prominent developmental changes in the network, and reveals the existence of communities of strongly connected, highly active neurons that display synchronous oscillations. Using this approach, one could distinguish healthy from diseased networks, or track the effects of therapeutic interventions.

Riera et al. (2014) describe an "electro-physiological microscope" with high spatial and temporal resolution. It consists of a 3-dimensional array of micro-electrodes, and a novel way of analyzing the current-source density data collected

by the array. Their method can localize single whisker barrels from event-related responses to a single whisker deflection, but can also provide information about the spatiotemporal dynamics of neuronal aggregates in several barrels, with the resolution of single neurons. Their method constitutes a significant advance over previous approaches, and could thus change the way the activity of cortical neurons is analyzed in the future.

## 4. NETWORK FUNCTIONALITY

## 4.1. The Importance of Being Synchronized

For the past several decades, theoreticians and experimentalists alike have focused on neuronal synchrony and on the important roles that it might play in brain function, from "The Binding Problem" in perception (Gray, 1999) to Consciousness (Crick and Koch, 1990; Melloni et al., 2007). For a fuller discussion and additional references, see Singer (2007). Several of the articles in this research topic illuminate the issue of synchrony from both physiological and computational perspectives.

Canavier et al. (2013) address the problem of how neurons can synchronize their responses with minimal time lag. They developed a graphical method for determining the effect of the phase response curve (PRC) shape on synchronization and illustrate it using type 1 PRCs, consisting of advances (delays) in response to excitation (inhibition). They showed that the skewness of the PRC affects synchrony. Their analysis of pairwise synchronization tendencies form a useful framework to understand the synchronization behavior of neurons within larger networks.

Konstantoudaki et al. (2014) explore the role of interneurons in the maintenance of a dynamic balance between excitation and inhibition, since changes in this balance have been identified in several neuropsychiatric diseases, such as schizophrenia. They constructed a pre-frontal-cortex (PFC) microcircuit, consisting of pyramidal neuron models and the three interneuron types described in the literature. Their simulations showed that generic somatic inhibition acts as a pacemaker of persistent activity, and that fast-spiking specific inhibition modulates the amplitude and synchrony of the pacemaker's output.

Nie et al. (2014) make use of Information Geometry (IG), which is based on the expansion of the joint probability distribution of an N-neuron system. They used two measures, the single-IG measure and the pairwise IG-measure to examine the activity of simulated interconnected neurons that exhibit oscillations. They considered two oscillatory mechanisms, externally driven oscillations and internally induced oscillations. For both mechanisms, they showed a linear relationship between the single-IG measure and the external input magnitude and a linear relationship between the pairwise-IG measure and the the sum of connection strengths between two neurons.

Cavallari et al. (2014) investigate the effect of employing current- or conductance-based synapses in models of neural networks, both of which have been widely used. They create comparable networks that use the two types of synapses, and compare their dynamics. They report that these two types of networks, which had comparable first-order statistics, showed profound differences in their second-order statistics of neural interactions, and in the modulation of these properties by external inputs. Thus, the second order statistics of the network dynamics depend strongly on the choice of synaptic model, a fact that modelers of neural networks will find very useful.

Thivierge et al. (2014) investigate synaptic motifs created by a relay network, where two populations of neurons communicating via a third relay population achieve synchronization. By employing models of neuronal dynamics, they demonstrate that the use of relay networks leads to the creation of a global attractor of activity that prevents neurons from being responsive to input stimuli. They overcome this limitation by introducing a selective gain inhibition mechanism which allows neurons to respond effectively to external stimuli. They present results to show that patterns of neural synchronization follow stimulus presentation, and that synchronization disappears after the stimulus is removed.

Chary and Kaplan (2014) investigate the role of synchrony in the functioning of reward circuits in the brain. Their computational study demonstrates that synchrony can have two opposing effects in networks that are sensitive to the correlation between stimulus and reward: weakly correlated inputs amplify short-term recall, but suppress long-term recall. Their main finding is that even weak stimulus-reward correlations can facilitate the short-term repetition of a pattern of neural activity, while blocking the long-term embedding of that pattern.

Latorre et al. (2013) implement a network model of the Inferior Olive (IO) to study its synchronization behavior, using electrically coupled conductance-based neurons. In the presence of stimuli, different rhythms are encoded in the spiking activity of the IO neurons that nevertheless remains constrained to a commensurate value of the subthreshold frequency. Moreover, the stimuli induced spatio-temporal patterns that reverberate for long periods. These results have implications beyond IO studies, and is related to tremor, migraine, and epilepsy where these modeling techniques could have a potentially significant impact.

## 4.2. Computation

Kaplan and Lansner (2014) address the issue of odor perception, and investigate the processing of odors through multiple processing stages within a hierarchical system. They use a largescale network model which spans olfactory receptor neurons (ORNs), three types of cells in the olfactory bulb, and three types of cortical cells in the piriform cortex. A competitive Hebbian– Bayesian learning algorithm is used to adjusting synaptic weights. Their model is able to perform robust concentration-invariant odor recognition.

Eguchi et al. (2014) use a detailed computational model of the early visual system in an attempt to bring our understanding of cortical color processing to a level thought to exist for orientation processing. They use information-theoretic measures, and train their model using natural images, in trying to understand how cells of similar color preference come to cluster together in the cortex. Like several other papers in this research topic they also explore the function of synchrony, and the role it might play in deciding what color is used in the visual stimulus.

## ACKNOWLEDGMENTS

We wish to acknowledge NIMBIOS (National Institute for Mathematical and Biological Synthesis, www.nimbios.org) at the University of Tennessee for providing us support in creating a Cortical Networks Working Group, composed of the following members: John Beggs, Guillermo A. Cecchi, Dmitri Chklovskii,

## REFERENCES


Jack Gallant, Maria N. Geffen, Judith Hirsch, Ehud Kaplan, Marcelo Magnasco, Dario Ringach, Ravi Rao, Sidarta Ribeiro and Youping Xiao (in alphabetical order). The direction and theme of the current research topic was shaped by two intensive workshops and discussions between these team members spread over 2 years. We are also grateful to the staff at Frontiers for their assistance in producing this research topic.


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2015 Rao, Cecchi and Kaplan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Attributed graph distance measure for automatic detection of attention deficit hyperactive disordered subjects

#### *Soumyabrata Dey1 \*, A. Ravishankar Rao2 and Mubarak Shah1*

*<sup>1</sup> Department of Electrical Engineering and Computer Science, Center for Research in Computer Vision, University of Central Florida, Orlando, FL, USA <sup>2</sup> Self, Consultant, Data Science, Yorktown Heights, USA*

#### *Edited by:*

*Ehud Kaplan, Mount Sinai Hospital, USA*

#### *Reviewed by:*

*Nivaldo Vasconcelos, Champalimaud Neuroscience Programme, Portugal Huiguang He, Chinese Academy of Sciences, China*

#### *\*Correspondence:*

*Soumyabrata Dey, Department of Electrical Engineering and Computer Science, Center for Research in Computer Vision, University of Central Florida, 4328 Scorpius St., Suite 245, Orlando, FL 32816-2365, USA e-mail: soumyabrata.dey@ knights.ucf.edu*

Attention Deficit Hyperactive Disorder (ADHD) is getting a lot of attention recently for two reasons. First, it is one of the most commonly found childhood disorders and second, the root cause of the problem is still unknown. Functional Magnetic Resonance Imaging (fMRI) data has become a popular tool for the analysis of ADHD, which is the focus of our current research. In this paper we propose a novel framework for the automatic classification of the ADHD subjects using their resting state fMRI (rs-fMRI) data of the brain. We construct brain functional connectivity networks for all the subjects. The nodes of the network are constructed with clusters of highly active voxels and edges between any pair of nodes represent the correlations between their average fMRI time series. The activity level of the voxels are measured based on the average power of their corresponding fMRI time-series. For each node of the networks, a local descriptor comprising of a set of attributes of the node is computed. Next, the Multi-Dimensional Scaling (MDS) technique is used to project all the subjects from the unknown graph-space to a low dimensional space based on their inter-graph distance measures. Finally, the Support Vector Machine (SVM) classifier is used on the low dimensional projected space for automatic classification of the ADHD subjects. Exhaustive experimental validation of the proposed method is performed using the data set released for the ADHD-200 competition. Our method shows promise as we achieve impressive classification accuracies on the training (70*.*49%) and test data sets (73*.*55%). Our results reveal that the detection rates are higher when classification is performed separately on the male and female groups of subjects.

**Keywords: attention deficit hyperactive disorder, functional magnetic resonance imaging, support vector machine, multidimensional scaling, attributed graph**

## **1. INTRODUCTION**

Attention Deficit Hyperactive Disorder (ADHD) is one of the most commonly found functional disorders affecting children. Around 5–10% of school aged children are diagnosed with ADHD (Biederman, 2005). In spite of all the efforts made in the studies of ADHD, the root cause of this problem is still unknown. No well known biological measure exists to date to detect ADHD. Instead, it is characterized by clinical symptoms such as inattention, impulsivity and hyperactivity all of which are subjective. In the proposed method we try to address the problem of automatic classification of the ADHD subjects from their rs-fMRI data alone. For this purpose we construct the resting state functional connectivity network of the brain and exploit the topological differences of the networks of the ADHD and control subjects for classifications. In the rest of the article, the words network and graph are used interchangeably with similar meaning.

Recently, fMRI has become very popular for brain activity related studies. Researchers use it for identifying the brain regions which are responsible for particular cognitive activities based on the correlation of input stimulus signal and captured brain fMRI signals (task-related fMRI). Also, it is used for better understanding of different brain functional diseases like dementia (Rombouts et al., 2009). Likewise, ADHD is also being studied under the light of structural and functional brain imaging techniques. Structural MRI (sMRI) analysis suggests that there are abnormalities in ADHD brains, specifically in the frontal lobes, basal ganglia, parietal lobe, occipital lobe, and cerebellum (Castellanos et al., 1996; Overmeyer et al., 2001; Sowell et al., 2003; Seidman et al., 2006). In another set of studies ADHD brains were analyzed using task-related fMRI data. Bush et al. (1999) found significant low activity in the anterior cingulate cortex when ADHD subjects were asked to perform the Counting Stroop during fMRI. Durston (2003)showed that the ADHD conditioned children have difficulties performing the go/nogo task and display decreased activity in the frontostriatal regions. Teicher et al. (2000) demonstrated that the boys with ADHD have higher T2 relaxation time in the putamen which is directly connected to a child's capacity to sit still. A third set of works was done using the resting state brain fMRI to locate any abnormalities in the Default Mode Network (DMN). Castellanos et al. (2008) performed Generalized Linear Model based regression analysis on the whole brain with respect to three frontal foci of DMN and found low negative correlated activity in precuneus/anterior cingulate cortex in ADHD subjects. Tian et al. (2006) found functional abnormalities in the dorsal anterior cingulate cortex, (Cao et al., 2006) showed decreased regional homogeneity in the frontal-striatal-cerebellar circuits but increased regional homogeneity in the occipital cortex among boys with ADHD, (Zang et al., 2007) verified decreased Amplitude of Low-Frequency Fluctuation (ALFF) in the right inferior frontal cortex, left sensorimotor cortex, bilateral cerebellum, and the vermis, as well as increased ALFF in the right anterior cingulate cortex, left sensorimotor cortex, and bilateral brainstem.

While studies of group level statistics may indicate the abnormal regions of ADHD patients, their use for automatic diagnosis is still under investigation. There have been relatively few investigations at the individual level of classification of the ADHD subjects. One such study was performed by Zhu et al. (2008) where ADHD subjects were classified based on the regional homogeneity of their fMRI data. In another work, bag-of-words framework was used by Solmaz et al. (2012) for the classification of the ADHD subjects. Recently, there was a global competition (ADHD-200) organized, involving researchers from different scientific disciplines, for automatic diagnosis of ADHD subjects as well as understanding the underlying pathophysiology. For this purpose the organizers released a large data-set containing rsfMRI data, sMRI data and phenotypic information of ADHD and control subjects. Different automatic classification methods were published using this data-set (Bohland et al., 2012; Brown et al., 2012; Chang et al., 2012; Cheng et al., 2012; Colby et al., 2012; Dai et al., 2012; Dey et al., 2012; Eloyan et al., 2012; Olivetti et al., 2012; Sato et al., 2012; Sidhu et al., 2012). Many of these approaches used some combination of rs-fMRI, sMRI and phenotypic data. Cortical thickness, gray matter probability, texture of structural brain images were some of the common sMRI features used for the classification. Regional homogeneity, and Fourier transformation of fMRI signal were some of the features used from functional images. Several studies computed functional networks from fMRI data and used different network statistics as features. Brown et al. (2012) showed that even the use of only phenotypic features can produce high classification accuracy. All of these works achieved classification accuracy higher than the chance factor.

As discussed, researchers have identified considerable differences between the ADHD and control groups while analyzing rs-fMRI data. This motivates us to use the rs-fMRI data of the ADHD-200 competition data set for the validation of our proposed classification algorithm. Use of data from other modalities like structural MRI and phenotypic information might improve the classification accuracies but our aim is to verify the effectiveness of rs-fMRI data only for solving the proposed problem. As shown in **Figure 1**, our method can be subdivided into three main parts. In the first part we construct the resting state brain functional connectivity networks for the subjects under consideration. The networks are modeled as attributed graphs where each node is assigned a signature. Attributed graphs are used previously in different works (Jouili and Tabbone, 2009; Xu et al., 2012). The signature of a node is a set of attributes which characterizes the node. The attribute set includes the degree of the node, the degree of the neighboring nodes, the power of the node, the power of the neighboring nodes and the physical location of the node. The power of a node is calculated by averaging the power of the fMRI time series of all the voxels comprising the node. In the second part we compute distances between all possible pairs of graphs. The distance computation for a pair of graphs is a two step process. In the first step distances for all the node pairs are computed based on their signature values. In the next step, all nodes of one graph are assigned to the nodes of the second graph such that the total matching cost is minimized. The Munkres algorithm is used for the node assignment problem (Munkres, 1957). In the last part the graphs are projected to a space of specified dimensions based on their distance measures. The MDS (Torgerson, 1952) method is used for this purpose. Finally, a Support Vector Machine (SVM) is used for the classification of ADHD subjects in the projected space. The main contribution of our work is to propose a novel automatic classification framework of ADHD subjects based on the topological differences of the functional brain connectivity networks of the ADHD and control groups of subjects. Unlike the other methods, which use functional brain networks for ADHD subject classification, we refrain from using network features. Instead we mapped the networks onto a low dimensional spatial configuration and perform classification on the projected space. We also provided physical interpretations of each of the dimensions of the projected space. We achieve impressive detection accuracies on training (70*.*49%) and test sets (73*.*55%). To the best of our knowledge, our average detection rate on the test sets outperforms the previous best results (69*.*59% by Dey et al., 2012).

The rest of the article is organized as follows. Data descriptions are provided in section 2.1. In section 2.2, we provide brief introduction of MDS. The main method is described in 2.3.1, 2.3.2, and 2.3.3 sections.

## **2. MATERIALS AND METHODS**

## **2.1. DATA**

The data, provided by Neuro Bureau for the ADHD 200 competition, is used for our study. Eight different centers contributed to the compilation of the whole data set, which makes it diverse as well as complex. In total it consists of 776 training and 197 test subjects. Different phenotypic information, such as age, gender, handedness, IQ, is also provided for each subject. The experimental validations of our proposed method are performed on the training and test data sets of 4 of the data centers - Kennedy Krieger Institute (KKI), Neuro Image Sample (NeuroImage), Oregon Health and Science University (OHSU) and Peking University (Peking). Also, based on the information provided with the phenotypic data, we excluded all those subjects from our study which have questionable functional image quality (*QCRest*<sup>1</sup> = 0 of the phenotypic data sheet). Consider **Table 1** for an overview of the data used in our study. Different data centers used different scanners and scanning parameters for capturing data. For example KKI and NeuroIMAGE used Siemens Trio 3 tesla scanner, OHSU used Siemens Magnetom TrioTim syngo MR B17 scanner and Peking used Siemens Magnetom TrioTim syngo MR B15 scanner. Some important scanning parameters used by the data centers are listed in **Table 2**. Also different data acquisition parameters are used by different data centers such as KKI and NeuroIMAGE captured data with subjects' eyes closed, OHSU

**FIGURE 1 | Flowchart of our proposed method. (A)** High power voxels are selected, **(B)** voxels belong to each region of interest of CC200 map are clustered together and represented by their cluster centers, **(C)** edges of the network are formed based on the correlations of average fMRI signals of the clusters, **(D,E)** inter network distances are computed in two steps. First, for a pair of networks a node to node distance matrix is

computed. Next, each node of the network with fewer node count is assigned to a node of the second network using Munkres algorithm such that total matching distance is minimized. **(F)** MDS is used to form a spatial configuration of the subjects on a low dimensional space based on the inter-graph distance measures, **(G)** classification is performed in the projected space.


**Table 1 | Summary of the training and test data-sets from four test centers which are used in our work.**

*The data was released for ADHD-200 global competition.*

and Peking asked their subjects to keep their eyes open. While OHSU showed a fixation cross at the screen, Peking didn't show anything. All research conducted by ADHD-200 data contributing sites were performed with local IRB approval, and contributed in compliance with local IRB protocols. In compliance with the Health Insurance Portability and Accountability Act(HIPAA) privacy rules, all data used for the experiments of this article are fully anonymized. The competition organizers made sure that the 18 patient identifiers as well as face information are removed.

For all our experiments we used the preprocessed rs-fMRI data released for the competition. The preprocessing is performed by the competition organizers using the AFNI Cox (1996) and FSL Jenkinson et al. (2012) tools and computed on Athena computer clusters at the Virginia Tech advance research computing center. All the fMRI scans are slice timing corrected, motion corrected to the first image of the time series, registered on a 4 × 4 × 4 mm voxel resolution Montreal Neurological Institute (MNI) space, filtered using a bandpass filter (0.009 Hz <f <0.08 Hz), and blurred with a 6-mm FWHM Gaussian filter. We used a binary mask, provided with each of the subjects, to find out the voxels which are inside the brain volume. All the fMRI data volumes are of size 49 × 58 × 47 voxels, but the number of samples across time varies among the data capturing centers. For further information about the data and preprocessing steps and how to access the freely

**Table 2 | Table lists the summary of scan parameters for all the data centers.**


available data we refer the interested readers to the following web document (NITRC, 2011).

## **2.2. MULTIDIMENSIONAL SCALING**

We provide a general overview of MDS for the sake of the completeness of the paper. MDS is a set of data analysis techniques that enables one to understand the key dimensions of the objects under investigation. The method and term were first introduce by Torgerson (1952). Given a set of objects and the proximities of each possible pairs of objects, MDS techniques can find a spatial configuration of the objects based on their proximities. Here, proximities suggest the overall dissimilarities or similarities of the objects being considered. Hence, MDS can be viewed as a method to project the objects from a space of unknown dimensions to a space of specified dimensions such a way that the original proximities of the objects are preserved as closely as possible. To state it formally, given *N* numbers of objects and a dissimilarity (or similarity) matrix *DNxN*, MDS projects the objects on a space of given dimensions in such a way that *D* − *Dp* is minimized. *Dp* is the distance matrix in the projected space.

Depending on how a dissimilarity (or similarity) matrix is computed, MDS can be subdivided into direct and indirect methods. While for the direct methods numerical dissimilarity value of each pair of objects can be directly computed, for the indirect methods the dissimilarity values need to be derived from other values like confusion data. MDS can be divided into classical and nonmetric classes depending on how the problem is solved. While the classical methods assume that the dissimilarity matrix contains exact distances of the objects, the nonmetric methods consider only the ordinal information of the object proximities. For more details on the MDS we refer the interested readers to Kruskal and Wish (1978). For our experiments we used a direct classical MDS technique.

## **2.3. METHOD**

The proposed method can be divided into three main parts such as network construction, graph distance computation and ADHD subject classification. The following sections describe each of the parts in details.

## *2.3.1. Network construction*

For all the subjects of the data set the resting state functional connectivity networks are computed. The following steps describe the network construction method and the concept is graphically explained in **Figures 1A–C**.

The first step of the network construction method is the selection of the candidate voxels which constitute the network. We observe that all the brain voxels do not contain valuable information and including irrelevant voxels can degrade the classification performance. This motivates us to select the voxels with high activity level which are more effective in modeling the functional connectivity networks and also in discriminating the ADHD and the control groups of subjects. We substantiate our observation by examining experimental data in Section 3, where we show that the inclusion of all the brain voxels in the construction of the network degrades classification performance. We consider the power of the fMRI time series of a voxel as the measure of its activity. The higher the power of a voxel, the higher is its activity level. For a discrete time series *T* = {*t*1*, t*2*,..., tn*}, the power can be computed as,

$$P(T) = \frac{1}{n} \sum\_{i=1}^{n} t\_i^2 \tag{1}$$

We then normalize the power values of all voxels between [0*,* 1]. The voxels are then ranked based on their power values. Finally, for the network construction we select the voxels ranked with 98 percentile or more.

The second step of the network construction method is to decide how to represent the nodes of the network. One easy solution is to assign every voxel to a node of the network. The problem of doing this is that it makes the size of the network very large, which is inefficient for further computational analysis. Also, the network constructed in this fashion is full of redundant information as the voxels in close spatial proximity have very similar functional activity patterns. For these reasons we use a functional regions of interests (ROIs) map, (CC200) proposed by Craddock et al. (2011), to construct the nodes of the network. The map is generated by parcellating whole brain resting state fMRI data into 190 spatially coherent regions of homogeneous functional connectivity (FC). We cluster all the selected voxels belong to the same ROIs and represent each of the clusters as a node of the network. The issue concerning the best resolution of ROIs which contains maximum information with minimum redundancy for the functional study of the brains is not addressed in this work.

In the third step we construct the edges of the network and compute the weights of the edges. We represent each node by the average fMRI time series of all the voxels comprising the node. Then, a correlation matrix is computed which contains correlation values of the fMRI time series of all possible pairs of the nodes in the network. For two nodes *m* and *n* with fMRI time series *mT* = {*m*1*, m*2*,..., mt*} and *nT* = {*n*1*, n*2*,..., nt*} respectively, the correlation value is computed as:

$$corr(m\_T, n\_T) = \frac{\left(t\sum\_{i=1}^t m\_i n\_i\right) - \left(\sum\_{i=1}^t m\_i\right)\left(\sum\_{i=1}^t n\_i\right)}{\sqrt{\left[t\sum\_{i=1}^t m\_i^2 - \left(\sum\_{i=1}^t m\_i\right)^2\right] \left[t\sum\_{i=1}^t n\_i^2 - \left(\sum\_{i=1}^t n\_i\right)^2\right]}},\ (2)$$

Note that the correlation values have range [−1*,* 1]. We empirically verified that the networks constructed with only positive correlation values generate better classification accuracies than the networks constructed with only negative correlation values or absolute correlation values. Hence, the experimental results reported use the networks with edges constructed with positive correlation values only. Also, we use a correlation threshold *corrTh* to remove all the edges from the network which have correlation values less than the threshold.

In the final step, we represent the network as an attributed graph where each node of the network is represented by a set of attributes. We call it the signature of a node. Given a node *n*, its signature is defined as:

$$Signature(n) = \left< \deg(n), \deg(\text{ngh}(n)), \text{pow}(n), \text{pow}(\text{ngh}(n)), \text{coord}(n) \right>,\tag{3}$$

where the functions, *deg*(*.*), *ngh*(*.*), *pow*(*.*), return sum of weights of all the edges connected, the nodes connected by an edge and the power of the input node respectively. *coord*(*.*) is the mean physical coordinate of all the voxels comprising the node.

## *2.3.2. Graph distance*

Once the functional networks are constructed for all of the subjects in the data set, we compute the distances of all possible pairs of networks as shown in **Figure 1D**. For a pair of networks distance computation is a two step process. In the first step we compute the distances of all the node pairs formed by selecting one node from each of the networks. Given two networks *G*<sup>1</sup> = (*V*1*, E*1) and *G*<sup>2</sup> = (*V*2*, E*2) and two nodes *v*<sup>1</sup> ∈ *V*<sup>1</sup> and *v*<sup>2</sup> ∈ *V*2, the distance between *v*<sup>1</sup> and *v*<sup>2</sup> is computed as the difference of their signatures:

$$dist(\nu\_1, \nu\_2) = \mathbf{W} \cdot \begin{bmatrix} d\_1, d\_2, d\_3, d\_4, d\_5 \end{bmatrix}^T,\tag{4}$$

where **W** = [0*.*2*,* 0*.*1*,* 0*.*2*,* 0*.*1*,* 0*.*4] is the weight vector and *d*1*, d*2*, d*3*, d*4*, d*<sup>5</sup> are the differences of the node degrees, the neighbor node degrees, the node powers, the neighbor node powers, and the physical locations of *v*<sup>1</sup> and *v*2. All the difference values are normalized between [0*,* 1] to enable proper comparison. The values for *d*<sup>1</sup> and *d*<sup>3</sup> are simply calculated by computing degree and power differences of *v*<sup>1</sup> and *v*<sup>2</sup> and dividing them respectively by the maximum degree and power encountered for any of the nodes in the training set. To compute *d*<sup>2</sup> first we sort the neighbor degrees in descending orders. The node with less number of neighbor nodes is zero padded at the end to make the size of the degree arrays same. Finally, we sum up the absolute differences of the array elements and divide the summed up value by (*maximumdegree* ∗ *size*(*degreearray*)). *d*<sup>4</sup> is computed in a similar fashion while power values are used instead of degrees. *d*<sup>5</sup> is calculated as follows:

$$d\_5 = \frac{1}{1 + \Re0e^{(|c\_1 - c\_2|)/4}},\tag{5}$$

where *c*<sup>1</sup> and *c*<sup>2</sup> are the physical coordinates of *v*<sup>1</sup> and *v*<sup>2</sup> respectively. This is a sigmoid curve which restricts the value of *d*<sup>5</sup> to the range [0*,* 1]. The parameters of the equation are intuitively determined in such a manner that the value of *d*<sup>5</sup> is close to zero when |*c*<sup>1</sup> − *c*2| = 0, low for the nodes in spatial locality and steeply increasing for the nodes which are further apart. The components of the weight vector *W* are determined intuitively considering the following criteria. First, we want to make sure that the nodes which are physically far apart should not match and therefore set the highest weight corresponding to the physical distances of the nodes. Next, we want to give the same importance to the degree and power distances of the nodes. Hence, the weights corresponding to the node degrees and power distances are assigned the same value so are the neighbor node degree and power distances. Finally, we assume that the importance of the node feature distances will be higher than the importance of the neighbor nodes feature distances and hence weight for the neighbor node distances are lower than the node distances. In general the distance of a pair of graphs should be calculated in such a way that the nodes from the nearby regions with similar degrees and powers and with similar neighbor nodes' degree and power distributions should match.

In the next step, we use the Munkres assignment algorithm Munkres (1957) to assign all the nodes of one network to the nodes of second network such a way that the total assignment cost is minimized. This assignment cost is considered as distance of the network pair. Note that the numbers of nodes for all the networks are not same. This is because when we select the high power voxels there are some ROIs from which no voxels are selected.

## *2.3.3. Classification*

When the subjects are modeled as graphs, they cannot be directly used for classification but need to be mapped onto a feature space. A common way to deal with this is to compute different network features which can be used for the classification (Zhu et al., 2008; Bohland et al., 2012; Dey et al., 2012). We took a different approach to solve this problem. As shown in **Figures 1E,F**, we use the direct classical MDS technique to project the networks in a space with specified dimensions. The MDS method takes the network distance matrix, computed in the previous part of our method, as input and produces a spatial configuration of the networks in the projected space. The number of dimensions of the projected space can also be specified in the MDS method. We got the best classification performances when we use number of dimensions as 2. All the results of our proposed method are generated on the 2 dimensional projected space.

The classification is performed in the projected space using the SVM Cortes and Vapnik (1995) with a polynomial kernel. We choose to use the SVM classifiers for the following reasons. First, the SVM can classify the data points from two classes, which are not easily separable in the feature space, by using a kernel trick to project the data points into a hyperspace where the separation is easy. Second, the SVM regresses the feature space without over fitting on the data by allowing miss classification with a penalty. Experimental results show that the classifiers perform better when trained separately on the male and female subjects. This indicates that there may be considerable differences in the functional connectivity networks of the male and female subject groups. Our result is in concordance of the work of Bálint et al. (2009) who showed that the male and female ADHD subjects have different levels of functioning.

## *2.3.4. Experimental setup*

The setups for all the different experiments performed are described in this section. Experiment results are listed in section 3.

For all our experiments we used MATLAB (version R2008b) implementations of the MDS and SVM. For the MDS, we used the function name *mdsscale* with the *criterion metricstress* and *MaxIter* = 100000. For the SVM, we used the functions named *svmtrain* (with polynomial kernel) and *svmpredict* to train the classifiers and test the detection accuracies respectively.

For all the training and test sets of all the data centers, three different sets of experiments are performed. While the first set of experiments is performed on all the subjects, the second and third sets of experiments are performed on the male and female groups separately. Please note that the classifiers are trained separately on the training and test sets of each data center. Hence, in total [ (4 *trainingsets* + 4*testsets*) ∗ 3 ] 24 different sets of experiments are performed. For the training sets, detection accuracies are achieved by leave one out cross validation method. For the test sets, the classifiers are trained on the subjects of the corresponding training sets and detections are performed on the test sets. For each of these sets of experiments we construct the networks by varying the *corrTh* from 0.30 to 0.90 with a step size of 0.10. The *corrTh* is explained in the section 2.3.1 while describing the network construction steps.

For the purpose of comparing our results we perform the same classification experiments using some standard graph features computed on the brain functional connectivity networks. The features are computed using the Brain Connectivity Toolbox (BCT) Rubinov and Sporns (2010), which contains a large selection of complex network measures commonly used for characterizing structural and functional brain connectivity data sets. The features we used are the degree, the topological overlap, the clustering coefficient, the local efficiency and the rich club coefficient. Following are the brief descriptions about the network features used:

• Degree of a node is the number of nodes in the network it is connected to by some edges.


Since each of the network features returns a feature vector whose size depends on the node count of the network, we had to make the node counts same for all the subjects to make the feature sizes same. For this reason we construct the networks in a little different way. Instead of using one power threshold value for selecting highly active voxels for the whole brain, we use separate power thresholds for each of the ROIs of CC200 map. For each of the ROIs, we select the voxels ranked 98 percentile or higher based on their power values. The rest of the network construction process is same as before. The experiments are also set up in the similar fashion as described for our proposed method.

To better understand the physical interpretations of each of the dimensions of the MDS projected space, we performed some analysis. First we compute some global feature values for each of the networks of the KKI training set. A brief description of the computed features is as follows:



**Table 3 | Summary of the results: table shows the best detection rates achieved (along with their specificities and sensitivities) on all the training sets using the proposed method.**

*The corrTh values are selected from the training sets where we achieve best detection rates. The rates on the test sets for the corresponding corrTh values are reported. The values under the heading "Male Female Separate" are computed by averaging the accuracies on the male and female groups.*


the network. The correlation value reported with *x* coordinates of female group is achieved when *powTH* = 0*.*85.

For each of the computed global features, two separate feature vectors are formed for the male and female group of subjects.

Please note here each feature vector represents a group of subjects (for e.g., the male and female groups) but not the individual subjects. Then the correlations of the feature vectors are computed with the *x* and *y* coordinates of the 2 dimensional space where networks are projected using the MDS method.

To show the importance of the high power voxel selection step we perform a set of experiments using our method but without the voxel selection step. Finally, we experimentally validate the effectiveness of the node attribute set used in out method. For this purpose, we compute the inter-graph distances using different subsets of the attribute set used. For each of the subsets, inter-graph distances are computed separately followed by the projection of the subjects to a low dimensional space using MDS and classification using SVM. It is not possible for us to compute results for all the possible subsets as there can be 31 different subsets for 5 attributes. Instead we start with one attribute and keep on adding attributes in the subsets. The results show that the classification accuracies steadily increase as we kept on adding attributes in the subset. Finally, we validate on all combinations of 4 attributes to show that even missing one of the attributes of our attribute set decreases the classification accuracy.

## **3. RESULTS**

The detection rates of our method, when classification is performed separately on the male and female subjects, are plotted in **Figure 2**. The plots show how the detection rates vary for the different data centers and with respect to different *corrTh* values. In **Table 3** we reported the best detection rates of our method along with the specificity and sensitivity values for all the training sets. The *corrTh* values corresponding to the best detection rates on the training sets are selected and used to get the detection rates for the test centers. One interesting fact is that in most of the cases we get better classification accuracies when experiments are performed on the male and female subjects separately. We achieve an average detection rate of 64*.*48% on the training data sets and an average detection rate of 62*.*81% on the test data sets when classification is performed on all the subjects and 70*.*49% on the training data sets and an average detection rate of 73*.*55% on the test data sets when classification is performed separately on the male and female subjects.

The detection rates of the classification experiments performed using the standard network features are shown in **Figure 3** along with the results of our method. The results are reported separately for each of the data canters as well as the average detection rates are mentioned. It can be seen in almost all of the cases our method performs better than the network features. Also, in average, none of the features performs better than our method when used separately on the male and female subjects. This justifies the need of a specialized method for the analysis of the brain functional problems like ADHD. Please note that we ignored the classification results if any of the specificity or sensitivity is zero. This implies that either all the subjects are classified as ADHD or control. This is why for some of the network features the detection accuracies are zero in **Figure 3**. **Figure 3** also shows the best detection rates of our method when no power threshold is applied for the voxel selection during the network construction step. The lower detection accuracies of these experiments compared to our results justify the importance of the voxel selection step.

**Figure 4** reports the results when different subsets of node attributes are used for the calculation of inter-graph distances. For each of the subsets the average classification accuracies on all the data centers are plotted in the Figure. The results reported are achieved when classification is performed separately on the male and female subject groups. As it can be seen, the best detection rates are achieved when we use all the attributes in the set. This justifies the importance of using all the attributes in the set for inter-graph distance calculation.

## **4. DISCUSSION**

In this work we propose a novel framework for automatic detection of the ADHD subjects using the rs-fMRI data of brain. For this purpose we construct the functional connectivity network of the brain and represented it as attributed graph. The first step of the network construction method is the efficient selection of the voxels which will be best to capture the functional activities of the brains with the minimum redundancy. We select the highly active voxels for the construction of the networks where voxel activity levels are measured based on the power of their fMRI time series. Often signal to noise ratio of low active voxel time series

group. The spaces are segmented during the SVM training phase.

are very high. Also, these noisy time series can have considerable correlations with each other which lead to the adding of spurious edges or changing the edge weights of the networks. The intuition behind selection of the highly active voxels is to reduce this noise which can affect the correlation weights of the network edges. As shown in the plots of **Figures 3A,B**, the voxel selection process in general helps to improve the classification scores. But, we have not experimentally verified what is the ideal power threshold value for this. We used a functional ROI map (CC200) to construct the nodes by clustering the selected voxels which belong to the same ROIs. The active voxel selection step along with the use of CC200 map helped us to reduce the computational cost of our algorithm by a great deal. Compared to around 28000 voxels per brain volume, the average node count of the constructed networks is around 60.

Next, we model the networks as attributed graphs where each node of the networks has its signature. These signatures of the nodes contain information about the local structures of the networks. Then, at the time of inter-graph distance computation step, the Munkres algorithm is used to match these local descriptors in a globally optimized fashion. To discourage the algorithm from matching two nodes which are far apart in the physical space, we uses the Euclidian distance of their coordinates as a parameter of the matching cost computation.

The inter-graph distance measures allow us to use the MDS technique to map the networks from an unknown space to a 2 dimensional projected space. **Figure 5** shows the spacial configuration of the subjects of the KKI training set when mapped to their projected space. As it can be seen, the ADHD subjects can be better segmented when the male and female groups are plotted separately compared to when all the subjects are plotted together. This fact is reflected in the experimental validations where we consistently get better results when classification is done separately on the male and female groups.

We perform an analysis to understand the physical interpretation of the different dimensions of the MDS projected space. For this purpose we computed the correlations of the different global features of the networks with their coordinates in the projected space. The correlation values are reported in **Table 4**. As it can be seen, the *x* coordinates of the projected spaces of the male and female groups are highly correlated with the density and rich club coefficient features and moderately correlated with the global efficiency. It should be noted that these three features capture different aspect of network edge structures. The last

**Table 4 | Correlations of the global features of the networks with the** *x* **and** *y* **dimensions of the projected spaces of the male and female groups.**


feature shows some correlation with the *y* coordinate of female group.

To justify the importance of a specialized method for analysis of the ADHD, we compared our results with some of the standard brain connectivity measures heavily used for functional analysis of the brain. As shown in **Figure 3** our method out performs the standard network features by a large margin. Only the topological overlap feature performs similar to our method on the training data sets.

**Figure 2** shows how detection rates vary with different correlation thresholds used for the network computation. It can be seen that the peaks of the detection rates are not same for the different data centers. There are two main potential reasons for this variation. First, there are variations in experimental protocols followed by the different data centers. Also, to capture the data different data centers used different scanner models and scanning parameters. Second, the subjects, participated in the different centers, have different age distributions. Mehnert et al. (2013) found changes of functional connectivity measures with age in human brain. The variation of detection rate patterns across the centers indicates that there is a need to follow a more standardize experimental procedure for the future studies.

To conclude, we develop a novel classification framework which is modeled in a computationally efficient fashion as we are able to drastically reduce the functional connectivity networks sizes by efficiently selecting voxels and clustering them. Also, our approach is able to produces impressive classification accuracies (70*.*49% on training data sets and 73*.*55% on test data sets) especially on the test sets where we get the better detection accuracies than any of the previously reported results (69*.*59% by Dey et al., 2012 was the previous best). For this purpose we construct the functional connectivity networks of the brains and use their internetwork distance measures to project them onto a 2 dimensional space. We provide physical interpretations of the dimensions of the projected space in our analysis. Also, we show the superior performance of our method over the standard network measures.

### **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 15 November 2013; accepted: 27 May 2014; published online: 16 June 2014. Citation: Dey S, Rao AR and Shah M (2014) Attributed graph distance measure for automatic detection of attention deficit hyperactive disordered subjects. Front. Neural Circuits 8:64. doi: 10.3389/fncir.2014.00064*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Dey, Rao and Shah. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## Engineering a thalamo-cortico-thalamic circuit on SpiNNaker: a preliminary study toward modeling sleep and wakefulness

#### *Basabdatta S. Bhattacharya1 \*, Cameron Patterson2, Francesco Galluppi 2, Simon J. Durrant <sup>3</sup> and Steve Furber <sup>2</sup>*

*<sup>1</sup> School of Engineering, Engineering Hub, University of Lincoln, Lincoln, Lincolnshire, UK*

*<sup>2</sup> School of Computer Science, APT Group, University of Manchester, Manchester, Lancashire, UK*

*<sup>3</sup> School of Psychology, Lincoln Sleep and Cognition Laboratory, University of Lincoln, Lincoln, Lincolnshire, UK*

#### *Edited by:*

*A. Ravishankar Rao, IBM Research, USA*

#### *Reviewed by:*

*Edward Lee Bartlett, Purdue University, USA Anastasia A. Ford, North Florida/South Georgia VA Medical Center, USA*

#### *\*Correspondence:*

*Basabdatta S. Bhattacharya, School of Engineering, Engineering Hub, University of Lincoln, Brayford Pool, Lincoln LN6 7TS, UK e-mail: basab@ieee.org*

We present a preliminary study of a thalamo-cortico-thalamic (TCT) implementation on SpiNNaker (Spiking Neural Network architecture), a brain inspired hardware platform designed to incorporate the inherent biological properties of parallelism, fault tolerance and energy efficiency. These attributes make SpiNNaker an ideal platform for simulating biologically plausible computational models. Our focus in this work is to design a TCT framework that can be simulated on SpiNNaker to mimic dynamical behavior similar to Electroencephalogram (EEG) time and power-spectra signatures in sleep-wake transition. The scale of the model is minimized for simplicity in this proof-of-concept study; thus the total number of spiking neurons is ≈1000 and represents a "minicolumn" of the thalamocortical tissue. All data on model structure, synaptic layout and parameters is inspired from previous studies and abstracted at a level that is appropriate to the aims of the current study as well as computationally suitable for model simulation on a small 4-chip SpiNNaker system. The initial results from selective deletion of synaptic connectivity parameters in the model show similarity with EEG power spectra characteristics of sleep and wakefulness. These observations provide a positive perspective and a basis for future implementation of a very large scale biologically plausible model of thalamo-cortico-thalamic interactivity—the essential brain circuit that regulates the biological sleep-wake cycle and associated EEG rhythms.

**Keywords: SpiNNaker, thalamo-cortico-thalamic circuit, computational model, sleep, Izhikevich model, synaptic connectivity, PyNN**

## **1. INTRODUCTION**

Computational models are being adopted at an increasing rate as a tool to investigate the cellular mechanisms of brain rhythms in both normal and pathological conditions (Aradi and Érdi, 2006; Breakspear et al., 2010; Terry et al., 2011). While computational resource is an obvious constraint in such endeavors, two further significant obstacles in mimicking the biology are parallelizing neuronal activity, and "de-syncing" the population activity from the master-clock of the computer. Our longer-term interest is in mimicking electroencephalogram (EEG) signatures of the sleep-wake cycle, by simulating biologically plausible computational models using biologically plausible computational techniques. In recent years the University of Manchester has been developing SpiNNaker (Spiking Neural Network architecture), a bespoke massively parallel machine to mimic the inherent parallelism of neuronal activity in real time (Furber et al., 2013). The brain-inspired parallel and asynchronous architecture of SpiNNaker permits biologically plausible computation of brain models—a feature that would otherwise rely on heavyweight software and its compilation on conventional Von-Neumann architectures, and yet achieve minimal parallelism. The study presented here is an initial attempt to design and implement a thalamo-cortico-thalamic (TCT) circuitry on the intrinsically parallel SpiNNaker, which can then be scaled up to mimic biologically plausible EEG signatures of the sleep-wake cycle. The purpose of this work is to demonstrate, as a proof of concept, that such a model can be implemented on SpiNNaker, and to investigate the benefits and drawbacks of this approach. It is not our intention here to produce a model which fully and correctly replicates all brain rhythms measured by EEG in regard to the TCT circuitry; capturing the complex dynamics involved in that system is beyond the scope of the current work.

Neuronal dynamics recorded in EEG, often termed brain rhythms (Buzsáki, 2006), are an inexpensive and popular means of correlating brain activity with its various functional states (Wright and Liley, 1996; Nunez, 2000). The feed-forward and feed-back circuitry between the thalamus and the cortex has long since been known to play a key role in modulating brain rhythms associated with the various sleep stages as well as the sleep-wake transition (Steriade et al., 1993; Steriade, 2003, 2005; Crunelli et al., 2011). Computational models of the TCT brain circuit have therefore been the basis for studying neuronal mechanisms in sleep (Lumer et al., 1997a; Hill and Tononi, 2005; Traub et al., 2005; Bojak et al., 2011; Olbrich et al., 2011; Robinson et al., 2011) as well as in conditions where the EEG is qualitatively similar to certain sleep stages such as epilepsy (Breakspear et al., 2006) and under anaesthesia (Hutt and Longtin, 2010). While all such models refer to a similar holistic structure of the thalamocortical circuit, the models' internal structure, simulation platforms and parameterizations are significantly diverse. Thus, a fundamental aspect in computational modeling of the brain is the level of abstraction; the level of biological detail incorporated in a model needs to be appropriate to the problem at hand. For example, Olbrich et al. (2011) has attempted a multi-scale (time) model architecture in sleep, while (Bojak et al., 2011) has stressed on multi-modal models. On the other hand, (Hill and Tononi, 2005) have based their model on that of Lumer et al. (1997a,b) and have looked into a multi-columnar model of the thalamocortical circuit to mimic brain rhythms of sleep and wakefulness as well as to understand memory consolidation during sleep (Nere et al., 2013).

Another key aspect is the source of experimental data for both model structure and parameterizations. Comprehensive data on synaptic connectivity in the mammalian visual cortex is available in the works of Binzegger et al. (2004); Douglas and Martin (2004) and Neymotin et al. (2011) with some estimation for parameters which were not available from physiological studies. Further, extensive physiological data on rodent and other mammalian lateral geniculate nucleus (LGN: the thalamic nucleus in the visual pathway) is available in Horn et al.(2000); Sherman and Guillery (2001); and Jones (2007). Based on these thalamic and cortical physiological datasets as well as DTI (Diffusion Tensor Imaging) data obtained from two human samples, Izhikevich and Edelmann (2008a) have presented a comprehensive TCT circuit using minimal parameter spiking neural models (Izhikevich, 2003) to mimic spiking population behavior. The SpiNNakerbased TCT model presented here is at the level of abstraction of the model in Izhikevich (2003), and has two modules viz. a thalamic module and a cortical module. The design and layout of the thalamic module is as in Bhattacharya et al. (2011) and is based on physiological data obtained from Sherman (2006). The cortical module layout and parameterizations are based on a previous implementation on SpiNNaker (Sharp et al., 2012) that was designed to test fast, stable and power-efficient performance on SpiNNaker when compared with other available platforms. The detailed modeling approach and parameterizations is covered in section 2. To the best of our knowledge, we are not aware of any prior instance of mimicking EEG signals using the SpiNNaker machine; similarly, this is the first instance of implementation of a TCT model within the SpiNNaker framework.

In section 3, we present the preliminary results from this study based on our observation of the membrane potential time-series and power spectra of the cell populations. Specifically, the output of the excitatory cells of the thalamus and the cortical layer 4 are studied as a part of the first set of results from the TCT model simulation on SpiNNaker. An average of three trial runs of the model with all parameters at their initial values showed the membrane potential of both cell populations as noisy time series outputs with the dominant frequency of oscillation within the alpha band (8–12 Hz), a characteristic of quiet wakefulness. Next, we performed preliminary engineering of the model parameters to induce a sleep-wake transitional behavior in the model. The particular case we examined, which is outlined in more detail in section 3, was that of disconnecting the thalamic reticular nucleus (TRN) cell population in the model. This was designed to alter the thalamo-cortico-thalamic loop, which is responsible for the maintenance of the quiet wakefulness alpha rhythm, and simulate the situation during sleep in which cortical areas become functionally disconnected (Massimini et al., 2005). It thus provides a good test of the neuronal dynamics of the model in a situation in which the real dynamics are reasonably well understood. In previous (Bhattacharya, 2013; Bhattacharya et al., 2013) as well as ongoing (unpublished) work, lumped parameter models of neuronal population of the thalamocortical circuits [also known as neural mass models (Marreiros et al., 2009)] have shown dependence on the TRN connectivity for mimicking qualitative dynamics as seen in EEG patterns of sleep and quiet wakefulness. Our results showed some important similarities with real sleep EEG time series data (also shown) when the TRN population is disconnected. However, significant differences with sleep power spectral data have also been observed; this suggests the model requires further tuning before it can fully capture sleep/wake thalamocortical dynamics.

It is important to note that the purpose of the work presented here is to design a working model structure of the TCT circuit on SpiNNaker such that the model dynamics show some similarity to known dynamics of sleep and wake EEG in terms of characteristic spectral power; the intention is not to present a fully tuned model or a detailed exploration of those dynamics. A discussion on the motivation of the current work, the drawbacks, the implications of the initial results presented and future work plans is provided in section 4.

## **2. MATERIALS AND METHODS**

In this section, we first give a brief background of the SpiNNaker architecture, followed by a detailed description of the TCT model and modeling methods adopted in this work. The simulation methods, and methods for observing results on the SpiNNaker platform are also outlined.

## **2.1. THE SpiNNaker MACHINE AND TOOL CHAIN**

## *2.1.1. The architecture*

The SpiNNaker project, led by the University of Manchester and its partners in academia and industry, aims to create a biologically inspired high performance computing architecture for the simulation of large real-time Spiking Neural Networks (Furber et al., 2006, 2013). It incorporates characteristics of fault-tolerance and power frugality, similar to those of the biological brain, whose low-power and resilient performance is achieved through extensive parallel computation.

A SpiNNaker system is formed by the interconnection of SpiNNaker chips and boards (**Figure 1**), each chip being a custom Application Specific Integrated Circuit (ASIC) containing 18 ARM processors—the likes of which are found in mobile telephones. Each processor is low-power in operation, but fully programmable, permitting each to execute arbitrary neural and synaptic models. Spikes emitted by a simulated neuron in operation are conveyed as short packets to efferent neurons using

a bespoke network on chip, and further afield to processors on neighboring chips using a network of connections which resiliently interconnect the chips to form the SpiNNaker machine.

**block from which larger systems will be constructed.**

The maximum number of chips in a SpiNNaker configuration is in excess of 65,000, and with 18 processors on each chip a machine can exceed one million processors. Even with the medium performance ARM processors used it is possible to simulate multiple neurons on each processor in real time, depending on their model complexity, potentially delivering many hundreds of millions of point-type neurons in a full deployment (Furber et al., 2006).

## *2.1.2. Programming SpiNNaker*

The selection of neuron and synaptic models and their interconnectivity is achieved by the user through a high-level modeling language. This flexible approach becomes increasingly important as networks grow in size, and it becomes impractical to specify each individual neuron and its connections—the network description is therefore made through multiple levels of hierarchy. The primary language used in the specification of Spiking Neural Networks to operate on SpiNNaker is PyNN (Davison et al., 2009), which is a popular description specification. Support of the PyNN library is enabled by a software tool-chain coined "PACMAN," which has been developed to take this high level description of the network and perform Partitioning And Configuration MANagement (Galluppi et al., 2012). For example a 10,000 neuron network is analyzed by PACMAN, and partitioned into chunks which are manageable for a single processor using the neuron model specified. If each processor is able to handle 100 neurons of that type, then the partition size necessitates 100 processors and the tools take care of this partitioning and the necessary inter-connectivity. The next stage involves allocating the physical processors to this task based on the topology of the target SpiNNaker machine, the loading of data to it, and the execution and control of the simulation.

## *2.1.3. Results recovery*

There are two main methods of accessing the results on SpiNNaker. Firstly PyNN may be used to direct the simulation to make recordings of parameters periodically, for example neuron membrane potentials over time; and after the simulation this information may be recovered, processed and plotted. Secondly, it is possible to recover data from the simulation whilst it is "inflight"—also requested through a PyNN parameter, for example to direct spike outputs to a "dummy" efferent neuron whose role is to collect and distribute spikes to an external receiver. This second method becomes particularly useful in simulations which run over an extended period, for example on a robot where a control loop is to be closed (Denk et al., 2013), or to simulate multiple channels of activity simultaneously, and to this end real-time visualization software (VisRT) has been developed (Patterson et al., 2012). In this study we make use of both methods, data is recovered post-simulation into MATLAB for analysis, and VisRT is used to gain an insight into the firing rates and rhythms seen in the simulations for EEG-type channel plots.

## **2.2. THE THALAMO-CORTICO-THALAMIC MODEL**

The TCT model has two modules: cortical and thalamic; all information on the model parameters are provided in **Tables 2**, **3**. The thalamic module consists of the thalamocortical relay (TCR) cells, the inhibitory interneurons (IN) and the thalamic reticular nucleus (TRN). The synaptic connectivity layout and values of the thalamic module cell populations are sourced from Horn et al. (2000); Sherman (2006); and Jones (2007) and are as in our previous work (Bhattacharya et al., 2011). The cortical module cell populations are as described previously in Sharp et al. (2012) and are further subdivided into layers 2–6. Layer 1 is ignored in keeping with standard practice due to sparsity of neurons in this layer. Similarly, layers 2 and 3 are treated as a single layer in keeping with models based on physiology of the mammalian visual cortex (Binzegger et al., 2009). Each cortical layer consists of pyramidal (PY), basket (B) and non-basket (NB) cell populations. Layer 4 has an additional cell population of spiny-stellate (SS) cells.

The number of neurons in each cell population of the thalamic and cortical modules are provided in **Tables 3B,C**, respectively. The data on the proportion of cells of each type in the cortical layers are scaled versions of Izhikevich and Edelmann (2008a) and Sharp et al. (2012), which in turn are inspired by data from visual cortex of the cat as provided in Binzegger et al. (2004) and Douglas and Martin (2004). Based on literature reporting physiological data, it is estimated in Hill and Tononi (2005) that a thalamocortical column containing 94 (i.e., ≈100) neurons cover a surface area of 1454μm2. The total number of cells in the TCT model is 1090 (i.e., ≈1000) and may therefore be thought to represent a column of interconnected neurons covering ≈0.15 mm<sup>2</sup> of thalamocortical tissue.

Each synaptic connectivity parameter between two cell populations has two attributes: (1) a probability of connection *P* indicating the absence of all-to-all intra- and inter-module connectivity; and (2) the weight of the synaptic connectivity *C*, expressed as a percentage of the total number of synapses made on an individual synaptic node on the post-synaptic cell. In the cortical module, all *P* are identical to previous work (see Table 2, in Sharp et al., 2012) to ensure stability and comparability during simulation on SpiNNaker; the reader may refer to this work for details on how the specific values were obtained. All values for *C* in the cortical module are as in Izhikevich and Edelmann (2008a) and Sharp et al. (2012). In the thalamic module, and for connections between thalamic and cortical cells, the connection probabilities *P* are arbitrarily set to 0.25 for the sake of simplicity in this study. The intra-thalamic and corticothalamic values for *C* are sourced from previous work (Bhattacharya et al., 2011), which in turn are based on Horn et al. (2000) and Jones (2007). The values of *C* for the thalamocortical efferents to the SS and B cells of Layer 4 are sourced from Binzegger et al. (2004).

The TCR and IN cells of the thalamic module in the TCT model are fed with a spike source that follows a Poisson distribution with a spiking rate of 25 Hz and an all-to-all connectivity. The inter-module connectivities i.e., connections between the cortical module and the thalamic modules as well as between the external input source and thalamic module have an induced delay simulated by a uniformly distributed random number generator in PyNN.

## **2.3. SPIKING DYNAMICS OF THE THALAMO-CORTICO-THALAMIC MODEL NEURONS**

Each neuron in the TCT model is an implementation of the spiking neuron model proposed in Izhikevich (2003), which is now a widely used template for modeling spiking neuron behavior due to its computational efficiency and rich dynamics, and is commonly referred to as the "Izhikevich model." Our longer-term objective is to use the Izhikevich model to implement an appropriate spiking behavior for the neurons in each population of the TCT model based on experimental observations in biology. An excellent demonstration of how a changing set of parameter values in the Izhikevich model can simulate the various spiking dynamics of thalamocortical neurons is provided in Izhikevich (2004). We have adopted three types of spiking behavior in the model:

## *2.3.1. Tonic spiking*

Tonic spiking refers to a continuous train of spikes in response to an external stimulus and is known to be adopted by a cell when it is communicating information (McCormick and Feeser, 1990); for example tonic spiking of the TCR cells of the LGN indicate that they are in a "driver" mode and are passing retinal information to the visual cortex (Sherman, 2005). The tonic mode of spiking can be further classified based on a (qualitative) characteristic frequency of firing in response to a stimulus: regular spiking (*RS*) and fast spiking (*FS*). A comparison of *RS* and *FS* dynamics simulated using Izhikevich's model and from *in vitro* recordings on thalamocortical neurons is demonstrated in Izhikevich and Edelmann (2008a) (Figure 10 in the Supplementary Material of the cited work). We follow this work and parameterize the PY, SS and TCR populations in the TCT model to adopt similar *RS* dynamics in response to stimuli, while the cortical B cells are parameterized to respond in an FS mode. It may be noted that all the cell populations displaying the *RS* mode are excitatory in nature, while the inhibitory B cell population respond in a *FS* mode. For simplicity, we adopt a similar spiking behavior for the inhibitory IN cell population of the thalamus.

## *2.3.2. Spike frequency adaptation*

This terminology is used to define spiking dynamics where the inter-spike interval is low at the onset of the stimulus but "adapts" with passing time and the spiking frequency decreases. The cortical NB cells are modeled in Izhikevich and Edelmann (2008b) to exhibit a low threshold spiking (*LTS*) behavior, which is a type of spike frequency adaptation dynamics. We follow this work and parameterize the TRN cells in the TCT model to respond in an LTS mode to a step stimulus.

## *2.3.3. Tonic bursting*

Bursting behavior in neural dynamics refers to a series of spikes in quick succession; tonic bursting would thus refer to a train of such bursts of spikes. The burst spiking mode of the inhibitory TRN cell population is believed to be centrally important in generating the synchronized oscillations observed in EEG during slow wave sleep (Golomb et al., 1994; Destexhe and Sejnowski, 2002). The TRN cell population in the TCT model is parameterized to respond in a tonic bursting mode.

All data used to parameterize the cell populations in the abovementioned spiking modes is provided in **Table 1** and based on the implementation of the Izhikevich model in Python by Galbraith (2011). The excitatory and inhibitory synaptic parameters are set by empirical study in PyNN corresponding to a set of parameters to simulate the desired spiking dynamics. The corresponding

#### **Table 1 | The parameter set corresponding to the spiking dynamics shown in Figure 2.**


*All parameters are based on those provided in Galbraith (2011) for simulation using Python software. The final parameter values are adjusted by empirical study to simulate similar qualitative spiking dynamics on SpiNNaker.*

dynamics of a single example neuron in a population in response to an excitatory or inhibitory stimulus is shown in **Figure 2**.

## **3. RESULTS**

A typical human EEG recording taken during quiet wakefulness and sleep (Durrant et al., 2013) is shown in **Figures 3A–D**. Sleep in birds and mammals is divided into REM (Rapid-Eye-Movement) and non-REM parts. Non-REM sleep is further divided into light/transitional sleep (N1), which makes up 5–10% of the night and is not considered functionally significant; normal sleep (N2; **Figure 3B**), which is characterized by the presence of spindles and K-complexes and is present for 40–50% of the night; slow wave sleep (N3/SWS; **Figure 3C**) which is the deepest form of sleep and characterized by the presence of high-amplitude lowfrequency ("slow") waves. REM sleep (**Figure 3D**) is characterized by a mixed frequency waveform, low muscle tone and rapid eye movements. Sleep EEG is classified into these different stages based on 30 s epochs according to standardized sleep scoring criteria (Rechtschaffen and Kales, 1968; Ancoli-Israel et al., 2007). As a complement to the characteristic waveforms, power spectral density also differs considerably between sleep stages (**Figure 3E**). In particular, spectral power in sleep and quiet wakefulness is generally analyzed in four bands: delta (1–4 Hz), theta (4–8 Hz), alpha (8–12 Hz), and sigma (sometimes called the spindle band; 12–16 Hz). Higher frequencies in the beta and gamma ranges are associated with active wakefulness and task completion and are not involved in identifying sleep or wake patterns; these bands are not considered further here. In **Figure 3E**, the power spectra in all the sleep stages (REM and non-REM) are dominated by the delta band. In contrast, the power spectra in quiet wakefulness is dominated by the alpha band.

In order to test the ability of the model to capture some basic neuronal dynamics, we ran simulations and compared the model output to the recorded EEG data in **Figure 3**. The average membrane potential of all neurons in each cell population of the TCT model is considered as the output membrane potential of the population. Although EEG is believed to represent dendritic post-synaptic potentials from pyramidal neurons in the cerebral cortex, the TCR cell output in thalamocortical population models have been shown to mimic alpha rhythmic and slow-wave EEG characteristics (da Silva et al., 1974; Suffczynski, ´ 2000; Bhattacharya et al., 2013). Along these lines, in this work, we focus on the TCR cells of the thalamic module and the main target of their efferents to the cortical module (Gil et al., 1999; Lee and Sherman, 2008) viz. the Pyramidal cells in Layer 4 (PY4). Recent studies (Crunelli et al., 2011; Crunelli and Hughes, 2012) have identified the central role of the inhibitory neurons of the TRN acting via the TCR neurons in generating both slow oscillations and spindles that characterize non-REM sleep. In previous work, we have shown the pivotal role of the TRN cell afferents in effecting a time-series bifurcation of the TCR cell output in a population model of the thalamocortical circuit (Bhattacharya et al., 2013). In this work, we present a preliminary test on the TCT model by studying the output time series and power spectra with all model parameters at their base values. We then compare

model (Izhikevich, 2003, 2004) simulated on the SpiNNaker chip using the

correspond to an excitatory (inhibitory) current stimulus applied between 250 and 750 ms during a 1000 ms simulation time.

this with the case when the TRN cell population is disconnected from the model.

The model is simulated on SpiNNaker for 30 s for each simulation at a resolution of 1 ms, and subsequently downsampled to 200 Hz. The mean membrane potential of the PY4 and TCR cell population are averaged across three simulation runs to improve the reliability of the results. A snapshot of the real-time visualization of the model simulation on SpiNNaker as seen using visRT is shown in **Figure 4**. The human EEG used for comparison is recorded at 200 Hz from an occipital electrode (O1) referenced against the contralateral mastoid. Sleep stages are independently classified by two experts with more than 90% agreement. Both human EEG and the model output are filtered between 1 and 16 Hz with a Butterworth bandpass filter of order 10 in order to focus on spectral bands of interest. Power spectral density is estimated using a Welch periodogram with 800 FFT points using a

**FIGURE 3 | EEG characteristics of human sleep and wake.** Quiet wakefulness is represented in panel **(A)** and is characterized by the presence of the alpha rhythm, which is absent during sleep (see the power spectra at the bottom of the figure). Normal sleep, often referred to as N2 in sleep literature, is represented in panel **(B)** and is characterized by the presence of spindles (A, circled in cyan) and K-complexes (B, circled in green). Slow wave sleep (SWS) is

represented in panel **(C)** and is characterized by high amplitude slow oscillations. REM sleep **(D)** has a mixed frequency pattern, and is additionally identified by the presence of eye movements and low muscle tone. The power spectra in the four bands involved in distinguishing wake and different stages of sleep **(E)** shows a greater delta power during the sleep stages, while quiet wakefulness has stronger alpha power. Data taken from Durrant et al. (2013).

Hamming window half the length of the sampling frequency and a 50% overlap.

The TCR time-series output with all model parameters maintained at basal values (**Figure 5A**) show a similarity with the EEG time series in quiet wakefulness (**Figure 3A**). The corresponding time series output of the PY4 cells are shown in **Figure 5C** and show a similarity with their main "driver" cells of the TCR, albeit with a larger amplitude of oscillation. It may be noted that the time series plots presented in **Figure 5** are unfiltered data sampled at 5 ms intervals (200 Hz). A power spectra analysis of both the TCR and PY4 outputs corresponding to basal parameters show a dominant frequency within the alpha band (**Figure 5E**), similar to the power spectra of quiet wakefulness shown in **Figure 3E**. Next we disconnect the TRN cell population from the TCT model by removing the connectivity from the TRN to the TCR and vice-versa (see **Table 3A**). We note a distinct bifurcation in both the TCR and PY4 time series output shown in **Figures 5B,D**, respectively with a reduced frequency of oscillation compared to the output corresponding to basal parameters; an increased amplitude of oscillation is also observed in the TCR output (**Figure 5B**). A comparison of the TCR time series with real EEG data show a resemblance with the SWS time series (**Figure 3C**). However, the frequency of the oscillatory activity in **Figures 5B,D** appears (on visual inspection) to be higher than that in **Figure 3C**. This observation is reflected in the power spectra of both TCR and PY4 cell populations corresponding to disconnection of the TRN, showing a dominant frequency within the theta band (not shown here). This is unlike the power spectra of SWS, which have a dominant frequency within the delta band. Further, we observe that the amplitude of oscillation in the PY4 output time series does not show any significant increase with TRN disconnection, which is not in agreement with the classic definition of EEG "slowing" (reduced frequency, higher amplitude).

Overall, and given the preliminary nature of this work, we would not expect the model parameters to be tuned to give a perfect replication of human EEG, and indeed we do see substantial differences between the two. The most important difference between the model output and human EEG at present is the lack of strong delta power with the TRN cells disconnected from the model, and this area should be prioritized for further research.

**FIGURE 5 | Sample of the time series outputs of the (A,B) TCR and the (C,D) PY4 cell populations for a period of 5 s, clipped arbitrarily between the 20th and the 25th s from the 30 s (unfiltered) signal and downsampled to 200 Hz.** A comparison with real EEG time series data of quiet wakefulness (**Figure 3A**) shows a similarity with the **(A)** TCR and **(C)** PY4 outputs when all model parameters are at their basal values. A comparison with real EEG time series data of SWS (**Figure 3C**) shows a similarity with the **(B)** TCR and **(D)** PY4 outputs when the TRN cell population is disconnected from the model. **(E)** The power spectra of the TCR and PY4 cell populations with all model parameters at their basal values. A dominant alpha rhythm is observed, similar to that in the real EEG power spectra of quiet wakefulness (**Figure 3E**). (The reader may kindly note that the results presented here is a preliminary attempt in studying the plausibility of simulating EEG rhythms in models developed on the SpiNNaker computer. At no point do we expect to see exact match of model results with real EEG data; rather, we do expect to identify differences between the two that will inform our ongoing work).


**Table 2 | The synaptic connectivity parameters between the cells of the cortical layers of the TCT model.**

*The cortex is classified into six layers based on the cell types and intra-areal connectivities. Of these, Layer 1 is known to be sparsely populated and is mainly associated with cortico-cortical connections and not considered in this work. Layers 2 and 3 are often treated as a single layer using the nomenclature L2/3 primarily due to a lack of marked boundary between the two "layers" in terms of the cell-types and spatial layout. The nomenclature of the cells in each layer are—PY: Pyramid cells; SS, Spiny Stellate cells; B, Basket cells; NB, Non-basket cells. The SS cells of Layer 4 and the PY cells of layers 5 and 6 send out dendritic projections to other layers and thus are indicated with the layer number as suffix within brackets. Each connectivity parameter between a pre-synaptic population (say K) to a post-synaptic population (say L) has two attributes and are placed as a 2-element column: the top number in the column is the synaptic connectivity "weight" C, which is expressed as a percentage of the total number of synaptic connections made by all pre-synaptic populations of L on the latter; the bottom number in the column is the probability P that a spike by K will be communicated to L. Values of the first attribute C are as in Izhikevich and Edelmann (2008a), and those of the second attribute P are as in Sharp et al. (2012). All "X" in the table indicate absence of synaptic connectivity between the respective cell populations. "From" refers to the pre-synaptic cells, and "To" refers to the post-synaptic cells.*

## **4. DISCUSSION**

Sleep and its biological relevance and mechanisms have been of interest in research (Rasch and Born, 2013) and beyond; a "healthy" sleep pattern have tremendous impact on daily activities (Mednick and Ehrman, 2006). Thus it is not surprising that sleep disturbances are a common accompaniment of several neurological and psychiatric disorders (Brown et al., 2012). Additionally, the time and frequency signatures of sleep electroencephalography (EEG) in neurological disorders often provide a better understanding of the disease conditions [for example in schizophrenia (Gardner et al., 2014); Alzheimer disease (Jonkman, 1997)]. Furthermore, rapid-eye-movement **Table 3 | (A) The "weight" of the synaptic connectivities between the thalamic and cortical module cells as well as between thalamic cell populations. The probability of connection for inter-module connectivity is 0.25 in the current model. The synaptic connections from the retina to the thalamic cells have an all-to-all connectivity. (B) The population of neurons of each type in the cortical module are mentioned in the first column and the cortical layers are mentioned in the top row. The cortical layer references within brackets (for Layers 5 and 6 and for the SS cells) indicate the dendritic arborization of the cells to these layers. An "X" indicates the lack of the cell type in the cortical layer. (C) The population of neurons of each type in the thalamic module.**

## **(A) CONNECTIVITY PARAMETERS: INTRA-THALAMIC, THALAMO-CORTICO-THALAMIC AND RETINO-THALAMIC**



(REM) sleep is thought to play a role in memory consolidation involving the non-hippocampal brain parts (Born et al., 2006). The thalamo-cortico-thalamic circuitry plays a key role in generating brain rhythms (Steriade et al., 1993; McCormick and Bal, 1997). Several studies on thalamocortical dynamics have used mesoscopic scale lumped parameter models to mimic EEG in healthy conditions (Robinson et al., 2002; Zavaglia et al., 2006; Deco et al., 2008; Modolo et al., 2013; Moran et al., 2013), as well as to investigate anomalous EEG in neurological disorders (Suffczynski et al., 2004; Roberts and Robinson, ´ 2008; Pons et al., 2010; de Haan et al., 2012). In recent research (Bhattacharya, 2013), which is along similar lines as in Lytton (1996); Erdi et al. (2006), the need for detailed synaptic mechanisms in thalamocortical lumped parameter models to facilitate biologically realistic mapping of model features is emphasized. While extended work on the lumped parameter model implementing synaptic dynamics remains ongoing, we believe it is necessary to have a parallel line of investigation using a population model comprising of network(s) of single neuron models (i.e., single-neuron-level population model as opposed to lumped parameter population models) that is similar in structure to the former. This gives a "two-scale" architecture to the thalamo-cortico-thalamic framework. The endeavor will be to use the framework for realistic simulation of EEG dynamics in sleepwake transition. Here, we have presented a preliminary study on inducing a transition from quiet wakefulness to a "slow wave" (higher amplitude, lower frequency) pattern in the model output, and have shown the similarity and dissimilarity of the model output with real EEG data of sleep and wakefulness; these are discussed further below.

The primary issue in building a single-neuronal-level population model is the deficiency in available computational resources in terms of implementing biologically plausible parallel and asynchronous information transmission and exchange within the model framework. Another key aspect is energy-efficiency whereby maximal information processing is carried out using minimal resources, a mechanism that allows biology to deal with massive amounts of data in a fast and power efficient manner. This necessitates specialized computational tools to provide a low-power, parallel asynchronous framework for building very-large-scale-biologically-plausible models (VLSBm). The SpiNNaker (Spiking Neural Network architecture) chip is a platform designed to occupy this space; it meets all of the above criteria for building VLSBm and has been tested to outperform current available software and hardware platforms when building a cortical model of spiking neural networks (Sharp et al., 2012).

In this work we have built a thalamo-cortico-thalamic spiking neural network for implementation on SpiNNaker. The miniframework consists of 1090 neurons to mimic approximately 0.15 mm2 of thalamocortical tissue. We have focussed on the thalamocortical relay (TCR) cells and the cortical Layer 4 pyramidal (PY4) cells; the layer 4 cells are known to be dominated by the sensory pathway input from the thalamus compared to inputs from other cortical areas (Gil et al., 1999). With all model parameters at their base values, the TCR time series output and its power spectra resembles the EEG characteristics of quiet wakefulness. Observation of the corresponding PY4 cell outputs indicate that the behavior of these cells are largely driven by the TCR cells. Next, we endeavored to vary specific model parameters to simulate non-rapid eye movement (non-REM) sleep stages. The thalamic reticular nucleus (TRN) neurons are implicated in playing a vital role in effecting slow wave oscillation in the EEG such as observed during slow wave sleep (SWS). To test this feature in the model, we disconnect all efferents from and afferents to the TRN cell population. We observe a distinct transition in the time series behavior of both the TCR and PY4 cells that resemble the EEG time series in SWS, albeit at a slightly higher frequency of oscillation (observed by visual inspection). This observation is reflected in the power spectra where the dominant frequency of oscillation for both population outputs are within the theta band, unlike the dominant delta band frequency seen in all stages of sleep EEG data. We speculate that the current disagreement in the power spectra of the SWS simulation on the TCT model may be addressed by dynamically changing the spiking behavior of the model cell populations (see below for further discussion on this). Furthermore, it will be interesting to observe how the intracortical afferents affect the PY4 cells in comparison to the TCR afferents (Destexhe, 2008; Lee and Sherman, 2008) and whether the model behavior conforms to experimental observations. Nonetheless, we note that the framework presented herein is a pilot study only, designed primarily to test the ability of the hardware to capture thalamocortical dynamics. We believe that the outcome from this study will provide a "basis" for simulating EEG signals on SpiNNaker-based computational models. Thus, at this stage, we do not attempt to simulate a true replication of the sleep-wake dynamics on the model. The larger goal of the work is to lay the foundations for building a VLSBm of thalamocortical interactivity to simulate biologically realistic sleep rhythms as observed in EEG. However, further testing and simulation on SpiNNaker will be required before scaling up the model for realistic simulation of EEG rhythms; we will take this up as an extension of the current work. Altogether, we believe this is a promising first demonstration of SpiNNaker as a platform for investigating thalamocortical circuits in humans.

A widespread current concern in the computational neuroscience community is the non-trivial task of populating the parameter space of computational models; the task gets harder with increasing model size as experimental data with definitive values for specific parameters are difficult to acquire. We have sourced appropriate model parameter values from Binzegger et al. (2004); Izhikevich and Edelmann (2008b); Bhattacharya et al. (2011); Galbraith (2011); and Sharp et al. (2012). Model layout and neuronal dynamics are from Sherman (2006) and Bhattacharya et al. (2011) and Izhikevich (2003, 2004), respectively. The absolute values of the model parameters often require appropriate scaling for the simulation platform, and a common approach to deal with this aspect has been to normalize all model parameters to a "simulator-friendly" scale. Along these lines, several assumptions and simplifications have been made in this study:

First, burst spiking dynamics of the thalamic cells that are crucial for generating slow wave oscillations (Jeanmonod et al., 1996; Magnin et al., 2005) are explored minimally. The thalamocortical relay (TCR) cells are tested for tonic spiking behavior in this work, which best align with the awake state of the brain. We speculate that the results reflect this behavioral mode of the TCR cells, clearly showing a resemblance with both timeseries and power spectra of EEG in quiet awake state. However, the TCR displays burst spiking dynamics during the stages of sleep. Similarly, the TRN cells are known to show rich spiking dynamics (e.g., rebound bursting, low threshold spiking) that underlie sleep-wake oscillatory activity. These variant dynamics of the TCR and TRN cells will be further investigated in our ongoing work. Thalamic interneurons are more problematic; there are to our knowledge no references in the modeling literature relating specifically to the spiking dynamics of the thalamic interneurons (Destexhe et al., 1998). However the cortical basket cells, which are also categorized as local interneurons depending on their function and dendritic arborization, are described in Izhikevich and Edelmann (2008a) using Fast Spiking (FS) dynamics. We have arbitrarily adopted this spiking behavior for the IN cells. Overall, much more detailed exploration and simulation of the individual thalamic cell spiking dynamics needs to be performed to preview the parameter space that would allow full replication of EEG in different sleep stages and the sleep-wake transition. It needs to be mentioned here that a high number of synaptic efferents from the thalamic interneurons are dendrodendritic (Cox and Sherman, 2000). However, this aspect does not affect the synaptic transmission in the TCT framework as it comprises of spiking neuron models, and does not take into account the detailed axonal and dendritic attributes related to spike transmission and reception.

Second, the Izhikevich model uses common excitatory and inhibitory synaptic parameters for all cell populations of excitatory and inhibitory types. This is a significant limitation and requires modification in future versions of the model to enable a direct comparison with the current lumped parameter models that include neurotransmitter and receptor dynamics.

Third, the neuronal population in the thalamus represents a loose estimate as no definitive data on the number of thalamic cells within a cortical column is available from literature. We preserve the (intra-thalamic) proportion of thalamic cells in the (Izhikevich and Edelmann, 2008a) thalamocortical model (only "specific nucleus" parameters are considered; the "non-specific nucleus" parameters are ignored), but scale this up by a factor of 102. This may be contrasted with a factor of 10 scaling of the number of cortical cells. Thus the model is designed to place increased emphasis on the thalamic behavior and its effects on cortical oscillations for our test purposes.

Fourth, our objective is to simulate EEG in sleep and quiet wakefulness. Thus, the simulated retinal input to the model needs to conform to discharge rates of the retinal spiking neurons during the resting state. In an early work on the cat retina (Kuffler, 1953), it is observed that the resting state discharge rate of a single retinal neuron is approximately 25 Hz. This is in agreement with the spike source rate provided as input to the TCT model in this work. However, in a relatively recent work (Robinson et al., 2004), it is estimated that the resting state firing rate of retinal input is 11 Hz, while in an alert awake state this is in the range 12–20 Hz. Thus, it would need further work to test these variations in experimental data and the effects on the model output in context to mimicking sleep-wake EEG.

Fifth, the probability of connection between the intra-thalamic cells as well as for the feedforward and feedback connections between the thalamus and the cortex is arbitrarily set at 0.25 by empirical study on SpiNNaker. This will need further attention and more detailed tuning in future work.

Finally, the conduction delay for thalamocortical and corticothalamic communication is implemented using a uniformly distributed function to generate a random delay. However, data acquired from physiology and tested on computational models is available in literature (Roberts and Robinson, 2008). This will be explored for implementation in future work.

In conclusion, we have presented a pilot study which involved building biologically plausible networks on a biologically plausible computational platform—SpiNNaker. The study examines the feasibility of simulating EEG rhythms of sleep and wakefulness by implementing a thalamo-cortico-thalamic framework. The longer-term aim is to build a VLSBm of thalamo-corticothalamic synaptic interactivity on SpiNNaker, which will then be validated with real EEG data collected during sleep (Durrant et al., 2013). The work presented here gives a preliminary study of this approach. Ongoing work to build a similar framework with the lumped parameter approach will provide a "multiscale" architecture to the model in both space and time. Together these models should provide new insights into the mechanisms which give rise to the rich thalamocortical dynamics seen in the human brain.

## **REFERENCES**


Hill, S., and Tononi, G. (2005). Modeling sleep and wakefulness in the thalamocortical system. *J. Neurophysiol.* 93, 1671–1698. doi: 10.1152/jn.00915.2004


gamma oscillations, sleep spindles and epileptogenic bursts. *J. Neurophysiol.* 93, 2194–2232. doi: 10.1152/jn.00983.2004


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 28 November 2013; accepted: 21 April 2014; published online: 20 May 2014. Citation: Bhattacharya BS, Patterson C, Galluppi F, Durrant SJ and Furber S (2014) Engineering a thalamo-cortico-thalamic circuit on SpiNNaker: a preliminary study toward modeling sleep and wakefulness. Front. Neural Circuits 8:46. doi: 10.3389/ fncir.2014.00046*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Bhattacharya, Patterson, Galluppi, Durrant and Furber. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## Synchrony can destabilize reward-sensitive networks

## *Michael Chary\* and Ehud Kaplan*

*Department of Neuroscience, Icahn School of Medicine Mount Sinai, Friedman Brain Institute, New York, NY, USA*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA Jens Kremkow, SUNY Optometry, USA*

#### *\*Correspondence:*

*Michael Chary, Department of Neuroscience, Icahn School of Medicine Mount Sinai, Friedman Brain Institute,* 50 *East* 98*th St. New York, NY,10029, USA e-mail: michael.chary@mssm.edu*

When exposed to rewarding stimuli, only some animals develop persistent craving. Others are resilient and do not. How the activity of neural populations relates to the development of persistent craving behavior is not fully understood. Previous computational studies suggest that synchrony helps a network embed certain patterns of activity, although the role of synchrony in reward-dependent learning has been less studied. Increased synchrony has been reported as a marker for both susceptibility and resilience to developing persistent craving. Here we use computational simulations to study the effect of reward salience on the ability of synchronous input to embed a new pattern of activity into a neural population. Our main finding is that weak stimulus-reward correlations can facilitate the short-term repetition of a pattern of neural activity, while blocking long-term embedding of that pattern. Interestingly, synchrony did not have this dual effect on all patterns, which suggests that synchrony is more effective at embedding some patterns of activity than others. Our results demonstrate that synchrony can have opposing effects in networks sensitive to the correlation structure of their inputs, in this case the correlation between stimulus and reward. This work contributes to an understanding of the interplay between synchrony and reward-dependent plasticity.

**Keywords: cortical networks and systems, synchrony code, computational models in psychiatry, plasticity and learning, substance abuse**

## **1. INTRODUCTION**

Synchrony refers to a coordinated pattern of network activity. Synchrony occurs between (i) action potentials, (ii) local field potentials, or (iii) action potentials and local field potentials. The latter two types of synchrony are frequently called coherence. Neural networks with strong recurrent connections can demonstrate synchronous activity that persists over seconds to minutes (Tetzlaff et al., 2012). Changing synaptic strengths allows that activity to persist over longer time scales (Holtmaat and Svoboda, 2009).

Synchrony between action potentials helps localize sounds (Joris et al., 1998), signal the direction of motion (Meister et al., 1995; Meister and Berry, 1999), and discriminate among odors (Stopfer et al., 1997; Tetzlaff et al., 2012).

When exposed to addictive substances, only some individuals develop an addiction or dependence (Ersche et al., 2010). Of those who become addicted or dependent, only some respond to treatment (Gawin, 1991). Alterations in activity-dependent learning in areas of the brain involved in reward processing are important in the pathogenesis of addictive disorders (Koob and Le Moal, 2005). Increased synchrony can predict intoxication (Li et al., 2011), resilience, susceptibility (Coullaut-Valera et al., 2014), or likelihood of relapse (Camchong et al., 2013), depending on in which brain region the synchrony manifests.

These observations suggest that many aspects of addiction can be understood as changes in the structure of synchronization of neural networks. To explore this, we study the stability of a pattern of activity in the face of different stimulus-reward inputs.

## **2. RESULTS**

## **2.1. SUMMARY OF MODEL**

Equation (1) describes the dynamics of a group of neurons, **v**. Those neurons interact linearly with each other according to the intrinsic connection matrix **M**, and receive input, **u**, weighted according to the feedforward connection matrix **W**. The weights in **W** depend on (i) the correlation between the stimulus, **u**, and network activity, **v**, denoted **u** ⊗ **v**, and (ii) the correlation between the the stimulus, **u**, and the reward associated with the stimulus, **r**, denoted **r** ⊗ **v**. The second line in Equation (1) is a linear differential equation in **M**, which means that it can only remove pairwise correlations.

The top line of Equation (1) describes the firing rate of a population of neurons. That firing rate decays in the absence of recurrent or feedforward input. The second line implements Hebbian modification of the feedforward weights, modulated the by the reward associated with the stimulus, **r**. The third line implements anti-Hebbian modification of the recurrent weights. Anti-Hebbian modification prevents the network from responding identically to inputs with the same amount of active units.

$$
\tau\_\mathbf{v} \frac{d\mathbf{v}}{dt} = -\mathbf{v} + \mathbf{M} \cdot (\tanh \mathbf{v}) + \mathbf{W} \cdot \mathbf{u}
$$

$$
\tau\_\mathbf{W} \frac{d\mathbf{W}}{dt} = \mathbf{K} \cdot \mathbf{W} \cdot \mathbf{u} \left(\mathbf{r} - \mathbf{v}\right)
$$

$$
\tau\_\mathbf{M} \frac{d\mathbf{M}}{dt} = \left(\mathbf{I} - \mathbf{M}\right) - \left(\mathbf{W} \cdot \mathbf{u}\right) \mathbf{v} \tag{1}
$$

The importance of correlations arises directly from the bottom two lines in Equation (1) because the outer product of two vectors can be interpreted as the cross-correlation between those two vectors. In this paper, we only consider 1-dimensional stimuli for simplicity. The dependence of the dynamics of connections among neurons on the correlation between stimulus activity and network activity allows patterns of network activity that are very far from **v**∞ to maintain stable connections between neurons.

Connections between units in the network stabilize, that is *d dt* **M** → 0, when the correlation between network activity, **v**, and the filtered version of the input, **W** · **u**, lies parallel to the deviation between the connection matrix, **M** and the identity matrix, **I**. Connections between the network and input stabilize, that is *d dt* **W** → 0, when network activity accurately predicts the reward, **r** = **v** or the neurons in the network become autonomous, **M** = **I** so **K** = **0**.

## **2.2. COMPUTATIONAL RESULTS**

## *2.2.1. Stimuli*

We model (crudely) the initiation, continuation, and cessation of drug use with three patterns of stimuli, exposure, chronic, and cessation, respectively (**Figure 1**, left). We combine these stimuli with two types of reward saliences, designed to model susceptible and resilient individuals (**Figure 1**, right). The reward associated with a stimulus is a log-Gaussian for susceptible individuals and a Gaussian for resilient individuals. A log-Gaussian function was chosen to reflect experimentally observed dynamics of positive reinforcement (Koob and Le Moal, 2005; Koob, 2013). A Gaussian function was chosen to model the slower and softer dynamics suggested to occur in resilient individuals (Ersche et al., 2010). We calculate the stimulus-reward patterns as the convolution of each combination of stimulus and reward (**Figure 2**).

**Figure 3** investigates the ability of our network to maintain a preset pattern in the face of different stimuli and different rewards associated with those stimuli. In that figure, all panels in a row share the same reward. All panels in a column share the same stimulus. Each panel has three components, a raster plot, the stimulus, and the reward associated with that stimulus. The middle column, in which the stimulus is tonic, shows the greatest deviation from the resting pattern. Each row of the raster indicates the firing pattern of a neuron, with black indicating an action potential and white indicating the absence of firing. The middle graph in each panel indicates the stimulus pattern. The bottom graph in each panel indicates the perceived reward.

**Figure 3** shows that susceptible networks are more able to maintain the preset pattern in the face of a chronic stimulus than resilient ones are; however, resilient networks can better maintain the present pattern once the stimulus stops. In the context of neural network computation, stability of our network in the face of different stimulus-reward patterns reflects (i) the incompatibility between the patterns the inputs would embed and the preset patterns embedded in the network, and (ii) the lower energy associated with the preset patterns which favors maintaining them. In the context of addiction, patterns that are stable in the face of input could model the lack of alteration of synaptic weights in resilient individuals or the perpetuation of destructive behaviors in susceptible individuals who develop substance dependence.

To quantify the similarity in patterns between two panels, we considered each of the *N* rows of each panel's raster to represent a vector. We calculated the similarity between two patterns, *a* and *b*, denoted by *qab*, as the average of the cosine of the angle, θ, between each corresponding rows Equation (2).

$$q\_{ab} = \frac{1}{N} \sum\_{n=1}^{N} \frac{\mathbf{v}\_{n,a} \cdot \mathbf{v}\_{n,b}}{||\mathbf{v}\_{n,a}|| \cdot ||\mathbf{v}\_{n,a}||}\tag{2}$$

**Figure 4** shows the result of applying Equation (2) to **Figure 2**. Changes greater than this magnitude are beyond the 85th percentile in the empiric cumulative distribution function created from randomly shufffling all rows in all rasters in **Figure 3**. This corresponds to a change in the cosine of the angle of more than 0.05. That is to say, the deeper blue the square, the more effective the stimulus-reward input was at embedding its pattern.

resilient. Columns denote different patterns of drug usage, initiation (exposure), chronic (continual use), or cessation. All patterns last for 200 time steps.

**stimulus-reward inputs.** Each panel shows the raster (top), stimulus (middle), and associated reward (bottom) for one of the six stimulus-reward patterns from **Figure 2**. The row (x-label of raster) indicates the reward pattern, susceptible or resilient. The column (y-label of raster) indicates the

indicates a neuron. The x-axis of the raster indicates time. A black mark is placed at the *it*th position if neuron *i* fired at time *t*. The simulations in all panels began with the same initial condition, being within the basin of attraction of **v**0.

This stability (resistance to embedding) is lowest with the most prolonged stimulus, chronic use, as shown by the deep blue colors in **Figure 4**. The impairment persists only for networks whose reward correlation follows a susceptible scheme. In the lowest three rows in the first column of **Figure 4**, the square corresponding to chronic (prolonged) exposure is deep blue, but the others are paler than their counterparts in the susceptible scheme. Interestingly, the susceptible network has a more profound negative reaction than the resilient network does

of each box in the heat map shows the circular mean of the cosine of the angles formed between each row of the corresponding panels in **Figure 3**. A row makes an angle of 0 with itself, which corresponds to a cosine value of 1. Cooler colors indicate more different patterns.

to initial exposure and sensation (bottom graph in the panels in **Figure 3**).

Susceptible networks exhibit more stable patterns of activity with continual exposure to a highly rewarding stimulus than do resilient networks (**Figure 5**). We calculated stability according to Equation (9) (see Materials and Methods). Taken with the impairment in recall, this suggests that, in susceptible networks, chronic use creates new fixed points while destabilizing existing ones. **Figure 6** shows that previously stable patterns become associated with higher energies in susceptible but not resilient networks after ceasing to be exposed to a highly rewarding stimulus.

## **3. DISCUSSION**

This paper discussed the ability of a computational model of neural population dynamics with activity-dependent plasticity to maintain preset patterns of activity in the face of different stimulus-reward patterns. The types of stimuli were chosen to model patterns of drug use. Rewards and stimuli were chosen to reflect the division into susceptible and resilient organisms, noted in the experimental and clinical literature.

We found that a tonic stimulus, modeling chronic exposure, was most effective in destabilizing the network. If the network perceived rewards according to Gaussian (resilient) dynamics it fully recovered. If it perceived rewards according to log-Gaussian (susceptible) dynamics, then it remained altered. The discontinuation of the tonic stimulus promoted unstable network activity in networks that follows log-Gaussian (susceptible) but not Gaussian (resilient) reward dynamics. Our computational results agree with experimental and clinical findings. Chronic but not acute use causes cognitive impairment for many drugs of abuse (Block et al., 2002; Lundqvist, 2005). These impairments persist

in some people even after cessation (Gouzoulis-Mayfrank et al., 2003). The chronic use of drugs of abuse impairs certain neurocognitive domains more than others (Bechara, 2005).

Simulating the relationship between synchrony and network activity may provide insight into the pathogenesis and treatment of functional brain disorders. It also suggests that certain patterns of deep brain stimulation may be more effective than others for a given pathology. For example, structured patterns of stimulation may be more effective for some neuropsychiatric disorders, while a noisier stimulus, similar to that used in electroconvulsive therapy, may be more appropriate for other disorders. In support of this postulate, the frequencies used in deep brain stimulation, even in the same region, vary with the disease being treated (McIntyre et al., 2004). Stimulation of the internal capsule and adjacent ventral striatum are effective for treating obsessive-compulsive disorder only at frequencies between 100 and 130 Hz (Greenberg et al., 2010). Tonic but not phasic stimulation of the medial prefrontal cortex at 100 Hz reverses a depressive phenotype in mice (Covington et al., 2010).

Future work, beyond addressing the caveats below, could investigate whether the stimulus-reward patterns used here induce similar effects in networks with different classes of embedded patterns. This network embedded patterns using a bivariate covariance rule. Many other schemes exist for embedding patterns, including those using multivariate covariance rules. Our model considered only the rewarding effects of drugs. A more realistic model could account for negative reinforcement of withdrawal, which may be more important in the maintenance of drug-seeking behavior (Koob, 2013).

## **3.1. CAVEATS**

The network constructed here grossly simplified the interactions in neural networks, assuming that (i) all units in the network interacted linearly, (ii) the dynamics of the network followed a Markov chain, and (iii) there is no learning of new memories. These assumptions limit how widely the conclusions of this paper apply. The assumption of linear interactions simplifies the analysis. However, neuromodulators, such as dopamine and acetylcholine, are important in learning and memory and

reward-dependent plasticity. Their effects on neural activity are non-linear. The Markov assumption simplifies simulation and allows the calculation of an energy function at the expense of making this network unable to manifest very slow correlations (Glauber, 1963; Kim and Nelson, 1999).

## **4. MATERIALS AND METHODS**

#### **4.1. OVERVIEW**

This section details the construction of a model neural network with (i) excitatory and inhibitory connections, (ii) external input, and (iii) the ability to recover prior patterns of activity. For more detail, refer to the Supplementary Material. All computer code used in the project are available in the GitHub repository synchrony.

**Figure 7** sketches a portion of the network with three neurons, *i*, *j*, and *k*. The matrix **M** contains the strength of connections between neurons. The matrix **W** contains the strength of connections between components of the input, **u**, and neurons in the network. Equation (3) describes the dynamics of the network. The equation inset in **Figure 7** is a version of Equation (3) for one neuron.

$$\tau\_\mathbf{v} \frac{d\mathbf{v}}{dt} = -\mathbf{v} + \mathbf{M} \cdot \mathbf{F}(\mathbf{v}) + \mathbf{W} \cdot \mathbf{u} \quad \mathbf{F}(\mathbf{v}) = \tanh \mathbf{v} \tag{3}$$

Equation (4) constructs a symmetric matrix, **M**, from a finite set of memories, **a** .

$$\mathbf{M} = \frac{1}{(1 - \alpha)\,\alpha \mid \left\{ \mathbf{a}^{\Omega} \right\} \mid} \sum\_{\{\mathbf{a}^{\Omega}\}} \left( \mathbf{a}\_{i}^{\Omega} - \alpha \mathbf{n} \right) \left( \mathbf{a}\_{i}^{\Omega} - \alpha \mathbf{n} \right) - \frac{1}{\alpha \mid \left\{ \mathbf{a}^{\Omega} \right\} \mid} \tag{4}$$

We introduce the terms Hebbian modification and anti-Hebbian modification to denote a strengthening or weakening of connections in the presence of correlated activity, respectively. Without an anti-Hebbian term in the dynamics of the recurrent connection matrix, **M**, each row of the feedforward weight matrix, **W** will come to lie parallel to the principal eigenvector of the input correlation matrix. This will make each target neuron respond identically. To break this redundancy we allow anti-Hebbian modification into the dynamics of **M**, using Equation (5) from Goodall (1960).

$$
\tau\_\mathbf{M} \frac{d\mathbf{M}}{dt} = \left(\mathbf{I} - \mathbf{M}\right) - \left(\mathbf{W} \cdot \mathbf{u}\right)\mathbf{v} \tag{5}
$$

## *4.1.1. Reward-dependent plasticity*

Dysregulation of brain areas that process rewards plays a role in the pathogenesis of addictive disorders (Everitt and Robbins, 2005). A simple way to account for the rewarding effects of a stimulus, **u**, is to make the connections between that stimulus and the network, **W**, dependent on the magnitude of that reward, *r*. Equation (6) shows for one neuron, *v*, the Rescorla-Wagner rule, a simple mathematical formulation of this concept (Rescorla and Wagner, 1972).

$$\begin{aligned} \nu &= \mathbf{w} \cdot \mathbf{u} \\ \mathbf{w} &\leftarrow \mathbf{w} + \varepsilon \delta \mathbf{u} \\ \delta &= r - \nu \end{aligned} \tag{6}$$

In Equation (6), **u** denotes input to the network. The vector **w** weights those inputs. The scalar, ε, represents the associability of the stimulus with the reward. The vector, *δ*, denotes the rewardprediction error. This name for *δ* arises from interpreting the second line in Equation (6) as a gradient descent rule that minimizes the quantity (*r* − *v*)2, which is the mean squared error between the actual reward, *r*, and the prediction, *v*.

Equation (7) modifies Equation (5) by incorporating Equation (6).

$$
\tau\_\mathbf{W} \frac{d\mathbf{W}}{dt} = \mathbf{K} \mathbf{W} \mathbf{u} \left(\mathbf{r} - \mathbf{v}\right) \tag{7}
$$

#### *4.1.2. Stability of memories*

If the state of any unit *i* in the network at some time *t* follows Equation (8), then network activity, **v**, evolves as a Markov chain (Glauber, 1963). Equation (8) assumes the activity of the *i*th unit, *vi* follows Equation (3).

$$P\left[\nu\_i(t+\Delta t) = 1\right] = \frac{1}{1+e^{-\nu\_i}}\tag{8}$$

A network of binary units updated according to Equation (8) is often called a Boltzmann machine because once the network has reached equilibrium, a Boltzmann distribution defines the probability that a pattern of network activity will occur (Hinton and Sejnowski, 1986). In a classical Boltzmann machine, one unit is randomly selected and updated at each time point.

Every pattern of activity in the network, **v**, has an energy, *E* (**v**), associated with it [top line of Equation (9)]. Patterns with lower energy are more stable, that is more likely to occur, because they are more likely to occur. The probability of a pattern, **v** occurring, increases as the energy associcated with that pattern, *E* (**v**), decrease [bottom line of Equation (9)] .

$$E\left(\mathbf{v}, \mathbf{u}\right) = -\left(\mathbf{u} \cdot \mathbf{W} \cdot \mathbf{v}\right) + \frac{1}{2}\left(\mathbf{v} \cdot \mathbf{M} \cdot \mathbf{v}\right)$$

$$\begin{aligned} \mathbb{P}[\mathbf{v}] &= \frac{e^{-E(\mathbf{v})}}{Z} \\ Z &= \sum\_{\{\mathbf{v}\}} e^{-E(\mathbf{v})} \end{aligned} \tag{9}$$

## **ACKNOWLEDGMENTS**

This work was supported by NIH grants EY-16224, NIMH-R21MH093868 and NIGMS-1P50GM071558 to Ehud Kaplan.

## **SUPPLEMENTARY MATERIAL**

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/journal/10.3389/fncir. 2014.00044/abstract

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 31 October 2013; accepted: 08 April 2014; published online: 30 April 2014. Citation: Chary M and Kaplan E (2014) Synchrony can destabilize reward-sensitive networks. Front. Neural Circuits 8:44. doi: 10.3389/fncir.2014.00044 This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Chary and Kaplan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## An investigation of Hebbian phase sequences as assembly graphs

#### *Daniel G. Almeida-Filho1, Vitor Lopes-dos-Santos 1, Nivaldo A. P. Vasconcelos 2,3, José G. V. Miranda4, Adriano B. L. Tort <sup>1</sup> and Sidarta Ribeiro1 \**

*<sup>1</sup> Brain Institute, Federal University of Rio Grande do Norte, Natal, Brazil*

*<sup>2</sup> Circuit Dynamics and Computation Laboratory, Champalimaud Neuroscience Programme, Lisbon, Portugal*

*<sup>3</sup> Universitary Center of Rio Grande do Norte , Natal, Brazil*

*<sup>4</sup> Physics Department, Federal University of Bahia, Salvador, Brazil*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Antonio C. Roque, Universidade de São Paulo, Brazil Fernando Montani, CONICET Argentina, Universidad Nacional de La Plata, Argentina*

#### *\*Correspondence:*

*Sidarta Ribeiro, Laboratory of Memory, Sleep and Dreams, Brain Institute, Federal University of Rio Grande do Norte, Av. Nascimento de Castro 2155, Natal, RN 59056-450, Brazil e-mail: sidartaribeiro@neuro.ufrn.br* Hebb proposed that synapses between neurons that fire synchronously are strengthened, forming cell assemblies and phase sequences. The former, on a shorter scale, are ensembles of synchronized cells that function transiently as a closed processing system; the latter, on a larger scale, correspond to the sequential activation of cell assemblies able to represent percepts and behaviors. Nowadays, the recording of large neuronal populations allows for the detection of multiple cell assemblies. Within Hebb's theory, the next logical step is the analysis of phase sequences. Here we detected phase sequences as consecutive assembly activation patterns, and then analyzed their graph attributes in relation to behavior. We investigated action potentials recorded from the adult rat hippocampus and neocortex before, during and after novel object exploration (experimental periods). Within assembly graphs, each assembly corresponded to a node, and each edge corresponded to the temporal sequence of consecutive node activations. The sum of all assembly activations was proportional to firing rates, but the activity of individual assemblies was not. Assembly repertoire was stable across experimental periods, suggesting that novel experience does not create new assemblies in the adult rat. Assembly graph attributes, on the other hand, varied significantly across behavioral states and experimental periods, and were separable enough to correctly classify experimental periods (Naïve Bayes classifier; maximum AUROCs ranging from 0.55 to 0.99) and behavioral states (waking, slow wave sleep, and rapid eye movement sleep; maximum AUROCs ranging from 0.64 to 0.98). Our findings agree with Hebb's view that assemblies correspond to primitive building blocks of representation, nearly unchanged in the adult, while phase sequences are labile across behavioral states and change after novel experience. The results are compatible with a role for phase sequences in behavior and cognition.

**Keywords: cell assembly, phase sequence, graph, sleep, learning and memory**

## **INTRODUCTION**

The firing synchronization of groups of neurons is a well-known phenomenon in the brain (Harris et al., 2003; Buzsáki, 2004; Harris, 2005; Canolty et al., 2010; Lopes-dos-Santos et al., 2011). According to the cell assembly hypothesis (Hebb, 1949), neurons transiently synchronize in order to form elementary units of information processing. Some reports have provided experimental evidence that assembly activity, i.e., the co-firing of assembly members, can be related to formation of memories and behavior (Wilson and McNaughton, 1994; Stopfer et al., 1997; Robbe et al., 2006; Peyrache et al., 2009; Liu et al., 2012; Ramirez et al., 2013). Furthermore, sensory or electrical stimulation able to synchronize neuronal firing in the millisecond scale has been shown to generate sequentially, in the minute to hour scale, synaptic potentiation, immediate-early gene expression, synaptic remodeling and dendritic sprouting (Chang et al., 1991; Bliss and Collingridge, 1993; Deisseroth et al., 1995; Klintsova and Greenough, 1999). In principle, this sequence of events satisfactorily explains why neurons that fire together wire together, and vice-versa. However, to date there is still a mechanistic hiatus between neuronal synchronization and the perception of complex stimuli, or the planning and execution of complex motor tasks.

The gap between cell assemblies and behavior was anticipated by Hebb (1949), who proposed that synchronized cell assemblies would evolve over time as *phase sequences*: "Any frequently repeated, particular stimulation will lead to the slow development of a 'cell-assembly,' a diffuse structure comprising cells in the cortex and diencephalon (and also, perhaps, in the basal ganglia of the cerebrum), capable of acting briefly as a closed system, delivering facilitation to other such systems and usually having a specific motor facilitation. A series of such events constitutes a 'phase sequence'—the thought process. Each assembly action may be aroused by a preceding assembly, by a sensory event, or—normally—by both."

For many years these ideas remained untestable, but in the past two decades, the detection and tracking of assemblies became feasible due to major improvements in multi-electrode recording techniques (Nicolelis et al., 2003; Buzsáki, 2004; Schrader et al., 2008), as well as the development of adequate mathematical frameworks for the identification of non-random synchronization (Berger et al., 2010; Denker et al., 2010; Peyrache et al., 2010; Lopes-dos-Santos et al., 2011, 2013). As a consequence, studies on assembly activity and learning were recently published (Peyrache et al., 2009; Benchenane et al., 2010); there were also demonstrations of information coding by the temporal sequence of neurons (Ikegaya et al., 2004; Ji and Wilson, 2006; Pastalkova et al., 2008; Peyrache et al., 2009; Dragoi and Tonegawa, 2010). The hippocampus, in particular, harbors assemblies activated by specific places or time intervals, forming representational sequences (Lee and Wilson, 2002; Macdonald et al., 2011; Kraus et al., 2013; Pfeiffer and Foster, 2013).

In the present work we aimed to advance the investigation of the next logical step in Hebbian theory, namely the detection of phase sequences as consecutive multi-assembly activation patterns. We also set out to investigate the relationship between phase sequences and cognitive behavior. The developed method was based on graph theory and it was applied to datasets comprising chronic extracellular spike recordings from the primary visual (V1) and somatosensory (S1) cortices, as well as the CA1 region of the hippocampus (HP), of rats subjected to a novel object exploration paradigm (Ribeiro et al., 2007).

## **MATERIALS AND METHODS**

## **EXPERIMENTAL PERIODS OF THE BEHAVIORAL PARADIGM**

We used data from five Long-Evans adult male rats (300–350 g) recorded before, during and after a novel object exploration paradigm (Ribeiro et al., 2007). The behavioral paradigm began with 1–2 h of recordings as a freely-behaving rat went through the wake-sleep cycle (PRE period). Next, the animal was allowed to explore 4 novel objects placed in the corners of the recording box for 20 min (EXP period). Finally, the objects were removed and the animal was recorded for an additional 1–4 h, freely traversing the wake-sleep cycle (POST period). Video recordings with infrared illumination were used to document behavior. The present study focused on the 1 h PRE and POST periods flanking EXP (**Figure 1A**).

## **MULTIELECTRODE ARRAY IMPLANTATION**

Briefly, the rats were anesthetized and surgically implanted with multielectrode arrays of tungsten microwires (35µm, 1.0–1.2 MOhm at 1 kHz). A screw implanted on the frontal portion of the skull served as recording ground. The arrays targeted HP, S1, and V1 in the left hemisphere stereotaxic coordinates in mm from Bregma with respect to the antero-posterior (AP), mediolateral (ML), and dorso-ventral (DV) axes (Paxinos and Watson, 1997): HP (AP: −2.80; ML: +1.5; DV: −3.30); S1 (AP: −3.00; ML: +5.5; DV: −1.40); V1 (AP: −7.30; ML: +4.00; DV: −1.30). DV measurements were taken with respect to the pial surface. Arrays comprised 16–32 microwires spaced at 250 mm intervals (4 × 4 arrays for S1 and V1, 2 × 16 array for HP). In S1 and V1, arrays were aimed at pyramidal layer V.

## **ELECTROPHYSIOLOGICAL RECORDINGS AND UNIT SORTING**

As described in detail in Ribeiro et al. (2007), action potentials (spikes) and local field potentials (LFP) were recorded with multielectrode arrays placed in the dorsal CA1 region and dentate gyrus of HP, in the barrel field of S1, and in V1. Animals were recorded after a 1-week recovery period following surgery. A 96-channel multineuron acquisition processor (MAP, Plexon Inc, Dallas, TX) was used for digital spike waveform discrimination and storage. Action potentials (spikes) were extracted from the high frequency band data and sorted into units using supervised online spike sorting (SortClient 2002, Plexon Inc.) associated with posterior offline validation (Offline Sorter 2.3, Plexon Inc). LFPs recorded from the same wires were pre-amplified, filtered, and digitized using a Digital Acquisition card (National Instruments, Austin, TX) and a MAP (Plexon Inc). Behaviors were recorded throughout the entire experiment under infrared illumination, by way of two CCD video cameras and a videocassette recorder. Video and neural recordings were synchronized with a millisecond-precision timer (model VTG-55; For-A, Tokyo, Japan). Within each region, the amount of units consisted of 42 HP, 33 S1 and 20 V1 for rat # 1, 59 HP, 23 S1 and 28 V1 for rat # 2, 34 HP, 25 S1 and 23 V1 for rat # 3, 39 HP, 27 S1 and 37 V1 for rat # 4 and 45 HP, 39 S1 and 42 V1 for rat # 5.

## **SORTING OF BEHAVIORAL STATES**

We used LFP data associated with a behavioral state sorting algorithm (Gervasoni et al., 2004) to classify the states with 1 s resolution. The algorithm is based on a two-dimensional state space defined by two spectral amplitude ratios calculated by dividing integrated spectral amplitudes at selected frequency bands. A scatter plot of the two chosen LFP spectral amplitude ratios (state-space) reveals distinct clusters that correspond to the three major wake-sleep states studied here: waking (WK), slow wave sleep (SWS), and rapid eye movement sleep (REM).

## **ASSEMBLY DETECTION**

A cell assembly is a subset of cells that somehow behave as a single entity. Here we assumed a linear model. More specifically, we defined the activity of a cell assembly as a weighted sum of the activity of individual units. In order to determine the weights of each neuron to each cell assembly we used a recently developed framework (Lopes-dos-Santos et al., 2013), which can be briefly described in four main steps:



$$AA\_b = \sum\_{i=1}^{N\_{\text{meas}}} \omega\_i z\_{ib} = W^T Z\_b,$$

where *AAb* is the assembly activity at time bin *b*, *Nneurons* is the total number of neurons, *wi* is the weight of neuron *i* in a specific assembly and *zib* is the z-scored activity of neuron *i* within bin *b*. We removed the contribution of single units firing alone (for instance, if a heavily-weighted neuron activated but others were silent, the assembly activity remained low).

**Figure 1B** shows an illustrative example of an activity matrix (top panel) along with the assembly activities estimated by the method. For more details, see (Lopes-dos-Santos et al., 2013).

## **RESULTS**

## **TIME BIN DETERMINATION**

We used an empirical approach to adequately choose the size of the time bins. First, we tested a wide range of bin sizes (2–256 ms) to investigate the relationship between bin size and number of detected assemblies. As shown in **Figure 2A**, we found an inverse relation between bin size and number of assemblies. We analyzed this closely and found that single assemblies detected with larger bin sizes could be split in two other assemblies when smaller bin sizes were used. The raster plot in **Figure 2B** shows the 20 most weighted units, sorted from heavier (top) to lighter (bottom), which comprise the patterns of assembly A (80% of the total weight). This assembly is one of the assemblies detected using

a 16 ms time bin in rat # 1 dataset, and its activity is shown in black (**Figure 2B**, bottom); while the activities of two assemblies detected using a 4 ms bin size (A and A--) are depicted in blue and green, respectively (**Figure 2B**, bottom).

To use a quantitative criterion to compare assembly composition, a Similarity Index (SI) was defined as the absolute value of the inner product between the assembly patterns (unitary vectors) of two given assemblies, varying from 0 to 1. Thus, if two assemblies attribute large weights to the same neurons, SI will be large; if assemblies are orthogonal, SI will be zero. We applied a permutation test in order to determine whether SIs were significantly above chance. This test consisted in shuffling the weights

weight of the correspondent neuron in the assembly pattern. Note that neurons participating in assembly A (bin 16 ms) were sorted into assemblies A and A-- (bin 4 ms), which can be active in sequence (black arrow and arrow head) or independently (red arrow). **(C)** Exploring similarities between assemblies. Panels show the histogram of SI values from 10,000 comparisons made by shuffling the neurons weights within assemblies to build a null hypothesis (bootstrap procedure). Red dashed line shows the threshold for significance at *p* = 0*.*01. Red circles depict the SI between A and A- (0.82, top), A and A-- (0.51, middle), and A and A-- (0.016, bottom). Note that assembly A is significantly similar to A and A-- (*SI* = 0*.*82 and 0.51, respectively). The SI between A and A- was small (*SI* = 0*.*016), indicating that, in addition to the fact that these assemblies have independent activity, they also have orthogonal membership. A and A- exhibit strong assembly activations at different time bins (panel **B**–arrows vs. arrow head). However, when 16 ms time bins were used, the activities of these assemblies were packed in the same time window, causing the merge of A and A-into A.

of each pattern across neurons, and then recalculating the SI. We ran 10,000 permutations in order to construct a null hypothesis distribution. Two patterns were regarded as *representations* of the same assembly if their original SI was larger than the 99th percentile of the null hypothesis distribution (i.e., *p* = 0*.*01). Using this process, we found that both A and A- were significantly similar to assembly A (**Figure 2C**). This indicates that units with larger weights in assembly A were split in two independent (*SI* = 0*.*016) assemblies A and A- comprising partially nonoverlapping sets of units (respective action potentials indicated by blue and green dots in the raster plot of **Figure 2B**, respectively). Considering that large bin sizes may conceal fast assembly sequences (**Figure 2B**), we chose the 5 ms bin as a compromise between a high temporal resolution and the need to avoid small bin sizes close to the neuronal refractory period.

## **SEARCHING FOR ASSEMBLIES IN DIFFERENT EXPERIMENTAL PERIODS**

After defining bin size, we focused on the assessment of the differences among assemblies detected using spike matrices from different experimental periods (PRE, EXP and POST). Our goal was to investigate whether the exposure to novel objects changes the assembly repertoire. At first we ignored sleep states and extracted assembly patterns from entire PRE, EXP and POST-WK periods (each one independently). Next, we used the SI to compare all assemblies between experimental periods.

We found little variation in the numbers of assemblies across different experimental periods (**Figure 3A**). Most animals showed a maintenance or minor decrease in the number of assemblies from PRE to EXP, except for rat # 2, which showed an increase of one assembly. From EXP to POST, the number of assemblies detected also dropped slightly, except for rat # 3, which showed a stable number of 10 assemblies per period. Rat # 1 showed the highest variance in the number of assemblies detected across periods, ranging from 13 in PRE to 10 in POST. **Figure 3B** illustrates the substantial similarity between assemblies detected in different experimental periods for rat # 2, which overall showed the largest number of assemblies. To assess assembly conservation across experimental periods, we then categorized the assemblies within each experimental period as showing unitary correspondence, non-unitary correspondence, or no correspondence. An assembly was considered to show unitary correspondence when it was significantly similar to only one assembly in each of its flanking experimental period(s) with *p <* 0*.*0001; non-unitary correspondence defined assemblies which showed more than one correspondence or, in the case of EXP, those with correspondence to one assembly from a flanking period but not with the other (e.g., correspondence with PRE but not with POST); the no-correspondence category comprised assemblies showing no significant correspondences. Group results across different experimental periods (**Figure 3C**) show that the number of assemblies exhibiting unitary correspondence was significantly higher than those showing non-unitary correspondence or no correspondence, including EXP which is flanked by two neighbor periods (Wilcoxon ranksum test, *p <* 0*.*05, Bonferroni corrected).

A comparison across experimental periods reveals that the percentage in PRE of assemblies with no correspondence was slightly elevated, while non-unitary correspondence was very minor. During EXP the percentage of non-unitary correspondences increased, while the percentage of unitary correspondences and no-correspondences decreased. This could represent the fact that EXP is flanked by two neighbor periods, while PRE and POST are flanked by only one. Another possible explanation is that the exposure to novel objects could have changed some assembly activation patterns, increasing their co-activations (see **Figure 6B**), and causing separate assemblies to be detected as one. This may decrease the SI, leading to non-significance between similar assemblies, and/or to significant similarity of one assembly with two or more assemblies from flanking periods, comprising significant but lower SIs. The POST period showed the highest

**FIGURE 3 | Cell assemblies are highly conserved across experimental periods. (A)** Number of assemblies detected using spike matrices from the different experimental periods. **(B)** SI values among assembly patterns of rat #2 across experimental periods. Assembly patterns were detected using a 5 ms bin size. Assembly labels were sorted to let highest values in the main diagonal. **(C)** For each experimental period, the panels show the percentages of assemblies within each of the categories defined by the number of significant correspondences between the assemblies of a given experimental period and the assemblies from flanking periods (from top to bottom, PRE, EXP and POST). Two assembly patterns were deemed correspondent if their SI was above a threshold set by a bootstrap procedure (*p* = 0*.*0001). The categories were defined as unitary correspondence, non-unitary correspondence and no correspondence, representing the percentage of assemblies within rats that showed, respectively, a single correspondence between flanking periods, two or more flanking correspondences, or no correspondence whatsoever. Note that the percentage of assemblies within the unitary correspondence category was considered significantly higher than the other categories for all experimental periods (Wilcoxon ranksum test, <sup>∗</sup>*p <* 0*.*05, Bonferroni corrected).

percentages of assemblies in the unitary correspondence category, with a very small percentage of assemblies in the non-unitary and no-correspondence categories. This indicates that the typically smaller number of assemblies in POST (**Figure 3A**) comprises a subset of assemblies that is essentially the same as in EXP. Across all animals, we found an average of only one EXP assembly per rat that showed no correspondence to any PRE assembly, and yet had correspondence with a POST assembly. This points to a very high conservation of assemblies across experimental periods, and rules out the possibility that new assemblies are formed within EXP and reverberate during POST. For this reason, we continued our investigation of assembly sequences by extracting the assembly patterns from a concatenated spike matrix of all WK intervals (PRE+EXP+POST), and then projecting the assembly activity over the entire recording, throughout the wake-sleep cycle. Using this approach, we detected 11, 18, 10, 13 and 13 assemblies for rats # 1 to # 5, respectively.

## **DETECTING ASSEMBLY ACTIVATIONS**

In order to improve the time resolution for the analysis of assembly activation sequences, we first re-binned the spike trains from each unit using 1 ms bins, and convolved the data with a Gaussian kernel (maximum = 1, 80% of the AUC within 5 ms windows). Then we projected the activity of all assemblies, and defined a threshold (for each assembly) as the 99th percentile of the distribution of activity values across time bins (**Figure 4A**, red lines). **Figure 4A** shows the activity of three exemplary assemblies (A, B, and C) from rat # 1, which above-threshold peaks are depicted by red, blue and green letters (assembly activations), respectively. Subsequent assembly activation was only considered after a "refractory" period of 3 ms elapsed.

## **CALCULATION OF ASSEMBLY GRAPH ATTRIBUTES**

We constructed the assembly activation sequence by labeling and concatenating assembly activations from different assemblies (**Figure 4A**, bottom). Graphs were built from this sequence, so that each assembly corresponded to a node, each edge corresponded to the temporal sequence of consecutive node activations, and the time intervals between two assembly activations were considered inter-activation intervals (IAI) (**Figure 4A**, bottom). The coactivation of two or more assemblies within the same time bin was represented as an additional node in the graph, whose label comprised the labels of the assemblies activated at the same time. For instance, if assemblies F and J displayed synchronous activation, a fourth node FJ was added to the graph, always in the alphabetical order (**Figure 4B**).

Two parameters shaped the graphs: maximum IAI and number of activations per graph (activation count). The maximum IAI parameter defined the threshold IAI within each graph, i.e., every time interval between assembly activations within a graph should be less than or equal to this maximum IAI. Seven different maximum IAI values ranging from 10 to 1000 ms were explored.

An initial assessment of the data varying only the maximum IAI criterion showed that, in general, the assembly graph attributes were proportional to the activation count in a graph (**Figure 4C**, median of absolute Pearson correlation indexes distribution = 0.74), while the duration (the interval between the first and last assembly activation within a graph) was not correlated to assembly graph attributes (**Figure 4C**, median of absolute Pearson correlation indexes distribution = 0.18).

A fixed number of assembly activations per graph was used to control for this variability in the graph attributes. Since the minimum activation count necessary to maximize the density of a graph (**Table 1**) is the square of the number of nodes –*Number of Assemblies*2, we evaluated seven values of activation count as percentages of *Number of Assemblies*<sup>2</sup> (10, 20, 50, 100, 120, 150 and 200%). The custom-made java software *Speechgraphs* (Mota et al., 2012; http://neuro*.*ufrn*.*br/softwares/ speechgraphs) was used to calculate 13 assembly graph attributes (**Table 1**).

## **CHANGES IN POPULATION RATE DO NOT EXPLAIN THE ACTIVITY OF INDIVIDUAL ASSEMBLIES**

The algorithm to algebraically define assembly activity was the squared linear combination of the firing rate of the units in a

**FIGURE 4 | Determination of sequences of cell assembly activations. (A)** 1.5 s interval showing activity of 3 assemblies (A–C) of rat # 1 (3 top panels). Thresholds are the 99th percentiles of the activity values for each assembly. Threshold-crossing peaks are considered assembly activations. Assembly activation sequence is defined as the series of activation across different assemblies within subjects; and the time interval between two subsequent activations is called inter-activation interval (IAI) (bottom panel). **(B)** Exemplary graph generated with assembly activations from the first WK

episode of rat # 1 during PRE. **(C)** Distribution of absolute Pearson correlation values between graph attributes and two other variables: activation count and graph duration. Graphs were generated using assembly activation sequences from behavioral states' episodes. Panel shows distribution of data from all episodes. Note that activation count was generally correlated with graph attribute values in our dataset (median = 0.74, 74% of correlations were significant with *p <* 0*.*05), while the graphs duration were not (median = 0.18, 8% of correlations were significant with *p <* 0*.*05).

#### **Table 1 | Graph attributes.**


given time bin (Lopes-dos-Santos et al., 2011, 2013). Hence, while assembly activity is dependent on population firing rate, it is not fully determined by it, because its projection also depends on the weight of each unit on that specific assembly.

A plethora of studies have shown that firing rate changes convey behavioral information (Adrian and Zotterman, 1926; Hubel and Wiesel, 1959; O'Keefe and Dostrovsky, 1971; Moritz et al., 2008); thus, it was first important to show that assembly activity is not just an epiphenomenon of population rate. To address this issue, we plotted the squared mean population rate against the mean of all assemblies' activity within each bin along the whole experiment for each rat (**Figure 5A** for rat # 1, dark red dots). The *R*<sup>2</sup> of the linear fit between these two variables was low for all animals (**Figure 5B**), indicating that they display a weak correlation. We then plotted the same squared mean of the population rate against the mean assembly activity projected using spike matrices with surrogated rates within each single bin (**Figure 5A**, dark green dots). This allowed us to vary one of the variables that define assembly activity (weights of each unit within each assembly), while keeping the other unchanged (population rate). This approach showed linear fits with even lower *R*<sup>2</sup> values (**Figure 5A**, light green line for rat # 1 and **Figure 5B** for all rats).

Next we investigated activity time-series of individual assemblies (**Figure 5C**, exemplary assembly from rat # 1). **Figure 5D** shows *R*<sup>2</sup> values for the linear fits from all individual assemblies as in **Figure 5C**, for all animals (real data—left; surrogated data—right). All values are very low, and become even lower when we use the surrogated dataset, including a statistically significant difference in *R*<sup>2</sup> values between real and surrogated datasets, for rats # 1 and # 5. (**Figure 5D**, asterisk, Wilcoxon signed-rank paired test, *p <* 0*.*05). Altogether, these results indicate that the activity of individual assemblies is not reducible to fluctuations of the population firing rate.

#### **ASSEMBLY ACTIVATION RATE AND COACTIVATIONS**

We analyzed assembly activation time-series (**Figure 6A**, exemplary plot from rat # 5) from all behavioral states (WK, SWS and REM) and experimental periods (PRE, EXP and POST). Considering all rats, we found that the assembly activation rate during WK was significantly higher in almost all the paired comparisons (18 out of 21) of experimental periods between behavioral states (gray lines with asterisk, *p <* 0*.*05, Wilcoxon ranksum test, bootstrap corrected). Moreover, in all rats the assembly activation rate during POST SWS was significantly higher than during PRE SWS (**Figure 6A**, exemplary plot from rat # 5, black line with asterisk), which suggests that the increase in firing rates after novel object exploration (Ribeiro et al., 2007) may underlie the elevated co-firing of assembly neurons. Interestingly, two out of the three rats that displayed REM during PRE and POST, showed elevated activation rate after the experience. Previous work with larger groups including the present dataset showed no significant firing rate change between PRE REM and POST REM (Ribeiro et al., 2007). The distribution of assembly coactivations followed the same pattern of the assembly activation rate, in which POST SWS displayed higher values than PRE SWS for all rats. The number of coactivations was also higher during WK than during sleep (**Figure 6B**, exemplary plot from rat # 5); with significant differences in 18 out of 21 possible comparisons.

**fluctuations. (A,C)** show exemplary panels from rat # 1 and **(B,D)** show group data. **(A)** Squared mean of the population rate and the mean of all assemblies' activity within each 1 ms bin (dark red dots). In order to scramble associative behavior and keep the firing rate fixed, we also plotted the mean population rate against the mean assemblies' activity projected using the spike matrix with neurons' labels surrogated within each time bin (dark green dots). Light red and green lines depict the least square linear fit for each color coded subset of points along with the correspondent coefficients of determination (*R*2). **(B)** Coefficient of determination distribution for all rats.

from rat # 1. **(D)** Shown are distributions of all rats *R*<sup>2</sup> values for the linear fits from the correlation between mean population rate and individual assemblies' activity (left) and mean population rate and individual assemblies' activity estimated from surrogated spike matrices (right). Note that both distributions exhibit very low *R*<sup>2</sup> values and that there is a decreasing trend from real to surrogated data, with significant difference for rats # 1 and # 5 ( <sup>∗</sup>*p <* 0*.*05, Wilcoxon signed-rank paired test).

## **GRAPH ANALYSIS**

We found that graph attributes varied significantly across behavioral states and experimental periods (**Figure 7**). We tested therefore whether a Naïve Bayes classifier could extract, from the assembly graph attributes, information enough to sort behavioral states and experimental periods (John and Langley, 1995). We used the java software *Weka* (http://www*.*cs*.*waikato*.*ac*.*nz/ ml/weka/) to perform the classifications and estimated their quality by the area under the receiver operating characteristic curve (AUROC). **Figures 8A,B** show that it was possible to sort behavioral states with very high quality of classification, particularly when WK and REM were compared (maximum AUROCs ranging from 0.78 to 0.98). WK and SWS could also be distinguished, at a somewhat lower level (maximum AUROCs ranging from 0.69 to 0.96). The poorest quality of classification was obtained by sorting SWS from REM (maximum AUROCs ranging from 0.64 to 0.78).

The classification quality across experimental periods was not as good as across behavioral states (median across rats 0.57 vs. 0.69, Wilcoxon ranksum test, *p <* 0*.*01), except for rat # 1. **Figures 8C,D** show that the maximum AUROC values for the comparisons between experimental periods ranged from 0.55 to 0.99, with distribution of all values yielding 0.52 and 0.67 as the first and third quartiles, compared to 0.58 and 0.84, as quartiles for the comparisons between behavioral states. We found a strong positive correlation between the AUROC of graph attributes and activation count for all the comparisons made (e.g., rats # 4 and # 1 in **Figures 8A,C**). One example of this correlation is shown on a plot of the AUROC values from the classification between PRE WK and PRE SWS vs. the activation count of the graphs of rat # 1, considering only values obtained using the 1000 ms maximum IAI (**Figure 8E**). The figure shows a positive correlation associated with an extremely strong linear fit (*R*<sup>2</sup> = 0*.*95) and a 1*.*2 × 10−<sup>3</sup> slope, in association with major variation in AUROC values (full range: 0.54–0.80). To test if this was a general effect of assembly count on AUROCs and to analyze the general effect of maximum IAI on AUROCs, we plotted the AUROCs vs. the activation counts along a constant maximum IAI; and

Behavioral states boxplots are color coded as red, blue, and green for WK, SWS and REM, respectively. Experimental periods (PRE, EXP and POST) are placed together and in chronological sequence within each behavioral state. Black lines with asterisks reflect significance between two different experimental periods within a given behavioral state. Gray lines with asterisks reflect significance between two different behavioral states within a given experimental period (*p <* 0*.*05, bootstrap corrected for multiple comparisons).

the AUROCs vs. the maximum IAIs considering a constant activation count for the panels from all rats. Note that activation count accounts for AUROC variability significantly more than the maximum IAI, except for rat # 1 (**Figure 8F**), according to a positive correlation (**Figure 8G**). It is important to note that there was no AUROC above 0.68 when we used maximum IAIs below 20 ms. Maximum AUROCs were obtained using each of the seven different activation counts explored.

## **DISCUSSION**

Our results show that assembly graphs comprising synchronized neuronal units recorded from the hippocampus and primary sensory cortices can be used to sort behavioral states (maximum AUROC values ranging from 0.64 to 0.98) and experimental periods (maximum AUROC values ranging from 0.55 to 0.99) before, during and after novel object exploration. This sorting is based on several attributes that reflect the structural properties of assembly graphs. At this point we do not know whether these attributes are informative due to a causal relationship with behavior, or as an epiphenomenon of some other underlying cause. In all, our investigation corroborates the notion that phase sequences, understood as specific patterns of assembly activations, reflect the different regimes of neural processing as animals traverse the wake-sleep cycle and acquire novel information about the environment.

Such interpretation of the results cannot be furthered without addressing the problem of the arbitrary definition of time scale for synchronous firing. As shown in **Figure 2A**, the number of assemblies detected decreases with bin size. We showed evidence that this may be due to the tight temporal association of assemblies detected using smaller bin sizes, which are detected as a single assembly when larger bin sizes are used. Our choice of bin size = 5 ms for the generation of assembly graphs, well within the potentiation window of spike time dependent plasticity (STDP) (Bi and Poo, 1998), represents a compromise between the number of assemblies detected and the need to avoid extremely low bin sizes near the neuronal refractory period.

Our results show that the repertoire of assemblies is almost unchanged across experimental periods, which suggests that novel experience does not create new assemblies in the hippocampus and primary sensory neocortex of the normal adult rat. Our finding is compatible with Hebb's hypothesis that assemblies correspond to the primitive building blocks of representations, being slowly formed across development but nearly unchanged in adulthood. The experience-dependent changes in the structure of assembly graphs, revealed by the use of a classifier, also corroborates the complementary Hebbian hypothesis that relevant information about concepts, percepts and behavior in general is coded at the level of multiple assembly activations, the so called phase sequences (Hebb, 1949).

We also showed that the activity of single assemblies cannot be reduced to the changes in firing rate. Changes in neuronal firing rates constitute well-known indexes of behavior (Adrian and Zotterman, 1926; Hubel and Wiesel, 1959; O'Keefe and Dostrovsky, 1971; Moritz et al., 2008). If phase sequences are indeed important to generate new neural representations, they should carry more specific information than firing rates. Since assemblies are subsets of neurons that function transiently as closed systems, the neurons related to a given perception or behavior should have their rates affected synchronously, so as to be detected as assemblies. The calculation of assemblies and the projection of their activity is a way to reduce the dimensionality of a population of neuronal units onto neuronal subsets which are likely related to behavior. Investigation of whether phase sequences carry more information than firing rates is ongoing.

The automatic sorting of behavioral states using the attributes of assembly graphs reached a very high level, but the sorting of experimental periods was substantially less accurate. The major behavioral states comprise markedly different physiological patterns in the brain (Noda et al., 1969; Vanderwolf, 1969; Hobson and McCarley, 1971; Gervasoni et al., 2004), likely not the case for the experimental periods investigated here. One possible cause for this difference may be the small amount of assemblies detected, due to the under-sampling of the neuronal units actually involved in novel object exploration.

**FIGURE 7 | Assembly graph attributes vary significantly across behavioral states and experimental periods.** Panels show the distribution of graph attributes' values from rat # 5, using 1 s maximum IAI and 169 activations/graph, for different behavioral states and experimental periods. As in **Figure 6**, behavioral states boxplots are color coded as red, blue, and green for WK, SWS and REM, respectively. Experimental periods (PRE, EXP and POST) are placed together and in chronological sequence within each behavioral state. Black lines with asterisks reflect significance between two different experimental periods within episodes of a given behavioral state (*p <* 0*.*05, Wilcoxon ranksum test, Bonferroni corrected). Gray lines with asterisks reflect significance between two different

behavioral states within a given experimental period. Note that nearly all the attributes sorted WK from SWS, during PRE or POST (except for L1 during PRE and L3 during POST). WK was significantly different from REM during PRE (12 attributes) and POST (11 attributes), SWS was significantly different from REM during POST (10 attributes), but no attribute could sort SWS and REM during PRE. Only one attribute was capable of sorting PRE from EXP within WK. When comparing PRE × POST within WK, 12 attributes could separate them. EXP WK graphs were detected as different from POST WK graphs by 3 attributes. PRE SWS could be sorted from POST SWS, and PRE REM could be sorted from POST REM, using any of the graph attributes studied.

**classification of experimental periods and behavioral states. (A,C)** The rows of each panel represent the graphs maximum IAI (within the graph, every IAI is less than or equal to the maximum IAI value), while the columns percentages of the squared number of assemblies. Color codes vary from 0 to 1 and represent the median AUROC of 50 classifications made for 20 *(Continued)*

#### **FIGURE 8 | Continued**

random graphs from each of the experimental periods compared using a Naïve Bayes classifier; e.g., 20 graphs from PRE WK compared with 20 graphs from EXP WK. In some cases of the parameter screening, we could not obtain the minimum 20 graphs necessary for the classification. For instance, it was impossible to generate one single graph comprising 200 activations (200% of *Number of Assemblies*<sup>2</sup> for rat # 3) within the 10 ms maximum IAI. These conditions were coded blue to indicate no classification. The maximum AUROC value of each panel is indicated. **(A,B)** Sorting of behavioral states. **(A)** Panels show the classification quality across different maximum IAI and activation count values for rat # 4. **(B)** Histograms of AUROC values as in panel **(A)** for all rats. Red line depicts the 0.6 AUROC value, which sets the lower bound for a good classification quality. WK and REM were well sorted by graph attributes, with maximum AUROC values ranging from 0.68 to 0.98 for all rats within both PRE and POST periods. The sorting of SWS and REM was substantially less accurate, with maximum AUROC = 0.78 during POST in rat # 5. The sorting between WK and REM was very good for all rats during PRE, (maximum AUROCs from 0.78 to 0.98). **(C,D)** Sorting of experimental periods. **(C)** Panels show the classification quality for rat # 1 across different maximum IAI and activation count values. **(D)** Histograms of AUROC values from the panels as in panel **(C)** for all rats. All the comparisons yielded maximum AUROCs ranging from 0.55 to 0.99. **(E)** Correlation between AUROC and activation count using a

It is important to point out that in the present study we assumed that the activity of a cell assembly could be described as a linear combination of the activity of individual neurons. While this simplification of the assembly model allows for the analysis of large neuronal populations, it also presents some potential caveats (Lopes-dos-Santos et al., 2013). In particular, strong nonlinear correlations between neurons may lead to spurious results, since both the determination of the number of assemblies and the extraction of assembly patterns are based on the linear model. Nevertheless, because this representation of assemblies is intuitive and straightforward, it is possible to verify the outcomes of the analysis; for instance, visual inspection of the raw data confirms that co-activations of assembly members correspond to peaks in assembly activity (see **Figure 2B**, also see examples employing similar linear methods in (Nicolelis et al., 1995; Peyrache et al., 2009, 2010; Benchenane et al., 2010; Lopes-dos-Santos et al., 2011, 2013). In principle, a non-linear method should be more robust and realistic, but we are not aware of any nonlinear method capable of extracting assembly composition from the ongoing activity of neuronal populations with dozens of neurons. An ideal method should also incorporate information on the physiology of specific cell types and neural circuits. Taken together, our results show that, despite any possible non-linear correlations that may exist among neurons, the linear ones carry relevant information that support a role for phase sequences in behavior and cognition. Future research shall include non-linear modeling and also consider a neural coding approach, in order to fully characterize the repertoires of phase sequences, and elucidate the role of specific graph attributes in the representation of contextual cues, sensory stimuli and motor behavior.

## **AUTHOR CONTRIBUTIONS**

Sidarta Ribeiro collected the data; Daniel G. Almeida-Filho, Nivaldo A. P. Vasconcelos, Vitor Lopes-dos-Santos, and Sidarta Ribeiro analyzed the data; Daniel G. Almeida-Filho prepared the figures; Daniel G. Almeida-Filho, Sidarta Ribeiro, Nivaldo A. P. 1000 ms maximum IAI from rat # 1 graphs comparing PRE WK and PRE SWS. The slope of the linear fit indicates that each single activation added to a graph, adds 0.0012 to the AUROC, with activation counts varying from 12 to 242 (AUROCs vary from 0.54 to 0.80). **(F)** Distribution of slopes of the linear fits between activation count and AUROC with fixed maximum IAI value (e.g., panel **E**); and between maximum IAIs and AUROC with fixed activation count. We used the AUROCs from all the comparisons and conditions (maximum IAI and activation counts) for all animals and considered only fits with three or more data points. The analysis shows that the maximum IAI contribution to the AUROC is around zero (mean across rats = 0.0024) and even negative, while the contribution of the activation count is divergent, with a clear majority of positive contributions (mean across rats = 0.097), yielding a significant difference between these two variables, except for rat # 1 **(G)** Distribution of Pearson correlations indexes for the comparisons in panel **(F)**. Note that activation count shows strong positive correlation with AUROCs (medians = 0.92, 0.93, 0.78, 0.80, and 0.91 for rats # 1 to # 5, respectively; 53% of the values with *p <* 0*.*05), while maximum IAIs are scattered, with values spanning the entire scale, and medians closer to zero or even negative for all rats (0.59, −0.33, −0.56, −0.39, and 0.12 for rats # 1 to # 5, respectively; 8% of the values with *p <* 0*.*05). Asterisks indicate significant differences between activation count and maximum IAI distributions of correlation values within the same animal.

Vasconcelos, Vitor Lopes-dos-Santos, Adriano B. L.Tort, and José G. V. Miranda wrote the manuscript.

## **ACKNOWLEDGMENTS**

Support was obtained from the Pew Latin American Fellows Program in the Biomedical Sciences, Financiadora de Estudos e Projetos (FINEP)—Grant 01.06.1092.00, Ministério da Ciência e Tecnologia e Inovação (MCTI), CNPq Universal 481351/2011-6, PQ 306604/2012-4, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), FAPERN/CNPq Pronem 003/2011, Capes SticAmSud, FAPESP Center for Neuromathematics (grant #2013/ 07699-0, São Paulo Research Foundation), and NIMBIOS working group "Multi-scale analysis of cortical networks." We thank N. B. Mota, P. Petrovitch, and R. Furtado for help with the SpeechGraphs software, A. Karla for administrative help, and D. Koshiyama for library support.

## **REFERENCES**


two-year-old rat hippocampus *in vitro*. *Neurobiol. Aging* 12, 517–522. doi: 10.1016/0197-4580(91)90082-U


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 20 December 2013; accepted: 19 March 2014; published online: 08 April 2014.*

*Citation: Almeida-Filho DG, Lopes-dos-Santos V, Vasconcelos NAP, Miranda JGV, Tort ABL and Ribeiro S (2014) An investigation of Hebbian phase sequences as assembly graphs. Front. Neural Circuits 8:34. doi: 10.3389/fncir.2014.00034 This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Almeida-Filho, Lopes-dos-Santos, Vasconcelos, Miranda, Tort and Ribeiro. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Dynamical criticality during induction of anesthesia in human ECoG recordings

#### *Leandro M. Alonso1 \*, Alex Proekt 2,3\*, Theodore H. Schwartz 4, Kane O. Pryor 2, Guillermo A. Cecchi <sup>5</sup> and Marcelo O. Magnasco1*

*<sup>1</sup> Center for Studies in Physics and Biology, The Rockefeller University, New York, NY, USA*

*<sup>2</sup> Department of Anesthesiology, Weill Cornell Medical College, New York, NY, USA*

*<sup>3</sup> Laboratory for Neurobiology and Behavior, The Rockefeller University, New York, NY, USA*

*<sup>4</sup> Department of Neurological Surgery, Weill Cornell Medical College, New York, NY, USA*

*<sup>5</sup> IBM, Thomas J. Watson Research Center, Yorktown Heights, NY, USA*

#### *Edited by:*

*A. Ravishankar Rao, IBM Research, USA*

#### *Reviewed by:*

*Enzo Tagliazucchi, Goethe University Frankfurt, Germany Uncheol Lee, University of Michigan Medical School, USA*

#### *\*Correspondence:*

*Leandro M. Alonso, Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Ave., New York, NY 10065, USA e-mail: leandrosmandes@ gmail.com; Alex Proekt, Department of Anesthesiology, Weill Cornell Medical College, 1300 York Ave., New York, NY 10065, USA e-mail: proekt@gmail.com*

**1. INTRODUCTION**

It has been suggested that neural systems operate in a critical regime similar to phase transitions in physics, given several computational desirable features of such states represented by the statistics of the thermodynamic variables (Chris, 1990). Evidence for statistical criticality is based on the observation that various aspects of neuronal activity such as avalanches observed in local field potentials and action potentials in tissue preparations and in animal models (Gireesh and Plenz, 2008; Ribeiro et al., 2010), as well as magneto-encephalography (MEG) and electro-corticography (ECoG) in human subjects (He et al., 2010; Shriki et al., 2013), exhibit long tailed-distributions well approximated by power laws. The critical regime provides important functional benefits; quantities such as dynamic range and information transmission are optimized near criticality (Shew and Plenz, 2013).

More recently, the dynamical aspect of criticality has been brought into focus, as a similarly desirable feature not fully captured by steady-state statistics such as avalanche size distributions (Magnasco et al., 2009; Chialvo, 2010; Mora and Bialek, 2011; Beggs and Timme, 2012); a perturbation in an extended dynamical system that is close to a critical point will neither decay nor explode, thus allowing for long range communication across the entire system. In contrast, if the system is far from criticality (therefore stable), perturbations damp out and no information integration takes place beyond the characteristic damping time scale (Tononi, 2008).

In this work we analyze electro-corticography (ECoG) recordings in human subjects during induction of anesthesia with propofol. We hypothesize that the decrease in responsiveness that defines the anesthetized state is concomitant with the stabilization of neuronal dynamics. To test this hypothesis, we performed a moving vector autoregressive analysis and quantified stability of neuronal dynamics using eigenmode decomposition of the autoregressive matrices, independently fitted to short sliding temporal windows. Consistent with the hypothesis we show that while the subject is awake, many modes of neuronal activity oscillations are found at the edge of instability. As the subject becomes anesthetized, we observe statistically significant increase in the stability of neuronal dynamics, most prominently observed for high frequency oscillations. Stabilization was not observed in phase randomized surrogates constructed to preserve the spectral signatures of each channel of neuronal activity. Thus, stability analysis offers a novel way of quantifying changes in neuronal activity that characterize loss of consciousness induced by general anesthetics.

**Keywords: criticality, anesthesia, ECoG, depth of anesthesia monitoring, consiousness, dynamical systems**

While models of self-organized criticality exhibit both dynamically and statistically critical behavior (Bak et al., 1987; Gil and Sornette, 1996), the two aspects of criticality are not necessarily related. The winnerless network provides an illuminating example: under very generic conditions, neural systems can display a phase space determined by heteroclynic orbits connecting saddle nodes (i.e., at least one unstable manifold), such that the resulting dynamics are quasi-periodic cycles over the nodes, without necessarily exhibiting statistically critical distributions (Rabinovich et al., 2001; Aguiar et al., 2011; Ashwin et al., 2011). A model connecting statistical and dynamical criticality in neural systems was proposed recently by Magnasco et al. (2009). They consider an abstract model in which the activity of a set of neurons is encoded in a *N*-dimensional vector *x* which evolves in time according to a *N* × *N* connectivity matrix *A*, characterized by its set of *N* eigenvalues {λ*n*}. By assuming anti-Hebbian dynamics for the connectivity matrix a very rich dynamical scenario emerges. The eigenvalues of the matrix *A* evolve toward the dynamically critical point *Re*(λ*n*) ≈ 0 ∀*n* and the solutions of the model exhibit complex spatio-temporal dynamics, as well as long tailed avalanche distributions and other signatures of statistical criticality. Consistent with this observation, experimental evidence of both statistical and dynamical criticality was reported in human ECoG recordings; however, the precise mechanism by which critical dynamics occur has not been investigated. The analysis showed that the eigenvalues crowd near the critical line, and moreover that task performance (finger tapping) implies a subtle but significant decrease in dynamical criticality, presumably because the modes related to motor execution impose higher stability (Solovey et al., 2012). Of note, signatures of statistical criticality were not strongly affected by task performance.

If indeed dynamical criticality is a useful feature of brain activity rather than an epiphenomenon, stability of neuronal dynamics ought to be modulated by the behavioral state of the subject. Here, we hypothesized that a particularly dramatic change in stability accompanies changes in the level of wakefulness (consciousness). When the brain is awake and displaying complex behavior its dynamical state ought to be close to a bifurcation point; marginally stable modes contribute to long range interactions across the system. Conversely when higher-order functions associated with wakefulness have been diminished and eventually completely disrupted by anesthetics, brain dynamics should exhibit more stability. In other words, anesthesia induction should lead to stabilization of brain dynamics.

Changes in the level of arousal (wakefulness) have been historically quantified using spectral analysis of neuronal activity. In this view, decrease in the level of wakefulness is reflected in the increase and prevalence of low frequency oscillations and the concurrent decrease in the high frequency oscillations reviewed in Brown et al. (2010). While this is true for some states of decreased arousal such as slow wave sleep, this association breaks down during other states in which arousal is similarly depressed such as rapid eye movement (REM) sleep for instance. Furthermore, state of general anesthesia can be characterized by different spectral signatures depending on the specific choice of anesthetic agent (Maksimow et al., 2006). This makes current modes of detecting the "depth of anesthesia" unreliable (Avidan et al., 2011).

Lack of clear association between changes in the spectral content of brain signals and level of arousal is not entirely surprising. It is likely that the overall level of wakefulness is a consequence of the interactions among many brain regions rather than any specific feature of neuronal activity observed at any one region taken in isolation. Therefore, more recent efforts have been aimed at detecting decreases in arousal using connectivity measures based on spectral coherence as well as mutual information and phase relationships among brain activity recorded simultaneously at multiple locations (Imas et al., 2005; Cimenser et al., 2011; Lee et al., 2012). While this connectivity analysis does suggest that integration of information between different brain regions may be decreased when the level of wakefulness is reduced, it is not trivial to relate changes in connectivity to the changes in global dynamics of the brain.

To address the dynamics, we fitted vector autoregressive (VAR) models to ECoG signals collected directly from the cortex of human subjects as they were gradually induced into the state of general anesthesia. These models were independently fit to short temporal windows with an arbitrarily large overlap. Thus, while we assume that the dynamics are locally linear and stationary over a short temporal window, on a longer time scale the dynamics are expected to be arbitrarily non-linear and non-stationary. This locally linear approximation allows us to quantify the changes in stability of brain activity in terms of temporal evolution of the distribution of eigenmodes of the fitted models. As previously reported (Solovey et al., 2012), we found a prevalence of critical eigenmodes across the entire recordings. However, the stability of the models shows statistically significant differences across different stages of induction. While the distribution of eigenvalues changes in non-trivial ways, high frequency modes become more damped as anesthesia is induced. Moreover, modes closer to criticality, regardless of frequency, show a gradual shift to stability spanning several drug volleys over approximately 20 min.

This work is organized as follows. In the next section we describe the induction protocol and the analysis method. We present our results in section 3. In section 4 we summarize and discuss our findings.

## **2. METHODS**

All experimental protocols were approved by the IRB at the Weill Cornell Medical College (protocol number 1106011763). After obtaining informed consent, three subjects undergoing surgical treatment for intractable epilepsy were enrolled in this study. Subdural electrode grids and strips (Ad-tech, Medical Instruments Corp., Racine, WI) were implanted for the purposes of localization of the epileptogenic loci. The location and the number of electrodes were determined by the clinical considerations (temporal lobe for all subjects in this study). After the initial implantation of the subdural electrodes, the subjects underwent video and EEG monitoring, duration of which was dictated solely by clinical considerations (1–2 weeks in these subjects). The recordings analyzed in this work were obtained during induction of anesthesia for the second craniotomy performed after completion of this observation period. During induction of anesthesia (see below), blood pressure, ECG, heart rate, pulse oxymetry, and end tidal carbon dioxide were monitored and maintained within normal limits. Patients were given supplemental oxygen via nasal cannula.

After obtaining baseline recordings (without any pre-medication) anesthesia was gradually induced using target controlled infusions of propofol using pharmacokinetic parameters derived by Schnider et al. (1999), administered using STANPUMP. Target propofol concentration was increased slowly while the level of sedation was accessed using responses to simple verbal commands. Propofol infusion continued until subjects lost the ability to respond to verbal commands. At this point additional propofol, opioids, and neuromuscular blockers were administered (at the discretion of the anesthesia provider) and trachea was intubated. Recordings were terminated at this point.

Recordings were obtained using SynAmps<sup>2</sup> (Neuroscan) using DC coupled recording. Data were acquired at 10 KHz. 64 channels of ECoG signals were acquired in each subject. While both conventional EEG and ECoG are thought to primarily reflect the sum of synchronized postsynaptic potentials of neurons in the vicinity of the electrode, the invasive nature of the ECoG signals allows for much greater signal to noise ratio and significantly improves spatial and temporal resolution of the signals.

No online filtering was performed. ECoG data was collected from three human subjects as they were induced into general anesthesia. For all subjects, the infusion started 60 s into the recording and the concentration of anesthetics was increased every 300 s. For **Subject 1**, propofol infusion started 60 s into the recording. 360 s into the recording the subject reports being awake. 510 s into the recording the subject no longer responds. 960 s into the recording the subject is given additional drugs and intubated. For **Subject 2**, propofol was incremented at 300, 600, and 900 s. At 660 s the subject opened eyes. 720 s into the recording the subject no longer responded. 1140 s into the recording subject was given additional drugs and was intubated. For **Subject 3**, propofol infusion started 60 s into the recording. The concentration was increased every 300 s and maintained constant before and after. 900 s into the recording the subject no longer responded to verbal commands or light taps on the shoulder. 1200 s into the recording subject was given additional drug and was intubated.

Data was bandpass filtered at 0.1 − 500 Hz and detrended in segments of 10 s. We applied notch filters at 60 ± 2, 120 ± 2, and 180 ± 2 Hz. Finally, the amplitude of the signal in each channel was normalized by its standard deviation. For our analysis data was partitioned in equally sized windows of τ = 200 ms centered every *tj* <sup>+</sup> <sup>1</sup> − *tj* = = 100 ms. In each window, we assumed that the dynamics is locally linear and fitted a vector auto regressive model (VAR) of order *p* = 1.

$$
\varphi\_{n+1} = A\varphi\_n + \mu\_n \tag{1}
$$

where *yn* are the fitted values, *A* ∈ R*N*×*<sup>N</sup>* is the matrix to be estimated and *un* is assumed to be white noise. Here, *yn* ∈ R*<sup>N</sup>* is a multivariate time series that represents the recorded activity in all channels at time *tn* and *A* corresponds to the LAG 1 correlation between channels. A comprehensive treatment of this model and its estimation can be found in Lütkepohl (2006). In this work we used a python implementation of Schnider's et al. algorithm to estimate *A* (Schneider and Arnold, 2001). This procedure yields a set of matrices *Aj* which govern the stability properties of the VAR model at time *tj*. In order to address changes in the stability of the fitted models we considered the distribution of the modulus of the eigenvalues at each time step. Also, since our underlying hypothesis corresponds to a continuum model we performed a transformation in order to obtain a correspondence between the eigenvalues of *Aj* and the timescales of the dynamics. Let λ*<sup>j</sup>* = ρ*je<sup>i</sup>*<sup>φ</sup> be the eigenvalue corresponding to the *j*-th mode, the frequency of the mode is given by

$$f\_{\vec{\jmath}} = \frac{\phi\_{\vec{\jmath}}}{2\pi \, dt} \tag{2}$$

while the growth rate (timescale) of the mode is given by

$$\pi\_{\dot{\jmath}} = \frac{\log(\rho\_{\dot{\jmath}})}{dt} \tag{3}$$

Here *dt* = <sup>1</sup> *Sf* <sup>=</sup> <sup>0</sup>.0001 s, where *Sf* is the sampling frequency of the recordings. A mode with eigenvalue λ is critical if

$$\|\lambda\| = 1\tag{4}$$

In practice however, we call a mode critical if λ ≈ 1 (thus τ ≈ 0 s). These are modes which are close to alternate their behavior between damping and growth (Strogatz, 2006).

The distributions so obtained were compared to the initial distribution (prior to induction) by means of two statistical tests. Kolmogorov-Smirnov (KS) tests the null hypothesis that the distributions are the same and yields the maximal difference of the cumulative distributions to quantify for the changes. Wilcoxon rank-sum (W) tests the null hypothesis that the distributions are the same against the alternative hypothesis that they are shifted and returns a *z*-value to account for the magnitude of the shift. If the values of the subsequent distributions are smaller than those of the reference distribution (awake state) then *z* > 0, therefore, an increase of the *z*-value indicates an increase of the stability.

We settled on a VAR-1 model because the main results related to the effect of anesthesia are robust for VAR-2 and VAR-3 models. We have explored window sizes ranging from 100 ms to 1 s and found no significant changes. Our method was tested against surrogate data obtained by phase randomization of the signal; for each channel we computed the Fourier transform of the signal, changed the phase value by a random number [drawn from a flat distribution in (0, 2π )] and transformed back to obtain the surrogated signals. Note that by construction this procedure preserves the power spectrum of each signal.

## **3. RESULTS**

We performed VAR analysis on three human subjects as they were induced into general anesthesia. Our primary focus was to detect changes in the distribution of the stability parameters λ*j* during induction of anesthesia. To quantify changes in the stability of the models we used two non-parametric statistical tests [Kolmogorov-Smirnov (KS) and Wilcoxon rank-sums(W)]. The results of this analysis are shown in **Figure 1A** [for each subject top row shows (KS) and bottom row shows (W)]. To improve visualization the results were smoothed using moving average windows of 10 s. The distribution of eigenmodes computed over different windows during the awake state fluctuates. To scale the observed differences in stability during induction of anesthesia by these spontaneous fluctuations, we computed the time average of both KS and W statistics over the awake period (1 min) and subtracted this value from the curves shown in **Figure 1A**. In all cases, the temporal average of the *p*-values behaves similarly to the KS-Z values. During the first minutes of the procedure we find that *p* ≈ 0.75, thus, the null hypothesis that the distributions are the same cannot be safely rejected. However, we find a drastic drop of the *p*-value concomitant with changes in KS-Z values. For the regions indicated in blue and green (**Figure 1A**), the average *p*-value of both tests are in the range of 0.2 − 0.3 suggesting that the distributions have changed. While the KS test simply indicates that the distributions of stability parameters during awake and anesthetized states are different, the increase in the *z*-values of the Wilcoxon test implies that λ*j* tends to decrease with induction of anesthesia, i.e., the dynamics is becoming more stable. Note that the change in the distribution of the stability parameter is not observed in phase randomized surrogates (red curves in **Figure 1A**). Thus, the observed changes in stability are not given by the spectral properties of neuronal activity.

Note that in general the eigenvalues of the autoregressive matrices fitted to the ECoG signals are complex numbers whose real and imaginary parts give rise to the timescale τ and frequency *f* of the corresponding eigenmode (see Equations 2, 3). While **Figure 1A** focused just on changes in the distribution of the stability parameters, **Figure 1B** shows changes in both the distribution of timescales (abscissa) and bulk frequencies (ordinate) treated independently. Time elapsed since the onset of experiment is color coded from red (awake) to blue (anesthetized). The

**FIGURE 1 | ECoG signals were recorded from three human subjects as they were induced into general anesthesia.** Data was locally fitted with VAR(1) models in windows of 200 ms every 100 ms (see methods). The linear stability of each model is compared to the awake state by means of two statistical tests. **(A)** top rows: Kolmogorov Smirnov test. For each model we plot the KS statistics of comparing the fitted distribution of eigenvalues against the awake state. **(A)** bottom rows: The distribution of time scales is compared using Wilcoxon test. Both quantities were averaged in time intervals of 10 s. The stability properties of locally fitted VAR(1) models change as the subjects undergo anesthesia. We defined three different segments (color rectangles) which were used for subsequent figures. **(B)** Changes in the frequency and stability of the eigenmodes. We compared the distributions of frequencies and time scales using a Wilcoxon test. In each figure, the vertical axis shows the *z*-value of comparing the frequency distributions whereas the horizontal axis shows the same test for the stability parameters distributions. The color code represents time elapsed since the beginning of the recording. In this representation all realizations yield qualitatively similar results: as the subjects are induced, the fitted frequencies shift to higher values at the same time they become more damped.

bulk evolution of the eigenmodes is consistent in all subjects: as induction progresses, modes shift to higher frequencies while they become more stable. To validate that the results obtained with the VAR-1 model are robust, we show in **Figure A1** (included as Appendix) the same analysis as in **Figure 1A** implemented with a VAR-3 model. As it can be seen, the changes in the distribution of eigenmodes are almost identical to those for VAR-1.

While **Figure 1B** suggests an increase in the bulk frequency and decrease in the time constant, this does not fully characterize the way in which anesthetics change the distribution of eigenvalues in the plane spanned by timescale and frequency. **Figure 2** shows how we represent the distributions of the eigenvalues of *Aj*. The vertical axes corresponds to frequencies plotted on a logarithmic (base 2) scale. Horizontal axes indicate the modes damping/growth timescale. The sign indicates whether the mode's amplitude is growing (positive) or decaying (negative). Histograms are color coded with blue indicating low occupancy to red indicating high occupancy. Note that the damping time and

The count in each bin is color-coded and the number of samples is *N* > 106. The frequency axis is in logarithmic scale (base-2). The arrows indicate points in the stability plane for which the qualitative dynamics of the corresponding mode is illustrated. The dynamics of each mode can be expressed as an oscillation of frequency *f* whose amplitude (red curves) is modulated by an exponential decay/growth (blue curve). Each solution is shown for 2/τ s. Note that for the points in the plane with non-zero count, a number of oscillations occur before the mode is damped out. For the case labeled with , the mode grows exponentially (i.e., it is super-critical).

frequency are not independent and modes with lower frequencies tend to have longer damping times, with slow oscillations found near the critical point (τ ≈ 0). Traces on the margin of the figure illustrate the dynamics for particular pairs of damping time and frequency. Note that the traces are plotted on the timescale commensurate to the damping time rather than on an absolute time scale. The inter-relationship between damping time and frequency assures that most modes located along the most densely populated ridge go through several complete cycles before being damped out, while the modes located to the left of the dominant ridge are damped out earlier and are thus less likely to carry out meaningful computations performed by the brain.

**Figure 3A** shows the distribution of eigenvalues in the plane introduced in **Figure 2** during three stages of the induction process (100 s segments shown in **Figure 1A**). In order to better resolve the distributions we performed a moving VAR analysis with *tj* <sup>+</sup> <sup>1</sup> − *tj* = 1 ms of spacing between adjacent windows. In order to visualize changes in the eigenvalue distributions we normalized the count value of each histogram by its maximum. Then, we used the normalized values in each bin to code for color in RGB space as indicated in the filled circles. **Figure 3B**, correspond to the superposition of such images. In this way, regions of the stability space that are similarly occupied in the three stages are coded in gray scale [with white corresponding to maximal occupancy (1,1,1)] and pure colors RGB correspond to values that are exclusive to the first, second and third stage respectively. A prominent feature shown by these panels is the shift of high-frequency eigenmodes toward

decay timescale of each mode. Figures correspond to 2D histograms of this quantities. The count in each bin is color coded and the number of samples is *N* ≈ 106. The frequency axis is on logarithmic (base 2) scale. **(A)** Distributions

increased stability. While the full worm-like distribution of eigenvalues changes in subtle ways, the left-ward shift in these frequencies is ubiquitous in all subjects. **Figures 4A,B** correspond to vertical and horizontal "slices" respectively of the histogram shown in **Figure 3A** for subject 1. **Figure 4C** shows details of how these distributions change for subject 1. We performed the same comparison as before but choosing a smaller frequency range for computing the histograms. The shift to damped states is more pronounced for modes with frequencies that are greater than 64 Hz.

Finally, we investigated how the distribution of the most critical modes is affected by induction. This was partially inspired by results previously reported in human ECoG, showing that differences between task and resting conditions can be detected precisely by changes in these populations (Solovey et al., 2012). We show in **Figure 5** the result of comparing the distribution of modes truncated to eigenvalues with damping constant above a given threshold close to criticality. For all subjects, the distributions show a gradual change in the stability of near-critical modes along the entire span of the induction process **Figure 5A**. This is somewhat surprising, as the induction process is controlled

red, pure green and pure blue correspond to eigenvalues that are only present in the first, second and third stage. The rainbow-like pattern indicates a shift of high frequency modes as they become more damped.

by discrete events of drug increase which notably affect the full eigenmode distribution.

## **4. CONCLUSIONS**

Dynamical systems theory indicates that systems that are capable of performing computations should have a large number of modes with marginal stability. In such a scenario an arbitrary perturbation will not decay or explode, thus allowing for information integration across the entire system. Previous work suggest that the brain might operate in a dynamically critical regime (Magnasco et al., 2009; Solovey et al., 2012). A simple model exhibiting complex spatio-temporal dynamics was recently proposed, in which statistically critical behavior emerges due to dynamical instabilities. Within this framework we tested the hypothesis that the stability properties of the system change as anesthesia is induced; specifically, we hypothesized that wakefulness is related to dynamical criticality while the anesthetized state corresponds to increased damping of the dynamics. To test this hypothesis we assumed locally linear dynamics estimated in short segments of the recordings using eigenmode decomposition of VAR models.

**FIGURE 4 | Detailed view of eigenmodes distributions for subject 1.** Colors indicate the segments indicated in **Figure 1A**. **(A)** Histogram in **Figure 3A** is sliced by restricting the analysis to frequency bands of 5 Hz centered around the indicated values. Note that the blue histograms are always to the left of the red ones indicating increased stability as anesthesia is induced. **(B)** Same analysis as **(A)** restricted to slices of 50 <sup>1</sup> *<sup>s</sup>* . Note the emergence of highly damped high frequency oscillations in the anesthetized

We found that as the subjects become anesthetized the linear stability of the ECoG recordings show significant changes which are efficiently tracked by non-parametric statistical methods. These markers are remarkably robust to changes in the way data is normalized (choice of filters, amplitude normalization, sampling frequency). Moreover, changes in this quantities were found to be consistent with the subjects behavior as reported by the medical team. This suggests that our indicators could be used to monitor depth of anesthesia.

Our results are also consistent with the criticality hypothesis: we found a prevalence of modes close to criticality across the whole induction procedure. However, as the subjects became anesthetized there were significant changes in the stability condition (blue). In both cases we plot count number on the scale (0.46000) **(A)** and (0.120000) for **(B)**. **(C)** Similarly to **Figure 3B**, we computed the superposition of histograms in a smaller frequency range for better visualization. The histograms were done with logarithmic binning and the frequency axis is logarithmic (base 2). The rainbow indicates a shift toward more damped states. The organization of stability undergoes non-trivial changes in low frequency bands (4–128 Hz).

properties of the fitted dynamics. These changes were examined closely in selected stages of the procedure and are visualized by the superposed histograms in **Figures 3B**, **4C**. This analysis revealed that changes in the stability exhibit much richer structure than a simple block shift to damping across all frequencies. Yet, we observe a consistent pattern in all three subjects; the eigenvalues of the fitted models shift toward higher frequencies and increased damping. This should be interpreted carefully; it is not necessarily the case that there is an increase of high frequency spectral content of the ECoG signals. Although there ought to be a relationship between a moving spectral analysis and the eigenmodes of a moving VAR analysis, this relationship may be complex.

The increase in the prevalence of eigenmodes characterized by high frequency (high gamma) may be seen as surprising given the well-known observation that the power of high frequency oscillations tends to decrease with some anesthetics including propofol. This result, however, ought to be interpreted carefully. The increase in the number of eigenmodes does not equate to the increase in power. For instance, there could be fractionation of a single correlated pattern of high frequency oscillations in the awake state into multiple mutually independent patterns of high frequency oscillations.

The finding that high frequency modes become more damped as the subject is anesthetized is to some extent reassuring. If we adopt the traditional view that high frequency activity is associated to cognitive processes our results are consistent with an appealing interpretation. The effect of the anesthetic procedure is to damp out high frequency activity while still allowing for low frequency modes to perform a function. Low frequency activity can then presumably be associated to the maintenance tasks which keep the subject alive.

A number of recent reports have been aimed at characterizing criticality as a universal feature in ECoG recordings (He et al., 2010), and as particularly relevant to differentiate wakefulness from sleep (Meisel et al., 2013; Tagliazucchi et al., 2013) (see also Ribeiro et al., 2010 for comparable results with action potential recordings). In this context, our results provide support for a consistent and theoretically founded interpretation of the relationship between criticality and wakefulness. While the theoretical model is not the focus of the present publication, it is interesting to note that it implies a specific and falsifiable prediction: the model achieves self-tuned criticality by means of plastic synaptic adaptation. It follows that blocking synaptic changes should result in a breakdown of criticality; similarly, the model should also be able to explain changes in criticality during the sleep cycle, given the concomitant changes in plasticity patterns (Ribeiro et al., 2007). This will be the subject of future publications, along with further validation of the stabilizing effect of anesthesia in animal models, effects of different anesthetic agents, larger number of subjects, recovery from anesthesia, and application of the methods to EEG recordings.

## **5. AUTHOR CONTRIBUTIONS**

Leandro M. Alonso, performed data analysis; Leandro M. Alonso, Alex Proekt, Guillermo A. Cecchi, andMarcelo O.Magnasco wrote the manuscript. Alex Proekt, Guillermo A. Cecchi, Marcelo O. Magnasco designed the experiments; Alex Proekt, Kane O. Pryor, and Theodore H. Schwartz performed the experiments; Leandro M. Alonso and Alex Proekt contributed equally to this work.

### **ACKNOWLEDGMENTS**

This work was funded by NSF grant EF-0928723 awarded to Marcelo O. Magnasco and by NIGMS (1K08GM106144-01) awarded to Alex Proekt

## **REFERENCES**


**Conflict of Interest Statement:** The Guest Associate Editor A. Ravishankar Rao declares that, despite being affiliated to the same institution as the author Guillermo A. Cecchi, the review process was handled objectively and no conflict of interest exists. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 27 November 2013; accepted: 24 February 2014; published online: 25 March 2014.*

*Citation: Alonso LM, Proekt A, Schwartz TH, Pryor KO, Cecchi GA and Magnasco MO (2014) Dynamical criticality during induction of anesthesia in human ECoG recordings. Front. Neural Circuits 8:20. doi: 10.3389/fncir.2014.00020 This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Alonso, Proekt, Schwartz, Pryor, Cecchi and Magnasco. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## **APPENDIX**

## Attractor dynamics in local neuronal networks

#### *Jean-Philippe Thivierge1 \*, Rosa Comas <sup>1</sup> and André Longtin2*

*<sup>1</sup> School of Psychology, University of Ottawa, ON, Canada*

*<sup>2</sup> Department of Physics, University of Ottawa, ON, Canada*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Gabriel B. Mindlin, University of Buenos Aires, Argentina Thibaud Olivier Taillefumier, Princeton University, USA*

*\*Correspondence:*

*Jean-Philippe Thivierge, School of Psychology, University of Ottawa, 136 Jean Jacques Lussier, Ottawa, ON K1N 6N5, Canada e-mail: jthivier@uottawa.ca*

Patterns of synaptic connectivity in various regions of the brain are characterized by the presence of synaptic motifs, defined as unidirectional and bidirectional synaptic contacts that follow a particular configuration and link together small groups of neurons. Recent computational work proposes that a relay network (two populations communicating via a third, relay population of neurons) can generate precise patterns of neural synchronization. Here, we employ two distinct models of neuronal dynamics and show that simulated neural circuits designed in this way are caught in a global attractor of activity that prevents neurons from modulating their response on the basis of incoming stimuli. To circumvent the emergence of a fixed global attractor, we propose a mechanism of selective gain inhibition that promotes flexible responses to external stimuli. We suggest that local neuronal circuits may employ this mechanism to generate precise patterns of neural synchronization whose transient nature delimits the occurrence of a brief stimulus.

**Keywords: computer simulations, attractor, synchronization, oscillations, spiking neurons, mean field**

## **INTRODUCTION**

The mammalian brain is composed of a complex network of synapses that permit the flow of electrochemical activity between populations of neurons. In the cerebral cortex, synaptic networks form a dense map whose cytoarchitecture has been studied extensively (Braitenberg and Schuz, 1998). Several factors influence the probability of local synaptic connections in cortex, including physical distance (Song et al., 2005), functional domains (sets of neurons that show similar response properties) (Mountcastle, 1997), and selective connectivity amongst similar cell types (Stepanyants et al., 2004). Another characteristic feature of cortical networks is the presence of synaptic motifs, defined as triplets (or, more generally, *n*-tuplets) of neurons whose synaptic pattern follows a particular configuration (Sporns and Kotter, 2004; Song et al., 2005; Roxin et al., 2008). These motifs provide the building blocks of connectivity at a given spatial scale, and have been explored in various contexts outside of brain connectivity, including gene regulation and other biological and artificial networks (Milo et al., 2004).

Motif configurations have been studied in the context of both local cortical networks *in vitro* (Song et al., 2005) and in the structural connectivity of macaque and cat cortex (Sporns and Kotter, 2004). In all instances, a subset of motifs reoccurs with higher-than-chance prevalence, suggesting a functional role in cortical information processing (Thivierge and Marcus, 2007). Simulated networks of neurons whose excitatory synapses follow a "relay" motif (**Figure 1A**)—the most frequent motif in primate visual cortex—exhibit synchronization with near-zero time lag (Traub et al., 1996; Vicente et al., 2008). This form of activity is reported in a spectrum of experiments including retinal ganglion cell recordings (Ackert et al., 2006), in cells of the lateral geniculate nucleus (Alonso et al., 1996), and in the electroreceptors of the weakly electric fish (Doiron et al., 2003). Zero-lag synchronization emerges because of the common input provided by the relay node to the two other nodes.

While computer simulations of a relay network suggest a substrate for the emergence of synchronization between neurons, these networks are limited in the scope of their behavior, and typically follow a limit cycle whose period is determined by the intrinsic properties of the model (Coombes et al., 2006; Kopelowitz et al., 2012; Viriyopase et al., 2012). This limit cycle has been shown to generalize to a large class of neuronal models that follow a relay motif (Grossberg, 1978). While cortical recordings show evidence of limit-cycle oscillations (Rodriguez et al., 1999), this behavior is typically transient in non-pathological states. Brain oscillations are usually restricted to short time periods, and remain coherent for only a limited number of cycles (Fries, 2005). Furthermore, transient neuronal responses themselves carry stimulus-relevant information in visual (Ackert et al., 2006) and olfactory (Mazor and Laurent, 2005; Geffen et al., 2009) processing. The question thus arises of how to generate transient, yet precise synchronization with connectivity that follows a relay motif, resisting the propensity of this motif to generate ongoing synchrony in a limit cycle. This question has received scant attention, despite many studies examining the impact of connectivity on simulated brain dynamics (Schuster et al., 1979; Cohen and Grossberg, 1983; Sporns and Kotter, 2004; Coombes et al., 2006; Thivierge and Marcus, 2007; Vicente et al., 2008; Goldman, 2009; Ostojic et al., 2009).

Here, we begin by examining neuronal activity in a simplified mean-field model that allows us to visualize global network activity using a phase plane plot, a graphical display of how nodes interact to produce patterns of activity. This model highlights the effect of key parameters in generating limit cycle activity, multistability, and stimulus encoding. We then turn to a second, more detailed model based on integrate-and-fire neurons, to show conditions under which a relay network leads to a strict limit

cycle, thus preventing the encoding of incoming stimuli. Finally, we describe a mechanism of selective gain inhibition (Vogels and Abbott, 2009) that promotes stimuli encoding by breaking up the functional interactions in relay networks. These results carry important functional implications on how connectivity constrains patterns of neuronal activity in synaptically-coupled networks.

trajectory has completed a full cycle (at 50 ms), it loops back onto itself and

## **MATERIALS AND METHODS**

#### **WILSON–COWAN MODEL**

repeats the process.

Our starting point is a simplified population model where the fundamental unit is a set of coupled noise-free Wilson–Cowan equations (Wilson and Cowan, 1972):

$$\begin{aligned} \varepsilon \frac{d\mathbf{x}}{dt} &= -\mathbf{x} + \theta \left( -\varkappa \mathbf{x}\_{\mathbf{r}} + \varkappa \mathbf{y}\_{\mathbf{r}} + \varkappa \mathbf{z}\_{\mathbf{r}} + I \right) \\ \varepsilon \frac{d\mathbf{y}}{dt} &= -\mathbf{y} + \theta \left( \alpha \varkappa \mathbf{x}\_{\mathbf{r}} - \varkappa \mathbf{y}\_{\mathbf{r}} + I \right) \\ \varepsilon \frac{d\mathbf{z}}{dt} &= -\mathbf{z} + \theta \left( \alpha \varkappa \mathbf{x}\_{\mathbf{r}} - \varkappa \mathbf{z}\_{\mathbf{r}} + I \right), \end{aligned} \tag{1}$$

where *x*, *y*, and *z* each represent the mean firing rate of a local population of neurons, *w* is a weighted connection, *I* is a constant input stimulus (set to zero by default), α is a free parameter (set to 1.0 by default), θ is a sigmoid function, θ (*x*) = 1/ - 1 + *e*−*<sup>x</sup>* , ε > 0 is a rate parameter that governs the speed at which the firing rate changes, and τ is a fixed synaptic transmission delay. Unless otherwise stated, connections are set to *w* = 103, leading to excitatory connections between populations and inhibitory self-connections. For illustration purposes only (and bearing in mind the limited biological correspondence of this simplified account), we draw an equivalence of 1 time-step = 0.1 ms of simulated activity. Unless otherwise stated, we introduce a delay of τ = 1.5 ms in synaptic transmission from one population to another. We employ an Euler method (integration step of 0.1) for the integration of Equation 1.

#### **POPULATIONS OF LEAKY INTEGRATE-AND-FIRE NEURONS**

In addition to the above mean-field model, we considered a network of integrate-and-fire neurons whose membrane potential is described by

$$\mathcal{L}\_m \frac{dV}{dt} = \left(V\_{\text{rest}} - V\right) + \mathcal{g}\_{\text{ex}} \left(E\_{\text{ex}} - V\right) + \mathcal{g}\_{\text{inh}} \left(E\_{\text{inh}} - V\right) \quad (2)$$

$$+ I\_{\text{syn}} + R \left(I\_{\text{ext}} + I\_{\text{tonic}}\right),$$

where *V*rest is the resting membrane potential, *gex* and *g*inh are synaptic conductances of excitation and inhibition, *Eex* and *E*inh are the reversal potentials of excitation and inhibition, *R* is a unitless scalar gain, *I*ext is an external current, *I*tonic is a tonic current, and *cm* is the membrane capacitance. The synaptic input *I*syn for a neuron *i* is given by

$$I\_{\text{syn},i} = \sum\_{j=1}^{N} w\_{ij} K\_j,\tag{3}$$

where *wij* is a synaptic weight from neuron *j* to neuron *i*, and *Kj* is the excitatory postsynaptic membrane potential of a neuron *j*:

$$K\_{\dot{f}} = V\_0 \sum\_{s=1}^{S} \exp\left(\frac{t\_s - t}{\tau\_{\text{fall}}}\right) - \exp\left(\frac{t\_s - t}{\tau\_{\text{rise}}}\right),\tag{4}$$

where *s* = 1,..,*S* indexes spike times and *V*<sup>0</sup> is a scaling factor. The rise and fall times of the postsynaptic membrane potential are given by τrise and τfall, respectively. A spike is triggered when the membrane potential (Equation 2) reaches its firing threshold from below. At that point, *V* is held at 40 mV for 1 ms, then reset to −70 mV for an absolute refractory period lasting 3 ms. In all numerical simulations, we imposed a fixed time delay on synaptic transmission (see parametric values below).

Some of the above model's parameters were designed to vary across the population of simulated neurons (Thivierge and Cisek, 2008). This was achieved by randomly drawing parametric values from a Gaussian distribution with σ = 0.33 times the mean. Means for these parameters were as follows: *gex* (0.8 nS), *g*inh (−1.5 nS), *E*ex (0 mV), *E*inh (−80 mV), τrise(3 ms), τfall (5 ms), firing threshold (−55 mV), resting potential (*V*rest = −60 mV) and synaptic delays (3 ms). Other parameters were constant across the entire population of neurons: *I*tonic (3.5 mV), *cm* (0.02), *R* (10), and *V*<sup>0</sup> (0.09).

Synaptic connectivity (*wij*) was configured to produce three distinct populations of neurons (with a total of 10,000 neurons per population), characterized by strong within-population interactions, and weaker between-population interactions. Both within- and between- population weights were drawn from a Gaussian distribution with mean of 100 nS (or −100 nS in the case of inhibitory neurons) and standard deviation of 0.33 times the mean. Twenty percent of connections were inhibitory. These connections were chosen randomly amongst all potential connections. Only a portion of all possible connections were present: the probability of a within-population connection between pairs of neurons was set to 0.9, while the probability of a betweenpopulation connection was set to 0.2. A cartoon illustration of three regions of neurons where between-population connections reflect a relay motif is shown in **Figure 1A**. Three populations of neurons are labeled by different colors, and arrows represent between-population connections.

## **RESULTS**

#### **NETWORK CONNECTIVITY AND MEAN-FIELD ACTIVITY**

In order to investigate limit cycle activity in interconnected networks, we performed simplified simulations using a Wilson– Cowan population model (Equation 1). Activity at each node of the network was approximated by a single equation that describes mean population behavior (**Figure 1A**, right). We simulated a relay network for 100 s, and displayed the resulting activity on a phase plane plot (**Figure 1B**). This plot relates all three nodes of the network at time-step *t* vs. *t* + 1, showing a trajectory of neuronal activity. A limit cycle on a phase plane is characterized by a closed loop that repeats itself by following the same trajectory over and over again. While these simulations are highly abstracted, and represent the Wilson–Cowan equation of Equation 1 for only a specific set of parameters and initial conditions, the resulting dynamics provide a clear illustration of the influence of network connectivity on ongoing dynamics, and are in line with previous work relating relay networks with the emergence of limit cycle activity (Coombes et al., 2006; Kopelowitz et al., 2012; Viriyopase et al., 2012). Zero-lag synchronization arises here because of bidirectional connections in the relay network, allowing two nodes (in blue and red) to coordinate their activity through a third node (in black) that serves as intermediary. In this way, zero-lag synchronization arises despite the absence of direct connections between the blue and red nodes.

In order to weaken (or remove) the limit cycle resulting from a relay network, it suffices to eliminate the influence of the relay node on the other two populations of the model (**Figure 2A**, left). This is done by setting α = 0 (Equation 1). In this scenario, connections are strictly feedforward, projecting onto the relay node without feedback. With this configuration, activity in two of the nodes (red and blue traces, **Figure 2A**, right) shows a periodic cycle; the third node (black trace, **Figure 2A**, right), however, shows no repeating pattern in terms of amplitude, even over extended periods of time. If we considered only the activity of the latter node, we might be led to conclude that the activity at that node is best described by random amplitude fluctuations. However, displaying the activity of the model in a phase plane reveals a hidden structure: while neuronal activity does not display a simple closed loop, it is constrained to a limited portion of the total space (**Figure 2B**). The activity of the model never repeats itself exactly over time, but follows an "orbit" that forms a well-defined pattern in the phase plane plot. Note that one can also change the input *I* in Equation 1 to bias the system out of a limit cycle (see Linear Stability Analysis below).

**FIGURE 2 | Activity becomes unstable in a relay network with no feedback connections. (A)** Left: illustration of a three-node network where connections are strictly feedforward (connections in gray are set to zero, i.e., α = 0 in Equation 1). Right: pattern of activity obtained for each of the three nodes in **(A)** over time. **(B)** Phase-plane of activity where nodes in **(A)** are simulated with a Wilson–Cowan model. **(C)** Duration (in ms) of stable cycles in a relay network with both feedback and feedforward connections between nodes. Each dot shows initial conditions for the relay node and the two outer nodes.

To evaluate the stability of limit cycle activity in relay networks, we let *An*(*t*) reflect the activity of a given node *n* at time-step *t*, and sought values of *d* for which

$$A\_n(t) = A\_n(t+kd) + s \tag{5}$$

where ε was set to four orders of magnitude below the resolution of the model (ε = 10−<sup>5</sup> μA) and *k* is an arbitrary constant integer. If a solution to *d* exists, the system is deemed periodic, and the value of *d* determines the duration of the period. While the exact value of this duration was dependent upon the initial conditions of the system, convergence to a limit cycle was observed across a range of starting points for *A*1, *A*2, and *A*<sup>3</sup> (**Figure 2C**). This result shows that a relay network consistently leads to a limit cycle, with the length of the cycle dependent upon the initial conditions of the system. The finding that the length of the limit cycle depends upon initial conditions of the model is consistent with the idea of multistability in models of neuronal activity (Foss et al., 1996). Accordingly, a range of stable solutions exist, and each solution can be reached by activating the model in a particular way.

We repeated the above analysis for a network with feedforward connectivity (**Figure 2A**, setting α = 0 in Equation 1) and found no solution to *d* across any configuration of initial conditions. A more formal analysis of stability and of the origins of chaotic behavior in relay and feedforward networks is presented below.

Two parameters of the Wilson–Cowan model bear a strong influence on its activity. The first of these parameters is the transmission delay between nodes (the amount of time elapsed before the activity at a given node influences the activity at another node). Shorter delays (below 78 ms) did not produce limit cycle activity (i.e., no solution to Equation 5 was found); above that value, changes in the value of delays did not markedly alter the shape of the limit cycle (**Figure 3A**). In a strictly feedforward network, a similar result was found: short delays (e.g., 10 ms) were not sufficient to generate trajectories in the phase plane plot (**Figure 3B**).

A second parameter playing a key role in network activity is the strength of the connection weights between populations. Low (*w* = 102) and high (*w* = 103) weights between populations yielded markedly different shapes of attractors (**Figure 4**). Of particular interest, a feedforward network generated a limit cycle when connection weights were low (e.g., *w* = 102). In this case, the red and blue nodes oscillate and transmit that oscillation in a weak form to the black node, which then also oscillates. Together, these results show that both transmission delays and weight magnitudes influence the production of attractors in the activity of the Wilson–Cowan model.

To further explore the route that goes from a limit cycle to a more complex form of activity, we examined the order parameter α that modulates the influence of feedback connections from the relay node (see Equation 1). With a value of α = 0.2 and greater, feedback connections are strong enough to produce a limit cycle behavior; below that value, however, weaker feedback results in more complex forms of activity (i.e., where no solution to Equation 5 was found) (**Figure 5**).

Next, we examined the response of a Wilson–Cowan model to a constant input injected into all three nodes. In different simulations, each lasting 100 s, we varied the intensity of input (from *I* = 0,. . . ,104). When connectivity followed a relay network, activity in the network increased in response to inputs ranging from *I* = 0 to *I* = 102, then saturated from *I* = 10<sup>2</sup> to

**FIGURE 3 | Influence of transmission delay on network activity.** (**A**; left panel) Illustration of a relay network with both feedforward and feedback connections. (right panel) Phase-plane plots of activity simulated with a

Wilson–Cowan model. Limit cycle activity emerges as a sharp transition between a delay of 78 ms and a delay of 79 ms. (**B**; left panel) Feedforward network. (right panel) Phase plane plots of activity.

*I* = 10<sup>4</sup> (**Figure 6A**). This result was found with both high (*w* = 103) and low (*w* = 102) connection strength. We compared these results with those obtained when injecting input into a network with feedforward connectivity. In this case, activity monotonically increased in response to inputs from *I* = 0 to *I* = 103, a broader range than that obtained with a relay network (**Figure 6B**). Upon close inspection, the difference in responses between the relay and feedforward networks is largely explained by the fact that the feedforward network exhibits lower activation under weak input (i.e., mean activation is low when *I* is small). To further probe the effect of input on network dynamics, we examined phase plane plots of activity, as described above. In relay networks, for all values of input tested, activity consistently yielded a limit cycle (i.e., where a solution to Equation 5 could be found) (**Figure 6C**). By contrast, in feedforward networks, activity yielded different patterns depending on the intensity of input. With weak input (*I* < 10), activity followed no repeating trajectory (**Figure 6D**); however, as the intensity of input increased, network activity settled into a limit cycle attractor. In sum, a strictly feedforward network led to a greater dynamical range of responses than a relay network; in addition, a feedforward network resulted in a different attractor depending on the strength of input, whereas a relay network always resulted in a limit cycle attractor.

In a final series of simulations, we considered a scenario where a relay network is embedded in a larger network of Wilson–Cowan nodes. We began by generating a sparse randomly connected network of 1000 nodes, where one node had a 1% probability of being connected to any given node in the network. Then, we selected three nodes at random and forced their connectivity to follow a relay network (**Figure 7A**). All connection weights, both within the relay network and outside of it, were set to *w* = 10<sup>3</sup> if a connection was present, and *w* = 0 otherwise (self-connections were set to *w* = −103). Examples of activity generated when a relay network was embedded in a larger network are shown in **Figure 7B**. The resulting pattern of activity can be described as a "noisy" limit cycle, where perturbations coming from activity in the surrounding network made the trajectory of the limit cycle deviate from its path. Here, embedding a relay network in a larger network did not result in a fundamentally different pattern of activity, but rather a perturbed version of the original pattern obtained when the relay network was simulated as a stand-alone network. Of course, increasing the density of connections within the larger network would lead to more pronounced perturbations, yet would result in a less plausible scenario from the point of view of cortical connectivity. Excitatory cortical cells receive only sparse afferents from other excitatory cells. The probability of contact between two neocortical excitatory cells that are 0.2–0.3 mm apart is estimated to be *p* < 0.1, and between two such cells that are more than 1 mm from each other, *p* < 0.01 (Braitenberg and Schuz, 1998; Song et al., 2005). Because nodes in the Wilson–Cowan are aimed at simulating populations of neurons rather than individual synaptic contacts, we rely on the latter probability as a point of comparison. Our simulations of three-node relay networks embedded in larger random networks show that patterns of activity are robust to the influence of ongoing activity generated from the surrounding network under reasonable conditions of connectivity.

#### **LINEAR STABILITY ANALYSIS**

The above simulations show that relay networks are prone to oscillations that are caught in a limit cycle, while feedforward networks generate more complex forms of activity that do not oscillate in a strict manner. Here, we derive a linear stability analysis that yields insight into the propensity of a relay network to oscillate compared to a feedforward network. We consider the three node (*x,y,z*) model of Equation 1, where outer nodes *y* and *z* (red and blue nodes, **Figure 1A**) project to a relay node *x* (black node in **Figure 1A**) with connection weight

**connectivity.** (**A**; left) Illustration of a relay network with feedback connections. (right) In a relay network, mean activation increases monotonically with the strength of input, but saturates for values of input greater than 102. Black and blue lines represent nodes of the network in **(A)** (activation of the red node overlaps with that of the blue node). Solid lines, weights of *w* = 1000. Dashed

connections. (right) In a feedforward network, mean activity increases in response to input, and does not saturate until the strength of input reaches 103. **(C)** In a relay network, activity follows a limit cycle regardless of the strength of input. **(D)** In a feedforward network, activity follows a limit cycle for stronger input (102) but not for weaker input (100).

*w* > 0. Likewise, node *x* projects back to nodes *y* and *z*, but with a connection strength αw. Here α is an adjustable parameter between 0 and 1; when α = 1, the three-node system embodies a relay network, while for α = 0 it represents a feedforward network. This formulation enables us to smoothly move between a relay and feedforward network. It also enables us to investigate all system parameters, combinations of parameters, and initial conditions.

The fixed points of the system in Equation 1, that is, the values of (*x,y,z*) for which the derivatives are zero, are given by solutions of the following non-linear equations:

$$\alpha^\* = \theta \left( -\alpha x^\* + \omega y^\* + \alpha z^\* + I \right) \tag{6a}$$

$$
\gamma^\* = \theta \left( \alpha a \alpha^\* - a \gamma^\* + I \right) \tag{6b}
$$

$$z^\* = \theta \left(\alpha a \alpha^\* - \alpha z^\* + I\right). \tag{6c}$$

To solve the above system, we first note that the solutions remain invariant upon interchanging *y* and *z*. The same can be said of the original system (Equation 1). This means that a solution

the surrounding network provide slight alterations to the trajectory. Three plots show patterns of activity obtained from different probabilities of connection between nodes in the surrounding network (1, 5, and 10%). Each run of the model lasted 100 s at a resolution of 0.1 ms.

(*x*(*t*),*y*(*t*),*z*(*t*)) will be the same as a solution (*x*(*t*),*z*(*t*),*y*(*t*)), provided that the variables *y* and *z* have the same initial values; in other words, *y*(*t*) tends to *z*(*t*) when time is large enough provided that *y*(0) = *z*(0). As for the fixed point, it will be such that *y*<sup>∗</sup> = *z*∗, that is, it will lie on the plane (*x*∗,*y*∗,*y*∗). Numerical simulations indeed reveal that this is the case, and also that solutions evolve to ones where *y* approaches *z* for a large range of differences between *y*(0) and *z*(0). However, solutions where *y*(0) differs significantly from *z*(0) can evolve such that both solutions are the same, but maintain a fixed time lag between them. One should note that the value of the input *I* determines the precise value of the fixed point. In other words, this system admits a simple rate coding where the system settles onto a fixed point "rate" that varies smoothly with the strength of the input. As we will see below, as certain parameters change, this system undergoes a "Hopf bifurcation," or in other words a transition between a stable fixed point—corresponding to a stable constant firing rate in the model—to stable limit cycle oscillations of the firing rate. Our paper mainly concerns the robustness of these oscillations.

In order to investigate the dynamical properties of the system, in particular what combinations of parameters lead to an equilibrium (i.e., a fixed point) or an oscillation, an important starting point is to non-dimensionalize the equations. This will be done here by scaling the time variable by the delay, leading to a new continuous dimensionless time *T* = *t*/τ that is counted in the number of delays (e.g., *T* = 5.677 means *t* = 5.677τ ). Further defining *k* = ε/τ , and new variables *X*(*T*) = *x*(*t*), *Y*(*T*) = *y*(*t*), and *Z*(*T*) = *z*(*t*), the model evolves according to:

$$k\frac{dX}{dT} = -X(T) + \theta \left[ -\omega X(T-1) + \omega Y(T-1) \right] \qquad \text{(7a)}$$

$$+ \left. \omega Z(T-1) + I \right| $$

$$k\frac{dY}{dT} = -Y(T) + \theta \left[\alpha\nu X(T-1) - \nu Y(T-1) + I\right] \tag{7b}$$

$$k\frac{dZ}{dT} = -Z(T) + \theta \left[\alpha\omega\varkappa X(T-1) - \varkappa Z(T-1) + I\right]. \tag{7c}$$

The fixed point (*X*∗,*Y*∗,*Z*∗) for this system is identical to that of Equations 6a–c above, with the substitution of *X,Y,Z* for *x,y,z.* While it is not possible to explicitly solve this transcendental system, our numerical simulations reveal that there is only one relevant fixed point (*X*∗,*Y*∗,*Z*∗). Investigating the linear stability of this fixed point will reveal how solutions behave near this point, and in particular, if bifurcations can occur between a stable equilibrium and a stable oscillation. This linearization is done using a multivariate Taylor expansion, keeping only the first order terms. We first move the origin (0,0,0) onto the fixed point (*X*∗,*Y*∗,*Z*∗) by a change of coordinates: *X* = *X* − *X*∗, *Y* = *Y* − *Y*∗, *Z* = *Z* − *Z*∗. The resulting linearized system is given by:

$$k\frac{dX'}{dT} = -X'(T) - \omega AX'(T-1) + \omega AY'(T-1)\qquad(8a)$$

$$+ \omega AZ'(T-1)$$

$$\delta Y' = \delta Y' \delta \qquad \omega'\mathbf{v}'(\pi - \mathbf{u}) = \omega'\mathbf{v}'(\pi - \mathbf{u})\qquad(11)$$

$$k\frac{dY}{dT} = -Y'(T) + \alpha \omega A'X'(T-1) - \omega A'Y'(T-1) \quad \text{(8b)}$$

$$k\frac{dZ'}{dT} = -Z'(T) + \alpha\omega A'X'(T-1) - \omega A'Z'(T-1),\quad(8c)$$

where *A* = *<sup>d</sup>*θ (*g*) *dg <sup>g</sup>*<sup>∗</sup> with *<sup>g</sup>*<sup>∗</sup> = −*wX*<sup>∗</sup> <sup>+</sup> *wY*<sup>∗</sup> <sup>+</sup> *wZ*<sup>∗</sup> <sup>+</sup> *<sup>I</sup>*, and *A* = *<sup>d</sup>*θ(*h*) *dh <sup>h</sup>*<sup>∗</sup> with *<sup>h</sup>*<sup>∗</sup> <sup>=</sup> <sup>α</sup>*wX*<sup>∗</sup> <sup>−</sup> *wY*<sup>∗</sup> <sup>+</sup> *<sup>I</sup>*. Both *<sup>A</sup>* and *<sup>A</sup>* are slopes of the firing function, and act as a feedback gain. One observation that can be made from the analysis thus far is that, in order to examine the linear properties of either the relay or the feedforward networks, the only important parameters are the ratio *k* and the products *wA* and *wA* .

A full analysis of this system is beyond the scope of our needs here, but we will make a few observations. First, this system can be simplified further by defining two new variables as the sum *S* and difference *D* of *Y* and *Z* , *S* = *Y* + *Z* and *D* = *Y* − *Z* . This yields the system

$$k\frac{dX'}{dT} = -X'(T) - \omega AX'(T-1) + \omega AS'(T-1) \qquad \text{(9a)}$$

$$k\frac{dS}{dT} = -S(T) - \omega A'S(T-1) + 2\alpha\omega A'X'(T-1) \quad (9b)$$

$$k\frac{dD}{dT} = -D(T) - \omega A'D(T-1).\tag{9c}$$

In the (*X* ,*S*,*D*) coordinates, it becomes apparent from Equation 9c that the difference between the activities of the two nodes *y* and *z* behaves independently of the variables *X* and *S*; these latter two variables, however, evolve in a coupled manner. It is known that, since *w* > 0, the difference *D* obeys linear delayed negative feedback dynamics (Erneux, 2009); consequently, if either (or both) the delay or the linear connection weight *wA* are sufficiently large, then the stable fixed point will continuously transition into a stable oscillation (also known as a stable limit cycle). In technical terms, this process is termed a supercritical Hopf bifurcation.

Assuming a relay network (α = 1) and the stable fixed point case, the activities at nodes *y* and *z* will be constant and equal after a transient period (leading to a trivial form of zero-lag synchrony). This is because *D*<sup>∗</sup> = 0 implies that *Y* and *Z* are at their equilibrium values of 0, that is, *Y* = *Y*<sup>∗</sup> and *Z* = *Z*<sup>∗</sup> (recall that *Y*<sup>∗</sup> = *Z*∗). This implies that one can effectively study the dynamics of the whole model by focusing on Equations 8a,b alone.

For the feedforward network (α = 0) and the stable fixed point case, it is clear already from Equations 8b to 8c that the behavior of *Y* and *Z* will be the same, up to a time shift that depends on their initial conditions. In fact, even considering the full dynamics in Equations 1a–c, it is seen that, for the feedforward network, both variables *y* and *z* have the same rule governing their evolution, but behave independently of each other. Further, *y* and *z* are merely a source of external forcing on node *x*. If the parameters are such that *y* and *z* tend to a fixed point *y*<sup>∗</sup> = *z*∗, then over long periods of time node *x* will receive an identical constant forcing from each of these nodes. Node *z* could be eliminated, and the weight of the connection from node *y* doubled—node *x* would not see the difference (the same holds true if replacing *y* by *z*).

Alternately, in a feedforward network with α = 0, the parameters can be such that *y* and *z* oscillate autonomously. Their sum in Equation 1a also oscillates at the same period, and qualitatively, the dynamics of node *x* amounts to a periodically-driven delaydifferential equation. The dynamics can be very rich in this case, with chaotic solutions and/or long transients, since the unforced system *x* can oscillate on its own, and this oscillation competes with the one imposed by the sum of *y* and *z*. This is the kind of solution we find in the feedforward network (see **Figure 2A**).

Coming back now to a relay network (α = 1), but this time with an oscillation for the difference variable *D* in Equation 9c, variables *Y* and *Z* will move close and away from each other periodically. This case also potentially leads to a complex solution. But for the parameters of interest in our study, the feedback from *x* to nodes *y* and *z* has a stabilizing effect, in the sense that the whole three-dimensional system usually settles on a limit cycle where all nodes oscillate at the same frequency.

One can carry out a linear stability analysis to find the regions of parameter space where a Hopf bifurcation occurs, using the reduced *X-S* system of Equations 9a,b. One first substitutes trial solutions *x*(*t*) = *xo*exp(λ*t*) and *y*(*t*) = *yo*exp(λ*t*) into Equations 9a,b, where λ = μ + *i*ω is a complex eigenvalue (note that we denote the angular frequency by ω, distinct from the coupling weight *w*). Assuming this solution is valid for arbitrary nonzero constant amplitudes *xo* and *yo*, and defining the effective feedback gains β = *wA* and β = *wA* , this yields the characteristic equation

$$(k\lambda + 1 + \beta e^{-\lambda})(k\lambda + 1 + \beta' e^{-\lambda}) = 2\alpha\beta\beta' e^{-2\lambda}.\qquad(10)$$

This equation admits an infinite number of complex conjugate roots (i.e., values of λ) corresponding to eigenvalues for the system of Equations 9a,b linearized around the fixed point. In order to find the conditions where the roots migrate from the left hand side to the right hand side of the complex plane (a characteristic of a Hopf bifurcation) we set the real part of the eigenvalue to zero: μ = 0, i.e., λ = *i*ω in Equation 10*.* The resulting two equations obtained by setting both the real and imaginary parts of this special form of Equation 10 equal to zero define a relationship between all the parameters and the frequency ω at the onset of oscillation.

With respect to the Hopf bifurcation, the feedforward case (α = 0) is well-documented (Erneux, 2009). In particular, a bifurcation occurs when increasing either the delay or β; the higher the one is, the smaller the required value of the other in order for the Hopf bifurcation to occur (if both parameters are high, the system is clearly in the oscillation regime). From the first factor on the left hand side of Equation 10, the frequency of the zero-amplitude solution born at the bifurcation is given by ω = β<sup>2</sup> − 1/*k*, with β being the first root of tan[ β<sup>2</sup> − 1/*k*] = − β<sup>2</sup> − 1. The same expressions but with β substituted for β apply to the second factor on the left hand side of Equation 10. Numerically, we find that the *X* system starts oscillating when the coupling strength is *w* ≈ 9.15 with the delay fixed at 1. At this onset, the *S* system still goes to a fixed point. This is so because, as the coupling strength *w* increases, the first factor acquires a purely imaginary root before the second factor does. This situation where *X* oscillates but *S* does not is possible because of the unidirectional coupling from *S* (i.e., *Y* and *Z*) to *X*.

In the relay case (α > 0), the analysis of the roots of Equation 10 is much more involved and beyond the scope of this paper. Numerically, we find that even for very small values of α, choosing *w* ≈ 9.15 as in the previous paragraph now yields an oscillation in *S*, and a larger amplitude oscillation in *X*. Both the *X* and *S* oscillations are at the same frequency, i.e., it is a global oscillation of the whole bi-directionally coupled system. In other words, a smaller delay or effective feedback gain *A* or *A* can then generate oscillatory activity. Thus, based on the transition between a fixed point and a limit cycle, the relay network is more prone to oscillate when compared to the feedforward network.

In summary, our stability analysis reveals that both the relay and feedforward networks can exhibit a Hopf bifurcation. Transitions to the limit cycle are favored by an increase in two parameters: the delay, or the product of the connection weight and the slope of the firing function evaluated at the fixed point (the parameters *k* and *I* also have an effect, but this was not explored here). Upon increasing α we find that the network is more prone to oscillate (c.f., **Figures 1A**, **2A**). In the feedforward case, when the outer nodes (red and blue nodes in **Figure 2A**) are in a limit cycle regime, the third node (black node in **Figure 2A**) produces complex dynamics via the interaction of its intrinsic oscillation and the unidirectional periodic forcing from the two outer nodes. This system can be largely understood with only one variable instead of three. By increasing the value of α, we transition from a feedforward to a relay network where the dynamics of the whole system settle onto a limit cycle.

Taken together, simulations and analysis of the Wilson–Cowan model show a key role of delays, connection strength, and relay connectivity on the ability of the model to generate limit cycle activity. Further, results show that a feedforward network yields a broader range of responses to stimuli than a relay network. While these results reveal that the elimination of feedback projections from the relay node to the outer nodes can break the network out of a limit cycle, it is unclear how this could be achieved in living synaptic networks. In the next section, we explore a mechanism by which selective inhibition of relay neurons provides a network with the ability to escape its limit cycle and modulate its activity in response to incoming stimuli.

## **A RELAY NETWORK WITH SPIKING NEURONS**

We now turn to a more detailed model of neuronal activity based on 30,000 integrate-and-fire neurons divided into three distinct populations, resulting in a global connectivity that followed a relay network (**Figure 8A**, see Materials and Methods) (Thivierge and Cisek, 2008, 2011; Rubinov et al., 2011). Simulated activity in this network (**Figure 8B**) shows the appearance of a global limit cycle, with two of the populations (in red and blue) exhibiting synchronization with near-zero time-lag (Vicente et al., 2008). This pattern of activity never faded away for as long as the simulation was carried out (in this case, 5 min). Extensive simulations revealed that the emergence of a limit cycle was not sensitive to initial conditions of the network. This result mirrors those obtained with the mean field model described above. Notice, however, that while the overall pattern of activity in the network follows a limit cycle, the precise spike times of individual neurons do not, because of intrinsic fluctuations in the model (no external noise was added, see Equation 2). In addition, notice that not all neurons are synchronized, and some of the neurons remain quiescent throughout.

While precise synchronization may convey information about the input to a neuronal circuit (Thivierge and Cisek, 2008), we argue that a strict limit cycle imposes severe constraints on the behavior of circuits in response to an incoming stimulus. Consider a periodic stimulus that is delivered at a fixed squarepulse voltage (width of 5 ms) across all neurons (**Figure 9A**). By varying the inter-stimulus interval and voltage intensity, we can examine conditions under which network activity is modulated by the incoming stimulus. Simulated network activity was generated for 30 s and a periodic stimulus was delivered during that entire time to all neurons. A network configured according to a relay network (**Figure 9B**) exhibited only marginal modulations in mean firing rate in relation to either the intensity (**Figure 9C**) or the frequency (**Figure 9D**) of stimulation. This rigid behavior is explained by the fact that a relay network is highly entrenched

black, and blue, see Materials and Methods). Arrows indicate the presence of between-population connections. **(B)** Spike raster of spontaneous activity for 100 neurons from the relay network in **(A)**, with gain inhibition set to its default value (*g*inh = 1.5 nS). **(C)** Influence of gain inhibition on mean firing rates across a whole network. Each value of the graph is obtained from a

inhibition for the other two populations was held at its default value. This increased gain inhibition reduces the excitatory coupling from the relay neurons (black) to other populations. **(D)** Spike raster for 100 neurons of relay network from **(A)**, with gain inhibition set to *ginh* = 2.25 nS (corresponding to a 50% increase from baseline).

varied according to the frequency (Hz) and amplitude (μA) of these pulses. **(B)** Network with a relay configuration. The network was composed of three subpopulations of neurons, each having gain inhibition set to *g*inh = 1.5 nS. **(C,D)** Response to external stimuli in a

"relay neurons") had gain inhibition set to *g*inh = 2.25 nS (a 50% increase from the default value). **(F,G)** Responses to external stimuli in a relay network with increased gain inhibition as shown in **(E)**. All simulations were run for a total of 30 s of activity. Vertical bars = SEM.

in limit cycle activity (**Figure 8B**); this activity cannot easily be dislodged from this attractor by incoming stimuli. Put differently, a system that has reached a state of global oscillation cannot easily be affected by external perturbations (Golubitsky et al., 2006).

## **SELECTIVE GAIN INHIBITION**

A synaptic configuration based on a relay network is highly prevalent in mammalian cortex (Sporns and Kotter, 2004; Song et al., 2005), yet the above simulations show that such a network promotes the emergence of a limit cycle where activity is largely unaffected by an incoming stimulus. To reconcile these observations, one possibility is that cortical neural circuits are capable of dynamically reconfiguring their pattern of functional interactions such that an architectural substrate based on a relay network could disengage from its strict limit cycle behavior and generate more flexible responses to incoming stimuli.

It is unclear, however, how biological circuits may be able to disengage from strict limit cycle activity. Under the reasoning that relay neurons (in black, **Figure 8A**) are responsible for driving zero-lag synchrony, we suggest that if we tune down the influence of that subpopulation, we may prevent the emergence of a global attractor. There are several ways in which this could be achieved; here, we describe one candidate mechanism based on selective gain inhibition (Vogels and Abbott, 2009). By tuning up the inhibitory gain (*g*inh, Equation 2) of relay neurons, we can selectively reduce activity in these neurons. In turn, less activity would flow from the relay neurons to other neurons in the model, thus altering the global patterns of neuronal activity.

To test the idea of selective gain inhibition, we simulated spontaneous activity in a network with a global connectivity based on a relay network, and, in different simulations, applied gradually increasing values of gain inhibition to relay neurons (**Figure 8C**). When gain inhibition was increased by 50% from its baseline value (from 1.5 to 2.25 nS), network activity was no longer characterized by synchronized activity (**Figure 8D** compared to **8B**). Spontaneous activity in this regime yielded an overall low firing rate (mean rate of 1.01 Hz, s.d. 0.44) and followed no strict repeating pattern over time. Importantly, a balance of gain inhibition was necessary: if inhibition was too low (<50% increase from baseline), network activity remained comparable to baseline (**Figure 8C**). Conversely, if gain inhibition was too high (100% increase from baseline), activity in the relay population vanished completely.

To examine the effect of selective gain inhibition on a network's response to an incoming stimulus, we began with a network whose global connectivity follows a relay network, as described earlier. Then, we increased gain inhibition by 50% in all of the relay neurons (**Figure 9E**). In response to increasing stimulus intensities, the two neuronal populations sending input to the relay neurons modulated their mean firing rate in a nearmonotonic fashion (**Figure 9F**). The same two populations also increased their firing rate in proportion to increased stimulus frequency (**Figure 9G**). A network with increased gain inhibition was thus able to modulate its firing rate based on an incoming stimulus, and did not remain stuck in a persistent state of activity. Put differently, increased inhibition in this circuit resulted in increased responsiveness to stimuli.

In follow-up simulations, we injected a network having selective gain inhibition (*g*inh = 2.25 nS) with a stimulus consisting of a constant current (30μA) lasting 2000 ms. After that time, the stimulus was reduced to 5μA and held constant for 2000 ms (**Figure 10A**). During presentation of the first stimulus (30μA), activity was highly synchronized and strongly periodic. As soon as the first stimulus ended and the second stimulus (5μA) began, the network became quiescent. Neurons thus produced a highly synchronized and periodic response to a stronger stimulation, and relatively little response to a weaker stimulation. This simulation shows the capacity of a network with selective gain inhibition to generate synchrony based on stimulus amplitude. Such transient responses would not be possible without selective gain inhibition, given that a network configured with a relay network follows a persistent limit cycle attractor (**Figure 8B**) and does not modulate its response to incoming stimuli (**Figures 9C,D**).

To further examine the transient synchronization of a network in response to a stimulus, we designed, as above, a relay network where we increased the gain inhibition of the relay population of neurons (**Figure 8A**, in black) by 50% from its baseline value (from 1.5 to 2.25 nS). We then injected a constant stimulus of 30μA into all neurons for a 10 s period. We computed the crosscorrelation between each pair of neurons during the stimulus presentation:

$$C\_{ij} = \frac{E\{\left[\mathbf{x}\_i(t) - E\_i\right] \left[\mathbf{x}\_j(t) - E\_j\right]\}}{\sqrt{E\left[\left[\mathbf{x}\_i(t) - E\_i\right]^2\right] E\left\{\left[\mathbf{x}\_j(t) - E\_j\right]^2\right\}}},\tag{11}$$

where *xi*(*t*) and *xj*(*t*) are the time-series of two given neurons having means *Ei* and *Ej*, respectively. Next, we obtained a crosscorrelogram of activity by taking the mean cross-correlation across all pairs of neurons. We found prominent zero-lag synchronization (**Figure 10B**, leftmost panel, vertical dashed line), as typical of activity for relay networks (Vicente et al., 2008). Hence, selective gain inhibition did not disrupt the capacity of a relay network to generate zero-lag synchronization. When we repeated the above simulation with a weaker input current (5μA), cross-correlations no longer displayed a prominent peak at zero time-lag as obtained with a stronger stimulation of 30 μA (**Figure 10B**, middle panel). Selective gain inhibition thus prevented a relay network from spontaneously generating zero-lag synchronization.

In a final series of simulations, we injected a 10 s input of various intensities (from 0 to 45μA) into a relay network with selective gain inhibition as described above. For each input intensity, we computed the mean zero-lag cross-correlation across

**FIGURE 10 | Transient synchronization in response to stimulation. (A)** Spike raster showing responses of 100 neurons from a relay network to a strong external current (*I*ext = 30μA, solid black line) followed by a weak current (*I*ext = 5μA, solid gray line). Gain inhibition of the relay neurons was set to *g*inh = 2.25 nS (50% higher than baseline) throughout the simulation.

**(B)** Mean cross-correlation of the network during presentation of a strong current (left panel) and a weak current (middle panel). Right panel: Mean cross-correlation as a function of external current. Black and gray arrows show a weak (*I*ext = 5μA) and strong (*I*ext = 30μA) current as simulated in **(A)**. Vertical dashed line: zero time-lag.

all pairs of neurons. This value increased as the input intensity was gradually amplified from 0 to 45μA (**Figure 10B**, rightmost panel, gray and black arrows), then remained stable from 30 to 45μA. The network thus modulated its degree of zero-lag synchronization in response to inputs of various current intensities, within a given range.

Taken together, our results show that selective gain inhibition can modulate the behavior of a relay network, such that the network can generate zero-lag synchronization in response to an incoming stimulus, yet does not remain stuck in a global attractor dominated by a fixed limit cycle.

## **DISCUSSION**

While there is a growing consensus that patterns of structural connections in the brain provide the backbone for a rich repertoire of activity (Bullmore and Sporns, 2009), here we argue using both simulations and mean-field analysis that a relay network imposes strict constraints on the types of dynamics produced by a network. Going further, simulation results using spiking neurons suggest that a mechanism of selective gain inhibition allows a network to modulate its patterns of activity and escape the rigid constraints imposed by synaptic connectivity, providing flexible and transient responses to an incoming stimulation.

While there are several examples of transient zero-lag synchronization in the central nervous system, a prominent one is found in the response of direction-sensitive (DS)—ON ganglion cells in the visual system (Ackert et al., 2006). In these cells, GABAergic inhibition forces activity to desynchronize following a transient phase of stimulus-induced zero-lag synchronization initiated by gap junction couplings between DS-ON and wide-field amacrine cells. While GABAergic inhibition suppresses zero-lag synchronization, it leaves intact the broad synchronization profile of cross-correlations at non-zero time lags. An analogous behavior was observed in our simulated spiking neurons, where selective gain inhibition suppresses stimulus-induced zero-lag synchronization (**Figure 10B**, leftmost panel) but leaves intact the broad profile of cross-correlations (**Figure 10B**, middle panel). Zero-lag synchronization amongst neighboring DS-ON cells is the product of shared excitation passing exclusively through an indirect gap junction coupling that operates through polyaxonal amacrine cells. Similarly, in simulations of spiking neurons, zero-lag synchronization emerges between two populations of neurons that are coupled exclusively through an indirect excitatory pathway involving a third population of neurons (**Figure 8A**).

The emergence of zero-lag synchronization through an indirect excitatory pathway has been reported in other computational work (Vicente et al., 2008); however, previous work did not address the question of how a network can transiently synchronize and desynchronize in response to stimulation. Using two different models of neuronal activity, we showed that patterns of activity in a relay network generally remain stuck in a strict limit cycle and are highly unresponsive to external stimuli. This limitation is particularly problematic given the high prevalence of relay networks in brain regions that play a central role in the integration of polysensory information, including dorsolateral prefrontal cortex, posterior cingulate cortex, and insula (Sporns et al., 2007). These regions, by their anatomical location and functional role, are expected to be highly responsive to input activation. Our simulation results provide a potential mechanism whereby a fixed anatomical substrate based on a relay network can, through selective gain inhibition, modulate its firing rate in response to an incoming stimulus. This mechanism is similar in essence to a recent gating network (Vogels and Abbott, 2009) where responses can be gated "on" by a command signal that disrupts the precise balance of excitation and inhibition. In our case, increased gain inhibition provides a way of breaking the fixed limit cycle attractor of a populations of neurons. In living systems, synapse-specific gain inhibition could be achieved by homeostatic mechanisms that dampen network reverberation, as evidenced in CA3 pyramidal cells (Kim and Tsien, 2008). It could also be achieved via cholinergic modulation, which performs cell-specific targeting and exhibits rapid response times (Ford et al., 2012; Taylor and Smith, 2012).

Zero-lag synchronization is proposed to play a number of functional roles in neuronal information processing. Synchronized activity may enhance the saliency of incoming stimuli, thus controlling the flow of information transmitted to downstream neurons. Zero-lag synchronization also provides an exquisite mechanism for precise temporal responses to rhythmic stimuli (Thivierge and Cisek, 2008, 2011), and may in itself constitute a unique channel for information transmission. Conceptually, a code based on synchronized action potentials necessitates a fewer number of presynaptic neurons to generate a postsynaptic response, and therefore allows for a greater number of input combinations than a code based on asynchronous activity (Stevens, 1994). In DS-ON ganglion cells, transient zero-lag synchronization is proposed to play a role in movement detection (Ackert et al., 2006), where a prominent synchronized/desynchronized response reinforces the presence of movement along a cell's preferred direction.

The transient synchronization of a neuronal population in response to a stimulus is supported by a range of experiments in cat cortex (Gray and Singer, 1989) as well as human electroencephalography (Rodriguez et al., 1999). A simulated network that generates synchronized oscillations only as long as a specific external signal is applied—and returns to a non-synchronized state once the signal is removed—is consistent with experiments where oscillations are observed only during the presence of a particular stimulation (Doiron et al., 2003; Ackert et al., 2006).

## **CONCLUSION AND FUTURE WORK**

Taken together, our simulation results show that a variety of factors—including patterns of synaptic connectivity, delays in synaptic transmission, synaptic efficacies, selective gain inhibition, and surrounding network activity—contribute to both spontaneous and evoked activity in local neuronal networks. These factors provide a panoply of constraints and degrees of freedom that shape the landscape of behaviors that emerge from the interaction of neurons in synaptic circuits of the brain. Future work could extend our results by investigating how connectivity schemes (e.g., allowing both excitatory and inhibitory connections) delimit the patterns of activity produced in local populations of neurons.

## **ACKNOWLEDGMENTS**

This research was supported by grants to Jean-Philippe Thivierge from the NAKFI Keck Future Initiatives, NSERC Discovery, and CIHR operating funds. André Longtin was funded by an NSERC Discovery Grant. Authors are thankful to Mikhail Rubinov for comments on an earlier draft of the manuscript. The authors declare no competing financial interests.

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 08 December 2013; accepted: 02 March 2014; published online: 20 March 2014.*

*Citation: Thivierge J-P, Comas R and Longtin A (2014) Attractor dynamics in local neuronal networks. Front. Neural Circuits 8:22. doi: 10.3389/fncir.2014.00022 This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Thivierge, Comas and Longtin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Color opponent receptive fields self-organize in a biophysical model of visual cortex via spike-timing dependent plasticity

#### *Akihiro Eguchi <sup>1</sup> \*, Samuel A. Neymotin2,3 and Simon M. Stringer <sup>1</sup>*

*<sup>1</sup> Oxford Centre for Theoretical Neuroscience and Artificial Intelligence, University of Oxford, Oxford, UK*

*<sup>2</sup> Department of Physiology and Pharmacology, Downstate Medical Center, State University of New York, New York, NY, USA*

*<sup>3</sup> Department of Neurobiology, Yale University School of Medicine, New Haven, CT, USA*

### *Edited by:*

*A. Ravishankar Rao, IBM Research, USA*

#### *Reviewed by:*

*James A. Bednar, University of Edinburgh, UK Yoonsuck Choe, Texas A&M University, USA*

#### *\*Correspondence:*

*Akihiro Eguchi, Department of Experimental Psychology, Oxford Centre for Theoretical Neuroscience and Artificial Intelligence, University of Oxford, South Parks Road, Oxford, Oxfordshire, OX1 3UD, UK e-mail: akihiro.eguchi@psy.ox.ac.uk*

Although many computational models have been proposed to explain orientation maps in primary visual cortex (V1), it is not yet known how similar clusters of color-selective neurons in macaque V1/V2 are connected and develop. In this work, we address the problem of understanding the cortical processing of color information with a possible mechanism of the development of the patchy distribution of color selectivity via computational modeling. Each color input is decomposed into a red, green, and blue representation and transmitted to the visual cortex via a simulated optic nerve in a luminance channel and red–green and blue–yellow opponent color channels. Our model of the early visual system consists of multiple topographically-arranged layers of excitatory and inhibitory neurons, with sparse intra-layer connectivity and feed-forward connectivity between layers. Layers are arranged based on anatomy of early visual pathways, and include a retina, lateral geniculate nucleus, and layered neocortex. Each neuron in the V1 output layer makes synaptic connections to neighboring neurons and receives the three types of signals in the different channels from the corresponding photoreceptor position. Synaptic weights are randomized and learned using spike-timing-dependent plasticity (STDP). After training with natural images, the neurons display heightened sensitivity to specific colors. Information-theoretic analysis reveals mutual information between particular stimuli and responses, and that the information reaches a maximum with fewer neurons in the higher layers, indicating that estimations of the input colors can be done using the output of fewer cells in the later stages of cortical processing. In addition, cells with similar color receptive fields form clusters. Analysis of spiking activity reveals increased firing synchrony between neurons when particular color inputs are presented or removed (ON-cell/OFF-cell).

**Keywords: brain modeling, visual cortex, neocortex, color, color selectivity, self-organizing color maps, self-organizing feature maps, STDP**

## **INTRODUCTION**

It has long been known that many neurons in primary visual cortex (V1) are tuned to exhibit preference to particular simple oriented line segments, forming orientation maps that capture the preferred orientation of neurons across the cortical surfaces (Hubel and Wiesel, 1962). Similarly, clusters of color-selective neurons in areas V1/V2 have been reported, as mapped with optical imaging and electrophysiological recordings (Landisman and Ts'O, 2002; Friedman et al., 2003; Xiao et al., 2003; Lu and Roe, 2008; Salzmann et al., 2012). While several computational studies have been conducted to explain the emergence of the orientation map (Somers et al., 1995; Choe and Miikkulainen, 1998; Paik and Ringach, 2011), only a few have been done over such patchy distribution of color selectivity within an area of V1/V2 (Bednar et al., 2005; Rao and Xiao, 2012). Barrow et al. (1996) have proposed a model for the formation of cortical blobs, regions in primary visual cortex that are densely stained by cytochrome oxidase (CO) (Livingstone and Hubel, 1984), using the Hebbian learning rule. This model reproduces receptive fields of neurons inside and outside CO blobs, and the results showed that neurons outside the blobs are selective for orientation while neurons inside the blobs are selective for color. However, the spatial organization of a large number of color-selective areas was not studied in their model. In this paper, we investigate the emergence of the spatial organization of color preference maps by developing a hierarchical neural network model that reflects anatomically faithful processing pathways and projections.

Physiological studies have shown that color information is first represented by the activity of specific types of photoreceptors and transmitted along specific fibers in the optic nerve (Komatsu, 1998). Visual signals leaving the eyes then reach the primary visual cortex via the lateral geniculate nucleus (LGN). LGN has multi-layered organization, and different color information is coded at specific layers (Chatterjee and Callaway, 2003). Although actual neural processing is not known, Komatsu and Goda (2009) theorized that a two-stage model can explain the transformation of color signal that takes place between photoreceptors and V1, resulting in forming the color selective neurons. At the first stage, signals from color opponent neurons are linearly summed with various combinations of weights, with the results rectified. This information is then propagated to neurons in the second stage where a further linear summation and rectification is performed.

Rao and Xiao (2012) have recently started investigating similar principles in computational simulations and successfully produced maps of orientation and color selectivity using anatomically realistic projections incorporating two color opponent channels and a luminance channel. However, this model used rate-coded neurons, which do not convey the precise times of action potentials or spikes emitted by cells. Various physiological studies have indicated that spiking dynamics can be important for the simulation and information processing (Sugase et al., 1999; Freiwald and Tsao, 2010). Although our current model does not investigate orientation selectivity, one of the aims of our study is to expand the focus in previous research (Bednar et al., 2005; Rao and Xiao, 2012) to see if it is possible to observe the spatial organization of color preference maps and spike-timing related phenomena such as ON/OFF selectivity using more physiologically realistic Hodgkin–Huxley (HH) neuron models via Spike-Timing Dependent Plasticity (STDP).

Many neural networks are implemented with rate-coded neuron since it is observed that the mean firing rates of sensory neurons are correlated with the intensity of the encoded stimulus feature. For example, it is widely viewed that the information sent to the visual cortex by the retinal ganglion cells are encoded by the mean firing rates of spike trains generated with a Poisson process. A theoretical study conducted by Rullen and Thorpe (2001) showed that rate codes are optimal for fast information transmission but cannot account for the efficiency of information transmission between the retina and the brain; however, temporal structure of the spike train can be efficiently used to maximize the information transfer rate. This could therefore be an important feature that contributes to the development of neurons tuned to specific features.

Another benefit of our approach is that the precise firing times of spiking HH neurons allow investigating the temporal dynamics of information processing. Such investigations could include determining the role of temporal processing of C1, C2, and L channels in LGN (Chatterjee and Callaway, 2003), and selective representation of different stimuli by neuronal population synchronization (Evans and Stringer, 2013). In addition, spiking neurons allow incorporation of biologically plausible learning rules, such as STDP. A number of experiments (Markram et al., 1997; Bi and Poo, 1998) have reported that synaptic strength changes depending on presynaptic and postsynaptic spike time, and this mechanism has been extensively studied from a theoretical point of view (Gerstner et al., 1996; Abbott and Nelson, 2000).

Meanwhile, similar to the orientation maps and color maps, physiological studies have shown that various brain areas manifest a small-world structure, characterized by the presence of highly clustered neurons (Yu et al., 2008), and the factors leading to this organization have been investigated in several theoretical works (Shin and Kim, 2006; Kato et al., 2007, 2009; Basalyga et al., 2011). In the present study, we were particularly interested in whether such small-world structures could evolve from a network whose weights were initialized randomly, after learning with natural images.

We speculated there would be difficulty in the development of such cells since the representation of color is more complex than oriented bars. However, with this model, we hypothesized that the response patterns of neurons in the output layer (layer 5 of V1) would develop heightened responses to specific colors solely due to learning taking place during exposure to multiple image patches extracted from natural images of indoor scenes used in Quattoni and Torralba (2009), as a result of integrating different color opponent signals that occurred at different levels of the network. We also hypothesized that the learning would allow for a distribution of neurons that were tuned to similar color input with spatial clustering, where neurons within the cluster had heightened synaptic weights, relative to neurons outside of the cluster.

## **MATERIALS AND METHODS**

#### **MODEL** *Architecture*

The model is composed of nine layers of neurons which are organized into five hierarchical areas: photoreceptor layers (R, G, B), lateral geniculate nucleus (LGN) layers (L, C1, C2), V1 layer 4 (L4), V1 layer 2/3 (L2/3), and V1 layer 5 (L5). The dimensions of each layer are shown in **Table 1**, and the total number of neurons is thus 5700.

Each color input presented to the network is first decomposed into an RGB representation (range: 0–1) in digital images to be consistent with the trichromatic color vision in primates as a result of S, M, and L cones (Rowe, 2002) (**Figure 1**). The degree of each input is represented as different spiking frequencies of photoreceptors with 10% of random noise. To be consistent with physiology, a stimulus that a human would perceive as red activates the green channel very strongly as well. The frequency of each cone is determined as follow:


Specific combinations of the decomposed color signals are then projected to cells in LGN. The projections reflect the physiological findings that reported different characteristics in different layers of LGN (Shapley et al., 1981). Specifically, as later studies

#### **Table 1 | Dimensions of each layer.**


revealed, different layers of LGN receive different visual information via optic nerves and show different functionality, forming a luminance channel (L) and two opponent color channels, comprising red–green (C1) and blue–yellow (C2) channels as follows (Casagrande, 1994; Goda et al., 2009; Rao and Xiao, 2012):


Physiological studies also report that while the MC and PC pathways project their output to V1 L4, the KC pathway terminates in V1 L2/3 (Chatterjee and Callaway, 2003), and many neurons in L2/3 project excitatory connections to the neurons in V1 L5 (Douglas and Martin, 2007). Our model incorporates this anatomical architecture (**Figure 1**). Physiological evidence indicates that there is heavy feedback from V1 to LGN (from layer 6) and the thalamic reticular nucleus is involved in both the feed-forward and feedback pathways, and data also suggests that retinal ganglion cells have widely different spatial extent; however, these are beyond the scope of this paper and are not explicitly modeled.

### *Synaptic connections*

Convergent connections are established to each neuron from a topologically corresponding region of the preceding layer, leading to an increase in the receptive field size of neurons through the visual processing areas, which reflects the known physiology of the primate ventral visual pathway (Pettet and Gilbert, 1992; Freeman and Simoncelli, 2011). While synaptic weights between the photoreceptor layers and LGN layers are kept static, the weights of other feed-forward connections are learned through visually guided learning.

Each feed-forward connection requires a 1 ms delay for signal transmission. Each neuron also establishes lateral short-range excitatory connections and long-range inhibitory connections, forming a Mexican-hat spatial profile (**Figure 2**). Whether this kind of lateral connectivity exists at the anatomical level is debatable (Martin, 2002; Kang et al., 2003; Hopf et al., 2006; Adesnik and Scanziani, 2010), since a detailed microcircuitry map at the neuron-to-neuron level is not currently available. However, we incorporated this architecture to (1) be consistent with a previous model by Rao and Xiao (2012) and (2) to abstract the function exhibited by this kind of architecture (Kang et al., 2003; Neymotin et al., 2011b). Further experimental work that details the wiring of cortical microcircuitry may reveal whether these considerations were justified (Alivisatos et al., 2013). The synaptic delay is 1 ms for the excitatory connections and 4 ms for the inhibitory connections.

## *Learning mechanism (STDP)*

While synaptic weights at the connections between photoreceptor layers and LGN layers were fixed, weights in all the other feedforward connections were plastic. Each synaptic weight in the model was learned using STDP, where Long-term potentiation (LTP) is caused if the pre-synaptic spike precedes the postsynaptic spike, and Long-term depression (LTD) is caused if the spike timing is in the opposite order. The degree of the modification depends on how close the two spikes are in time (Bi and Poo, 1998) as follows:

$$
\Delta\omega = \begin{cases}
\text{LR} \times \exp\left(\frac{-(t\_{\text{post}} - t\_{\text{pre}})}{p\_{\text{tun}}}\right) \dot{x} (t\_{\text{post}} - t\_{\text{pre}}) > 0 \\
\end{cases} \tag{1}
$$

**connectivities.** Activations of adjacent cells in the preceding layer are transmitted to a topologically corresponding cell in the following layer. Tiles filled with red represent cells that receive excitatory lateral connections while tiles filled with blue represent cells that receive inhibitory lateral connections, forming a Mexican-hat spatial profile.

where LR is a learning rate, *t*pre is the time when presynaptic cell becomes activated, *t*post is the time when postsynaptic cell becomes activated, and *p*tau/*d*tau controls the range of the influence. The curve generated by this function is show in **Figure 3**. Weights are originally randomly assigned within a fixed range, and after every iteration, weights in the same layers are normalized so that the mean of all the values are always kept in the middle of the pre-specified range, and also to prevent runaway excitation (Neymotin et al., 2011a, 2013; Rowan and Neymotin, 2013). Neurophysiological evidence for synaptic weight normalization is provided by Royer and Paré (2003).

## *Neuron model*

Model neurons utilized the standard parallel conductance model with Hodgkin–Huxley dynamics for generating action potentials. Neurons consisted of a single compartment (diameter of 30μm, length of 10μm, axial resistivity of 100 cm). The rate of change of a neuron's voltage (V) was represented as −*Cm dV dt* = *g*pas(*v* − *e*leak) + *i*syn + *iNa* + *iK*, where *Cm* is the capacitive density (10μF/cm2), *i*syn is the summed synaptic current, and *iNa* and *iK* represent the *Na*<sup>+</sup> and *K*<sup>+</sup> currents from the Hodgkin–Huxley channels. *g*pas represents the leak conductance (0.001 nS), which was associated with a reversal potential, *e*leak, of 0 mV.

Synapses were modeled using an instantaneous rise of conductance, followed by exponential decay with specified time-constant, τ. For excitatory synapses, we utilized AMPA synapses (τ = 5 ms, *e*rev = 0 mV), while for inhibitory GABA synapses (τ = 10 and *e*rev = −80). Synaptic currents followed *i*syn = *g*(*v* − *e*rev), where *v* is the membrane potential, and *e*rev is the reversal potential associated with the synapse.

## *Software*

Simulations were run using the NEURON simulation environment with the Python interpreter, multithreaded over 16–32 threads (Hines and Carnevale, 2001; Carnevale and Hines, 2009; Hines et al., 2009). Simulation is posted on ModelDB (https://senselab.med.yale.edu/ModelDB/ShowModel.

(*p*tau = 17 ms) used for spike-timing dependent plasticity where the equation is given in Equation (1) **(B)** Function without STDP: the synaptic weights are potentiated whenever both pre and post synaptic neurons become activated during the training time for 300 ms.

asp?model=152197) (Hines et al., 2004). Simulations were run on Linux on a 2.93 GHz 16-core Intel Xeon CPU X5670. A 300 ms simulation ran in approximately 30 s.

## **DATA ANALYSIS METHODS** *Clustering*

In order to quantify the degree of clustering of the activations in the network, a clustering coefficient *C* is calculated based on the responses among different color inputs at every training iteration as follows (modified from Kato et al., 2007):

$$C = \frac{1}{n \text{Cells} \times n \text{Stime}} \sum\_{s=1}^{n \text{Stime}} \sum\_{i=1}^{n \text{Cells}} C\_{s,i} \tag{2}$$

$$C\_{s,i} = \frac{\sum\_{\substack{s,i\\l=1\ m=l+1}}^{k\_{s,i}} \sum\_{m=l+1}^{k\_{s,i}} \text{(FR}\_{s,l} \times \text{FR}\_{s,m})}{\sum\_{ki} \text{C}\_2} \tag{3}$$

where *n*Cells is the number of neurons in a network; *n*Stims is the number of stimuli during the testing; FR*s*,*<sup>i</sup>* is the firing rates of the cell *i* when exposed to a stimulus *s*; *ki* sets the nearby neurons from the *i*-th neuron for the analysis. We use 9 (3 × 3) for the *k* value.

## *Single-cell information*

A single cell information measure was applied to individual cells to measure how much information is available from the responses of a single cell about which color input is present. The amount of color specific information that a certain cell transmits is calculated from the following formula:

$$I(s,\vec{R}) = \sum\_{r \in \vec{R}} P(r|s) \log\_2 \frac{P(r|s)}{P(r)} \tag{4}$$

Here *s* is a particular color and *R* is the set of responses of a cell to the set of color stimuli, which are composed of eight colors slightly varied the RGB values of original color by ±1%. This is based on the assumption that the same set of tuned cells will still respond to slightly variant colors and is to well differentiate the tuned cells from randomly responding cells. The maximum information that an ideally developed cell could carry is given by the formula:

$$\text{Maximum cell information} = \log\_2(n \times p) \text{ bits} \tag{5}$$

As eight different sets of colors (combination of 0 and 1 for each RGB value) are used in this analysis, the maximum information could be carried in this analysis is 3.

## *Multiple-cell information*

A multiple-cell information measure was used to quantify the network's ability to tell which stimulus is currently exposed to the network based on the set of responses, *R*, of a sub-population of cells, *C*-, as following formula with details given by Rolls and Milward (2000).

$$I\_{\vec{C}}\left(\mathcal{S},\mathcal{S}'\right) = \sum\_{s,s'} P\left(s,s'\right) \log\_2 \frac{P\left(s,s'\right)}{P(s)P\left(s'\right)}\tag{6}$$

$$P\left(s'\right) = \sum\_{s \in S} P\left(s'|R\_{\widehat{C}}(s)\right) \times P\left(R\_{\widehat{C}}(s)\right) \tag{7}$$

$$P\left(s,s'\right) = P\left(s'|R\_{\vec{C}}(s)\right) \times P\left(R\_{\vec{C}}(s)\right) \tag{8}$$

Here, *S* represents the set of the stimuli presented to the networks, and *C* defines the set of cells used in the analysis, which had as single cells the most information about which color input was present. From the set of cells *C*-, the firing responses *RC*- (*R* = *r*(*c*)|*c* ∈ *C*-) to each color in *S* are used as the basis for the Bayesian decoding procedure as follows:

$$P\left(s'|R\_{\vec{C}}\right) = \frac{P\left(s'\right)\prod\_{c\in\vec{C}}P\left(R\_{\epsilon}(s')|s'\right)}{\sum\_{s'' \in \vec{S}}P\left(s''\right)\prod\_{c\in\vec{C}}P\left(R\_{\epsilon}(s'')|s''\right)}\tag{9}$$

$$P\left(R\_{\mathfrak{c}}(\mathsf{s})\right)|s'\rangle = \frac{\sum\_{t=1}^{n\text{Trans}} \mathsf{p}df\left(R\_{\mathfrak{c}}(\mathsf{s},t), \bar{R}\_{\mathfrak{c}}\left(s'\right), \text{SD}\_{\mathfrak{c}}\left(s'\right)\right)}{n\text{Trans}} \tag{10}$$

where *n* Trans defines the number of possible transforms; in this case, similar but slightly different colors, and pdf computes the probability density function at firing response of a subset of cells when exposed to a stimulus *s* at *t*th transforms using the normal distribution with their mean and standard deviation.

## **RESULTS**

The results described in this study used a network model trained with various small color image patches extracted from original natural images of indoor scenes used in Quattoni and Torralba (2009). The size of the photoreceptor layer in our model is 10 × 10 pixels while the size of original images was an average of 504.1 × 658.4 pixels (112 images). The training sessionconsisted of 2000 iterations, where 2000 different 10 × 10 image patches were extracted from the set of images. This was designed as an abstraction of natural viewing, where eyes saccade, and the activation of photoreceptors corresponds to visual inputs bounded by their range of view.

#### **LEARNING PRODUCES SPATIAL CLUSTERING**

During the training, synaptic efficacy between each of two layers progressed from a uniform distribution at the initial state toward a binary distribution where only a limited number of synaptic connections were highly strengthened or weakened (**Figure 4**). This convergence toward an bimodal equilibrium state is consistent with other self-organizing modeling work with STDP (Song et al., 2000; Kato et al., 2009; Basalyga et al., 2011). Contrary, physiological studies have shown that synaptic weights tend to have unimodal distributions with a positive skew (Barbour et al., 2007). Barbour et al. (2007) raised a possible reconciliation with the bimodal distributions of modeling with such experimental data, given that the dendritic distribution of synaptic weights may have a wide range of values, due to electrotonic filtering effects. However, in order to explore this possibility, further investigation will be required.

Investigation into the firing count of each neuron to different color inputs shows that the weight convergence resulted in development of clustered responses in the networks (**Figure 5**). A comparison between the results with the weight distribution plots in **Figure 4** shows that even though the average weight was kept constant, neuronal firing activity became more prominent and deviated after the training; it was sparse (average rate of 2.165 Hz with standard deviation of 0.874) prior to learning, but after 2000 iterations of 300 ms exposure to image patches extracted from natural indoor images, the network developed different clustered firing patterns of neurons (average rate of 3.966 Hz with standard deviation of 1.169) for eight different

color inputs (red, orange, yellow, green, aqua, blue, purple, and pink).

In **Figures 5A,B**, the boundaries of peaks of firing counts for seven different colors (red, orange, yellow, green, aqua, blue, and purple) before and after training are plotted. The result shows that the training resulted in developing color selective clustered responses. Normalized firing activity of seven neurons in V1 L5 were recorded and plotted in **Figure 5C**. These results failed to show a clear spatial shift of the activation with gradual change of color inputs as reported in Xiao et al. (2003); however, the results revealed gradual changes of firing patterns according to changes of input colors, which is partially consistent with the physiological findings. This also shows that some cells show higher selectivity than others at responding to similar colors. This is likely due to the fact that the color representation takes a specific combination of three continuous values of RGB. Depending on the trained weights, activations of some neurons may only be influenced by one or two of the three values, and the activation patterns also vary due to different combinations of those values and influences from other nearby neurons.

We calculated a clustering coefficient [*C*; Equations (3, 3)] to assess the effectiveness of training in producing spatial clustering within the network. **Figure 6** shows *C* of V1 L4, V1 L2/3, V1 L5, as well as of V1 L5 trained with Hebb-like learning rule, plotted as a function of training iteration. The result demonstrates that the networks trained with STDP rule gradually increases clustering coefficients as training proceeds while the network trained with Hebb-like learning rule remains relatively low clustering coefficient.

The emergence of clustering may be explained by the lateral excitatory connections described in section 2.1.2. When a specific neuron becomes activated, the signal is propagated to

respectively. Additionally, the clustering coefficients of V1 L5 when the network was implemented with Hebb-like rule is plotted by the line with asterisk markers. The result demonstrates that the networks trained with STDP rule gradually increases clustering coefficients as training proceeds while the network trained with Hebb-like learning rule remains relatively low clustering coefficient.

the neighboring neurons making them more likely to become activated as well. Once the neighboring cell reaches a threshold and becomes activated, synaptic connections convergent onto the cell from recently activated cells in the preceding layer become strengthened via STDP. Repetitions of this process are likely to be the cause of the development of the clustered responses of cells. This phenomenon should be prominent only among nearby cells because of lateral propagation delays and long-range lateral inhibition.

The precise temporal dependence of STDP is crucial for the clustering learning process. Activation of neurons are laterally propagated within layers but with a specified delay. Therefore, temporal differences of the activations between the source in the preceding layer and the targets in the following layers become large as the signal is propagated. As a result, the degree of LTP decays as the differences become large, and LTD is turned on if the post-synaptic activation timing becomes closer to the next presynaptic activation, thus forming the distinct clustering responses in the networks.

In order to confirm the importance of spike-timing in forming color receptive field clustering, we ran a control simulation, using a Hebbian plasticity synaptic learning rule, which does not take into account the timing of pre- and post-synaptic neuronal spiking (**Figure 3B**). After learning with this Hebbian plasticity rule, the clustering coefficient value remained low (**Figure 6** lines with asterisks) relative to the results in the network trained with STDP. This underlines the importance of STDP in developing clustering in our model.

In addition, our model shows that the clustering coefficient in higher layers tended to be larger. This observation makes us expect information to gradually change in the different layers, and this assumption has been confirmed in the next section.

## **SELECTIVITY OF THE RESPONSES**

In order to identify how the learned connectivity shaped output neuron sensitivity to stimuli, the techniques of Shannon's information theory were employed (Rolls and Treves, 1998). If the responses *r* of a neuron carry a high level of information about the presence of a particular color stimulus *s*, this implies that the neuron will respond selectively to the presence of that color. Two information measures were used to assess the ability of the network to develop neurons that are selective to the presence of a particular color by measuring single cell and population information (see sections 2.2.2, 2.2.3). Since eight different sets of colors (red, orange, yellow, green, aqua, blue, purple, and pink) are used in this analysis, the maximum information carried in this analysis is 3 bits.

**Figure 7A** shows the single cell information analysis as plotted in rank order according to maximum information each cell carries for a specific stimulus. The results compare the information distribution of each layer in the trained network and of the final layer (V1 L5) in the untrained and trained network. The results demonstrate that neurons in the trained network generally carry more single-cell information.

While useful in assessing the tuning properties of a particular neuron, the single-cell information measure cannot provide mutuality of the responses; if all cells learned to respond to the same color input (according to the single-cell measure) then there would be relatively little information available about the whole set of color stimuli *S*. To address this issue, we used a multiple-cell information measure, which assesses the amount of information that is available about the whole set of color inputs from a population of neurons (see section 2.2.3).

In **Figure 7B**, the multiple cell information measures are plotted according to the number of cells used in the analysis. The result shows that the trained network conveys more color specific information than the untrained network. More interestingly, we found that the amount of color specific mutual information reaches a maximum with fewer neurons in the higher layers: 13 neurons in L4, 10 neurons in L2/3, and 8 neurons in L5. This analysis indicates that estimations of the input colors can be done using the output of fewer cells in the later stages of cortical processing.

More precisely, the total amount of mutual information (across a layer) can not increase through further processing as the Data Processing Inequality (DPI) states—it can only be preserved or lost. In other words, if all the information from all cells in each of the two layers was added up, it will decrease in the higher layer. However, our specific information measure explained in section 2.2.3 can increase for particular cells, as they become more selective throughout the layers. In this case, some of that information has shifted into different cells, and so all stimuli can now be represented with fewer neurons, allowing for fewer required cells to convey maximum information. Our information measure therefore improves, showing that the cells are becoming more tuned, even though the total information in the layer has decreased.

## **ON- AND OFF-CELLS**

The firing pattern of each cell in response to turning a stimulus ON and OFF was also investigated. During this testing procedure, eight different colors (red, orange, yellow, green, aqua, blue, purple, and pink) are presented for 240 ms, followed by 60 ms of no visual input presentation, and the voltage level of each neuron is recorded. In order to find if any neuron developed ON/OFF sensitivities during training with similar properties to those found in V1/V2 *in vivo* (Michael, 1978; Friedman et al., 2003), from each recorded voltage dynamics, the 30 neurons which responded the most during the first 60 ms and the last 60 ms were selected to be plotted in **Figure 8**. Similar to the physiological findings, we found both ON- and OFF-cells for each different color input, where populations of neurons showed a burst of firing just after a presentation or removal of a color input.

Also, further analysis revealed that some of those cells displayed the temporal color opponent property as reported in Friedman et al. (2003). **Figure 9A** shows two types of such cells: Red-ON/Green-OFF cells and Yellow-ON/Blue-OFF cells.

**FIGURE 7 | Single-cell/Multiple-cell information analysis of V1 L5.** Solid lines represents trained networks while dotted line represents naive network. **(A)** The single cell information measure are plotted in rank order according to how much information they carry. The result show that the maximum information each cell carry drops rapidly in the naive network while most of the cells in the trained network carry relatively higher amount of information. **(B)** The multiple cell information measures are plotted according to the

**vigorously when color input is presented or removed, from each experiment.** The color bars under each raster plot represent times at which

colors are presented to the neurons (each color is presented for 240 ms and removed). From these results, we found that many neurons exhibit the characteristics of ON/OFF-cells in the trained network.

In order to find such cells, we first identify 100 cells that show Red-ON (or Yellow-ON) property, and then chose 30 cells from the subset that show Green-OFF (or Blue-OFF) property.

green is removed. Figure on the bottom shows 30 neurons that responds

Neurons in the layers are exposed to different colors in natural images during the training, so the development of ON-cells which exhibit specific responses to specific inputs can be explained with standard feed-forward competitive learning principles (Rolls and Treves, 1998). In contrast, the development of OFF-cells are due to the lateral inhibitory connections emitted by ON-cells: suppose there are ON-cells that were tuned to the color red. If red is presented to the network, these ON cells become activated making surrounding cells that receive inhibitory synaptic connections from the ON cells less likely to become activated. When the color input is removed, ON-cells stop activating. As a result, the surrounding cells are no longer suppressed by the ON-cells, demonstrating their being OFF-cells.

However, the question is where the OFF-cells receive excitatory input to enable them to remain activated after the removal of the color input. In other words, there should be some mechanism where ON-cells immediately stop receiving excitatory input while OFF-cells keep receiving excitatory input, even after the removal of the color input. This may be caused by the differences in firing timing of different input cells as explained in **Figure 10**.

In our model, the maximum activation frequency of input cells was set to 40 Hz (25 ms interspike interval), which is gamma oscillations which are widespread in the visual cortex. Also, different input cells have different randomly determined delays from the input cell receiving color input to its firing, which is reflected in their firing timings. As shown in **Figure 10**, suppose the spike timing of an input cell A is 24 ms earlier than another input cell B. This means that there is at most 24 ms difference between the final spike timing of cell A and the timing of cell B before the removal of the color input. This 24 ms difference will result in giving a chance for the OFF-cell that receives most of the inputs from the input cells such as B to become activated after an ON-cell that happens to receive most of the inputs from the input cells such as A stops activating inhibitory signals.

In order to confirm the importance of the delay for the development of such ON/OFF cells, we have also trained the same network without randomly determined delays from the input cell receiving color input to its firing timings. **Figure 9B** shows the

**FIGURE 10 | Diagrams to explain the speculated cause of the differences in durations of firing activities of different neurons after removal of color inputs.** In this experiment, the maximum activation frequency of input cells was set to 40 Hz, and different input cells have different delays in the firing timings. Firing activation timing of input cell A is 1 ms later than another input cell B. This means there are 24 ms differences in the last activation before the removal of the color input. This difference will result in giving a chance for the OFF-cell that receives most of the inputs from the input cells such as B to become activated after an ON-cell that happens to receive most of the inputs from the input cells such as A stops releasing inhibitory signals.

firing activities of each 30 neurons selected by the same procedure used to find Red-ON/Green-OFF cells and Yellow-ON/Blue-OFF cells earlier. The results show that in the network that employed inputs without randomized delays, we failed to find Green-OFF and Blue-OFF cells within each subset of 100 Red-ON cells and Yellow-ON cells. This result indicates that the randomized delay plays an important role for the development of the OFF cells.

In animal V1, much of the ON and OFF component of the responses are thought to be inherited from similar properties of LGN and RGC cells. Therefore, we are not expecting that onset and offset transients arise in V1 alone. However, our results suggested the possibility of multiple mechanisms that impact the firing times of these cells.

## **DISCUSSION**

In this study, we have developed a model of early visual processing of colors including the pathway beginning at photoreceptors and terminating in the fifth layer of V1. We have incorporated anatomically accurate projections of signals between layers and the biologically plausible learning of synaptic weights based on STDP using Hodgkin–Huxley models of neuronal dynamics.

We have successfully shown that the networks gradually develop clustered firing activity of neurons during training (section 3.1). Information analysis based on averaged firing rates of each neuron also confirmed development of neuronal color selectivity after the training (section 3.2). Our results also indicated that populations of neurons can provide reliable predictions of the input color presented to the retina. Interestingly, the color information measure by multiple-cell information analysis rises more rapidly with fewer cells from L4 → L2/3 → L5, suggesting that layered neocortical architecture may enable it to *boost* important information. We also found that if the synaptic weights in the network were learned via a Hebbian plasticity rule, the level of clustering coefficient remained low relative to the results in the network trained with STDP.

However, the question is why other models without STDP, including the model by Rao and Xiao (2012), show similar types of clustering merely due to Mexican-hat connectivity. One possibility would be that in many hierarchical unsupervised neural network models, each layer is trained separately in turn. This is important for synaptic connectivities in higher layers to be appropriately tuned. However, in our model, all the synaptic connectivities are learned simultaneously, which may be more realistic. The implication would be that STDP may allow a network to learn connectivities more flexibly without the traditional *greedy* method of teaching one layer at a time. We propose this hypothesis because adding another dimension of timing via STDP allows the synaptic weights to be dynamically updated in realtime whereas rate coded neurons depend on averaged firing rates within pre-specified time windows.

Furthermore, investigating neuronal voltage dynamics revealed the presence of both ON-cells and OFF-cells, which respond maximally immediately after presentation or removal of a particular color input. These results led us to hypothesize that the emergence of OFF-cells was caused by different spike timing delays from input cells (section 3.3).

The role of neuronal synchrony in color processing is still an open question particularly since our model demonstrates that information analysis based on firing rates can successfully predict the color input. However, while the network was trained with various color input in natural images, in this analysis, the network was tested only with eight clearly distinct colors, and in order to accurately decode the subtle differences between similar colors, synchrony and its timing may play an important role for the representations at least in our proposing mechanism. In addition, the importance of timing delays in the creation of ON/OFF cells suggests rate codes alone may not be sufficient in visual system development.

## **ROLE OF SPIKE-TIMING DELAYS IN CREATING ON/OFF CELLS**

The mechanism of the emergence of OFF-cells due to spike timing delays allows us to propose a possible *in vivo* mechanism of the development of the ON/OFF-cell that is also combined with the R/G opponency shown in **Figure 9**. As shown in **Figure 11**, we suppose there is a simplified network that consists of three cells in the LGN layers and two cells in output layer (*R*ON/*G*OFF cell and its neighboring cell N). In this schematic, LGN cells consist of a C1 (R/G opponent) cell and two L (monochrome) cells. In addition, one of the L cells, L1, has a delayed Green input (see details in **Figure 10**).

When the color red is presented to the network (**Figure 11A**), all three cells in the LGN become activated, and the *R*ON/*G*OFF output cell that receives excitatory inputs from the C1 cell and one L cell (L1) becomes highly activated. When the red input is removed (**Figure 11B**), only L cells become slightly activated due to the delayed connection, which does not have a large influence on the *R*ON/*G*OFF cell.

When Green color input is presented to the same network (**Figure 11C**), the L cells become activated. Subsequently, the N cell in the output layer that sends an inhibitory signal to the *R*ON/*G*OFF cell becomes activated as well. Because of the inhibition, the *R*ON/*G*OFF cell does not become highly activated even though it receives excitatory input from the preceding L1 cell. When the color input is removed (**Figure 11D**), the L1 cell that has the delayed connection from the Green cell is kept activated, which causes the *R*ON/*G*OFF cell to become activated.

Similarly, a possible mechanism of the ON/OFF-cell that is combined with Y/B opponency is provided in **Figure 12**. In the figure, we suppose a simplified network consists of three C2 (Y/B opponent) cells in the LGN layer and three cells in the output layer (*Y*ON/*B*OFF, N1, and N2). Each cell in the output layer receives excitatory input from one C2 cell (C21, C22, and C23) cell). N2 cell establishes inhibitory connection to N1, and the N1 establishes an excitatory connection to the target cell, *Y*ON/*B*OFF. In this network, C22 cell establishes the delayed connections discussed above (in **Figure 10**) from the R and G cells, and C23 cell establishes delayed connections from all R, G, and B cells.

As shown in **Figure 12A**, when the color yellow is presented, all C2 cells become activated. As a result, the target cell, *Y*ON/*B*OFF, should become highly activated by receiving excitatory input from the preceding C21 cell. In addition, the target cell *Y*ON/*B*OFF receives some excitation from N1 cell. When the color input is removed (**Figure 12B**), due to the delayed connections, C22 and C23 cells are kept active for an interval, but not C21. As a result, the target cell, *Y*ON/*B*OFF would not get highly activated.

When the color blue is presented (**Figure 12C**), none of the C2 cells would become activated, leading to no activation of the target cell. On the other hand, when the color input is removed (**Figure 12D**), C22 cell becomes activated to some degree due to the delayed connection from R and G with their weak activations caused by the color blue. This leads to the activation of N1 that establishes excitatory connection to the target cell, *Y*ON/*B*OFF. In this way, it is possible to provide a possible dynamical mechanism of ON/OFF-cells that involves color opponency.

In order to test the hypothesized architectures above, we have modeled the simple networks using the same set of neurons used in our computational model and recorded firing activity of each neuron for 300 ms (240 ms of color input presentation followed by 60 ms of no color input presentation) (**Figure 13**). The results show that the same target neuron exhibits characteristics of both *R*ON/*G*OFF, and *Y*ON/*B*OFF firing activity. However, the result also showed that those responses are not observed immediately after the presentation or removal of the color input. In other words, there is still activity in ON-cells after the stimulus is turned off. Also, OFF-cells show responses when the stimulus is turned on. These effects are due to the transitional delay of signals. However, as shown in **Figures 8**, **9**, the population activity shows a more clear ON/OFF response.

## **POTENTIAL LIMITATIONS**

Although our model predicts that spike timing is important for the effective development of color selectivity, our model did not investigate development of orientation selectivity, which is known to coexist with color selectivity, as investigated in previous models (Barrow et al., 1996; Rao and Xiao, 2012). Therefore, in future work it will be important to model co-development of both color and orientation selectivity. A different limitation of our model is that the representation of color input was based on simplified input cells that detect digital RGB values. To investigate more realistic mechanisms of development, biologically-accurate architectures of the various types of retinal cells that are involved in the process should be implemented.

## **CONVERGENCE OF APPROACHES**

Our model of the early visual system displays convergence between the fields of computational neuroscience and artificial neural networks (ANNs). Computational neuroscience has

**FIGURE 13 | (A)** Firing activity of each neuron in the simple network described in **Figure 11**. Left sub-panels show the activity when color input of red, RGB(1, 0, 0), is presented while right sub-panels show the activity when color input of green, RGB(0, 1, 0) is presented, both for 240 ms. In the figures on the top, the activity of C1, L1, and L2 are plotted with blue, green, and red color, respectively. The figure on the middle plots the activity of the neighboring cell, N, and the figure on the bottom plots the activity of the *R*ON/*G*OFF cell. **(B)** Firing activity of each

traditionally attempted to understand neuronal dynamics by building models by using known biological detail without forcing an explicit engineered goal. ANNs, which emerged from the field of artificial intelligence, have stressed an approach that aims to develop systems displaying intelligence by constraining the system design to a specified goal, while taking inspiration from biological systems (Hinton et al., 2006).

Recent developments in ANNs, including *deep learning*, a technique drawing inspiration from neurobiology, have made significant progress in recent years (Hinton et al., 2006) improving performance on visual information processing (Lee et al., 2009). Progress has also been made by training recurrent neural networks to perform extremely well on difficult, specialized classes of problems, such as handwritten character recognition (Graves and Schmidhuber, 2008). Related developments have also started focusing on investigations into utilizing brain-inspired informatics to improve the intelligence of current technologies (Eguchi et al., 2013). However, currently, even the best machine learning algorithms have difficulty in matching human performance in recognizing arbitrary classes of complex visual stimuli. Basic research in neurobiology, combined with utilization of biological detail in computer models, is therefore needed to enable further improvements in machine learning. Improved understanding of how the brain circuitry represents and processes visual information may inspire new classes of visual processing algorithms. We have used this approach to design our model, which allows correlation of its neuronal dynamics with electrophysiological data, takes into account known neuroanatomy, and uses a biologically

shows the activity when color input of yellow, RGB(1, 1, 0), is presented while right-subpanel shows the activity when color input of blue, RGB(0, 0, 1), is presented, both for 240 ms. In the top panels, the activities of C21, C22, and C23 are plotted with blue, green, and red color, respectively, and in the middle panels, the activities of N1 and N2 are plotted with blue and green color, respectively. The bottom panels display the activity of the *Y*ON/*B*OFF cell.

plausible learning rule (Markram et al., 1997), and therefore takes a step toward improved understanding of *in vivo* brain dynamics.

## **NEOCORTICAL ARCHITECTURE**

One of the basic goals of neuroscience is to elucidate the mechanisms by which the structure of the brain leads to its function (Shepherd, 2004). This depends on a careful study of neuroanatomy as well as functional measures *in vivo* (Weiler et al., 2008). The importance of changes in microcircuitry is underscored with experimental studies that have shown how alterations in cortical connectivity can lead to diseases, such as autism (Qiu et al., 2011). Since it is not possible to measure the state of all neurons it is important to combine computer modeling with known neurophysiological circuitry data (Lytton, 2008). Following this approach in our model allows us to make predictions on the function and development of several features observed in visual cortex *in vivo*.

Our model suggests that *in vivo*, the process of development of color clustering is more likely to initiate in earlier layers (L4) of V1. This may be testable via electrophysiological methods applied during different stages of development. Our model is also consistent with more general implications, suggesting that through a process of development, each layer of neocortex may learn to enhance important signals as they progress within the microcircuitry. Although initial synaptic weights in our model were randomly distributed, visual information and STDP allowed the feed-forward projections of the neocortex to learn the color information as the signals flowed in successive layers. In our model, the color information progressed from L4 → L2/3 → L5. Although L4 is the input layer into V1, the final output layer (L5) had the highest information content about the color stimuli. Further experiments will be needed to elucidate the role that individual layers play in shaping the information coding capacity of the neocortex.

Prior modeling (Stringer and Rolls, 2002; Rolls and Stringer, 2006; Dura-Bernal et al., 2012) and experiments (Hung et al., 2005) have shown the importance of the feed-forward architecture of the visual cortex ventral stream for object recognition. Although our work makes use of the feed-forward architecture of cortical areas, it also takes into account additional details of wiring, including recurrent connectivity. As more microcircuitry data becomes available, it will be possible to refine our model further (Alivisatos et al., 2013). Part of this process will involve combined experimental/computational approaches. For example, Hung et al. (2005) studied the ventral visual pathway with the aim of understanding how object recognition takes place by building pattern recognition algorithms that utilize inferotemporal cortex neuronal spiking information to assess both object category and identity. In the future it will be possible to extend our model to use similar techniques to quantify performance in object recognition that is based on accurate color processing.

## **ACKNOWLEDGMENTS**

The authors would like to thank Erik De Schutter (Okinawa Institute of Science and Technology) for organizing the Okinawa Computational Neuroscience Course 2013, where some of this research was conducted; Michael Hines (Yale) and Ted Carnevale (Yale) for NEURON simulator support; Tom Morse (Yale) for ModelDB support.

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 22 October 2013; accepted: 17 February 2014; published online: 12 March 2014.*

*Citation: Eguchi A, Neymotin SA and Stringer SM (2014) Color opponent receptive fields self-organize in a biophysical model of visual cortex via spike-timing dependent plasticity. Front. Neural Circuits 8:16. doi: 10.3389/fncir.2014.00016*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Eguchi, Neymotin and Stringer. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Comparison of the dynamics of neural interactions between current-based and conductance-based integrate-and-fire recurrent networks

#### *Stefano Cavallari <sup>1</sup> \*, Stefano Panzeri 1,2 and Alberto Mazzoni 1,3*

*<sup>1</sup> Center for Neuroscience and Cognitive Systems@UniTn, Istituto Italiano di Tecnologia, Rovereto, Italy*

*<sup>2</sup> Max Planck Institute for Biological Cybernetics, Tübingen, Germany*

*<sup>3</sup> The BioRobotics Institute, Scuola Superiore Sant'Anna, Pisa, Italy*

#### *Edited by:*

*Ehud Kaplan, The Mount Sinai School of Medicine, USA*

#### *Reviewed by:*

*Alex Casti, Fairleigh Dickinson University, USA Sam Neymotin, State University of New York, USA*

#### *\*Correspondence:*

*Stefano Cavallari, Center for Neuroscience and Cognitive Systems@UniTn, Istituto Italiano di Tecnologia, Corso Bettini 31, 38068 Rovereto (TN), Italy e-mail: stefano.cavallari@iit.it*

Models of networks of Leaky Integrate-and-Fire (LIF) neurons are a widely used tool for theoretical investigations of brain function. These models have been used both with current- and conductance-based synapses. However, the differences in the dynamics expressed by these two approaches have been so far mainly studied at the single neuron level. To investigate how these synaptic models affect network activity, we compared the single neuron and neural population dynamics of conductance-based networks (COBNs) and current-based networks (CUBNs) of LIF neurons. These networks were endowed with sparse excitatory and inhibitory recurrent connections, and were tested in conditions including both low- and high-conductance states. We developed a novel procedure to obtain comparable networks by properly tuning the synaptic parameters not shared by the models. The so defined comparable networks displayed an excellent and robust match of first order statistics (average single neuron firing rates and average frequency spectrum of network activity). However, these comparable networks showed profound differences in the second order statistics of neural population interactions and in the modulation of these properties by external inputs. The correlation between inhibitory and excitatory synaptic currents and the cross-neuron correlation between synaptic inputs, membrane potentials and spike trains were stronger and more stimulus-modulated in the COBN. Because of these properties, the spike train correlation carried more information about the strength of the input in the COBN, although the firing rates were equally informative in both network models. Moreover, the network activity of COBN showed stronger synchronization in the gamma band, and spectral information about the input higher and spread over a broader range of frequencies. These results suggest that the second order statistics of network dynamics depend strongly on the choice of synaptic model.

**Keywords: recurrent neural network, integrate-and-fire neurons, current based neuron models, conductance based neuron models, spike correlation, local field potentials, correlation analysis, information encoding**

## **INTRODUCTION**

Networks of Leaky Integrate-and-Fire (LIF) neurons are a key tool for the theoretical investigation of the dynamics of neural circuits. Models of LIF networks express a wide range of dynamical behaviors that resemble several of the dynamical states observed in cortical recordings (see Brunel, 2013 for a recent review). An advantage of LIF networks over network models that summarize neural population dynamics with only the density of population activity, such as neural mass models (Deco et al., 2008), is that LIF networks include the dynamics of individual neurons. Therefore LIF networks can be used to investigate phenomena, such as the relationships among spikes of different neurons, that are not directly accessible to simplified mass models of network dynamics.

A basic choice when designing a LIF network is whether the synaptic model is voltage-dependent (conductance-based model) or voltage-independent (current-based model). In the former case the synaptic current depends on the driving force, while this does not happen in the current-based model. Current-based LIF models are popular because of their relative simplicity (see e.g., Brunel, 2013) and they have the key advantage of facilitating the derivation of analytical closed-form solutions. Thus currentbased synapses are convenient for developing mean field models (Grabska-Barwinska and Latham, 2013), event-based models (Touboul and Faugeras, 2011), or firing rate models (Helias et al., 2010; Ostojic and Brunel, 2011; Schaffer et al., 2013), as well as in studies examining the stability of neural states (Babadi and Abbott, 2010; Mongillo et al., 2012). Moreover, current-based models are often adopted, because of their simplicity, to investigate numerically network-scale phenomena (Memmesheimer, 2010; Renart and Van Rossum, 2012; Gutig et al., 2013; Lim and Goldman, 2013; Zhang et al., 2013). On the other hand, conductance-based models are also widely used because they are more biophysically grounded (Kuhn et al., 2004; Meffin et al., 2004). In particular, only conductance-based neurons can reproduce the fact that when the synaptic input is intense, cortical neurons display a three- to fivefold decrease in membrane input resistance (thus they enter a high-conductance state), as observed in intracellular recordings *in vivo* (Destexhe et al., 2003). However, an added complication of conductance-based models is that their differential equations can only be evaluated numerically or approximated analytically (Rudolph-Lilith et al., 2012) rather than being fully analytically treatable.

Despite the widespread use of both types of models, the differences in the network dynamics that they generate has not been yet fully understood. Previous studies comparing conductance- and current-based LIF models focused mostly on the individual neuron dynamics (Kuhn et al., 2004; Meffin et al., 2004; Richardson, 2004). Here we extended these previous works by investigating the network level consequences of the synaptic model choice. In particular, we investigated which aspects of network dynamics are independent of the choice of the specific synaptic model, and which are not. Understanding this point is crucial for fully evaluating the costs and implications of adopting a specific synaptic model.

We compared the dynamics of two sparse recurrent excitatoryinhibitory LIF networks, a conductance-based network (COBN) with conductance-based synapses, and a current-based network (CUBN) with current-based synapses. To properly compare the two networks, we set to equal values all the common parameters (including the connectivity matrix). Building on previous works (La Camera et al., 2004; Meffin et al., 2004), we devised a novel algorithm to obtain two comparable networks by properly tuning the synaptic conductance values of the COBN given the set of values of synaptic efficacies of the CUBN. Since the differences between the dynamics of the two synaptic models depend on the fluctuations of the driving force (i.e., of the membrane potential), they should be close to zero when the synaptic activity is low. Thus, when decreasing the background synaptic activity, the Post-Synaptic Currents (PSCs) of the two models should become more and more similar. Consequently, our procedure calibrated the conductances so that PSCs became exactly equal in the limit of zero synaptic input (see Methods). Then we investigated whether this procedure could generate COBNs and CUBNs with matching average single neuron stationary firing rates under a reasonably wide range of parameters and network stimulation conditions. We then studied how comparable conductance- and currentbased networks differed in more complex characterizations of population dynamics, such as the cross-neuron correlations of membrane potential (MP), input current and spike train, as well as the spectrum of network fluctuations. The latter was investigated not only for total average firing rates, but also for the simulated Local Field Potential (LFP) computed from the massed synaptic activity of the networks (Mazzoni et al., 2008). To study the spectrum of network fluctuations it is useful to use a LFP model (rather than a massed spike rate) mainly because cortical rhythms are more easily measured in experiments by recording LFPs rather than the spike rate (Buzsaki et al., 2012; Einevoll et al., 2013); therefore this quantification makes the models more directly comparable to experimental observations. We then quantified how the external inputs modulate the firing rate, the LFP spectrum and the spike train correlation by using information theory (Quian Quiroga and Panzeri, 2009; Crumiller et al., 2011). Finally, we discuss the similarities and differences of COBN and CUBN against recent experimental observations of dynamics of cortical network correlations (Lampl et al., 1999; Kohn and Smith, 2005; De La Rocha et al., 2007; Okun and Lampl, 2008; Ecker et al., 2010; Renart et al., 2010).

## **METHODS**

## **NETWORK STRUCTURE AND EXTERNAL INPUTS**

We considered two networks of LIF neurons with identical architecture and injected with identical external inputs. The only difference between the two networks was in the synaptic model: one was composed by neurons with conductance-based synapses and the other by neurons with current-based synapses (see subsection "Single neuron models" in Methods). The network structure was the same one used in a previous work (Mazzoni et al., 2008), to which we refer for a full description. Briefly, each network was composed of 5000 neurons. Eighty percent of the neurons were excitatory, that is their projections onto other neurons formed AMPA-like excitatory synapses, while the remaining 20% were inhibitory, that is their projections formed (A-type) GABA-like inhibitory synapses. The 4:1 ratio is compatible with anatomical observations (Braitenberg and SchüZ, 1991). The network had random connectivity with a probability of directed connection between each pair of neurons of 0.2 (Sjostrom et al., 2001; Holmgren et al., 2003), thus any neuron in the network received on average 200 synaptic contacts from inhibitory neurons and 800 from excitatory neurons (see Supplementary Figure 1). Both populations received a noisy excitatory external input taken to represent the activity from thalamocortical afferents, with inhibitory neurons receiving stronger inputs than excitatory neurons. This simulated external input was implemented as a series of spike times that activated excitatory synapses with the same kinetics as recurrent AMPA synapses, but different strengths (see **Tables 1**, **2**).

The input spike trains activating the model thalamocortical synapses were generated by a Poisson process, with a time-varying rate, νext(*t*), identical for all neurons. Note that this implied that the variance of the inputs across neurons increased with the input rate. νext(*t*) was given by the positive part of the superposition of a "signal," νsignal(*t*), and a "noise" component, *n*(*t*):

$$\mathbb{I}\upsilon\_{\text{ext}}(t) = [\upsilon\_{\text{signal}}(t) + n(t)]\_{+} \tag{1}$$

The separation of signal and noise in the input spike rate was to reproduce the classical experimental design in which a given sensory stimulus is presented many times, with each presentation (or "trial") eliciting different responses due to variations in intrinsic network dynamics from presentation to presentation. We achieved this by identifying the external stimulus with the signal term,νsignal(*t*), (which was thus exactly the same across all trials of the same stimulus) and by using a noise term, *n*(*t*), generated (as explained below) independently in each trial.

In this study we used three kinds of external signals. For the majority of the simulations we used constant stimuli, νsignal(*t*) = ν0, (with ν<sup>0</sup> ranging from 1.5 to 6 spikes/ms). In a second set of simulations we used periodic stimuli made by superimposing a constant baseline term to a sinusoid: νsignal(*t*) = *A* sin(2π*ft*) + ν0, where *A* = 0.6 spikes/ms; *f* ranged from 2 to 16 Hz in **Figure 12** and from 2 to 150 Hz in **Figure 13** and ν<sup>0</sup> was set to 1.5 (respectively 5) spikes/ms when studying the low- (respectively high-) conductance state. We also used a timevarying signal that reproduced the time course of Multi Unit Activity recorded from the LGN of an anaesthetized macaque during binocular presentation of commercially available color movies (Belitski et al., 2008). This latter dynamical stimulus, called "naturalistic", is fully described and characterized in (Mazzoni et al., 2008) to which we refer for further details. For the purposes of the present work, it is useful to remind that this naturalistic signal was a slow signal dominated by frequencies below 4 Hz.

The noise component of the stimuli, *n*(*t*), was generated by an Ornstein-Uhlenbeck (OU) process with zero mean:

$$
\pi\_n \frac{dn(t)}{dt} = -n(t) + \sigma\_n(\sqrt{2\pi\_n})\eta(t),\tag{2}
$$

where σ<sup>2</sup> *<sup>n</sup>* = 0.16 spikes/ms is the variance of the noise, and η(*t*) is a Gaussian white noise. The time constant τ*<sup>n</sup>* was set to 16 ms to have a cut-off frequency of 10 Hz. Note that the trial-to-trial differences in the stochastic process generated by Equation 2 were the first and largest source of trial-to-trial variability in the model, the second and last being the fact that each neuron received an independent realization of the Poisson process with rate νext(*t*).

In a specific set of control stimulations (Supplementary Figure 4), instead of the OU process described above, we used a Gaussian white noise with the same variance. Note that, for low frequencies, the power spectrum of the OU process was higher than the one of the white noise.

#### **SINGLE NEURON MODELS**

Both inhibitory and excitatory neurons were modeled as LIF neurons (Tuckwell, 1988). The leak MP, *V*leak, was set to −70 mV, the spike threshold, *V*threshold, to −52 mV and the reset potential, *V*reset, to −59 mV. The absolute refractory period was set to 2 ms for excitatory neurons and to 1 ms for inhibitory neurons (Brunel and Wang, 2003). The equation for the sub-threshold dynamic of the MP of i-th neuron had the following form:

$$
\pi\_m \frac{dV^i(t)}{dt} = -V^i(t) + V\_{\text{leak}} - \frac{I\_{\text{tot}}^i(t)}{\text{gleak}},\tag{3}
$$

where τ*<sup>m</sup>* is the membrane time constant (20 and 10 ms for excitatory and inhibitory neurons respectively), *g*leak is the leak membrane conductance (25 nS and 20 nS for excitatory and inhibitory neurons respectively) (Brunel and Wang, 2003) and *I<sup>i</sup>* tot (*t*) is the total synaptic input current. The latter was given by the sum of all the synaptic inputs entering the i-th neuron:

$$I\_{\text{tot}}^{\dot{i}}(t) = \sum\_{N\_{\text{(i, AMPAcc)}}} I\_{\text{AMPAcc}}^{\dot{i}}(t) + \sum\_{N\_{\text{(i, GAOA)}}} I\_{\text{GABA}}^{\dot{i}}(t) + I\_{\text{AMPAcc}}^{\dot{i}}(t), \tag{4}$$

the value of *N*(i, AMPArec) (respectively *N*(i, GABA)) being the set of excitatory (respectively inhibitory) neurons projecting into the i-th neuron, and *I<sup>i</sup>* AMPArec(*t*), *<sup>I</sup><sup>i</sup>* GABA(*t*), *<sup>I</sup><sup>i</sup>* AMPAext(*t*) the different synaptic inputs entering the *i*-th neuron from: recurrent AMPA, GABA, and external AMPA synapses respectively.

The difference between current- and conductance-based synapses lied in the definition of these synaptic input currents *I*syn. For the current-based model:

$$I\_{\rm syn}^{\rm CUBN}(t) = J\_{\rm syn} s\_{\rm syn}(t),\tag{5}$$

where *J*syn are the synaptic efficacies (see **Table 1**) and *s*syn(*t*) a function that models the synaptic kinetics (see below).

In the conductance-based model the synaptic input currents depended also on the MP, *V*(*t*):

$$I\_{\rm syn}^{\rm COBN}(t) = \mathcal{g}\_{\rm syn} s\_{\rm syn}(t)(V(t) - V\_{\rm syn}),\tag{6}$$

where *g*syn and *V*syn are respectively the conductance and the reversal potential of the synapse; the term (*V*(*t*) − *V*syn) is the driving force of the synaptic current. The values of the parameters *g*syn in Equation 6 were computed as described in the subsection "Procedure to determine comparable COBN and CUBN models." The reference values of reversal potentials and synaptic conductances are displayed in **Table 2**. In **Figures 6C,D** and **7D** these values were varied to test the robustness of our results.

The same function *s*syn(*t*) described the time course of the synaptic currents in both models; it depended both on the synapse type and on the kind of neuron receiving the input. Every time a pre-synaptic spike occurred at time *t* <sup>∗</sup>, *s*syn(*t*) of the postsynaptic neuron was incremented by an amount described by a

#### **Table 1 | Synaptic efficacies of the current-based network.**

**Current-based network**


#### **Table 2 | Reference values of the synaptic parameters of the conductance-based model.**

**Conductance-based network**


delayed difference of exponentials (Brunel and Wang, 2003):

$$
\Delta s\_{\rm syn}(t) = \frac{\mathfrak{r}\_m}{\mathfrak{r}\_d - \mathfrak{r}\_r} \left[ \exp\left( -\frac{t - \mathfrak{r}\_l - t^\*}{\mathfrak{r}\_d} \right) - \exp\left( -\frac{t - \mathfrak{r}\_l - t^\*}{\mathfrak{r}\_r} \right) \right], \tag{7}
$$

where the latency τ*l*, the rise time τ*<sup>r</sup>* and the decay time τ*<sup>d</sup>* are shown in **Table 3**.

A useful parameter for conductance-based neuron analysis is the effective membrane time constant τeff. Following a standard procedure we computed the total effective membrane conductance for the *i*-th neuron as:

$$\mathcal{g}\_{\text{tot}}^{\dot{i}}(t) = \mathcal{g}\_{\text{leak}} + \sum\_{N(\text{i, AMPAcc})} \mathcal{g}\_{\text{AMPAcc}} \boldsymbol{s}\_{\text{AMPAcc}}^{\dot{i}}(t) \tag{8}$$

$$+ \sum\_{N(\text{i, GABA})} \mathcal{g}\_{\text{GABA}} \boldsymbol{s}\_{\text{GABA}}^{\dot{i}}(t) + \mathcal{g}\_{\text{AMPAcxt}} \boldsymbol{s}\_{\text{AMPAcxt}}^{\dot{i}}(t),$$

and we rewrote Equation 3 as follows:

$$\pi\_{\rm eff}^i(t)\frac{dV^i(t)}{dt} = -V^i(t) + \frac{\text{gleak }V\_{\rm leak} + \sum\_{N(\rm i, syn)} \text{gsyn }s\_{\rm syn}^i(t) \, V\_{\rm syn}}{g\_{\rm tot}^i(t)} \tag{9}$$

$$\text{where } \mathsf{r}\_{\text{eff}}^{\dot{l}}(t) = \frac{\mathsf{r}\_{m} \,\text{g}\_{\text{leak}}}{\operatorname{g}\_{\text{tot}}^{\dot{l}}(t)} \tag{10}$$

is the effective membrane time constant and "syn" indicates: recurrent AMPA; GABA; external AMPA. In particular, for the *i*-th neuron, the effective AMPA conductance is defined as *<sup>N</sup>*(i, AMPArec) *g*AMPArec *s i* AMPArec(*t*) + *g*AMPAext *s i* AMPAext(*t*) and the effective GABA conductance as *<sup>N</sup>*(i, GABA) *g*GABA *s i* GABA(*t*) (see **Figure 3**).

#### **NUMERICAL METHODS**

Network simulations were done using a finite difference integration scheme based on the second-order Runge Kutta algorithm (Press et al., 1992), also known as the midpoint method, with time step *t* = 0.05 ms.

The noise, *n*(*t*), was obtained from Equation 2 by implementing an exact numerical simulation of the Ornstein-Uhlenbeck process (Gillespie, 1996). The temporal durations of the simulations varied from 4.5 s to 100.5 s, and they are specified in the figure captions. The regimes we investigated displayed average firing rates relatively low (0.4–13 Hz), thus, when computing the Inter-Spike Interval (ISI) and the pairwise spike train correlation, we used the longest simulation times (25.5 and 100.5 s) to obtain larger spike datasets. Since we studied stationary responses, the first 500 ms of the simulations were never included in any analysis. Analysis and simulations (the latter implemented using MEX file) were performed in Matlab. Both COBN and CUBN model source codes are available as Supplemental Material to this paper and on


the ModelDB sharing repository (http://senselab.med.yale.edu/ ModelDB/ShowModel.asp?model=152539) with accession number 152539.

## **SPECTRAL ANALYSIS**

To compute the power spectrum we used the Fast Fourier Transform with the Welch method (pwelch function in Matlab), dividing the time window under investigation into eight subwindows with 50% overlap.

For the entrainment analysis showed in **Figure 13** in case of periodic inputs with frequency *f*, we bandpassed the LFP at the correspondent frequency *f* with a Kaiser filter with zero phase lag and 2 Hz bandwidth, very small passband ripple (0.05 dB) and high stopband attenuation (60 dB). We extracted then the instantaneous phase by means of the Hilbert transform of the signal. To quantify entrainment, we computed the phase coherence between the phase of the input signal and of the LFP at the corresponding frequency (Mormann et al., 2000). Phase coherence, which we computed using the CircStat toolbox (Berens, 2009), ranges from zero (no relationships between phases) to 1 (perfect phase locking between the two signals).

#### **COMPUTATION OF SIMULATED LOCAL FIELD POTENTIAL**

We computed from network activity the LFP by using a procedure that has been proposed in previous works (Mazzoni et al., 2008, 2010, 2011), to which we refer for full details. The procedure is summarized and motivated in the following. LFPs are experimentally obtained by low-pass filtering the extracellularly recorded neural signal, and are thought to reflect to a first approximation the current flow due to synaptic activity around the tip of the recording electrode (Buzsaki et al., 2012). Thus, we computed the simulated LFP as the difference between the sum of the GABA currents and the sum of the AMPA currents (both external and recurrent) that enter all excitatory neurons. This quantity was then divided by the leak membrane conductance to obtain units of mV.

This simple recipe was motivated by two well-known geometrical properties of cortical circuits. First, AMPA synapses tend to be apical, i.e., they contact the dendrites away from the soma, while GABA synapses tend to be peri-somatic, i.e., they contact the soma or the dendrites close to the soma. Because of this spatial arrangement, the sink and sources resulting from the activation of both AMPA and GABA synapses will tend to produce in the extracellular field a dipole oriented from apical dendrites toward soma; hence we computed the LFP by subtracting the AMPA currents from the GABA currents (divided by the leak membrane conductance). Second, pyramidal neurons contribute more than interneurons to generation of LFPs in cortex because their apical dendrites are organized in an approximate open field configuration, whereas the organization of dendrites of interneurons is arranged to a first approximation in a close field configuration (Lorente De No, 1947; Murakami and Okada, 2006; Linden et al., 2011). Therefore we computed LFPs by considering only input currents to excitatory neurons (taken here to correspond to cortical pyramidal neurons). This model, though simple, proved to be an effective way to generate a realistic LFP signal that match many characteristics of LFPs in sensory cortex (Mazzoni et al., 2008, 2010, 2011).

### **PROCEDURE TO DETERMINE COMPARABLE CURRENT- AND CONDUCTANCE-BASED NETWORKS**

As mentioned above all the parameters that were directly shared between the two models were set equal; also the connectivity matrix was the same in the CUBN and in the COBN. The starting point of our comparison was to completely define the CUBN, by specifying the synaptic efficacies, *J*syn (reported in **Table 1**), as well as the values of the common set of parameters. Then, we computed the synaptic parameters of the COBN that made it comparable to the given CUBN. To simplify the problem, we first set the reversal potentials of the COBN to biophysically plausible values: *V*AMPA = 0 mV and *V*GABA = −80 mV (as reference values, but we also tested other values, see **Figures 6C,D**, **7D**). The "free" parameters left to set were now only the COBN conductances (*g*syn in Equation 6).

The procedure used to obtain the conductance values leading to comparable COBN and CUBN is illustrated in **Figure 1** and described in the following. Consistent with the fact that the effective membrane time constant of the COBN is equal to the membrane time constant of the CUBN only in absence of synaptic input (see Equation 10), we set the conductances of each synapse type to obtain the same PSCs as in the corresponding currentbased synapse in the limit of no synaptic activity. Explicitly, for each synapse type:

$$\mathcal{g}\_{\rm syn} = \frac{J\_{\rm syn}}{(\langle V \rangle\_{\rm pop} - V\_{\rm syn})},\tag{11}$$

where *V*pop was the average (over time and neurons) MP of excitatory and inhibitory populations obtained from network simulation of 4.5 s with a constant external input of 1.5 (spikes/ms)/cell. This last value was chosen because it was the lowest stimulus used throughout the paper, i.e., the one that induced the lowest synaptic activity. Since *V*pop depended on *g*syn, we determined both values numerically and recursively. We used as first guess the average MP obtained with the CUBN, we computed the associated conductances with Equation 11, we ran a COBN simulation with those conductances and then we used the resulting *V*pop to compute the updated conductances, until *V*pop (and consequently the conductances) reached a stable value (see **Figure 1**). Note that convergence was very fast: stability within a tolerance on average MPs of 0.01 mV was achieved usually in less than 10 steps. By using Equation 11, we rewrote the Equation 6 as follows:

$$I\_{\rm syn}^{\rm COBN}(t) = J\_{\rm syn} s\_{\rm syn}(t) \left[ 1 + \frac{V(t) - \langle V \rangle\_{\rm pop}}{\langle V \rangle\_{\rm pop} - V\_{\rm syn}} \right]. \tag{12}$$

Comparing Equation 12 with Equation 5 it is clear that the synaptic currents of the two networks are the same only when *V*(*t*) = *V*pop, that is in the limit of no synaptic input.

Conductance-based neurons can undergo transitions from low- to high-conductance states (Destexhe et al., 2001) and the simulations performed in this work included both states. However, current-based neurons cannot undergo such transitions and their membrane time constant is close to the effective membrane time constant of conductance-based neurons in a

The flowchart illustrates the iterative algorithm we used to set the synaptic conductances, *g*syn,such in a way to obtain a COBN comparable with the given CUBN. The two networks shared all the common parameters, so, once the CUBN was given, the synaptic conductances depended only on the synaptic reversal potentials of the COBN, *V*syn.

low-conductance state (see **Figure 3A**). Therefore, the correspondence between the two models that we defined is consistent with the physiologically-meaningful requirement that the differences between the two synaptic models decrease with synaptic activity (Destexhe et al., 2003).

## **COMPUTATION OF THE AVERAGE POST-SYNAPTIC POTENTIALS IN THE CONDUCTANCE-BASED NETWORK**

Modeling the synaptic input as conductance transients produces an activity-dependent increase of membrane conductance (that is a reduction of effective membrane time constant, see Equation 10) which attenuates and shortens the Post-Synaptic Potentials (PSPs) (Destexhe and Pare, 1999). In order to extract the average (activity-dependent) PSPs of the COBN we used a procedure similar to the one used in (Kumar et al., 2008): for each synapse type (see **Table 2**) we randomly selected 300 neurons from the network and we made a copy of them. These "cloned" neurons received the synaptic input of the original ones and had exactly the same spiking activity. The only difference with respect to the original is that the cloned neurons received an extra spike, from the synapse under investigation, each 100 ms (except for the first 500 ms), for a total of 100 PSPs for each cloned neuron (i.e., simulations lasted 10.5 s). We subtracted then the MP of the original neurons from the one of the cloned neurons and, by doing a spike triggered average over time and selected neurons, we obtained the average effective PSP.

## **COMPUTATION OF CORRELATIONS AMONG SIGNALS IN THE NETWORKS**

We quantified the effects of the choice of the synaptic model on the cross-neuron correlation in time. We computed the crossneuron pairwise Pearson's correlation coefficient of the time course of AMPA currents and of GABA currents entering the neurons, MPs and spike trains. The spike trains were binned in non-overlapping time windows of 5 ms and their correlation coefficients were averaged over all neuron pairs of the network (**Figures 10A–C**). Time courses of the other variables were expressed with the original time steps of 0.05 ms and the correlation was estimated averaging the correlation coefficients over all neurons' pairs obtained from two randomly selected subpopulations of 200 excitatory and 200 inhibitory neurons (**Figure 9**).

We measured also the average correlation between the time course of AMPA and GABA currents entering each single neuron. In particular, we computed the normalized cross-correlation between AMPA and GABA currents entering each neuron belonging to the two subpopulations of 200 neurons above mentioned. Then we averaged (over the neurons) the peak value and the peak position, i.e., the time lag for which the correlation was strongest (**Figure 8**).

## **COMPUTATION OF INFORMATION ABOUT THE EXTERNAL INPUTS**

We studied how networks encoded external stimuli by means of mutual information between stimulus and response (that we will simply call information in the manuscript) (Shannon, 1948). The information that a set of responses, *R*, carries about a set of stimuli, *S*, is given by:

$$I(S;R) = \sum\_{s \in S} P(s) \sum\_{r \in R} P(r|s) \log\_2 \frac{P(r|s)}{P(r)},\tag{13}$$

where *P*(*s*) is the probability of presentation of the stimulus *s*, *P*(*r*) the probability of observing the response *r*, and *P*(*r*|*s*) the probability of observing *r* when *s* is presented.

As explained above, we used three kinds of external input signals: constant input (**Figures 2**–**11**), periodic input (**Figures 12**, **13**) and a naturalistic input (**Figure 14**). In the constant input case, each input rate, ν0, was considered a different stimulus (with simulations lasting 25.5 s), while, for the periodic stimuli, each stimulus corresponds to a frequency *f* (with simulations lasting 10.5 s). In the naturalistic case, the stimulus presentation time (80 s) was divided into 2 s long non-overlapping windows and each window was considered as a different "stimulus" for the information calculation, following the procedure described in (Belitski et al., 2008). We discarded an interval at the beginning of the simulations (500 ms both for constant and periodic case and 2 s for the naturalistic case) to avoid artifacts due to initial conditions. When computing information we considered three different response sets *R*: the average network firing rate, the average cross-neuron spike train correlation, and the LFP power of each single frequency (Belitski et al., 2008) in the (1–150) Hz range. To facilitate the sampling of response probabilities, the whole range of response values was divided into six consecutive intervals. Each of these intervals contained the same number of responses (i.e., they were equi-populated). All the responses belonging to a given interval assumed then the same interval-specific discrete value. In summary, we discretized the responses into six equi-populated bins. Then conditional probabilities *P*(*r*|*s*) were evaluated empirically by using the results from 50 trials per each stimulus *s*. We corrected information estimations for the limited sampling bias (Panzeri et al., 2007) by using the "quadratic extrapolation procedure" described in Strong et al. (1998) implemented in the Information Breakdown Toolbox (Magri et al., 2009).

## **RESULTS**

We investigated the differences in the dynamics of neural populations between conductance-based LIF networks (COBNs) and current-based LIF networks (CUBNs), with particular emphasis in understanding how the neural population activity of these two types of network is modulated by external inputs. We first introduced an iterative procedure to determine synaptic parameter values so that the CUBN and the COBN were placed on a fair common ground, and could therefore be legitimately compared. We then analyzed similarities and differences of single neuron dynamics and of interactions among neurons in the two networks as a function of strength and nature of the external stimuli.

## **DETERMINING SYNAPTIC PARAMETER VALUES TO BUILD COMPARABLE CURRENT- AND CONDUCTANCE-BASED NETWORKS**

A necessary requirement to compare the activity of two different network models is to define a meaningful and sound correspondence between them. Our first step was thus to define a procedure to achieve comparable networks (see Methods for details). In brief, we set all the common parameters to exactly equal—and biologically plausible—values in both models. In this way the two models differed only because of the different synaptic model adopted: voltage-independent for CUBN (see Equation 5) and voltage-dependent for COBN (see Equation 6). In particular, the expression of the Post-Synaptic Currents (PSCs) in the COBN depended on conductances *g*syn and on reversal potentials (*V*AMPA and *V*GABA), while in the CUBN the PSCs depended only on synaptic efficacies *J*syn. We set *V*AMPA and *V*GABA at 0 and −80 mV respectively (but importantly our results were robust to changes

**FIGURE 2 | Individual synaptic events.** Dynamics of single synaptic events on excitatory neurons (see Methods). Results were qualitatively very similar when considering synaptic inputs impinging on inhibitory neurons (see "PSP peak amplitude" in Supplementary Table 1). **(A,B)** Shape of Post-synaptic Currents (PSCs, top) for individual synaptic events in case of recurrent AMPA **(A)** and GABA **(B)** connection (thalamic AMPA case is not shown because it is qualitatively very similar to the recurrent AMPA case). The origin of the time axis corresponds to the arriving time of the spike. Green lines represent the kinetics in current-based neurons, which is independent from background synaptic activity. Dashed blue lines indicate the kinetics of an isolated conductance-based neuron (thus without background activity), having starting membrane potential equal to *V* exc = −58.8 mV, that is the average potential of the excitatory neurons of the network when the external input signal is 1.5 (spikes/ms)/cell. Red lines indicate the average PSCs in

in these parameters, see **Figures 6C,D**, **7D**). We then used an iterative algorithm (detailed in Methods and illustrated in **Figure 1**) to set the values of the conductances *g*syn of the COBN in such a way to obtain a COBN comparable to the CUBN with the given synaptic efficacies *J*syn.

The PSCs and the Post-Synaptic Potentials (PSPs) of recurrent AMPA and GABA synapses in the comparable networks are shown in **Figures 2A,B,D,E** for three different cases: current-based synapse, conductance-based synapse of a single neuron without background synaptic activity and conductancebased synapse of neurons embedded in the COBN network (that thus received background synaptic activity). The post-synaptic kinetics of conductance-based neurons is activity dependent. The terms that mediate this dependency are: the driving force (see Equation 6) and the increase of the total effective membrane conductance (see Equation 8). Both these terms tend to reduce the post-synaptic stimulus, but the PSCs are affected only by the driving force, while the PSPs by both the driving force and the effective membrane conductance. To understand how these two terms shape the post-synaptic stimulus, it is important to compare post-synaptic responses of conductance-based neurons, with and without background activity. Firstly, we compared PSCs and PSPs of the current-based synapse with those of the conductance-based synapse in the absence of background conductance-based neurons embedded in the network (thus with background activity) when the external input signal is 1.5 (spikes/ms)/cell (see Methods for details). Blue and green lines are superimposed in **(A)**. **(C)** Absolute average values of the PSC peaks as a function of the external input rate for neurons embedded in the network. Results are relative to recurrent AMPA (red) external AMPA (green), and GABA (blue) synapses for current- (thick lines) and conductance-based (thin lines with markers) neurons. Shaded areas for the conductance-based neurons correspond to the standard deviation across neurons (for AMPA connections the shaded areas are not visible because they are too small). **(D–F)** Same as **(A–C)** for Post-Synaptic Potentials (PSPs). PSPs are more relatively affected by the choice of the synaptic model with respect to the PSCs, because, in the COBN, the PSCs depend on the driving force, while the PSPs both on the driving force and on the effective membrane time constant.

activity. In this condition the shape of excitatory PSCs and PSPs was almost identical for the two models when considering AMPA synapses (**Figures 2A,D**), while, for GABA synapses, differences between the two models were visible (**Figures 2B,E**). This asymmetry was due to the fact that the value of the average MP (see figure caption) was much closer to the reversal potential of GABA synapses than to the one of AMPA synapses (see Equation 12). Consequently the relative reduction of driving force during the post-synaptic event was higher for GABA synapses, provoking a stronger reduction of both PSCs and PSPs, with respect to the AMPA synapses (**Figures 2B,E**). Moreover, the PSPs of fast synapses (that is synapses with short τdecay) are less affected by synaptic bombardment (Koch, 1999; Kuhn et al., 2004), so, being the AMPA τdecay shorter than the GABA ones (see **Table 3**), the asymmetry was even stronger when looking at the PSPs (**Figures 2D,E**). Secondly, we considered the conductance-based neurons embedded in the COBN and we found that in this case both AMPA and GABA synapses displayed a reduction in the amplitude and in the timescale, because the background network activity affected the time course of the MP (thus of the driving force) and increased the total effective membrane conductance.

As stated above, differences between the two synaptic models were expected to increase with input strength because the background synaptic activity increases. We measured this effect

(red) and GABA (blue) conductances on excitatory neurons as a function of the external input rate. Results show that the COBN goes from low- to high-conductance states in the range of external stimuli considered. Same color code as **(A)**. Shaded areas represent standard deviation across neurons [in **(A)** for inhibitory time constant and in **(B)** for AMPA conductances they are not visible because too small]. Values are computed from a simulation of 10.5 s per stimulus and are averaged over time and neurons.

by injecting in the network constant inputs ranging from 1.5 to 6 (spikes/ms)/cell. **Figures 2C,F** show the amplitude of the different PCSs and PSPs as a function of the external input rate. Note that the PSCs (**Figure 2C**) and PSPs (**Figure 2F**) in the CUBN were activity-independent by construction, while, in the COBN, both PSCs and PSPs decreased substantially when input rate was increased; furthermore the relative reduction was the strongest for the slowest PSPs of GABA synapses (as stated above). Supplementary Table 1 reports average PSP amplitude values on both inhibitory and excitatory neurons.

neurons are shown for reference as thick lines. Results show that conductance-based membrane timescale is much faster than current-based one and that it decreases with input strength. **(B)** Average effective AMPA

**Figure 2** shows that, in the COBN, PSPs were not only smaller but also faster than in the CUBN, consistently with previous results (Kuhn et al., 2004; Meffin et al., 2004). This reflected the decrease of the effective membrane time constant, τeff, of the COBN, whose average value is shown in **Figure 3A** as a function of the input rate. When injecting stimuli with high input rates, we found that for both neuron populations the effective time constant, τeff, was in the 1–5 ms range, matching experimental observations relative to the high-conductance states (Destexhe et al., 2003).

We then asked how the effective conductances associated with the AMPA and GABA currents varied in the COBN as a function of the input rate. We found (**Figure 3B**) that the average conductances grew linearly with input rate, as observed in single neuron case (Kuhn et al., 2004). Crucially, for high input rates, the relative conductances *g*AMPA/*g*leak and *g*GABA/*g*leak displayed values respectively close to 1 and 3.5, in the range of those found experimentally in high-conductance states (Destexhe et al., 2003). This suggested that our input range was suited to investigate the whole continuum going from low- to high-conductance states.

## **AVERAGE SINGLE NEURON PROPERTIES**

After having examined the properties of PSPs and conductances in the two comparable networks, we began investigating how these properties affect the dynamics of neural activity in the networks. To gain some visual intuition about this, we plotted (**Figure 4**) example traces of how variables reflecting single neuron and network activity evolve over time for the two types of network both in the low- and high-conductance state. The overall spike rate of individual neurons was similar for the two networks in both low- and high-conductance state (compare **Figures 4A** with **4C** and **Figures 4B** with **4D**) suggesting that the level of network firing was only mildly dependent on the synaptic model adopted. On the other hand, single neuron MP traces were similar in the two networks in the low-conductance regime (compare **Figures 4E** with **4G**), but different in many aspects in the high-conductance regime (compare **Figures 4F** with **4H**). In particular, in the high-conductance state, the COBN MPs had rapid gamma-range variations which were correlated across neurons and whose amplitude was more prominent than that of the gamma oscillations in the CUBN MPs, suggesting that the oscillation regime in the high-conductance state was tighter in the COBN than in the CUBN. Finally, we considered the traces of the LFP (which can potentially capture both supra- and subthreshold massed neural dynamics). LFP traces were relatively similar across networks in the low-conductance state (**Figure 4I**). However, there was an interesting qualitative difference in the LFP traces in the high-conductance state: the COBN LFP had transient peaks of very high amplitude, which were not observed in the CUBN. At fixed level of overall firing rate, the amplitude of the LFP is modulated by the relative timing of the synaptic events contributing to it. Therefore this observation suggests that

the COBN may undergo larger fluctuations in synchronization than the CUBN. The visual inspection of example traces suggests that, while some network properties such as overall firing rate are consistently close in the two networks, other more subtle aspects of network dynamics (such as the ability of the network to transiently synchronize its activity) may not be entirely equivalent in the two networks, especially in the high-conductance state. In the following we will systematically quantify this intuition.

An important feature of the models is the dynamics of the average (over time and neurons) of the total synaptic input current *I*tot (Equation 4). We observed in both networks (**Figure 5A**) an increase of *I*tot with the input rate (Pearson correlation test, *p* < 10<sup>−</sup>5). However, *I*tot was significantly higher for the CUBN over all inspected inputs (*t*-test *p* << 10<sup>−</sup>10). The net input current *I*tot was also less modulated by the input rate in the COBN: the difference between the current (divided by the leak membrane conductance) at maximum and minimum input was 1 mV for COBN and 15 mV for CUBN. Even if the firing rate was very similar in the two networks (see **Figure 6A**), average GABA currents were weaker in COBN, while average AMPA currents were very similar (see **Figure 5B**). This discrepancy in the dynamics of the net input current was due to the fact that individual PSCs of GABA currents were more affected (i.e., reduced) by the change from CUBN to COBN with respect to the AMPA PSCs, as pointed out in **Figure 2**. Note also that in the case of external AMPA current, the spike trains that activated the synapses (more precisely the function *s*(*t*) in Equations 5 and 6) are exactly the same in the two models, while they were different for the other currents.

Consistent with the sample traces shown in **Figures 4G,H**, the average MP of the CUBN decreased steeply when we increased the input (−15 mV between maximum and minimum input, **Figure 5D**). This is due to the fact that, in the CUBN, the net input current strongly increased when increasing the external inputs (**Figure 5A**). Conversely, and consistently with the sample traces in **Figures 4E,F**, the decrease in COBN MP was smaller (−2 mV between maximum and minimum input, **Figure 5D**), consistent with previous results (Meffin et al., 2004). It is important to note that an increase of the input current led to an increase the voltage fluctuations in both models. However in the COBN, it caused also a concomitant increase of the membrane conductance, which in turn decreased the membrane voltage fluctuations. The dynamics of MP in COBN thus resulted from the competition between these two effects, which overall produced a suppression of both fluctuations and mean of the MP (Kuhn et al., 2004; Meffin et al., 2004; Richardson, 2004). We found that, for external inputs higher than 2 (spikes/ms)/cell, there was a linear relation (*R*<sup>2</sup> = 0.98, *p* << 10<sup>−</sup>10) between the ratio of the average MP changes induced by the external inputs in the two networks and the effective membrane time constant of the COBN (see **Figure 5E**). This result confirmed and extended what found for a single neuron model in a high-conductance state in

Richardson (2004). Shaded areas in **Figures 5A,D** indicate standard deviation across neurons, and show that the cross-neuron variability in both net input currents and MP was much larger in the CUBN than in the COBN, suggesting a more coherent activity for the latter (see subsection "Correlations among neurons**"**).

When we looked at the variability over time of the input currents, we found that it grew almost linearly and with very similar values for both COBN and CUBN (**Figure 5C**), while the increase of the variability over time of the MP was much more pronounced in the CUBN than in the COBN (**Figure 5F**). This result is still consistent with the suppression of voltage fluctuations typical of conductance-based model with respect to the current-based one.

In sum, our findings so far confirmed that dynamics previously observed in simpler conditions were valid also over a more extended range of conditions, proved that the range of input rates considered encompassed both low- and high-conductance regimes, and highlighted some of the differences between the dynamics of COBNs and CUBNs.

results are obtained from 50 trials of 4.5 s per stimulus, while for the panel **(B)** we used a single trial of 100.5 s per stimulus (see Methods). Results show that the two models have similar firing rates over the whole input range. This agreement is stable over a wide range of network parameters. On the other hand, the CV of the ISI increases with the input rate in the CUBN, while it does not in the COBN.

#### **FIRING RATE MODULATIONS**

Having established a procedure that computes comparable CUBN and COBN parameters, and having investigated the synaptic responses in these comparable networks, we next compared the average firing rates of single neurons in the two networks, and studied how they are modulated by the strength of the input to the networks.

AMPA and GABA reversal potentials. The relative difference is averaged over the whole stimuli set ranging from 1.5 to 6 (spikes/ms)/cell. Green arrow

We considered individually the excitatory and inhibitory neural populations since they fired at very different rates (Brunel and Wang, 2003). **Figure 6A** shows the way inhibitory and excitatory firing rates increase with the input rate in the two networks. Consistently with the qualitatively intuition gained form the visual inspection of the raster plots in **Figures 4A–D**, we found that the discrepancies between COBN and CUBN firing rates were extremely small (average difference over external inputs of 10%), though significant (*t*-test *p* < 0.05 except for excitatory neurons with external input rates greater than 4 spikes/ms). This shows that the algorithm used to set comparable networks produces networks whose neurons have similar average firing rates with a similar dependence on the input strength, both in low- and high-conductance states.

To verify if the agreement of the firing rate in the two comparable networks was robustly achieved over a wide range of parameters, we computed the COBN synaptic conductances for a set of 20 different COBN networks (obtained by using the setting procedure illustrated in **Figure 1** with 20 different combinations of the synaptic reversal potentials, *V*AMPA, ranging from 0 to −20 mV, and *V*GABA, ranging from −75 to −90 mV). We then computed the average firing rates for each resulting network. We found that even when *V*AMPA was −20 mV and *V*GABA −75 mV, and hence the discrepancies between the two models were stronger, the excitatory neurons firing rate differed between COBN and CUBN at most by 25%, but usually the difference was much smaller, on the order of 10% (**Figure 6C**). Note that, given the very low firing rate of excitatory neurons, the relative difference corresponded always to small values of absolute difference (<0.4 spikes/ms). The difference in the firing rate of the inhibitory neurons between COBN and CUBN were of the order of 10% for all reversal potentials combinations inspected (**Figure 6D**).

These results show that our procedure determines COBNs with firing rates similar to the compared CUBN for a wide range of parameters. In current-based neurons the firing rate is modulated only by the increase in the MP fluctuations (**Figure 5F**),

6 (spikes/ms)/cell with steps of 0.5 (spikes/ms)/cell) as a function of AMPA and GABA reversal potentials. Green arrow indicates reference values (see **Table 2**). **(E)** Modulation of the LFP gamma peak power for the two networks. Power modulation is defined as the difference of the power of a frequency at a given input signal and its power at the input signal of

COBN and CUBN (see Methods). Results are computed by using 50 trials of 4.5 s per stimulus and show that (i) the gamma peak position across stimuli is similar for the two networks and this agreement is robust to change in the network parameters, (ii) the amplitude of the peak power is more modulated in the COBN because of the stronger fluctuations of the synaptic currents at the network level.

while in conductance-based neurons, the firing rate activity is the result of two different competing effects: the shortening of the timescales (**Figure 3A**) and the increase of the membrane fluctuations (**Figure 5F**), that tend to facilitate the firing activity, and the increase of the effective membrane conductance, that acts in the opposite direction (**Figure 3B**) (Kuhn et al., 2004; Meffin et al., 2004; Richardson, 2004). It is therefore quite interesting that these underlying different dynamics compensate to produce, in the two corresponding network models, very similar firing rates over a wide range of inputs and parameters.

We then considered how the coefficient of variation (CV) of the inter-spike interval (ISI) changed with the strength of the input rate. We found (**Figure 6B**) that the two networks showed a very different dependence of CV on input rates. The ISI CV of neurons of the COBN was close to one for all considered input rates (indicating near-Poisson firing statistics). In contrast, in CUBN, the ISI CV was higher than 1 (i.e., the firing was more variable than that of a Poisson process) and increased with the input rate, reaching values up to 1.33 and 1.16 for inhibitory neurons and excitatory neurons respectively, confirming results of

neurons. **(B)** Cross correlation average peak position. This measure quantify how much AMPA inputs lags behind GABA ones. Same color code as **(A)**. Results are computed by using a simulation of 10.5 s per stimulus and show that (i) correlation between recurrent AMPA and GABA input currents is stronger in the COBN than in the CUBN, (ii) input correlation decreases monotonously with input rate in COBN, while it does not in CUBN, (iii) GABA inputs lags behind AMPA inputs by few milliseconds in both networks.

(Meffin et al., 2004). The difference between the CVs of neurons in COBN and CUBN was highly significant (*t*-test, *p* < 10<sup>−</sup>7) for all input rates above 1.5 spikes/ms. The larger ISI CV of neurons in COBN was consistent with our finding of larger MP fluctuations in time in the COBN (**Figure 5F**). ISI CV values were within the experimentally observed range 0.5–1.5 (Maimon and Assad, 2009) for both networks, but only the COBN reproduced the experimental result that the ISI CV of cortical neurons is not affected by the firing rate (Maimon and Assad, 2009).

The discrepancy between the similarity of the firing rates and the dissimilarity of the ISI CVs suggests that the first-order statistics of the two networks were close to match, but the second order statistics differed significantly.

#### **SPECTRAL MODULATIONS IN SIMULATED LOCAL FIELD POTENTIALS**

We investigated then the differences in the spectral modulations of network activity, as measured by the simulated LFP and by the total excitatory and inhibitory firing rate generated by the two networks. LFP models can offer interesting insights into the dynamics of cortical networks (Einevoll et al., 2013) because they offer an insight in both supra- and sub-threshold dynamics that can be compared with experimental recordings; however the differences in LFPs computed from networks with either currentor conductance-based synapses have not been investigated yet. We expected significant differences to arise because, as detailed above, the sub-threshold dynamics of COBNs and CUBNs were quite different.

The dependence of LFP spectrum on the input rate (**Figures 7A,B**) shows that, consistent with previous results (Brunel and Wang, 2003; Mazzoni et al., 2008, 2011), both networks develops gamma range (30–100 Hz) oscillations that become stronger and faster as the input is increased. **Figures 4I,J** illustrate this effect in the time domain. **Figures 7A,B** show the LFP input rate-driven modulation in COBN and CUBN. The dependence of response to variations in input rate in the two networks was qualitatively similar. There was no modulation for frequencies below 5 Hz (Pearson correlation test, *p* > 0.1); there was strong modulation in the gamma band and above (Pearson correlation test, *p* < 0.01). The difference between the position of the COBN and CUBN gamma peak was always below 5 Hz (**Figure 7C**). For comparison, we also computed the power spectrum of the total firing rate of excitatory or inhibitory neurons (**Figure 7C**). The spectral peaks of COBN and CUBN were very close also in this case.

We tested the robustness of the agreement between spectral peaks of CUBNs and COBNs by measuring the average (over stimuli) gamma-peak distance between the two networks for different AMPA and GABA reversal potentials (similarly to what was done in the analysis represented in **Figures 6C,D**), and we found that the two networks always displayed almost identical positions of the gamma frequency peaks (**Figure 7D**).

Note that we did not build the comparable networks to obtain robustly similar firing rates and similar dominant frequencies in the gamma band, as we used other constraints to select comparable parameters. The equivalence and robustness of rates and gamma peaks arose from network dynamics, and, in particular, the robustness corroborates the notion that our procedure indeed produces a meaningful comparison. We also tested other kinds of procedures to set the COBN synaptic conductances, *g*syn, given the CUBN synaptic efficacies, *J*syn. In particular we define *g*syn such in a way to maximize the similarity of PSCs (in one case) or PSPs (in another case) between the two networks at the single neuron level, to compensate for the post-synaptic stimulus reduction that is peculiar of the COBN with respect to the CUBN

membrane potential (MP) time courses of pairs of excitatory neurons as a function of the external input rate. While in the COBN the MP correlation increases with input rate, the opposite occurs in the CUBN. Shaded areas correspond to standard deviation across neuron pairs. Results are computed by using a simulation of 10.5 s per stimulus and show that in COBN the cross-neuron correlations between membrane potentials and between input currents are stronger than in CUBN.

(**Figure 2**). When using these procedures the results were both less robust to change in the synaptic reversal potentials and less similar between CUBN and COBN (data not shown).

On the other hand, differences between the LFP spectra of the two networks are also apparent in **Figures 7A,B**. First, the COBN gamma peak was larger and was modulated by the input rate in a much stronger way than the CUBN gamma peak (**Figure 7E**). Given the fact that the net input current in the COBN was smaller (**Figure 5A**) and also fluctuated slightly less than in CUBN (**Figure 5C**), at first we found this result surprising. However, the phenomenon can be understood after measuring the AMPA and GABA fluctuations. As reported in **Figure 7F**, the size of recurrent AMPA and GABA current fluctuations was larger in COBN than in CUBN, and the difference increased with the input rate. Indeed, while the simultaneous increases of AMPA and GABA fluctuations compensated each other in the COBN net input current (**Figures 5A,B**), the contributions of these two currents to the computed LFP have the same sign (see Methods), and this led to a stronger spectral peak in the COBN. Second, the CUBN displayed a broad LFP spectral peak in the high gamma region (>60 Hz), and small fluctuations in the low gamma region (<60 Hz), while, in the COBN, for inputs greater than 3 (spikes/ms)/cell there was a sharp peak in the high gamma band and also a pronounced plateau in the low gamma. Third, since the power associated with this plateau was modulated by the input rate, for the COBN all frequencies above 20 Hz were significantly modulated, while in the CUBN significant modulation occurred only for frequencies above 60 Hz. As we will see in the next section, the narrower gamma peak indicates a stronger synchronization in the COBN than in the CUBN, while the stronger modulation in the gamma power makes the amount of information conveyed by the COBN larger than in the CUBN (see "Information about external inputs" subsection).

For both networks, the spectra of the total firing rate were qualitatively very similar to the spectra of the LFP for all input rates considered (data not shown). Therefore all the aforementioned differences were present also when comparing the COBN and CUBN total firing rate power spectra.

#### **CORRELATION BETWEEN AMPA AND GABA CURRENTS**

The correlation between AMPA and GABA synaptic currents is known to play a very important role in determining the network dynamics and in particular the spike train variability (Isaacson and Scanziani, 2011). A negative correlation of AMPA and GABA input currents leads to sparse and uncorrelated firing events, while positive values lead to strong bursty synchronized events (Renart et al., 2010). We thus compared the cross correlation between recurrent AMPA and GABA currents impinging on single neurons in COBN and CUBN. We found that the correlation between GABA and AMPA inputs was stronger (i.e., more negative) in the COBN for all external input rates (**Figure 8A**). Moreover, in both networks, AMPA currents led GABA currents with lags shorter than 5 ms, of the order of those observed in (Okun and Lampl, 2008). However, for all external input rates, AMPA-GABA lags were smaller in the COBN (**Figure 8B**). Although **Figure 8** shows results only for excitatory neurons, similar results held for inhibitory neurons (Supplementary Figure 2). Finally, these results held also when using as external noise a white noise process instead of an Ornstein-Uhlenbeck process (see Supplementary Figure 4C).

## **CROSS-NEURON CORRELATIONS**

The fact that the cross-neuron variability in average current inputs and MPs was much smaller (**Figures 5A,D**) and high gamma frequency peaks were narrower in the COBN (**Figures 7A,B**) suggested that the activity was more coherent

in the COBN than in the CUBN. This view was further corroborated by the finding that the sum of the recurrent currents was larger in the COBN (**Figure 7F**) and suggested that, in this network, input currents may be more correlated across different neurons.

We verified this hypothesis by measuring the average Pearson correlation coefficient between the time evolution of the recurrent AMPA and of the GABA input currents over neuron pairs (see Methods), **Figure 9A** shows that for both AMPA and GABA currents the average cross-neuron correlation coefficient was indeed significantly stronger (*t*-test, *p* << 10<sup>−</sup>10) in the COBN for all external input rates. **Figure 9A** shows also that, in the COBN, the cross-neuron correlation grew with the external input rate for both currents (Pearson correlation test, *p* < 10<sup>−</sup>5). In the CUBN the AMPA currents were linearly correlated to the input rate (Pearson correlation test, *p* < 0.05), while GABA currents varied with the input rate in a non-monotonic way. However, if we used white noise, instead of the Ornstein-Uhlenbeck noise (see Methods), the cross-neuron current correlation was again higher in the COBN (*t*-test, *p* << 10<sup>−</sup>10), but grew monotonously with the input rate for both networks (Pearson correlation test, *p* < 10<sup>−</sup>5), as shown in Supplementary Figure 4A. The increase in the difference between the cross-neuron current correlation in COBN and CUBN with the input rate (**Figure 9A**) led to the increase of the difference in AMPA and GABA total fluctuations in the two networks, shown in **Figure 7F**. To fully appreciate the key role played by correlations note that, if the correlations were similar in COBN and CUBN, fluctuations would be expected to be larger in CUBN since the firing rate was similar for the two networks (**Figure 6A**) and the single PSC amplitude was larger for the CUBN (**Figure 2**). Cross-neuron correlation of the input currents should be reflected in cross-neuron MP correlation. The previously shown sample traces of the MP of neuron pairs (**Figures 4E,H**) suggested that the correlation was indeed similar for COBN and CUBN in the low-conductance state, but much stronger for the COBN in the high-conductance state. We thus analyzed the average correlation of the MP time courses of pairs of excitatory neurons (**Figure 9B**). Over the whole external input range considered, MP correlation in the COBN was significantly stronger than in the CUBN (*t*-test, *p* << 10<sup>−</sup>10). Cross-neuron MP correlation in the COBN increased with external input rate (Pearson correlation test, *p* < 10<sup>−</sup>8), while it was only mildly affected in the CUBN (Pearson correlation test, *p* < 0.02). These results held for all considered neuron

pairs (Supplementary Figure 3) and also when considering white noise, instead of Ornstein-Uhlenbeck noise (Supplementary Figure 4B).

We finally computed the cross-neuron spike train correlation. We expected it to be related to the MP correlation displayed in **Figure 9B**, even if, since both networks were in a fluctuationdriven state, the spike train correlation should be close to zero (Brunel and Wang, 2003; Renart et al., 2010). We found indeed a very low average spike train correlation (**Figures 10A–C**) such that, for low input rates, a significant fraction of pairs displayed negative correlation (**Figure 10D**). However, in the CUBN, the spike train correlation was weaker and less sensitive to input rate changes than in the COBN (see **Figures 10A–C** and compare **Figures 10D,E**). These results did not change if we injected white noise, instead of Ornstein-Uhlenbeck noise, in the network (Supplementary Figure 4D).

## **INFORMATION ABOUT EXTERNAL INPUTS**

In the previous subsections we investigated how the average level of spike rate, LFP and spike train correlation depends on the external input to the network, finding a more pronounced stimulus modulation of LFP gamma power and of cross-neural correlation in COBN. To quantify these stimulus modulations of network activity, we computed the mutual information between the stimuli to the network and various aspects of network activity (see Methods for details).

We first measured the information carried by the average firing rate, both of excitatory and inhibitory neurons, in the two networks by using constant stimuli in the range 1.5–3 (spikes/ms)/cell with steps of 0.1 (spikes/ms)/cell. We found that, consistently with the results shown in **Figure 6A**, the information carried by the average firing rate had the same value of 2.3 bits for both neural populations in both network models. Given that the modulation of spike train correlation with external input is greater in the COBN than in the CUBN, we expected that also the mutual information between the spike train correlation and the input rate was greater in the COBN than in the CUBN. Indeed this was the case: information in spike train correlation was much larger in the COBN (1.6 and 2.0 bits for excitatory and inhibitory neurons respectively) than in the CUBN (1.4 and 0.9 bits for excitatory and inhibitory neurons respectively).

We measured then the information content of the LFP power spectrum. The LFP power spectrum averaged over all the presented constant stimuli was higher for the COBN than for the CUBN for all frequencies above 15 Hz (**Figure 11A**). We found that, at all frequencies above 20 Hz, the COBN LFP spectrum carried more information about input rate than the CUBN LFP spectrum (**Figure 11C**). Most notably, the peak information increased by about 20%, and the (20–45) Hz frequency range was informative in the COBN, but not in the CUBN. We repeated the analysis considering the power spectra of the total inhibitory and excitatory firing rate in the two networks. Excitatory neurons in the

COBN had stronger power than excitatory neurons in the CUBN for all frequencies (**Figure 11B**, note that here the y-scale is linear, while in 11A is logarithmic) and showed a secondary peak at about 20 Hz. For inhibitory neurons, instead, the COBN power spectrum was higher only for frequencies above 15 Hz, as in the LFP.

So far we have investigated only the information carried about the strength of a time-independent input to the network. In a previous work on CUBN (Mazzoni et al., 2008) it has been shown that when the input to the CUBN is dominated by low frequency fluctuation, the network oscillations (captured by both LFP and massed firing rate measures) form two largely independent frequency information channels. A gamma-range information channel is generated by recurrent interactions of inhibitory and excitatory neurons and conveys information about the mean input rate. A low-frequency information channel is generated by entrainment of the low frequency network activity to the slow fluctuations of the input stimulus and carries information about the stimulus time course on such slow time scales. We wanted to test how these two information channels, developed when presenting the network with time-varying stimuli, depended on the choice of the synaptic model.

To investigate this point, we injected into the two networks periodic stimuli with fixed amplitude and frequency varying between 2 and 16 Hz. These input frequencies below 16 Hz were taken to represent the slow naturalistic fluctuations present in natural input signals (Luo and Poeppel, 2007; Chandrasekaran et al., 2010; Gross et al., 2013). Since we wanted to investigate potential differences between models separately in low- and highconductance states, we generated two kinds of input signals: a low-input regime (corresponding to a low-conductance state) and a high-input regime (corresponding to a high-conductance state). Thus the periodic input was made of a sinusoidal signal at a given frequency superimposed to a constant baseline that was set to a low value (ν<sup>0</sup> = 1.5 spikes/ms) to induce a low-conductance state and to a high value (ν<sup>0</sup> = 5 spikes/ms) to induce a highconductance state. The amplitude of the sinusoidal component of the input was 0.6 spikes/ms across all simulations. Results are reported in **Figure 12**.

We examined first the low-conductance state (left column of **Figure 12**). We considered the LFP power spectra of the two networks in response to periodic stimuli of different frequencies (**Figures 12A,C**). With respect to the previously examined constant input case (**Figures 7A,B**), the LFP power spectrum of both

frequency *f* from 2 to 150 Hz. **(A,B)** Average (over trials) coherence between the phase of the input signal, with frequency *f*, and the phase of the LFP bandpassed in the corresponding frequency range (*f* − 1, *f* + 1) Hz (see Methods for details). Note that the phase coherence lies in the interval (0, 1). Data are obtained from 50 trials of 10.5 s per stimulus; shaded areas

networks had an additional high narrow peak exactly at the same frequency of the periodic input. This peak signaled the entrainment of the network to the periodic input (Mazzoni et al., 2008). The ability of the two networks to entrain their dynamics to the low-frequency stimuli suggested that the power of the LFP at such low frequencies could discriminate which of these periodic inputs was being presented. We tested this suggestion quantitatively by using mutual information, and we found that the slow LFP frequencies conveyed indeed information about the stimuli, approximately in the same amount in both networks (**Figure 12E**). Note that, in the low-conductance state, there was also a slight modulation with the input frequencies of the power in the gamma band (40–70) Hz, with slightly lower gamma power for stimuli of faster frequency (**Figures 12A,C**). These modulations of gamma-range power resulted in moderate amounts of stimulus information in the same range, (40–70) Hz, (**Figure 12E**), and were likely due to the time taken by the networks to develop gamma oscillations following the very low input values occurring at the trough of the sinusoidal input.

We then investigated the high-conductance state (right column of **Figure 12**). **Figures 12B,D** shows that entrainment of both networks to low frequencies (signaled by the high narrow peak of LFP spectrum at the same frequency as the input) occurred

represent standard deviations across trials. Blue lines display results from COBN and red lines from CUBN. **(C,D)** LFP power spectrum in the COBN as a function of some selected external signal frequencies. The power spectrum is averaged over 50 trials. **(D)** Same color code as in **(C)**. **(E,F)** Same as **(C,D)** for the CUBN. In the low-conductance state both networks entrain very well to the external stimulus, whereas in the high-conductance regime the COBN entrains less well than the CUBN in the middle and in the highest frequency regimes.

strongly in the high-conductance state. The information about which of these periodic inputs was being presented, carried by the low frequency LFP power, was still identical in the two networks (**Figure 12F**). Moreover, and consistently with the above results obtained with constant inputs (**Figures 7A,B**), the gamma peak in the high-conductance states was much stronger and narrower in the COBN than in the CUBN. Probably because of this, the COBN (but not the CUBN) developed beats of the low-frequency peaks into the frequency range around 100 Hz (inset **Figure 12B**). Since the low-frequency peak varied with the input, these beats led to an amount of information in the COBN LFP power around 100 Hz. The moderate gamma-range information peak, observed in the (40–70) Hz range for the low-conductance state (**Figure 12E**), was absent in both networks for the high-conductance regime (**Figure 12F**), because the input rate was always high at any time point. Thus gamma oscillations in the range (80–94) Hz were always strong, with relatively small fluctuations over time, leading to not discernable modulation across the set of input frequencies considered (**Figures 12B,D**).

We then investigated the ability of the network to entrain to a wider range of input frequencies, in particular including frequencies as fast as or faster than the gamma oscillations intrinsically generated by the network. We did so by testing the network

**FIGURE 14 | Spectral information relative to naturalistic stimuli.** Information carried by LFP power spectrum (left column) and population firing rates power spectra (right column) about intervals of naturalistic stimulation based on LGN recordings in monkeys watching a movie. Recording time (80 s) is divided into 40 intervals, considered as different stimuli and the information is computed over 50 trials (see Methods for details). **(A)** Average power spectrum of LFP over the entire naturalistic input for COBN and CUBN (thin line with markers and thick line respectively). **(B)** Average power spectrum for the total firing rate of excitatory and inhibitory neurons (red and blue respectively) for the two networks. Same line code as in **(A)**. **(C)** Spectral information carried by

with periodic stimuli over the 2–150 Hz range of input frequencies (**Figure 13**). Again, to investigate differences between models separately in low- and high-conductance regimes, we generated two kinds of input signals that only differed for the value of the baseline, as described above. We quantified entrainment by computing the coherence between the phase of the input signal and the phase of the LFP bandpassed in a narrow band (with 2 Hz bandwidth) centered at the frequency of the periodic input. In the low-conductance state both networks were strongly entrained to the input over the whole range of frequencies examined, as indicated by the high phase coherence (**Figure 13A**). However, when injecting the same input frequencies with the highest baseline (i.e., making the network operate in a high-conductance state), the behavior of the two networks was very different. The CUBN could still entrain extremely well over the entire input frequency range tested. The COBN entrained extremely well to inputs in the (80–94) Hz input frequency range, but less well to inputs with frequency between 16 Hz and 80 Hz, and above 94 Hz. The reason for the presence in the COBN of frequency regions with lower phase coherence (and thus less accurate entrainment to the

LFP (see Methods for details). Same color code as in **(A)**. In the inset, it is shown the difference between COBN and CUBN information in the low frequency band. **(D)** Spectral information carried by total excitatory and inhibitory firing rates. Same color code as **(B)**. In the inset, it is shown the difference between COBN and CUBN information in the low frequency band. Results show that, also considering complex stimuli, the information relative to the mean value of the input [that here is the information carried by the frequencies above the delta band, (1–4) Hz] is higher and carried on a broader range of frequencies in the COBN, both in LFP and in firing rates. The information conveyed by delta band frequencies is instead almost identical in the two networks.

periodic input) may be because, in the high-conductance state, the COBN had stronger internally generated recurrent oscillations (of higher power than the CUBN, see **Figures 13D,F**) whose dynamics likely did not interfere constructively with the dynamics of the entrainment to the input. This resulted in peaks of less high amplitude in the COBN LFP spectrum at the exact frequency of the periodic input (**Figures 13D,F**). It is interesting to note that the COBN still entrained very well in the (80–94) Hz input frequency range (**Figure 13B**), despite this was also the frequency range exhibiting the strongest recurrent oscillations. Indeed, this range coincided with the peak amplitude of the internally generated gamma oscillations (**Figure 12B**). The ability of the network to entrain well in this gamma range can be understood by observing that this was also the range more strongly modulated by the input rate (**Figure 7A**). Thus, due to their particularly strong responsiveness to the input, external and internal oscillation in this range could interfere constructively, resulting in large peaks of the network LFP at the input frequency (**Figure 13D**).

To study the differences in the responses of the two networks to stimuli more complex and more biologically relevant than periodic functions, we finally compared the information carried by the LFP and firing rate spectra in COBN and CUBN when using the naturalistic time-varying inputs. We injected then in the networks naturalistic stimuli based on MUA recordings from the LGN of an anaesthetized macaque presented with a commercial 80 s color movie clip. The average LFP and total firing rate power spectra for both networks with this set of stimuli are displayed respectively in **Figures 14A** and **B**. All these spectra had higher power at low frequencies (as the input signal had), and the gamma peaks were low because the average stimulus rates were in the range 1.2–2 spikes/ms. We computed information about which part of the time-varying naturalistic signal was being presented (see Methods for details). We found that both LFP and firing rates spectra carried more information in the COBN than in the CUBN, for all frequencies (**Figures 14C,D**). The difference in spectral information between COBN and CUBN for frequencies below 5 Hz was almost zero for the LFP and very low for the firing rates (see insets of **Figures 14C,D**).

Our findings therefore confirm that the two independent information channels (one in the low frequencies due to the entrainment to the input, and one in the gamma band due to internally generated oscillations), which were previously reported for the CUBN (Mazzoni et al., 2008), also exist in the COBN. Moreover, our results show that the information about the input conveyed by low frequencies, both in low- and high-conductance states, does not depend on the details of the synaptic model adopted, while the information encoded in the gamma range is larger in the COBN than in the CUBN.

## **DISCUSSION**

Here we compared in detail the neural population dynamics of LIF networks with either current-based or conductance-based neuron models. The comparison of network dynamics was made on networks with all shared parameters set to an equal common value, and with model-specific synaptic parameters set by a novel recursive procedure that makes COBN and CUBN directly comparable. Our main result was that, although average firing rates and peak frequency of gamma oscillations in such comparable networks were very similar over a wide range of parameters, other aspects of neural population dynamics (such as shape of oscillation spectra or cross-neuron correlation) were significantly different between CUBN and COBN. In particular, oscillation spectra, gamma synchronization and cross-neuron correlation were more markedly modulated by the external input in COBN than in CUBN. The significance of these findings, and their relationship with both theoretical and experimental literature, is discussed in the following.

## **ESTABLISHING COMPARABLE NETWORKS**

The first contribution of the work presented here was to provide a new recursive algorithm to determine the COBN conductance values that correspond to a given set of CUBN synaptic efficacies in networks that have identical values for all the shared parameters. We found that this procedure was able to build two networks displaying relatively small differences, both in the average firing rates and in the gamma frequency peak position, for an input range sufficiently large to encompass both low- and high-conductance states (Destexhe et al., 2003). The relationship of our new procedure with the previous work we built on is discussed in the following.

In a previous work addressing the issue of building equivalent CUBN and COBN models (La Camera et al., 2004), the authors discarded the approach of setting synaptic conductances at fixed average MP (i.e., the one we used in this work) stating that "Although this might work for a single input, it does not work for all inputs in a large pool (results not shown)." La Camera and colleagues proposed instead to build equivalent networks by making both inhibitory and excitatory connectivity free parameters, so that the optimal equivalence was obtained when the CUBN had twice the excitatory and half the inhibitory connectivity of the COBN. Differently from this procedure, in our work all the common parameters of the two networks were identical, including the connectivity matrix. This, in our view, has the advantage that differences in network dynamics can be more directly imputed to changes in model synaptic dynamics. Meffin et al. (2004) determined the value of the conductances starting from a "fixed rough estimate of the average MP" set as the midpoint between threshold and reset potential. The difference with our work is that we used directly the actual average value of the MP of the neurons of each population. Note that there is a discrepancy between the two values since the true average MP was equal or slightly below the reset potential (**Figure 5D**). In extensive initial simulations, we found that using the average MP, rather than the midpoint between threshold and reset potential, made it much easier for the comparable networks to exhibit very close firing rates and gamma spectral peaks (results not shown).

In summary, the comparable networks established with our procedure exhibited average firing rate and position of the peak of the LFP power spectrum that were both similar across network models and were relatively robust to changes in the synaptic reversal potentials. In our view this strengthens the value and usefulness of the setting procedure introduced.

## **COMPARISON BETWEEN SYNAPTIC MODELS**

Previous seminal papers (Kuhn et al., 2004; Meffin et al., 2004; Richardson, 2004) compared the firing rate and MP of conductance- and current-based LIF neurons. Our findings, summarized in Supplementary Table 1, confirmed the main results of these previous works, and extended them in several ways. Our main contribution was to extend the comparison to include other aspects of neural population dynamics. In particular, we considered the effect of the synaptic models on the spectrum of network activity, on the cross-neuron correlations and on the stimulus modulation of these different network features. The significance of these advances is discussed in more detail below.

## **CORRELATION IN THE NETWORKS**

Spike trains of different neurons were more correlated in the COBN than in the CUBN, with the correlation difference increasing with the external input rate. The fact that the COBN spike train correlation was more strongly modulated by the input rate led to the fact that spike train correlation carried more information in the COBN.

In our networks, the neurons received inputs from the same simulated external pool and this led to values of shared input that were likely higher than those shared by pairs of cortical neurons recorded from different electrodes. However, in the COBN, the dependence of correlation on the network stimuli resembled qualitatively the one observed in real experiments, more than in the CUBN. First, the positive correlation between firing rate intensity and spike train correlation is often observed in neurophysiological experiments, (Kohn and Smith, 2005), and this behavior is only reproduced by the COBN. Further, MP of cortical neurons (Lampl et al., 1999) (but see also Yu and Ferster, 2010) are more correlated when they receive an input triggering a stronger response (i.e., having an higher contrast/the correct orientation). This resembles the dynamics displayed here by the COBN, but not by the CUBN. Moreover, in several experiments (see Isaacson and Scanziani, 2011 and references therein), the correlation between AMPA and GABA synaptic inputs is stronger the more intense is the stimulus, consistent with the COBN dynamics shown in **Figure 8A**.

The high values of correlation that we found in the COBN might, at first sight, look different from those of Renart et al. (2010) in which a conductance-based LIF network, with a structure similar to the one considered here, displayed a much smaller MP correlation thanks to the decorrelation due to a precise balance between excitation and inhibition. In other words, in that work, AMPA-GABA correlation and cross-neuron MP correlation were described as mutually exclusive. We think that the reason for the difference between their results and those obtained in our work is the crucial assumption of Renart et al. (2010) that AMPA and GABA timescales are identical. In a supplemental analysis the authors showed indeed that, when AMPA synapses were made progressively faster than GABA, the negative feedback was not fast enough to compensate for excitation and hence to decorrelate the neurons; the network became then more synchronized. When in Renart et al. (2010) the authors considered the case in which τ*rE* = 2 ms and τ*rI* = 5 ms (very close to our values, see **Table 3**), the correlation between GABA and AMPA currents reached values above 0.5, coherent with our results (**Figure 8A**).

#### **FREQUENCY SPECTRA OF NETWORK ACTIVITY**

We also compared the frequency spectra of the network activity in COBN and in CUBN. A marked difference was in the larger amount of information and stronger stimulus modulation of the gamma range for COBN. This, in our view, may be explained as follows. When increasing the external input rate, we observed an increase of the cross-neuron spike train correlation in the COBN, which was associated with an increase of the cross-neuron correlation of the synaptic currents (both AMPA and GABA). This caused a stronger modulation of the COBN currents and consequently of the LFP gamma peak. The stronger modulation of the gamma band in turn contributed to the fact that, both when time-constant and time-varying inputs were injected, the COBN carried more information than the CUBN in the gamma band.

Neurophysiological recordings of LFP spectra modulation in visual cortex during stimulation with various kinds of visual stimuli (Henrie and Shapley, 2005; Belitski et al., 2008) reported much broader gamma peaks than the ones we found for COBNs. The width of gamma peaks reported in cortical data was more similar to the broad gamma peak generated by CUBN rather than to the sharp peak generated by the COBN. We hypothesize that the sharpness of the COBN gamma peak may be over-emphasized by the lack of neuron-to-neuron heterogeneity in the specific network models implemented here. Introducing a small degree of variability in neuronal parameters could decrease the correlation in COBN while keeping it stimulus-dependent. An important point for future research is to understand how heterogeneities in network parameters differentially affect COBN and CUBN dynamics.

A final point worth discussing is that the COBN, unlike the CUBN, showed considerable amounts of information about input strength in the LFP power in the frequency range 15– 25 Hz. Notably, the power of real visual cortical LFPs (Belitski et al., 2008) also did not carry information in this frequency range. Belitski and coworkers hypothesized that the 15–25 Hz LFP frequency region related mainly to stimulus-independent neuromodulation. The additive contribution to the LFP of fluctuations generated by a stimulus-unrelated system would potentially cancel out the information generated by the network in this frequency range.

## **ACKNOWLEDGMENTS**

We are grateful to P. Salvagnini and C. Magri for useful advice on the software implementation of our simulations, and to D. Chicharro for many useful discussions and a precious feedback on an earlier version of the manuscript. We acknowledge the financial support of the SI-CODE project of the Future and Emerging Technologies (FET) program within the Seventh Framework Programme for Research of the European Commission, under FET–Open Grant number: FP7-284553, and of the European Community's Seventh Framework Programme FP7/2007-2013 under Grant agreement number PITN-GA-2011-2900.

## **SUPPLEMENTARY MATERIAL**

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/journal/10.3389/fncir. 2014.00012/abstract

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 10 November 2013; accepted: 07 February 2014; published online: 05 March 2014.*

*Citation: Cavallari S, Panzeri S and Mazzoni A (2014) Comparison of the dynamics of neural interactions between current-based and conductance-based integrate-and-fire recurrent networks. Front. Neural Circuits 8:12. doi: 10.3389/fncir.2014.00012 This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Cavallari, Panzeri and Mazzoni. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Information-geometric measures estimate neural interactions during oscillatory brain states

#### *Yimin Nie1, Jean-Marc Fellous <sup>2</sup> and Masami Tatsuno1 \**

*<sup>1</sup> Department of Neuroscience, Canadian Centre for Behavioural Neuroscience, University of Lethbridge, Lethbridge, AB, Canada <sup>2</sup> Department of Psychology, Program in Applied Mathematics, University of Arizona, Tucson, AZ, USA*

#### *Edited by:*

*A. Ravishankar Rao, IBM Research, USA*

#### *Reviewed by:*

*Carmen C. Canavier, LSU Health Sciences Center, USA Till R. Schneider, University Medical Center Hamburg-Eppendorf, Germany*

#### *\*Correspondence:*

*Masami Tatsuno, Department of Neuroscience, Canadian Centre for Behavioural Neuroscience, The University of Lethbridge, 4401 University Dr W, Lethbridge, AB T1K 3M4, Canada e-mail: tatsuno@uleth.ca*

The characterization of functional network structures among multiple neurons is essential to understanding neural information processing. Information geometry (IG), a theory developed for investigating a space of probability distributions has recently been applied to spike-train analysis and has provided robust estimations of neural interactions. Although neural firing in the equilibrium state is often assumed in these studies, in reality, neural activity is non-stationary. The brain exhibits various oscillations depending on cognitive demands or when an animal is asleep. Therefore, the investigation of the IG measures during oscillatory network states is important for testing how the IG method can be applied to real neural data. Using model networks of binary neurons or more realistic spiking neurons, we studied how the single- and pairwise-IG measures were influenced by oscillatory neural activity. Two general oscillatory mechanisms, externally driven oscillations and internally induced oscillations, were considered. In both mechanisms, we found that the single-IG measure was linearly related to the magnitude of the external input, and that the pairwise-IG measure was linearly related to the sum of connection strengths between two neurons. We also observed that the pairwise-IG measure was not dependent on the oscillation frequency. These results are consistent with the previous findings that were obtained under the equilibrium conditions. Therefore, we demonstrate that the IG method provides useful insights into neural interactions under the oscillatory condition that can often be observed in the real brain.

**Keywords: information geometry, spikes, spiking neuron model, oscillation, neural networks**

## **INTRODUCTION**

The dynamics of neural interactions have been conjectured to play an important role in neural information processing. One way to investigate the neural interactions is to record multi-neuronal firing activity from a freely behaving animal, and analyze the correlations between individual units. In past decades, electrophysiological studies have significantly been advanced by the use of multi-electrode recording techniques (Wilson and McNaughton, 1993; Chapin et al., 1999; Kudrimoti et al., 1999; Laubach et al., 2000; Hoffman and McNaughton, 2002; Buzsaki, 2004; Tatsuno et al., 2006; Euston et al., 2007; Davidson et al., 2009; Peyrache et al., 2009b; Dragoi and Tonegawa, 2011, 2013). In order to analyze such high-dimensional multi-neuronal datasets, a number of statistical methods have also been developed (Gerstein and Perkel, 1969; Abeles and Gerstein, 1988; Aertsen et al., 1989; Zhang et al., 1998; Panzeri and Schultz, 2001; Grun et al., 2002a,b; Brown et al., 2004; Fellous et al., 2004; Czanner et al., 2005; Shimazaki and Shinomoto, 2007; Amari, 2009; Gilestro et al., 2009; Peyrache et al., 2009a; Shimokawa and Shinomoto, 2009; Lopes-Dos-Santos et al., 2011). Recently, a method based on information geometry (IG) has been applied to the analysis of neural data (Amari and Nagaoka, 2000; Amari, 2001; Nakahara and Amari, 2002; Amari et al., 2003; Tatsuno and Okada, 2004; Eleuteri et al., 2005; Ikeda, 2005; Miura et al., 2006; Nakahara et al., 2006; Gilestro et al., 2009; Tatsuno et al., 2009; Ince et al., 2010; Lovette et al., 2011; Nie and Tatsuno, 2012). It has been demonstrated that IG provides a powerful statistical tool for analyzing spiking data. Some of the advantages of IG approach include the orthogonal decomposition of neural interactions (Amari, 2001; Nakahara and Amari, 2002), and its direct relationship to underlying connections (Tatsuno and Okada, 2004; Tatsuno et al., 2009; Nie and Tatsuno, 2012); the single-IG measure is related to the amount of external inputs and the pairwise-IG measure is related to the amount of direct neural interactions between two neurons.

These IG properties were often investigated under the assumption that the network is in an equilibrium state. However, in the brain, the equilibrium assumption does not hold true. Instead, the brain undergoes a variety of non-equilibrium states such as oscillations. For example, the slow-wave oscillation (∼1 Hz) was discovered during non-REM sleep (Steriade et al., 1993; Crunelli and Hughes, 2010), and evidence suggests that it plays an important role in memory consolidation (Huber et al., 2004; Stickgold, 2005; Diekelmann and Born, 2010). The theta (6– 10 Hz) rhythm is a prominent coherent oscillation observed in the hippocampus, and its surrounding area during rat spatial navigation (Vanderwolf, 1969; Bland, 1986; Buzsaki, 2002). The theta rhythm has also been observed in various human neocortical areas during the delay period of working memory tasks (Raghavachari et al., 2001; Meltzer et al., 2008). The beta (15–30 Hz) oscillation is conjectured to play a key role in action preparation and inhibitory control in the motor system (Baker et al., 1997). The gamma (30–80 Hz) oscillation has been shown to play a role in the integration of sensory information (Gray et al., 1989; Singer and Gray, 1995). The fast hippocampal sharp wave ripples (100–200 Hz) were also reported during an animal's awake immobility and slow-wave sleep (Buzsaki et al., 1992). Therefore, it is important to investigate if the IG measures can be applied to neural data under oscillatory conditions.

In this study, we investigated how the single- and pairwise-IG measures are influenced by a network oscillation. Under an equilibrium assumption, previous studies have shown that the single- and pairwise-IG measures provide a robust estimation of the magnitude of external input and direct neural interactions (Tatsuno et al., 2009; Nie and Tatsuno, 2012). We also focused on these IG measures in this study because the external inputs and intrinsic neural interactions are the two main factors for characterizing network dynamics. For the oscillation mechanisms, we have considered two representative cases; one is an external driven oscillation where a network is influenced by external oscillatory inputs. The other is an internally induced oscillation where interactions between excitatory and inhibitory neuron populations produce an oscillation. By computer simulations using simple binary model neurons or more biologically plausible spiking neurons, we investigated whether the properties of the IG measures that were established with the equilibrium condition still hold true under oscillatory network states.

In section Methods, we briefly introduce an informationgeometric analysis of neural spikes. In section Results, we describe the model and network structure used in the numerical simulation. In section Discussion, the simulation results for both externally driven and internally induced oscillations are described in detail. In section Acknowledgments, we summarize our findings and discuss future directions of research on this topic.

## **METHODS**

## **INFORMATION-GEOMETRIC METHOD**

We briefly introduce the information-geometric method for spiking data analysis (for details see Amari and Nagaoka, 2000). Generally, in an *N*-neuron system, the state of *i*-th*(i* = 1*,..., N)* neuron is represented by a binary random variable *xi*, where *xi* = 1 or 0 representing neuronal firing or silence, respectively. The joint probability distribution of the *N*-neuron system can be described by a fully expanded *N*-th order log-linear model (LLM)

$$\log p\_{\mathbf{x}\_1 \mathbf{x}\_2 \cdots \mathbf{x}\_N} = \sum\_i \theta\_i^{(N, \ N)} \mathbf{x}\_i + \sum\_{i < j} \theta\_{ij}^{(N, \ N)} \mathbf{x}\_i \mathbf{x}\_j + \cdots$$

$$+ \theta\_{12, \cdots, N}^{(N, \ N)} \mathbf{x}\_1 \mathbf{x}\_2 \cdots \mathbf{x}\_N - \Psi(\theta)^{(N, \ N)}, \qquad (1)$$

where θ *(N, N) ij,*···*<sup>m</sup> (*1 ≤ *m* ≤ *N)* represents the *m*-neuron interaction and ψ*(*θ*) (N, <sup>N</sup>)* with θ = θ *(N, N) <sup>i</sup> ,* θ *(N, N) ij ,...,* θ *(N, N)* 12*,*···*N* is a normalization constant such that *px*1*x*<sup>2</sup> ··· *xN* = 1. The first and the second superscripts in θ *(N, N) ij,*···*<sup>m</sup>* represent the order of the LLM and the number of neurons in the system. We use θ *(N, N) <sup>i</sup>* , θ *(N, N) ij* , and θ *(N, N) ij,*···*<sup>m</sup>* to describe the single-IG measure, the pairwise-IG measure and the *m*-neuron IG measure with the *N*-th order LLM for a *N*-neuron system, respectively, (Nie and Tatsuno, 2012). The joint probability of *N* neurons is calculated by

$$p\_{\mathbf{x}\_1 \mathbf{x}\_2, \dots, \mathbf{x}\_N} = \frac{c\_{\mathbf{x}\_1 \mathbf{x}\_2, \dots, \mathbf{x}\_N}}{\sum\_{\mathbf{x}\_1 \mathbf{x}\_2, \dots, \mathbf{x}\_N} c\_{\mathbf{x}\_1 \mathbf{x}\_2, \dots, \mathbf{x}\_N}},\tag{2}$$

where *cx*1*x*2*,...,x*<sup>N</sup> is the count of events *(X*<sup>1</sup> = *x*1*, X*<sup>2</sup> = *x*2*,..., X*<sup>N</sup> = *x*N*)* that occur.

However, in reality, it is difficult to calculate the statistical information from all neurons in a large network. Therefore, the partially expanded LLM is often used for the estimation of neuronal interactions. The partially expanded *k*-th order LLM in an *N*-neuron network is given by

$$\log p\_{\mathbf{x}\_1 \mathbf{x}\_2 \cdots \mathbf{x}\_k \ \ast \ \cdots \ \ast} = \sum\_i \theta\_i^{(k,N)} \mathbf{x}\_i + \sum\_{i
$$+ \theta\_{12,\cdots,k}^{(k,N)} \mathbf{x}\_1 \mathbf{x}\_2 \cdots \mathbf{x}\_k - \Psi(\mathbf{\theta})^{(k,N)} \qquad (3)$$
$$

where **θ** = θ *(k, N) <sup>i</sup> ,* θ *(k, N) ij ,...,* θ *(k, N)* 12*,*··· *k* . The first few terms of **θ** and normalization factor are given as follows:

$$\begin{aligned} \theta\_i^{(k,N)} &= \log \frac{\mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_i = 1, \dots, \mathbf{x}\_k = 0, \* \dots \* }{\mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_k = 0, \* \dots \*}, \\\\ &\quad \mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_i = 1, \dots, \mathbf{x}\_j = 1, \dots, \mathbf{x}\_k = 0, \* \dots \* \\\ \theta\_{ij}^{(k,N)} &= \log \frac{\mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_k = 0, \* \dots \* }{\mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_j = 0, \dots, \mathbf{x}\_k = 0, \* \dots \* } \\\ &\quad \mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_i = 0, \dots, \mathbf{x}\_j = 1, \dots, \mathbf{x}\_k = 0, \* \dots \* \dots \dots \\\ &\quad \psi^{(k,N)} = -\log \mathcal{P}\mathbf{x}\_1 = 0, \dots, \mathbf{x}\_k = 0, \* \dots \*, \end{aligned} \tag{4}$$

where ' ∗···∗ , represents the marginalization over the (*N* − *k*) neurons.

The single-IG measure θ *(k, N) <sup>i</sup>* and the pairwise-IG measure θ *(k, N) ij* are the two main focuses in this study because θ *(k, N) <sup>i</sup>* is related to the amount of external inputs and θ *(k, N) ij* is related to the amount of direct neural interactions between two neurons (Tatsuno and Okada, 2004; Tatsuno et al., 2009; Nie and Tatsuno, 2012). Using a network of simple binary neurons, and the assumption of an equilibrium state, the previous study has shown that the single-IG measure θ *(*2*, N) <sup>i</sup>* and the pairwise-IG measure θ *(*2*, N) ij* with the 2nd-order LLM are related to the network parameters as

$$
\theta\_i^{(2,\ N)} \propto 2h\_i + O\left(\frac{1}{N}\right), \quad \theta\_{ij}^{(2,\ N)} \propto (I\_{ij} + I\_{ji}) + O\left(\frac{1}{N}\right), \tag{5}
$$

where *hi* represents the magnitude of constant external input to a neuron *i*, and *Jij (Jji)* is the connection weight from a neuron *j* to *i* (from a neuron *i* to *j*), respectively, (Tatsuno et al., 2009). If a network receives correlated inputs, the relationship for θ *(*2*, N) ij* in Equation 5 does not hold true anymore. However, we have also shown that θ *(k, N) ij* with the higher k-th order LLM provides a better estimation of neural interactions (Nie and Tatsuno, 2012). For example, θ *(*4*, N) ij* was shown to have the relationship

$$\left(\Theta\_{ij}^{(4,\ N)} \propto (J\_{ij} + J\_{ji})\right),\tag{6}$$

within approximately a 10% error if the number of neurons *N* is ∼ 10<sup>3</sup> or larger; a typical size of network in a cortical column (Urban et al., 2001). We have also confirmed that the relationship θ *(*4*, N) <sup>i</sup>* α2*hi* holds true within approximately a 10% error (unpublished data).

These properties could be useful for the field of neuroscience because the IG measures can estimate the changes of underlying network parameters (*hi* and *Jij*) separately, while other correlation measures have not yet been shown to have such a property (Amari, 2009). However, these results were derived under the equilibrium limit, and little is known if the similar relationship holds under the oscillatory condition.

## **NEURON MODEL AND NETWORK STRUCTURE**

#### *Neuron model*

We investigated the influence of oscillations using a network of simple binary neurons with stochastic dynamics (Ginzburg and Sompolinsky, 1994) and biologically plausible spiking neurons (Izhikevich, 2003). Using simple binary model neurons, we first investigated whether the property of the IG measures that were shown under the equilibrium condition also held true for the oscillatory condition. We then extended our investigation to more realistic spiking neurons.

For a binary model neuron, the transition between the binary states is given by the transition rate *w* as

$$\begin{aligned} \varkappa(\mathbf{x}\_{i} = 0 \to \mathbf{x}\_{i} = 1) &= \frac{\mathbf{g}(u\_{i})}{\mathfrak{r}\_{0}}, \\ \varkappa(\mathbf{x}\_{i} = 1 \to \mathbf{x}\_{i} = 0) &= \frac{1 - \mathfrak{g}(u\_{i})}{\mathfrak{r}\_{0}}, \\ \varkappa(\mathbf{x}\_{i} = 0 \to \mathbf{x}\_{i} = 0) &= 1 - \varkappa(0 \to 1), \\ \varkappa(\mathbf{x}\_{i} = 1 \to \mathbf{x}\_{i} = 1) &= 1 - \varkappa(1 \to 0), \end{aligned} \tag{7}$$

where τ<sup>0</sup> is a microscopic characteristic time and *ui* represents the total input to the *i*-th neuron.

$$\lg(u\_i) = \frac{1 + \tanh(u\_i - m)}{2} \tag{8}$$

is the sigmoidal function in the bounded interval [0, 1] where *m* is a parameter controlling the firing probability of a model neuron.

For a biologically more plausible neuron model, we adopted the Izhikevich model because it is known to be computationally efficient and biologically plausible (Izhikevich, 2003). The Izhikevich model reduced the complex dynamics of the Hodgkin– Huxley (HH) neuronal models to two coupled differential equations as

$$\frac{dV\_i}{dt} = 0.04V\_i^2 + 5V\_i + 140 - U\_i + I\_i,\\ \frac{dU\_i}{dt} = a\_i(b\_i V\_i - U\_i). \tag{9}$$

Here the variable *Vi* represents the membrane potential of neuron *i*, and *Ui* represents a membrane recovery variable which correlates with the activation of *K*<sup>+</sup> ionic currents and inactivation of *Na*<sup>+</sup> [for detail see (Izhikevich, 2003)]. *Ui* and *Vi* are reset after a spike: if *Vi* ≥ 30 mV, then *Vi* ← *ci, Ui* ← *Ui* + *di*. *Ii* represents a total input to neuron *i*; *ai, bi, ci, di* are dimensionless adjustable parameters which are usually taken as *(ai, bi)* = *(*0*.*02*,* 0*.*2*)* and *(ci, di)* = *(*−65*,* 8*)* + *(*15*,* −6*)r*<sup>2</sup> *<sup>i</sup>* for excitatory neurons,*(ai, bi)* = *(*0*.*02*,* 0*.*25*)* + *(*0*.*08*,* −0*.*05*ri)* and *(ci, di)* = *(*−65*,* 2*)* for inhibitory neurons. *ri* is a uniformly distributed random variable on the interval [0, 1] (Izhikevich, 2003).

### *Network structure*

We considered two mechanisms for generating oscillatory network states; one is the oscillation driven by external inputs (**Figure 1A**), and the other is the oscillation induced by the intrinsic interaction between excitatory and inhibitory neuron populations (**Figure 1B**). The former mechanism can be a model for hippocampal theta oscillation in which the projections from the medial septum to the hippocampus play a central role (Dragoi et al., 1999). The latter structure where excitatory and inhibitory neuron pools interact is widely observed in cortical areas (Buzsaki and Wang, 2012). It can be a model for cortical oscillations (such as in a gamma-range) that rely on the interplay between excitatory and inhibitory neuron pools.

In the first scenario (externally driven oscillation, **Figure 1A**), a sinusoidal external input *hi(t)* = *h*<sup>0</sup> sin*(*ω*t* + ϕ*i)* for the *i*-th neuron was used to generate oscillatory states in a network, where *h*0, ω, and *upvarphii* represent the amplitude, angular speed, and phase of sinusoidal signals, respectively. Note that *h*<sup>0</sup> and ω are common to all neurons, but ϕ*<sup>i</sup>* can be different for individual neurons. The explicit expression of an input signal allows one to produce different network oscillations systematically. For the binary neuron model, the total input to the *i*-th neuron is written as,

$$u\_i(t) = \sum\_j J\_{i\bar{j}} \mathbf{x}\_j(t) + h\_i(t). \tag{10}$$

where *Jij* represents a connection weight from the *j*-th neuron to the *i*-th neuron. The neuronal state *xi(t)* was then updated following the transition rate *w* in Equation 7. Note that model neurons are identical, whether they are excitatory or inhibitory.

For the Izhikevich model in the first scenario, we considered a population of excitatory neurons. Although it has been demonstrated that a network of excitatory neurons can synchronize, a network of Izhikevich neurons that were connected in this particular way cannot produce an intrinsic oscillation (Mirollo and Strogatz, 1990; Hansel et al., 1995). This allows us to investigate the relationship between the IG measures and an externally driven oscillation in a more realistic setting. The total input to an Izhikevich neuron *i* is given by,

$$I\_i^E(t) = \sum\_j J\_{ij}^{EE} s\_j^E(t) + h\_i(t), \tag{11}$$

where *JEE* represent positive weights between excitatory neurons and *s E <sup>j</sup>* = δ*(t* − *t f <sup>j</sup> )* is the delta function representing the existence

**oscillations. (A)** Oscillation is generated by an external oscillatory input (externally driven oscillation). A neuron *i* in the network of *N* neurons with recurrent connections *Jij* receives a sinusoidal external input *hi* = *h*<sup>0</sup> sin*(*ω*t* + ϕ*i)* where *h*0, ω, and ϕ*<sup>i</sup>* represent the amplitude, angular speed, and phase of the sinusoidal input, respectively. **(B)** Oscillation is generated by the interaction between excitatory and

are connected by positive connections *JEE* , inhibitory neurons are connected by negative connections *JII* , inhibitory neurons receive positive connections *JIE* from excitatory neurons, and excitatory neurons receive negative connections *JEI* from inhibitory neurons. In addition, excitatory and inhibitory neurons receive external constant input *H<sup>E</sup> <sup>i</sup>* and *<sup>H</sup><sup>I</sup> k* , respectively.

of a spike emitted from an excitatory neuron *j* at time *t f <sup>j</sup>* . The neuronal state was then updated by Equation 9 and the associated reset dynamics. In the numerical simulation, we used a small bin width of 1 ms so that it would contain no more than one spike.

**Figures 2A–C** show example spike trains and multi-unit activity of binary model neurons driven by external sinusoidal inputs of 1, 6, and 100 Hz oscillations, respectively. Izhikevich neurons also exhibited very similar activity (data not shown). It can be clearly seen that neural activity is entrained to external input. Using these two models, we investigated how the IG measures are affected by externally driven oscillations.

In the second scenario (internally induced oscillation, **Figure 1B**), interaction between excitatory and inhibitory neuron pools generates an oscillation. For the binary neuron model, the total input to the *i*-th excitatory neuron and the *k*-th inhibitory neuron are written as,

$$u\_i^E(t) = \sum\_j J\_{ij}^{EE} \mathbf{x}\_j^E(t) + \sum\_j J\_{ij}^{EI} \mathbf{x}\_j^E(t) + H\_i^E,$$

$$u\_k^I(t) = \sum\_j J\_{kj}^{II} \mathbf{x}\_j^I(t) + \sum\_j J\_{kj}^{IE} \mathbf{x}\_j^I(t) + H\_k^I,\tag{12}$$

where *JII, JIE*, and *JEI* represent negative weights between inhibitory neurons, positive weights from excitatory neurons to inhibitory neurons, and negative weights from inhibitory neurons to excitatory neurons, respectively. The excitatory and inhibitory neurons receive constant external inputs *H<sup>E</sup> i* and *H<sup>I</sup> <sup>i</sup>* , and maintain sustained oscillatory activity. The neuronal state was updated following the transition rate *w* in Equation 7.

For the Izhikevich model in the second scenario, a similar relationship exists for the total inputs for the *i*-th excitatory neuron and the *k*-th inhibitory neuron,

$$I\_i^E(t) = \sum\_j I\_{ij}^{EE} s\_j^E(t) + \sum\_j I\_{ij}^{EI} s\_j^E(t) + H\_i^E,$$

$$I\_k^I(t) = \sum\_j I\_{kj}^{II} s\_j^I(t) + \sum\_j I\_{kj}^{IE} s\_j^I(t) + H\_k^I. \tag{13}$$

The neuronal state was then updated following Equation 9 and the associated reset dynamics. **Figures 2D,E** provide examples of spike trains and multi-unit activity of Izhikevich neurons that exhibited ∼6 and 40 Hz oscillations, respectively. Binary neurons also exhibited a very similar activity (data not shown). Neural activity was synchronized, but the degree of entrainment was weaker than the externally driven mechanisms. Using these two models, we investigated how the IG measures were influenced by the internally induced oscillation.

## **RESULTS**

## **EXTERNALLY DRIVEN OSCILLATION**

We investigated the relationship between the IG measures, θ *(*4*, N) i* and θ *(*4*, N) ij* , and the connection weights, *(Jij* + *Jji)*, using a network of 1000 binary neurons and 1000 Izhikevich neurons. We focused on the IG measures with 4th-order LLM because they have been shown to estimate connection weights (Nie and Tatsuno, 2012) and external inputs (unpublished data) within a 10% error under an equilibrium assumption. In the simulation, we kept the amplitude of external input at a value such that the overall network firing probability is relatively low (*pxi* ∼ 0*.*1). Connection weights were set to the order of 1*/N* to prevent saturation of neuronal activity. For a binary neuron model, we used *Jij* = 1*/N* + ε*ij* where ε*ij* is a random variable from a normal distribution *N(*0*,* 1*/N)* with a mean of 0 and the standard deviation of 1*/* <sup>√</sup>*N*. For the Izhikevich model, we restricted the simulations to a pool of only excitatory neurons to ensure that no internally induced oscillation occurred. The connection weights were

assigned as *JEE Ij* <sup>=</sup> <sup>1</sup>*/<sup>N</sup>* <sup>+</sup> <sup>ε</sup> *ij* where ε *ij* is a random variable following uniform distribution *U(*0*,* 1*/N)* within the interval of [0*,* 1*/N*]. θ *(*4*, N) <sup>i</sup>* and θ *(*4*, N) ij* were calculated by 10<sup>6</sup> updates of the network. With the time resolution of 1 ms, the simulation corresponds to ∼15 min of recordings. To obtain the mean and variances of the IG measures, we performed 100 independent simulations. Error bars in the figure represent the standard error of mean (SEM).

We investigated the oscillation frequencies that have often been observed in the brain; slow oscillation (∼1 Hz), theta oscillation (6–10 Hz), and ripple oscillation (100–200 Hz). The left column of (**Figures 3A,D,G**) shows the results for the slow oscillation. The multi-unit activity of the binary neurons exhibits a slow oscillation of the frequency of external input (**Figure 3A**) and the neurons were entrained to this frequency (**Figure 2A**). The spiking activity of Izhikevich neurons also showed almost identical activity (data not shown). To investigate how θ *(*4*, N) <sup>i</sup>* and θ *(*4*, N) ij* are related to the change of connection weights, we systematically modified the sum of connection weights between two neurons (1 and 2) from −9*/N* to 9*/N*. Due to the randomness

neurons with a 100-Hz oscillatory modulation is shown. **(D)** Relationship between the pairwise IG measure *(*θ *(*4*,* 1000*)* <sup>12</sup> *)* and the sum of connection weights (*J*<sup>12</sup> + *J*21*)* under a 1-Hz oscillation. Black and gray lines represent the simulation results by binary neurons and Izhikevich neurons, respectively. **(E)** Relationship between the pairwise IG measure (θ *(*4*,* 1000*)* <sup>12</sup> *)* and the sum of

between the single IG measure *(*θ *(*4*,* 1000*)* <sup>1</sup> *)* and the sum of connection weights (*J*<sup>12</sup> + *J*21*)* under a 6-Hz oscillation. **(I)** Relationship between the single IG measure *(*θ *(*4*,* 1000*)* <sup>1</sup> *)* and the sum of connection weights *(J*<sup>12</sup> + *J*21*)* under a 100-Hz oscillation.

of the connectivity, focusing the neurons (1 and 2) did not affect the generality. We found that θ *(*4*, N)* <sup>12</sup> was linearly related to the sum of the connection weights, and that the values of θ *(*4*, N)* <sup>12</sup> for both the binary and Izhikevich models were very close (**Figure 3D**, black line for a binary model and gray line for the Izhikevich model). On the other hand, θ *(*4*, N)* <sup>1</sup> and θ *(*4*, N)* <sup>2</sup> were independent from the change of synaptic weights (**Figure 3G**). These results are consistent with the previous findings under the equilibrium assumption; showing that IG measures can also provide useful insights in conditions where the network oscillates. The middle and right columns of **Figure 3** show the results for theta oscillations (**Figures 3B,E,H**) and ripple oscillations (**Figures 3C,F,I**), respectively. We found that the relationship between the IG measures and connection weights was robust against a different frequency of external inputs. This confirmed that the IG measures can also provide useful information for externally driven theta and ripple oscillations.

To further investigate if the robust property of the IG measures for slow, theta, and ripple oscillations holds true for other frequencies, we varied the frequency over 1–200 Hz, the range that can be typically observed in the brain. We set *(J*<sup>12</sup> + *J*21*)* = 2*J*. **Figure 4** shows that θ *(*4*, N)* <sup>12</sup> and θ *(*4*, N)* <sup>1</sup> did not depend on

oscillation frequencies (**Figures 4A,C** for a binary model, and **Figures 4B,D** for Izhikevich model). The results confirmed that the IG measures would be useful for neural data analysis when the brain exhibits a variety of oscillations depending on cognitive demands and the sleep stages.

The previous analyses (**Figures 3**, **4**) were performed under the zero relative phase difference between two neurons *i* and *j*, namely δϕ*ij* = ϕ*<sup>i</sup>* − ϕ*<sup>j</sup>* = 0. This corresponds to the synchronous neural firings that were depicted in **Figures 2A–C**. Neurons can, however, exhibit phase differences. For instance, sequential neural activity was observed in the natural and anesthetized brain states (Lee and Wilson, 2002; Euston et al., 2007; Luczak et al., 2007; Bermudez Contreras et al., 2013). Therefore, we calculated IG measures with phase differences. **Figure 5** shows the results of the 6-Hz simulations in which the phase difference between sinusoidal inputs to the neurons 1 and 2 was set to π*/*6 (**Figures 5A,C**) and π*/*2 (**Figures 5B,D**). The rest of the neuron pairs have random phases in the range of [0, 2π]. **Figures 5A,B** show that θ *(*4*,* 1000*)* <sup>12</sup> is linearly related to the sum of synaptic weights, suggesting that the relationship observed in zero phase difference condition also holds for the non-zero phase difference condition. Similarly, **Figures 5C,D** show that θ *(*4*,* 1000*)* <sup>1</sup> does not depend on the connection weights, even when neurons fire with phase differences. By comparing these results with **Figures 3E,H** where there was no phase difference, we also found that phase difference produced the shift of the actual values of IG measures. This suggests that if the phase relationship drastically changes between the two recording epochs, the values of the IG measures cannot be directly comparable. However, if their difference is not large or if phase difference can be estimated beforehand, we could use the information for adjusting the IG values. We also confirmed that these relationships held true for slow (1 Hz) and ripple (100 Hz) frequencies (data not shown).

So far we have focused on the relationship between the IG measures and the connection weights. Another important parameter is the magnitude of sinusoidal input *h*0. Therefore, we have analyzed how θ *(*4*,* 1000*)* <sup>1</sup> and θ *(*4*,* 1000*)* <sup>12</sup> are related to *h*0. **Figure 6** shows the result when the external sinusoidal input has a frequency of 6 Hz (theta oscillation). We found that θ *(*4*,* 1000*)* <sup>12</sup> was nearly independent from the change of *h*<sup>0</sup> (**Figure 6A**), but θ *(*4*,* 1000*)* <sup>1</sup> was almost linearly related to it (**Figure 6B**). We also confirmed that almost identical relationship holds true for other frequencies such as slow oscillation and ripple oscillation if there is no phase difference. For non-zero phases between neurons, we observed that the IG values were shifted, like the case for the IG values and connection weights, but that the same linear and independent relationship in **Figure 6** was sustained. The results under the oscillatory condition are consistent with the previous findings under the equilibrium condition; θ *(*4*, N) <sup>i</sup>* was linearly related to the magnitude of the constant input and that θ *(*4*, N) ij* was almost independent from it (Nie and Tatsuno, 2012). The investigation here provides further evidence that θ *(*4*, N) <sup>i</sup>* is useful for the estimation of the magnitude of external input.

**FIGURE 5 | Relationship between the IG measures and the sum of connection weights for non-zero phase differences.** An external oscillation mechanism was used, and the oscillation frequency was set to 6 Hz. **(A)** Relationship between the pairwise IG measure θ *(*4*,* 1000*)* <sup>12</sup> and the sum of connection weights (*J*<sup>12</sup> + *J*21) for the phase difference of δϕ<sup>12</sup> = π*/*6. Black and gray curves represent the simulation by binary neurons and Izhikevich neurons, respectively. **(B)** Relationship between the pairwise IG measure

 θ *(*4*,* 1000*)* <sup>12</sup> and the sum of connection weights, *(J*<sup>12</sup> + *J*21*)* for the phase difference of δϕ<sup>12</sup> = π*/*2. **(C)** Relationship between the single-IG measure θ *(*4*,* 1000*)* 1 and the sum of connection weights (*J*<sup>12</sup> + *J*21*)* for the phase difference of δϕ<sup>12</sup> = π*/*6. **(D)** Relationship between the single-IG measure θ *(*4*,* 1000*)* 1 and the sum of connection weights (*J*<sup>12</sup> + *J*21*)* for the phase difference of δϕ<sup>12</sup> = π*/*2.

In summary, we investigated how the IG measures were influenced by an externally driven oscillation. Using a simple binary neuron model, and a more realistic Izhikevich model, we found that θ *(*4*, N) ij* had a linear relationship with the sum of the connection weights, and that it was almost independent from the magnitude of a sinusoidal input. In contrast, θ *(*4*, N) <sup>i</sup>* was almost independent from the connection weights, but was linearly related to the magnitude of the sinusoidal input. These properties were not affected by the frequency of the oscillations or the relative phase differences between neurons.

#### **INTERNALLY INDUCED OSCILLATION**

As another mechanism for generating an oscillatory network behavior, we also investigated the interactions between excitatory and inhibitory neuron pools. We analyzed the network structure in **Figure 1B** by simple binary model neurons and Izhikevich neurons. Unlike the first oscillation mechanism, where an oscillation frequency and phase differences could be explicitly controlled, it was not easy to generate an oscillation with desired parameters. However, we were able to generate two examples that were often observed in the brain. **Figures 7A,B** show multi-unit activity corresponding to theta frequency (∼8 Hz) and gamma frequency (∼40 Hz), respectively. The same examples with a raster plot were also depicted in **Figures 2D,E**. To avoid saturation in neural activity, we have set the connection weights to the order of 1*/N*. For a theta oscillation, we set the connection as *JEE ij* <sup>=</sup> *<sup>J</sup>*<sup>1</sup> · *ij, <sup>J</sup>IE ij* = 5*J*<sup>1</sup> · *ij, <sup>J</sup>II ij* = −*J*<sup>2</sup> · *ij* and *<sup>J</sup>EI ij* = −*J*<sup>2</sup> · *ij* where *J*<sup>1</sup> = 1*/Ne, J*<sup>2</sup> = 1*/Ni*, and *ij* is a random variable following a uniform distribution *U(*0*,* 1*)* within the interval [0,1]. For a gamma oscillation, we used *JEE ij* <sup>=</sup> *<sup>J</sup>*<sup>1</sup> · *ij, <sup>J</sup>IE ij* <sup>=</sup> <sup>6</sup>*J*<sup>1</sup> · *ij, <sup>J</sup>II ij* = −*J*<sup>2</sup> · *ij* and *JEI ij* = −2*J*2· *ij*. The stronger *<sup>J</sup>IE* was necessary to induce oscillation (Adini et al., 1997). The external constant inputs to excitatory and inhibitory neurons were set as *H<sup>E</sup> <sup>i</sup>* <sup>=</sup> <sup>0</sup>*.*05 and *<sup>H</sup><sup>I</sup> <sup>i</sup>* = 0*.*02, respectively. The simulation of 10<sup>6</sup> update was performed with *Ne* = 1000 excitatory neurons and *Ni* = 250 inhibitory neurons. The mean and variance was estimated using 100 independent simulations.

To investigate the relationship between the IG measures and the sum of connection weights, without losing the generality, we modified *(J*<sup>12</sup> + *J*21*)* between the neurons (1 and 2) in the range of [−9*J,* 9*J*]. Firstly, we focused on the connections within the excitatory neuron population and within the inhibitory neuron population. In other words, both connections, *J*<sup>12</sup> and *J*21, were positive for the range of *(J*<sup>12</sup> + *J*21*)* ≥ 0 and both were negative for *(J*<sup>12</sup> + *J*21*) <* 0. **Figures 7C,E** show the relationship between θ *(*4*,* 1250*)* <sup>12</sup> and θ *(*4*,* 1250*)* <sup>1</sup> , and the sum of connection strengths *(J*<sup>12</sup> + *J*21*)* under the theta oscillation. The results clearly show that θ *(*4*,* 1250*)* <sup>12</sup> is linearly related to the sum of the connection weights and that θ *(*4*,* 1250*)* <sup>1</sup> was independent from the modulation of the connection weights. Furthermore, the dependency of the IG measures on the connection weights was continuous in both positive and negative ranges. This suggests that IG measures can be applicable to both positive and negative connections. **Figures 7D,F** show results for gamma oscillation. Similar results were obtained for both θ *(*4*,* 1250*)* <sup>12</sup> and θ *(*4*,* 1250*)* <sup>1</sup> .

Secondly, we investigated the interaction between excitatory and inhibitory neurons. Namely, we selected the neuron 1 from the excitatory neuron pool and the neuron 2 from the inhibitory neuron pool. The sum of connection weights was modified from −9*J* to 9*J*. **Figures 8A,B** are the same with **Figures 7A,B**, showing the multi-unit activity for theta and gamma oscillation, respectively. **Figures 8C,E** show the relationship between the IG measures and the sum of the connection weights under the theta oscillation. Similarly, **Figures 8D,F** are for gamma oscillation. The results show that the linear dependency of θ *(*4*,* 1250*)* <sup>12</sup> on the sum of the connection weights holds true for an excitatory and inhibitory neuron pair. We also found that θ *(*4*,* 1250*)* <sup>1</sup> had almost no relationship with the sum of connection weight. For the relationship between the IG measures and the magnitude of constant input *H<sup>E</sup> i* and *H<sup>I</sup> <sup>i</sup>* , we confirmed that θ *(*4*,* 1250*)* <sup>1</sup> was linearly related to their magnitude, but θ *(*4*,* 1250*)* <sup>12</sup> was independent from them (data not shown).

In summary, for internally generated oscillations, we demonstrated that the relationship between the IG measures and the connection weights that were found under equilibrium assumption also held true.

## **DISCUSSION**

Previous studies have shown that the IG measures provided useful information about network structures (Tatsuno and Okada, 2004; Tatsuno et al., 2009; Nie and Tatsuno, 2012). Specifically, the single-IG measure θ *(*4*, N) <sup>i</sup>* was related to the magnitude of external constant input, and the pairwise-IG measure θ *(*4*, N) ij* was related to the sum of the connection strengths. Although these studies were conducted under the equilibrium assumption, the real neural signals exhibit various oscillations depending on cognitive demand of the task or the state of the brain. Therefore, we studied the relationship between the IG measures and the neural network parameters under oscillatory network states.

We have considered two general oscillation mechanisms; one was the oscillation driven by external input, and the other was the oscillation induced internally due to interactions between excitatory and inhibitory neuron pools. Numerical simulation was performed by the network of a simple binary neuron model and the Izhikevich neuron model. The former model was used so as to compare the results with that of previous studies, and the latter was used to investigate the relationship with more realistic model neurons.

For the external oscillation, our investigation showed that θ *(*4*, N) ij* was linearly related to the sum of the connection strengths, and that θ *(*4*, N) <sup>i</sup>* was independent from it over a wide range of frequency from 1 to 200 Hz. We also showed that the relationship holds true when there are phase differences between neurons. In addition, we demonstrated that θ *(*4*, N) <sup>i</sup>* was almost linearly related to the magnitude of sinusoidal input, but that θ *(*4*, N) ij* was almost independent from it. For the internally induced oscillation, we have also confirmed that θ *(*4*, N) ij* was linearly related to the sum of the connection strengths, and that θ *(*4*, N) <sup>i</sup>* was independent from it. We have also shown that the same relationship holds true for any neuron pairs (within excitatory population, within inhibitory population, and across excitatory and inhibitory populations).

In summary, this study and previous studies have demonstrated that the IG measure provides useful information for analyzing neural circuits; not only for the equilibrium condition, but also for the oscillatory condition. The single-IG measure is useful for estimating the relative strength of external inputs. In addition, the single-IG measure is better than using the change in firing rate because the firing rate can be modulated both by the change in synaptic coupling strength and the magnitude of external inputs. Studies show that the appropriately selected single-IG measure is capable of estimating the external inputs with relatively small influence from synaptic interactions. Similarly, the

they were negative for *(J*<sup>12</sup> + *J*21*) <* 0. **(A)** Average firing probability of 1250 Izhikevich neurons with approximately a 6-Hz oscillatory oscillation is shown. **(B)** Average firing probability of 1250 Izhikevich neurons with approximately a 40-Hz oscillatory oscillation is shown. **(C)** Relationship between the pairwise-IG measure (θ *(*4*,* 1000*)* <sup>12</sup> *)* and the sum of connection weights

difference is kept the same, the IG measures can provide useful insights into network structures regardless of the oscillation frequencies.

weights (*J*<sup>12</sup> + *J*21*)* under a 6-Hz oscillation. **(F)** Relationship between the

<sup>1</sup> *)* and the sum of connection

<sup>1</sup> *)* and the sum of connection weights (*J*<sup>12</sup> + *J*21*)*

between the single-IG measure *(*θ

*(*4*,* 1000*)*

single-IG measure *(*θ

under a 40-Hz oscillation.

In the study of memory consolidation, one of the key questions is to understand how the changes in synaptic connections are related to learning and memory formation. Evidence suggests that neural activity during slow-wave sleep plays an important role in learning (Diekelmann and Born, 2010). Specifically, there is increasing evidence supporting the hypothesis that replay of neural activity during subsequent sleep is positively correlated with memory formation (Pavlides and Winson, 1989; Wilson and McNaughton, 1994; Kudrimoti et al., 1999; Lee and Wilson, 2002; Euston et al., 2007; Girardeau et al., 2009; Peyrache et al., 2009b; Ego-Stengel and Wilson, 2010). However, the direct information about synaptic change is not available from multi-unit recordings of a freely behaving animal because spikes and local field potentials are the two main observables. In this study, we showed that θ *(*4*, N) ij* was linearly related to the sum of the connection weights,

pairwise-IG measure can provide more direct information about the synaptic interactions between neurons than other correlation measures (Amari, 2009). It has been also shown that the pairwise-IG measure is statistically independent from the change in firing rate and that it provides pure neural interactions (Amari, 2001; Nakahara and Amari, 2002). Together with the findings in this study, the pairwise-IG measure is a very useful measure to study direct neural interactions between neurons.

This study suggests that the actual values of the IG measures depend on the mechanisms of oscillation. For an externally driven oscillation, θ *(*4*, N)* <sup>12</sup> ∼ 0*.*2 was obtained for *(J*<sup>12</sup> + *J*21*)* ∼ 1*/N*. For an internally induced oscillation, the same connection strength produced θ *(*4*, N)* <sup>12</sup> ∼ 0*.*002. Within the same oscillation mechanism, the selection of model neurons (binary model or Izhikevich model), or a small difference in network parameters such as phase differences also produced a difference in the actual value of the IG measures. Nonetheless, as long as the network is in one of the oscillation mechanisms, and the phase

was selected from the excitatory neuron pool and Neuron 2 was selected from the inhibitory neuron pool. In other words, one of the connections in *J*<sup>12</sup> + *J*<sup>21</sup> was positive and the other was negative. **(A)** Average firing probability of 1250 Izhikevich neurons with approximately a 6-Hz oscillatory oscillation is shown. **(B)** Average firing probability of 1250 Izhikevich neurons with approximately a 40-Hz oscillatory oscillation is shown. **(C)** Relationship between the pairwise-IG measure (θ *(*4*,* 1000*)* <sup>12</sup> *)*

and that θ *(*4*, N) <sup>i</sup>* was linearly related to the magnitude of external inputs, even under the oscillatory conditions. We have also verified these relationships not only with a simple binary model neuron, but also with a more realistic spiking model neuron. This finding would allow us to analyze neural activity during slowwave sleep before and after the task; θ *(*4*, N) ij* would be a good measure for the change of connection weight, and θ *(*4*, N) <sup>i</sup>* for the magnitude of background input that would be influenced by local field potentials. By comparing the relative change of θ *(*4*, N) ij* between slow-wave sleeps before and after the task, and the strength of memory replay/improvement of behavior performance, the IG measure may provide a way to estimate the relationship between the synaptic modification and memory formation without having direct access to information of synaptic change.

As a related approach to the IG method, the maximum entropy (MaxEnt) has attracted much attention recently (Schneidman

pairwise-IG measure (θ <sup>12</sup> *)* and the sum of connection weights (*J*<sup>12</sup> + *J*21*)* under a 6-Hz oscillation. **(E)** Relationship between the single-IG measure *(*θ *(*4*,* 1000*)* <sup>1</sup> *)* and the sum of connection weights (*J*<sup>12</sup> + *J*21*)* under a 6-Hz oscillation. **(F)** Relationship between the single-IG measure *(*θ *(*4*,* 1000*)* <sup>1</sup> ) and the sum of connection weights (*J*<sup>12</sup> + *J*21) under a 40-Hz oscillation.

et al., 2006; Tang et al., 2008; Tyler et al., 2012). The philosophy of the MaxEnt approach is not to assume anything other than what we know from the data. For example, if firing rate and pairwise correlation are the only information we have, the distribution with maximum entropy is given as the Boltzmann distribution,

$$P^{(2)}\left\{\mathbf{x}\right\} = \frac{1}{Z} \exp\left(\sum\_{i} h\_i' \mathbf{x}\_i + \sum\_{i$$

where *h <sup>i</sup>* is a bias term for the neuron *<sup>i</sup>*, *<sup>J</sup> ij* is the symmetric coupling strength between neurons *i* and *j*, and the partition function *Z* is given by,

$$Z = \sum\_{\{\mathbf{x}\}} \exp\left(\sum\_{i} h'\_i \mathbf{x}\_i + \sum\_{i$$

We see that the MaxEnt is equivalent to the IG with the 2nd-order LLM,

$$\log p\_{\mathbf{x}\_1 \mathbf{x}\_2 \ast \dots \ast} = \sum\_i \theta\_i^{(2,N)} \mathbf{x}\_i + \sum\_{i < j} \theta\_{ij}^{(2,N)} \mathbf{x}\_i \mathbf{x}\_j - \psi(\mathbf{e})^{(2,N)},\tag{16}$$

where the relationship between the parameters are given as,

$$\theta\_i^{(2,\,\,N)} = h'\_i,\\ \theta\_{i\bar{j}}^{(2,\,\,N)} = J'\_{i\bar{\jmath}}, \,\psi(\mathbf{\theta})^{(2,\,\,N)} = \log Z. \tag{17}$$

As was discussed in Tatsuno and Okada (2004) and in Tatsuno et al. (2009), it is possible to relate these IG measures, θ *(*2*, N) <sup>i</sup>* and θ *(*2*, N) ij* , to the network structure even for a network with asymmetric connections (Equation 5). However, under the influence of correlated inputs, we have also shown that the relationship in Equation 5 broke down, and that it was necessary to use the IG measures with the higher-order LLM such as the 4th-order (Equation 6) (Nie and Tatsuno, 2012). In other words, it was necessary to take into account neural activity of two additional neurons to estimate the direct neural interaction between neuron *i* and *j*. In summary, we see that the MaxEnt approach and the IG method are closely related. In addition, we also see that the MaxEnt can be considered a part of the IG method that provides a more general analysis framework for the space of the probability distributions.

In this study, we used the synchronous neural activity for estimating the direct neural interaction as the form of *(Jij* + *Jji)*. However, in the real learning processes such as sequential learning, it is possible that synaptic modification occurs differently for each direction; e.g., *Jij* increases, while *Jji* decreases. The proposed method is not able to estimate the directed synaptic change. As one possible remedy for this difficulty, calculation of the pairwise-IG measure using the time-lagged spiking activity between neurons was suggested (Tatsuno and Okada, 2004; Nie and Tatsuno, 2012). Another limitation of the present study is not including the effect of delay; e.g., axonal conduction delay or synaptic transmission delay. It is possible that these delays dramatically change the firing patterns as well as increase a variety of coexisting patterns (Izhikevich, 2006). Little is known about the relationship between the IG measures and direct neural interactions with conduction delay. In addition, it has not been clear how IG measures with more neuronal interactions such as triplewise-IG measures θ *(k, N) ijk* or quadruple-IG measures θ *(k, N) ijkl* behave under oscillatory conditions. It would be interesting to extend the current study to include more neuronal interactions.

Despite these limitations, the IG method is one of the most promising statistical tools for spike train analysis (Amari, 2001; Nakahara and Amari, 2002). Its direct relationship with the network parameters would provide useful information for the estimation of structural changes (Tatsuno and Okada, 2004; Tatsuno et al., 2009; Nie and Tatsuno, 2012). We hope that an advancement of novel analysis methods including IG will lead to a break-through finding in neuroscience.

## **ACKNOWLEDGMENTS**

We are grateful to Karim Ali for his support with the numerical simulations and to Amanda Mauthe-Kaddoura for proofreading the manuscript. This work was supported by Alberta Innovates Technology Futures (SCH001), the National Science Foundation (CRCNS-1010172), and Alberta Innovates Health Solutions (Polaris Award).

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 09 November 2013; accepted: 04 February 2014; published online: 24 February 2014.*

*Citation: Nie Y, Fellous J-M and Tatsuno M (2014) Information-geometric measures estimate neural interactions during oscillatory brain states. Front. Neural Circuits 8:11. doi: 10.3389/fncir.2014.00011*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Nie, Fellous and Tatsuno. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# A spiking neural network model of self-organized pattern recognition in the early mammalian olfactory system

## *Bernhard A. Kaplan1,2\* and Anders Lansner 1,2,3*

*<sup>1</sup> Department of Computational Biology, School of Computer Science and Communication, Royal Institute of Technology, Stockholm, Sweden*

*<sup>2</sup> Stockholm Brain Institute, Karolinska Institute, Stockholm, Sweden*

*<sup>3</sup> Department of Numerical Analysis and Computer Science, Stockholm University, Stockholm, Sweden*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Donald A. Wilson, New York University School of Medicine, USA Leandro M. Alonso, The Rockefeller University, USA*

#### *\*Correspondence:*

*Bernhard A. Kaplan, Department of Computational Biology, School of Computer Science and Communication, Royal Institute of Technology, KTH, SE-100 44 Stockholm, Sweden e-mail: bkaplan@kth.se*

Olfactory sensory information passes through several processing stages before an odor percept emerges. The question how the olfactory system learns to create odor representations linking those different levels and how it learns to connect and discriminate between them is largely unresolved. We present a large-scale network model with single and multi-compartmental Hodgkin–Huxley type model neurons representing olfactory receptor neurons (ORNs) in the epithelium, periglomerular cells, mitral/tufted cells and granule cells in the olfactory bulb (OB), and three types of cortical cells in the piriform cortex (PC). Odor patterns are calculated based on affinities between ORNs and odor stimuli derived from physico-chemical descriptors of behaviorally relevant real-world odorants. The properties of ORNs were tuned to show saturated response curves with increasing concentration as seen in experiments. On the level of the OB we explored the possibility of using a fuzzy concentration interval code, which was implemented through dendro-dendritic inhibition leading to winner-take-all like dynamics between mitral/tufted cells belonging to the same glomerulus. The connectivity from mitral/tufted cells to PC neurons was self-organized from a mutual information measure and by using a competitive Hebbian–Bayesian learning algorithm based on the response patterns of mitral/tufted cells to different odors yielding a distributed feed-forward projection to the PC. The PC was implemented as a modular attractor network with a recurrent connectivity that was likewise organized through Hebbian–Bayesian learning. We demonstrate the functionality of the model in a one-sniff-learning and recognition task on a set of 50 odorants. Furthermore, we study its robustness against noise on the receptor level and its ability to perform concentration invariant odor recognition. Moreover, we investigate the pattern completion capabilities of the system and rivalry dynamics for odor mixtures.

**Keywords: pattern recognition, olfactory bulb, piriform cortex, large-scale neuromorphic systems, spiking neural network, BCPNN, concentration invariance, pattern rivalry**

## **1. INTRODUCTION**

The major task of the olfactory system is to perform recognition of odors which is essential for survival by identifying edibility or danger. An odor evokes spatio-temporal patterns of activity in different stages of the olfactory hierarchy. The crucial mechanisms involved in odor object recognition are widely unknown, which is mainly due to the complexity of interactions and the transformations of information occurring between the different stages. In order to study the mechanisms embedded in the olfactory system, a system-level approach is required, comprising the three major levels of the early olfactory hierarchy including the epithelium, where the stimulus enters the nervous system, the olfactory bulb (OB) where the first transformation happens, and the piriform cortex (PC) which integrates and stores the information relevant for odor recognition (Gottfried, 2010; Wilson and Sullivan, 2011), and decision making (Gire et al., 2013). The OB and the PC and the connectivity between the two are crucial components for solving pattern recognition tasks, however experiments are only beginning to shed light on the possible connectivity principles. Neurons in the PC receive convergent synaptic input from different glomeruli (Apicella et al., 2010), but the question as to which principles underlie the connectivity between OB and PC is not yet resolved.

In this study we try to bridge the gap between the biophysics seen from a detailed perspective and the organization principles on a system level. Here, we present a model which is able to recognize artificial odor patterns in a self-organized manner using a Hebbian–Bayesian learning rule and ideas inspired from machine learning implemented on a biophysically detailed substrate. We will first embed our study in the context of existing literature, before we will explain the goals and hypotheses of our study.

## **1.1. CONTEXT AND OVERVIEW OF EXISTING PRIMARY LITERATURE**

The olfactory system has long been a model system to study memory formation (Haberly and Bower, 1989; Brennan et al.,

**Abbreviations:** OR, olfactory receptor; ORN, olfactory receptor neuron; OB, olfactory bulb; MT, mitral or tufted cell; PG, periglomerular cell; PC, piriform cortex; PYR, pyramidal neurons; AMPA, α-amino-3-hydroxy-5-methyl-4 isoxazolepropionic acid; NMDA, N-methyl-D-aspartate; GABA, γ-aminobutyric acid; WTA, winner-take-all operation; MC, minicolumn; HC, hypercolumn; VQ, vector-quantization; MDS, multi-dimensional scaling.

1990), object recognition (Davis and Eichenbaum, 1991) and pattern completion (Barnes et al., 2008). Computational modeling of the olfactory system began with the work by Rall et al. (1966) and continued to complement experimental research by testing hypotheses under controlled conditions and by connecting behavior with the underlying mechanisms.

Many studies focus on a single component of the lower levels of olfactory processing hierarchy, e.g. the OR responses (Hopfield, 1999), the epithelium (Simões-de Souza and Roque, 2004b; Sandström et al., 2009a), the OB or subparts thereof (Anton et al., 1991; Davison et al., 2003; Sandström et al., 2007; Brea et al., 2009; Linster and Cleland, 2009; Li and Cleland, 2013; Yu et al., 2013). There have only been few studies that attempt to model multiple parts of the olfactory pathways, for example, the study by Simões-de Souza and Roque (2004a) combines epithelium and OB. Modeling work on the PC can have a high level of detail (Wilson and Bower, 1992; Vanier, 2001) and describes the PC as a content-addressable memory system that is optimized for storing synaptic representations of odors through Hebbian learning (Barkai et al., 1994), yet often lacks a fair representation of the lower parts of the sensory pathway and the interactions in between. On the intermediate scale, Freeman's K-sets (Freeman and Erwin, 2008) have been used to model pattern recognition with chaotic dynamics (Yao and Freeman, 1990; Li et al., 2005), but this approach does not explain how connectivity emerges and misses lower parts as well. More recently, computational studies connect function with self-organization mechanisms and emergent connectivity in the OB (Migliore et al., 2007; Linster and Cleland, 2010; Migliore et al., 2010). The model by Li and Hertz (2000) involves both OB and PC and is based on rather abstract, oscillatory units and recognition works on the basis of temporal characteristics, which is argued for by other studies as well (Hopfield, 1991, 1995; Margrie and Schaefer, 2003; Schaefer and Margrie, 2012; Brody and Hopfield, 2003). Whether the temporal coding is crucial for recognition is up for debate and we will come back to this question in the discussion. Linster et al. (2009) presents a small scale model comprising simple models of olfactory receptor neurons (ORNs), MT, PG, granule and PYR cells to study response habituation effects based on synaptic adaptation and potentiation in PC for single odor patterns. The study offers a comparison with behavioral data, but lacks the generic pattern recognition capabilities which we are addressing in this study.

There exist a number of studies on classification and recognition in the insect olfactory system (Huerta et al., 2004; Nowotny et al., 2005; Schmuker and Schneider, 2007; Schmuker et al., 2011). The study by Nowotny et al. (2005) uses an approach similar to ours, by transforming the combinatorial code in the antennal lobe (the equivalent of the OB in insects) into a higher dimensional space and applying Hebbian learning with mutual inhibition in the mushroom body (the PC equivalent in insects). An improved understanding of the olfactory system through modeling also lead to substantial advances in machine olfaction (Gutierrez-Osuna, 2002; Pearce et al., 2006; Raman et al., 2011).

## **1.2. PURPOSE OF THIS STUDY**

As we have outlined above, most existing models either use an abstract description with components far away from the biological substrate or have a high level of detail but lack other relevant system components leading to an incomplete picture of the olfactory system. Furthermore, the role of the different components from a computational perspective is still under debate, for example whether most of the transformations involved in pattern recognition take place in the OB or rather in the PC and how the interactions between the two is organized is unknown. What is lacking is a generic computational model capable of behavioral relevant functions like pattern recognition which involves the ability to self-organize and which is able to run in a biophysically plausible setting. In this work, we are trying to make a first step toward filling this gap by presenting—to the best of our knowledge—the first functional biophysical model of the olfactory system integrating the first three stages on a high level of detail.

The goal of this paper is threefold. First, we propose a generic approach for neural information processing that generates the connectivity from the OB to the PC and within the PC by means of self-organization and competitive learning. More generally, we model the activity dependent formation of connectivity between sensory layers and cortical memory systems as well as the recurrent long-range intra-cortical connectivity. Second, we show that a biophysically plausible implementation of this approach in the context of olfaction is feasible. Third, we prove the functionality of our concept and the spiking implementation in a number of pattern recognition tasks and study the system's behavior therein.

Our model is based on an abstract generic model for cortical information processing (Lansner et al., 2009; Persaud et al., 2013) which offers a recursively applicable algorithm to generate functional connectivity within and between processing stages and is realized as a multi-layer spiking neural network. Furthermore, we explore the possibility of an OB model making use of a concentration interval code in the mitral (MT) cell layer to serve as input to an attractor network model of the PC, and we investigate the behavior of the system in the five following tasks. First, we show the functionality in a pattern recognition task for 50 artificial odor patterns. Second, we test the ability of the system to recognize odors at different concentrations and propose a solution to the concentration invariance problem (Cleland et al., 2011) in olfaction. In the third task we challenge the system with noisy patterns mimicking impure odors. The fourth task shows the system's pattern completion capabilities by testing with incomplete patterns of different sparsity, and the fifth task is to distinguish between different mixtures of learned patterns.

## **1.3. MAIN HYPOTHESES**

We will now explain the main computational hypotheses on which the model is based, name important experimental findings supporting these and explain the implementation in section 2. Hereby we move the olfactory pathway along from the receptor level to the cortex.

## *1.3.1. Activity dependent connectivity from epithelium to bulb*

Each ORN expresses only one olfactory receptor (OR) (Buck and Axel, 1991), and each odorant activates a broad range of ORNs involving different ORs (Firestein, 2001). ORNs expressing the same OR (in the following named an ORN-family) have different sensitivities to the same odorant and show dose-response curves with activation thresholds and saturation points covering a broad dynamic range (Grosmaitre et al., 2006). An ORN-family projects to only one or two glomeruli (Vassar et al., 1994; Mombaerts et al., 1996). We extend these principles by adding our first hypothesis which affects the connectivity from ORNs to OB. We assume that axons from one ORN-family undergo an activitydependent sorting process when connecting to the dendritic trees of MT and PG neurons in the same glomerulus. This assumption extends the chemoaffinity hypothesis (Sperry, 1963) and applies the existing idea that activity and experience is involved in the axon growth process (Gill and Pearce, 2003; Tozaki et al., 2004; Kerr and Belluscio, 2006; Imai and Sakano, 2008; Sakano, 2010; Mori and Sakano, 2011) to the local axon sorting process (Zhao and Reed, 2001; Serizawa et al., 2006; Takeuchi et al., 2010) and thereby shapes the response properties of MT cells. This activity-dependent sorting activates MT belonging to the same glomerulus as a function of the average firing rate of the convergent ORNs, an idea picked up earlier by Anton et al. (1991); Cleland and Linster (2005). A previous study has shown that activity dependent sorting can lead to map formation in the OB which could have perceptual advantages (Auffarth et al., 2011). We are using axon sorting mechanisms that are possibly active within an ORN-family to implement our second hypothesis, a concentration interval code in the OB.

## *1.3.2. Concentration coding*

The concentration interval coding hypothesis assumes that each MT cell has one preferred concentration of an odor to which it responds maximally (Sandström et al., 2009b) and we will explain in detail in section 2 how these two hypotheses are used to implement a fuzzy concentration interval code in the OB. This hypothesis is inspired by the idea of neuronal tuning which assumes that neuronal responses are tuned to specific inputs through experience and rules for optimally covering the stimulus space have been studied (Zhang and Sejnowski, 1999; Brown and Bäcker, 2006). Cells coding for an interval of a certain stimulus dimension have been found in many sensory systems. For example, just to name a few examples, in vision there exist interval codes for orientation (Hubel and Wiesel, 1962; Schoups et al., 2001; Li et al., 2012) and direction (Albright, 1984), in the auditory system for pitch (Bendor and Wang, 2005), and position, direction, speed (Poirier et al., 1997), in hippocampus place or grid cells show strong responses to their preferred position (Moser et al., 2008), and in the motor system neurons are tuned to end positions of movements and other parameters (Aflalo and Graziano, 2006).

The interval coding strategy can be used to encode variables in a probabilistic way, as tuning curves of individual neurons overlap and the value encoded by a population of units can be decoded in a Bayesian optimal sense (Ma et al., 2006). This "fuzzy" coding is related to the concept of Gaussian Mixture Models (GMMs), a generic probabilistic model capable of representing arbitrary densities which makes this coding suitable for unsupervised classification algorithms. GMMs are well-established for coding in learning and classification systems for complex stimuli, e.g. speaker recognition (Reynolds et al., 2000), person identification (Stylianou et al., 2005), and image classification (Permuter et al., 2003).

One of the canonical computations believed to be performed by lower sensory areas is decorrelation (Cleland, 2010; Linster and Cleland, 2010), which we assume to be performed in the concentration domain by MT cells receiving input from the same glomerulus (so called sister MT cells). We thereby assume that cells connected to one glomerulus operate as functional modules making use of the columnar organization as revealed by a viral tracer study (Willhite et al., 2006). In this study, we apply this idea to encode odorant concentration in a fuzzy manner by MT cells and explore the possibility of such a code in a functional model for self-organized pattern learning. The advantage of this coding scheme is that odor identity and concentration can be represented at the same time without relying on precise spike timings.

Whether mitral cells do exhibit a concentration interval code or not is not fully resolved, due to contrary indications from different experiments and the complex temporal dynamics of alternating excitation and inhibition (Chaput et al., 1992) and their sensitivity to concentration (Chalansonnet and Chaput, 1998). Experiments by Tan et al. (2010) show that at least in some glomeruli mitral cells do not exhibit a concentration interval code as we propose here. Other studies, in contrast, report nonmonotonous firing rates for increasing concentrations in mice (Reinken and Schmidt, 1986), rats (Wellis et al., 1989), and hamsters (Meredith, 1986). The study by Egana et al. (2005) suggests that sister MT cells often exhibit very different response characteristics in terms of increase in firing rate due to odor exposure and their respiratory-related temporal patterns. Likewise, it has been shown that sister MT cells show non-redundant temporal behavior (Dhawale et al., 2010) and it has been suggested that the reason for that might be found on the circuit level. Bozza et al. (2004) used an imaging technique showing the synaptic vesicle fusion in ORNs targeting glomeruli and found different concentration-response relationships for different glomeruli. The most sensitive glomeruli to 2-hexanone showed saturated response curves at an intermediate concentration (see Figure 5E in Bozza et al., 2004), thus providing non-monotonous input into some glomeruli which could possibly explain the different experimental indications mentioned above. The response characteristics of bulbar neurons have been studied mostly in anesthetized animals, but recent experiments by Kato et al. (2012) show that mitral and granule cell react differently toward anesthesia, and odor representations are different in awake and anesthetized states. Hence, MT cell odor responses might be more narrowly tuned in unanesthetized animals and strongly depend on the behavioral context (Shipley et al., 2008). Here, we explore the possibility of this hypothetical coding scheme in a biophysically detailed model and explore the capability for concentration coding in a functional context from a systems level perspective. An alternative idea, which is not mutually exclusive to the concentration interval coding hypothesis, is that MT cells code odor concentration and odor identity by the spike latency within a sniff (Margrie and Schaefer, 2003; Schaefer and Margrie, 2012). We will discuss the spike latency coding hypothesis in section 4 in the context of our results.

## *1.3.3. Rate-based hebbian learning from OB to PC*

Our next hypotheses concern the mechanisms underlying projections from OB to PC. First, we assume that learning is ratebased and hence primarily taking place on a coarser time-scale than e.g. spike-timing dependent plasticity usually modeled on a timescale of milliseconds, but use the response of the OB to odorant patterns over one long sniff (modeled as one long inhalation leading to a stimulus of ∼400 ms and simulated for 1600 ms). Furthermore, we do not regard learning mechanisms active within the OB, e.g. MT responses changing with exposure (Fletcher and Wilson, 2003), generation of granule cells (Mandairon et al., 2006) and disregard the dynamics of the odor afterimage (Patterson et al., 2013). We assume that the main component in olfactory learning is how projections from OB to PC and within PC are created and that pattern recognition is based on the activity evoked through these afferent fibers terminating in the PC and the recurrent activity within the PC. In order to organize the connectivity from OB to PC we use the mutual information of normalized individual mitral cells responses and a competitive correlation-based learning mechanism, which is used as input to the Bayesian Confidence Propagation Neural Network (BCPNN) algorithm (Lansner and Ekeberg, 1989; Lansner et al., 2009). Similar implementations thereof have been applied in various setups (Sandberg et al., 2002; Lansner et al., 2003, 2009; Auffarth et al., 2011; Persaud et al., 2013).

Oscillations are a prominent phenomenon in the olfactory system. In this study, we do not study oscillations, as they do not play a crucial role within our framework for the pattern recognition tasks we consider and, because according to our hypothesis, learning takes place on larger time-scales than oscillations do occur. Hence, oscillatory signatures have not been analyzed in this study, but can be found in modular network of very similar type as ours as studied by Lundqvist et al. (2010, 2011).

## *1.3.4. Olfactory cortex as an attractor memory system*

Another important component of the olfactory memory system is the recurrent connectivity within the PC. The association fiber network prominent in PC is regarded as the substrate for a content addressable and distributed memory system (see Haberly, 2001; Wilson et al., 2006; Wilson and Sullivan, 2011 for reviews). Our cortex model is inspired by the idea that the olfactory cortex acts like other associative cortices in the sense that it learns to create and distinguish sparse and distributed representations of odor patterns, and is able to associate simpler odor patterns with each other to form abstract complex odor objects (Haberly, 2001; Wilson and Sullivan, 2011). Attractor networks have been proven to be an effective model to explain memory formation and retrieval (Amit, 1992; Hasselmo and McClelland, 1999) and other brain functions (see e.g. Rolls, 2008) and are one approach to implement higher cognitive functions like holistic perception in biophysically detailed simulations (Lansner, 2009). Inspired by previous models, we see the cortex as a crucial part in the pattern classification process and derive the projections from OB to OC and the recurrent cortical connectivity with the help of the BCPNN algorithm (Fransén and Lansner, 1998; Sandberg et al., 2002; Lansner et al., 2009).

## **1.4. PRINCIPLE APPROACH**

This study explores the possibility to apply a generic, recursive approach to a self-organized pattern recognition system on a biophysical substrate resembling the mammalian olfactory system. Despite the fact that the PC is a three-layered paleocortex, we assume the PC to work in a similar way as other sensory and association cortices with regard to memory formation. In the model design and choice of parameters, we put emphasis on functional implications and on a qualitative match to the biological substrate rather than an accurate quantitative agreement between simulations and experimental data. Thus, our approach should not be seen as realistic in all detail, but rather be regarded as explorative and plausible toward bridging the gap between system-level computations and biophysical detail. We use numerical simulations of single and multi-compartment neuron models described by the Hodgkin–Huxley formalism and apply rate-based learning rules to derive functional connectivity to support pattern recognition. We used this family of neuron models, for several reasons. First, there already exists a number of neuron model implementations for the most prominent bulbar and cortical cell types that are relevant for our approach and ready to use with the NEURON simulator (Hines and Carnevale, 1997). Second, neuron models that were not implemented at the beginning of the studies could be adapted from existing neuron models (see **Table 1** for a brief overview of neuron types). Third, network models in NEURON are easily parallelizable and hence can be extended to larger scales and offer the possibility for future refinements and extensions, e.g. if more biophysical realism is desired.

## **2. MATERIALS AND METHODS**

## **2.1. NEURON AND SYNAPSE MODELS AND CHOICE OF PARAMETERS**

In order to model a multi-layered network with a reasonable level of detail, one has to fill several gaps by making assumptions because many aspects and parameters of the real system are not known. We have tried to use realistic parameters wherever possible, but as the primary goal of this paper is to present a holistic architecture implementing a high-level task with a spiking neural network, we had to reduce this goal at several points to achieve the desired function.

## *2.1.1. Neuron types*

For all simulations we use neuron models described by the Hodgkin–Huxley formalism, an overview of the used neuron models is shown in **Table 1**. Our principle approach was to use existent neuron models without modification if possible and to adapt existing neuron models if the desired function required changes. For ORNs we have extended an existing singlecompartmental neuron model described in Pospischil et al. (2008) by adding a time-dependent input current to model the odor stimulus, a low- and a high-threshold Calcium current and a Calcium activated Potassium channel to provide adaptation mechanisms to guarantee saturating dose-response curves. The ORN channel conductances have been tuned so that the model shows plausible dose response curves for a family or ORNs, i.e. different response onsets depending on the sensitivity and saturating output rates for high stimulus concentrations.



In the OB, we use three multi-compartmental cell types: MT cells, granule cells and PG neurons. As in other studies we model mitral and external tufted cells as one neuron type, as our focus lies in the projection from both neuron types to the cortex. Neuron models for MT and granule cells are identical to those in the study by Davison et al. (2003). MT cells have compartments for glomerular dendrite, primary dendrite, soma and secondary dendrite connecting to granule cells. Granule cells have compartments for their soma, peripheral and deep dendrites. In the absence of a neuron model for PG cells at the beginning of our study, we used the same neuron model for PG as for granule cells using their peripheral dendrite for interactions with ORNs and MT cells and dendrodendritic interactions to convey PG output to MT cells.

The PC model contains one excitatory adapting neuron type (PYR), a fast-spiking inhibitory interneuron [in the following called basket cell (Ekstrand et al., 2001)] and a regular spiking non-pyramidal (RSNP) neuron (all adapted from Pospischil et al. (2008). The BCPNN algorithm as described later gives bias values for each cortical module, which can be interpreted as intrinsic excitability implemented as an inhibitory A-type Potassium current (Bergel, 2010) added to RSNP and PYR neurons.

## *2.1.2. Synapse models*

Excitatory synapses are realized through exponential currents mediated by AMPA receptors with a time constant of 10 ms and NMDA receptors implemented as in Davison et al. (2003), which models a Magnesium block and operates at a longer time constant (≈150 ms). Inhibitory synapses only have one time scale and are modeled as exponential currents mediated by GABA receptors with a time constant of 20 ms.

## *2.1.3. Choice of parameters*

One set of parameters determines the network size that needs to be adapted to the number of patterns the system is trained with. These are the number of glomeruli (equal to the number of ORs), the number of HCs and the number of MCs per HC. We have not explored the number of ORs, HCs and MCs required to successfully learn a given number of patterns, because this would be out of the scope of this paper and should be studied with a less detailed model.

The BCPNN algorithm yields the connectivity between the OB and OC and within the OC as "abstract weights". Hence, these parameters are estimated by BCPNN, whereas the translation into biophysical weights is done with the help of free scaling parameters that were chosen to yield biophysically plausible synaptic conductance values in the order of a few nS. Furthermore, there exists a large set of model parameters (on the order of 70) controlling various aspects, like the individual cell models (cell morphology, ion channel conductances, background noise), connectivity parameters from ORNs to OB and within the OB. A subpart of these have been tuned by hand to achieve the desired behavior.

Because of the complexity of this model and the immense number of parameters involved, we omit a list of parameters here, but refer to the existing literature and the simulation code, which is available on request. As already mentioned, the focus of this study is to implement a functional model operating on multiple stages and not to build a precisely matched counterpart of the biological substrate. Hence, we decided to choose parameters to fulfill functional requirements as this is our primary goal. In combination with the small size of the networks compared to real systems this might have lead to unrealistic values in some cases. Furthermore, the vast amount of parameters would make a parameter sensitivity analysis extremely complex and computationally intensive and as the parameter space is very high-dimensional, it is likely that many different operating regimes could be found.

Simulations were performed with the NEURON simulator (Hines and Carnevale, 1997) on a Cray XE6 system using 96– 120 cores. For setting up simulation preparation, connectivity and analysis of results we used python with the modules numpy (Oliphant, 2007), scipy (Jones et al., 2001–2013) and orange (Demšar et al., 2013). Figures and data visualization were done using matplotlib (Hunter, 2007) and Inkscape (Andler et al., 2004–2014). Cell parameters were identical for all neurons of the same type. To account for natural variability all weights were randomly modified by 10%, the initial membrane voltage was drawn from a normal distribution with mean −70 mV, and standard deviation 5 mV, and each neuron (except readout neurons) received Poisson spike trains as background noise to model both network effects and stochastic opening and closing of ion channels.

## **2.2. ODOR INPUT PATTERNS**

In order to decide how strong each family of ORNs (each expressing one OR and targeting only one glomerulus) gets activated by an artificial odor pattern, we derive a distribution of odorant-OR affinities based on real-world data. Haddad et al. (2008) presented an optimized set of 32 physico-chemical descriptors which could account for variability in neural responses of ORNs and glomeruli in different species for different sets of odorants. This gives a 32-dimensional space, in which the 447 odorants they provide can be described. In short, we place virtual ORs in this 32-dimensional space as centroids resulting from clustering the odorants, calculate the Euclidean distance between the virtual ORs and real-world odorants, and based on this distance we obtain the affinity between the OR-odorant pair. This approach is inspired by the odotope theory Shepherd (1987); Mori (1995), which suggest that the molecular shape of an odorant and the molecular preference of an OR determine the OR response. This idea implicates that spatial proximity of ORs in this multidimensional space implies similar molecular receptive ranges of the ORs. This idea is currently debated because not only functional groups of odor molecules, but also the vibrational energy spectrum of molecules does play a role in determining OR responses (Franco et al., 2011; Gabler et al., 2013). Nevertheless, for simplicity we chose the odotope theory as a guiding principle to generate artificial odor patterns. It should be emphasized that the pattern recognition capability of our system is not constrained to this way of generating artificial odor patterns. Despite the fact that our virtual ORs lack a direct biological correspondence, the presented approach of interpreting ORs as centroids after clustering the odor space seems plausible, assuming that ORs could have specialized to code for parts of the olfactory world. The study by Geisler and Diehl (2002) suggests that perceptual systems are designed for encoding natural stimuli in an optimal way. Nei et al. (2008) suggest that variations in chemosensory receptor gene repertoires among species can be explained to a large extent by the adaptation of organisms to different environments. In the following, we describe the details of our approach inspired by these ideas.

The ORs were chosen to be the centroids of clusters in the odor space computed by the k-means clustering algorithm (Hartigan and Wong, 1979). As the distances between ORs and odorants are based on the results of the clustering procedure and hence depend strongly on the number of ORs to be put in the odor space and the random initial conditions, we have pooled distance distributions for different numbers of ORs over 100 trials. The motivation behind this approach is to get a picture of the real-world odor space and to derive a generic way to generate arbitrary numbers of virtual odor patterns that share the same characteristics in terms of odorant-OR distances as real odors could have based on the odotope idea described above.

For each number of ORs (centroids) we fitted a trimodal normal distribution to the obtained distance distributions, as it resembled the distribution reasonably well (see **Figure 2**) and observed that the fit parameters did not change qualitatively for distributions when 20–66 centroids were used to cluster the odor space. For more than 66 centroids, the k-means algorithm could often not converge because of too many centroids populating the odorant space and leaving centroids without odorants in their proximity. Hence, we used the averaged fit parameters of the distance distribution for 20–66 centroids to obtain a method to draw distances between artificial odorants and ORs, which gives us an average distance distribution *D* between real world odorants and virtual ORs. The activation pattern of an odorant was generated by first randomly choosing *n*activated ORs that do show a response given the system is exposed to that odorant in a noise-free environment (how noisy patterns are generated will be explained below). For each pattern we chose a random integer *n*activated to be between 30% and 50% of all receptors, as this is in the range of what has been reported experimentally (Ma et al., 2012). For each activated odorant *i* and OR *j* a distance *di*,*<sup>j</sup>* was sampled from *D* and transformed into an affinity *Ai*,*<sup>j</sup>* by applying this transformation function:

$$\mathcal{A}\_{i,\ j} = \exp\left(-\frac{d\_{i,\ j}^2}{(E[\mathcal{D}])^2}\right) \tag{1}$$

where *<sup>E</sup>*[*D*] = <sup>7</sup>.7 is the expected value for distances sampled from the distribution *D* as shown in **Figures 2A,B**. We chose this transformation function in order to have a strong influence of the distance between odorant and OR in the space determined by Haddad et al. (2008) and to obtain a population of affinity values covering the whole range between 0 and 1 even for small sample sizes of odorant-receptor pairs as in our model simulations. An example set of 50 patterns for 40 ORs is shown in **Figure 4A**.

The perception in noisy environments was modeled by modifying each element in the affinity matrix *A* resembling an odorant-receptor pair to *<sup>A</sup>* :

$$\mathcal{A}\_{i,j}^{'} = \max(0, \min(1, \mathcal{A}\_{i,j} + \text{rnd}(-\sigma, \sigma)))\tag{2}$$

where rnd(−σ, σ) stands for a random number uniformly distributed between −σ and σ, σ stands for the strength of noise. By this means affinities are constrained to the interval between 0 and 1. The idea behind this approach is that in noisy environments, other odors unrelated to the original odor pattern might be present which is represented by having new non-zero elements in *A*, whereas existing OR responses might be suppressed at the same time. For simplicity we have not considered the partly competitive and non-linear interactions between odorants and receptors (Rospars et al., 2008) when a receptor could react to several present odorants.

## **2.3. THE OLFACTORY EPITHELIUM**

The epithelium has been modeled as a population of ORNs without taking the spatial dimension into account. For simplicity, ORNs have been modeled as single-compartment Hodgkin– Huxley neurons with the goal to have a variety of saturating dose-response curves, similarly to experimental studies (see e.g. Rospars et al., 2000, 2003, 2008). An odorant stimulus is modeled as an input current as shown in **Figure 2**, either as a single puff stimulating ORNs for ∼450 ms or as a sequence of four briefer sniffs with a frequency of ∼4 Hz. The maximum input current into one ORN is determined by the product between the affinity of the OR expressed by the ORN family to the respective odor and by the maximum excitatory conductance determined by the physiology of cell, which could be the cell size, number of expressed ion channels or the number of receptors on the cilium of the cell. This product of affinity between an OR and an odor, which influences the individual ORN response, can be seen as the fraction of activated receptors or opened ion channels exciting the ORN. This fraction of activated receptors (OAV for odor activity value) can be translated into a concentration *c* or dose (without considering physical units) by applying *c* = OAV/(1 − OAV). Consequently, affinity values (OAV) values are constrained to be between 0 and 1.

We assume here that ORNs expressing the same OR do not have a single value for the maximum conductance, but rather a distribution based on the profound differences in response kinetics as seen in the experimental studies (Rospars et al., 2003; Grosmaitre et al., 2006) and described by statistical population models (Sandström et al., 2009a; Grémiaux et al., 2012). **Figure 2C** shows the responses of two example receptor neurons to excitatory stimuli. In the simulations presented throughout the study, our model contains 40 populations, each expressing a different OR and comprising 800 neurons that project onto one glomerulus but could be scaled up to include more ORs or more ORNs.

## **2.4. THE OLFACTORY BULB**

We will first describe the pathways in the OB model and explain the connectivity from OE to OB afterwards. Our model of the OB is intended to include the most prominent processing pathways and several inter- and intraglomerular interactions. The leading idea behind the synaptic organization in our OB model is to implement the hypothesized concentration interval code by MT cells within one glomerular module. As a basis for this we assume a columnar organization spanning different layers of the OB as reported by Willhite et al. (2006). For this purpose, we implement a soft winner-take-all (WTA) circuit within one glomerular module with feed-forward excitation provided by ORNs through axo-dendritic synapses, serial and reciprocal dendro-dendritic synapses between MT and PG cells and reciprocal synapses between MT and granule cells. MT cells receive direct excitation from ORNs via AMPA and NMDA receptors (Ennis et al., 1996) on their glomerular compartment resembling fast and graded monosynaptic input (Najac et al., 2011). A part of the interneurons situated in the glomerular layer (≈20% of the PG cells) also receive direct input from ORNs (Shepherd and Greer, 1998; Hayar et al., 2004; Toida, 2008). Inspired by the differences in dendritic arborization of PG cells reported by Toida (2008) we have implemented four types of PG cells that differ in their synaptic organization. **Figure 1** shows a schematic of the connectivity within one glomerular module in the OB model described in the following. One type of PG cells (marked with PG\_S1 in **Figure 1**, in Toida (2008) they are called TH-ir or type 1 neurons, as they contain the dopamine-synthesizing enzyme tyrosine hydroxylase) gets direct input from ORNs and makes a serial inhibitory (or in physiological reports often called symmetrical) synapse to MT cells. The second type of PG neurons (still being an TH-ir neuron, marked with PG\_S2 in **Figure 1**) additionally receives dendrodendritic excitatory input from a nearby MT cell, but inhibitis *another* MT cell as reported by Toida (2008). The third type of PG neurons (PG\_R1, in Toida (2008) called type 2 neurons, CB-ir neurons as they contain calbindin-d28k, or CR-ir as they contain calretinin) lie deeper in the glomerular layer and show a different arborization pattern. These neurons form "typical" reciprocal dendro-dendritic synapses with MT cells and do not receive direct input from ORNs. The fourth type of PG neurons we implement PG\_R2 has in addition to reciprocal synapses with MT neurons also inhibitory connections to other MT cells. As a rough physiological constraint we have set the number of reciprocal synapses in the glomerular layer to be about 25% (according to Shepherd and Greer, 1998).

Arevian et al. (2007) reported that lateral inhibition between MT cells with correlated activity is enhanced. We interpret this behavior as another aspect of a WTA mechanism between MT cells and use the dendritic arborization patterns of PG cells as one mechanism to implement this. Another possible mechanism underlying this lateral inhibition is the prominent dendrodendritic inhibition between MT cells and granule cells. Granule cells make two types of reciprocal synapses, one with mitral cells from one glomerulus, the other type with MT cells from all glomeruli in the OB, hence providing interglomerular inhibition (Urban and Sakmann, 2002). As our interest lies in the function of the system, the synapse strengths have not been matched to experimental data, but have been tuned so that MT cells show the hypothesized concentration interval code within a glomerular module.

Several studies have pointed out the importance of autoreceptors in MT cells (Montague and Greer, 1999; Friedman and Strowbridge, 2000; Salin et al., 2001; Schoppa and Westbrook, 2002). We have implemented excitatory AMPA and NMDA autoreceptors on the primary dendrites and NMDA autoreceptors on the secondary dendrites of MT cells to facilitate the hypothesized WTA mechanism between MT cells through self-excitation. Despite the fact the PG cells do connect with other glomeruli, presumably via short-axon and external tufted cells we have not included this type of cells and connections here to not increase the complexity of the model even further as we wanted to explore the possibility of the concentration interval code via WTA mechanisms. Likewise, for the sake of simplicity, our OB model makes no assumptions about chemotopy in the layout of glomeruli, i.e. there is no spatial organization for glomeruli. With regard to cell populations, we have 8 mitral cells per glomerulus, 20 PG cells per MT cell, 100 ORNs per MT cell, and 200 granule cells per MT cell.

Results shown in the following are based on an OB model with 40 glomeruli, i.e. 32,000 ORNs, 320 MT cells, 6400 PG cells, and 32,000 granule cells.

## *2.4.1. Connectivity from epithelium to OB*

When connecting ORNs expressing one receptor that project to a single glomerulus, we follow the hypothesis that activity dependent axon guidance mechanisms are involved in order to create the concentration interval code in the MT population. For this purpose, we order ORNs within one family by their sensitivity and divide them into a number of different groups, each group exciting one target MT cell and inhibiting another MT cell receiving excitatory input from the next less sensitive ORN group. For example, the most sensitive ORNs respond to an odorant already at a low level of activation and activate their corresponding MT cell. The same MT cell receives inhibitory input from PG\_S1

currents based on the affinity between odorant and OR and on the ORN sensitivity. ORNs expressing the same OR make excitatory connections with PG and MT cells in one glomeruli. PG cells show different dendritic arborization patterns and interact with MT cells of the same glomerulus through serial synapses (blue line with dot) and reciprocal synapses (green). MT cells have AMPA and NMDA auto-receptors (shown in red) on their distal primary and secondary dendrites providing self-excitation. Granule cells connect with MT cells through reciprocal synapses. MT and granule cells interact across glomerular modules throughout the OB granule cell layer. MT cells have afferent projections to excitatory pyramidal (PYR) and inhibitory

from distinct glomeruli. The PC has a modular attractor memory structure with pre-wirde (non-plastic) connections from RSNP cells to PYR neurons in their respective minicolumn (MC), between PYR within one MC, from PYR to basket cells, between basket cells and feed-back inhibition from basket cells to PYR. The learned connectivity in PC includes connections from PYR to RSNP and PYR cells in other MC and vice versa providing long-range connectivity. Connections from PYR to readout neurons are learned as well. ONL, olfactory nerve layer; Glom, glomerular layer; EPL, external plexiform layer; MBL, mitral cell body layer; GL, granule cell layer. Colors represent odorants, ORN family, cell type or odor identity, respectively.

neurons that get activated by the next less sensitive group of ORNs and hence receives the equivalent of the difference of the two response curves from these two ORN groups (see **Figures 2**, **3** for clarification). Because of this difference in response curves exciting the MT cell, we achieve the hypothesized interval code. This effect is amplified by the inhibition each MT cell receives from PG\_S2 and PG\_R2 neurons (see **Figure 2**). The intra-glomerular inhibition provided by PG\_S2, PG\_R2 neurons and granule cells leads to an approximate normalization of MT activity, i.e. the output rate of a glomerulus stays approximately constant over a wide range of concentration (see **Figure 3**).

## **2.5. THE PIRIFORM CORTEX**

Guided by the hypothesis that the PC acts like an attractor network when learning and retrieving odor patterns, we implement the PC based on previous work as a modular attractor network (Lundqvist et al., 2006; Lansner, 2009; Lundqvist et al., 2010). Despite the fact that a modular structure based on stimulus preference comparable to orientation columns in V1, for example, has not been observed in olfactory cortices, we explore the possibility that a modular network structure as an organization principle could be involved in tasks like pattern recognition, completion and rivalry. The basic structure of our PC model consists of several computational modules [in the following called hypercolumns (HC)], each consisting of several minicolumns (MC) with 30 excitatory and 4 inhibitory cells respectively (see **Figure 1** for a schematic). This modular structure has been chosen for two main reasons. First, we wanted to reflect the BCPNN algorithm as closely as possible in a spiking network in order to achieve the desired computational capabilities through attractor dynamics with soft WTA-like inhibition. Second, the modular structure including recurrent inhibition through basket cells (as

**FIGURE 2 | (A)** Distribution of distances *D* between virtual ORs and real-world odorants in a high-dimensional physico-chemical descriptor space taken from Haddad et al. (2008). Distances are obtained by clustering the multidimensional odor space a with k-means clustering algorithm, treating the resulting centroids as virtual ORs and averaging the Euclidean distances between ORs and odors over 100 trials. The red solid line shows the fit of a superposition of three normal distributions (light green, yellow, black dotted lines) to the mean distance distribution averaged over multiple clustering trials with number of ORs ranging from 20 to 66. Before the fitting, the distance distribution has been normalized

by the number of ORs (centroids). The y-axis shows the normalized number of occurrence pooled over 100 trials, x-axis shows the Euclidean distance *d* in odorant space. **(B)** Affinity distribution from which affinities between odorants and receptor pairs are drawn. The y-axis shows the probability to draw an affinity given on the x-axis. The affinity distribution has been obtained by transforming the distance distribution with the given function. **(C)** Odor input (upper panel) and example membrane potentials (bottom) of an ORN to two different kind of stimuli, odor puff (in blue) and sniffing (black). The blue membrane trace in response to an odor puff is shifted by +10 mV for visibility.

described below) is required to balance excitation in the system and hence plays an important role in shaping the dynamics toward biologically plausible regimes.

Our PC model comprises three cell types that are modeled as single-compartment Hodgkin–Huxley neurons all taken from Pospischil et al. (2008). Excitatory pyramidal cells (PYR) receive input from MT cells belonging to different glomeruli (Apicella et al., 2010) and can be associated with seminlunar, superficial and deep pyramidal cells (see e.g. Bekkers and Suzuki, 2013 for a recent review of cells in the PC). Similarly to the model in Lundqvist et al. (2006), PYR cells connect to other PYR cells within the same MC with a probability of 25% and to basket cells in the same HC with a probability of 70%. Basket cells receive excitatory input from PYR cells only and connect to PYR cells in the same HC with a probability of 70% and hence provide strong feedback inhibition to PYR cells imposing a soft winner-take-all like competition among MCs belonging to the same HC. RSNP neurons receive excitation from MT cells and from PYR cells. RSNP cells project to PYR neurons belonging to the same MC with a probability of 70%. The results shown in this study are from simulations of 12 HCs with 30 MCs each, giving 10, 800 PYR, 1440 RSNP, and 2160 basket cells, as we have 6 basket cells per minicolumn.

### *2.5.1. Connectivity between OB and PC*

The connectivity from the OB to PC is derived based on the mutual information between MT cells and the BCPNN algorithm, similar to previous models (Johansson and Lansner, 2006; Lansner et al., 2009). Connections are not derived on a cell-tocell basis, but target units in the PC that are represented by MCs consisting of 30 neurons each. After the weights from MT cells to MCs have been computed they will be translated into cell-tocell connections as described in section 2.5.3. For this purpose we simulate the responses of the epithelium and OB for 1600 ms to *Np* = 50 different random artificial odor patterns and use the MT cell responses to calculate their mutual information.

First, the *N*MT mitral cell responses to the *Np* pattern presentations are transformed into probabilities of activation *pi*. This is done by normalizing the number of spikes *f <sup>k</sup> <sup>i</sup>* fired by mitral cell *i* during pattern *k* by dividing through the sum of spikes fired during all *Np* patterns:

$$f\_i^{k'} = \frac{f\_i^k}{\sum\_{k}^{N\_\rho} f\_i^k} \tag{3}$$

Furthermore, we apply a half-normalization to each glomerular unit, i.e. if the summed normalized activity during one pattern in one glomerulus is higher than one, it is normalized to one:

$$\xi\_i^k = \begin{cases} f\_i^{k'} / \sum\_i^q f\_i^{k'} & \text{if: } \sum\_i^q f\_i^{k'} > 1, \\ f\_i^{k'} & \text{otherwise} \end{cases} \tag{4}$$

The indices *i* and *q* stand for the MT cells belonging to one glomerulus. This half-normalization is applied because we interpret MT cells as probabilistic sensors and the normalized activities within one glomerulus as probabilities of measuring the presence of a certain feature. As MT cells code for concentration this would correspond to the probability of sensing an odorant at the corresponding concentration. This is why the normalized activities must not sum up to a value above one.

Based on the normalized activation probabilities *pi* and probabilities for joint activation *pi*,*<sup>j</sup>* are obtained by:

$$p\_i = \frac{\sum\_{k}^{N\_p} \xi\_i^k}{N\_p} \tag{5}$$

**FIGURE 3 | Top:** ORN response curves from one family of ORNs expressing the same OR. Colors indicate groups within this ORN family which project to different target MT and PG cells. Each group contains 100 ORNs. Output rates were measured over one simulation run of 1600 ms, including the stimulation from one long sniff of ∼400 ms **Bottom:** Mitral cell response curve averaged over 10 trials, error bars indicate the standard deviation. Colors correspond to the source group of ORNs providing excitatory input.

$$p\_{i,j} = \frac{\sum\_{k}^{N\_p} \xi\_i^k \xi\_j^k}{N\_p} \tag{6}$$

Then the mutual information *Ii*,*<sup>j</sup>* and joint entropy *Ei*,*<sup>j</sup>* between mitral cells is calculated as follows:

$$I\_{i,j} = \begin{cases} p\_{i,j} \log(\frac{p\_{i,j}}{p\_i p\_j}) & \text{if } p\_i \cdot p\_j \neq 0 \text{ and } p\_{i,j} \neq 0\\ 0 & \text{otherwise} \end{cases} \tag{7}$$

$$E\_{i,j} = \begin{cases} -p\_{i,j} \log p\_{i,j} & \text{if } p\_{i,j} \neq 0\\ 0 & \text{otherwise} \end{cases} \tag{8}$$

From these two quantities the mutual information distance measure is defined as:

$$D\_{i,j} = \begin{cases} 1 - \frac{I\_{i,j}}{E\_{i,j}} & \text{if: } E\_{i,j} \neq 0\\ 1 & \text{otherwise} \end{cases} \tag{9}$$

In order to decide which MT connects to which HC in the PC, we apply a multi-dimensional scaling algorithm (MDS) to the distances *Di*, *<sup>j</sup>* into three dimensions implemented by the Python-Orange software package (Demšar et al., 2013). The mapping between MT cells and cortical HCs is achieved by doing a k-means clustering as vector quantization (VQ), resulting as HC being the centroids to a number of MT cells in the three-dimensional mutual information space. The VQ is repeated until no HC is empty, i.e. each HC gets input from at least one MT cell, ignoring MT cells that were silent during all patterns. This MT-HC mapping can be modified by allowing each MT cell to connect not only to one HC, but to the *m* nearest centroids or HCs. If not stated otherwise, we have used *m* = 4 for our simulations. A second VQ is applied to each HC to distribute the different patterns among the MCs in one HC to derive their specific response properties. This is done by building a new multidimensional MTresponse space in which each mitral cell assigned to the target HC represents one dimension and each pattern represents a Euclidean vector. The normalized MT cell activation ξ *<sup>k</sup> <sup>i</sup>* gives the magnitude for vector *k* in dimension *i*. The result of this second VQ maps patterns to the different MCs in a HC and gives a binary activation *Np* × (*N*HC · *N*MC) matrix containing information during which patterns a MC is activated by its source MT cells. This binary activation matrix is used in the next step as postsynaptic activation matrix ζ. Finally, the weights between MT cells and MCs are calculated based on the BCPNN algorithm:

$$\mathcal{W}\_{i,j} = \begin{cases} \log \frac{p\_{i,j}}{p\_{i\mid p\_j}} & \text{if: } p\_i \neq 0 \text{ and } p\_j \neq 0\\ \log 1/N\_{\mathcal{P}} & \text{if: } N\_{\mathcal{P}} \neq 0 \text{ and } p\_{i,j} = 0\\ 0 & \text{otherwise} \end{cases} \tag{10}$$

where *pi* is the normalized pre-synaptic activation probability of MT cells, *pj* = *Np <sup>k</sup>* <sup>ζ</sup>*<sup>k</sup> j Np* is the probability of activation of MC *j* and ζ*k <sup>j</sup>* is an element from the binary activation matrix of MC *j* in pattern *k*, i.e. the information if the MC has been assigned to pattern *k* in the second VQ as described above.

*2.5.2. Recurrent connectivity in PC and pattern recognition readout* As before, we compute connections with the help of the BCPNN algorithm and regard MCs as elementary units and derive longrange connections between MCs belonging to different HCs based on their probability of activation in an abstract sense. The previous step gave us the projections *wi*,*<sup>j</sup>* from MT cells to MCs which will now be used to calculate the responses of an abstract MC as follows. First, a MC *j* receives input *s k <sup>j</sup>* from MT cells during pattern *k*:

$$s\_j^k = \sum\_{i}^{N\_{\rm MT}} \omega\_{i,j} \xi\_i^k \tag{11}$$

This input or support is combined with the bias β*<sup>j</sup>* of that MC:

$$\beta\_{\hat{j}} = \begin{cases} \log(p\_{\hat{j}}) & \text{if: } p\_{\hat{j}} > 0 \\ \log(1/N\_{\hat{p}}^2) & \text{otherwise} \end{cases} \tag{12}$$

$$o\_j^k = \begin{cases} \exp\left(\beta\_j + s\_j^k\right) & \text{if: } s\_j^k > 0\\ 0 & \text{otherwise} \end{cases} \tag{13}$$

As for MT cells that code with their normalized activity for the presence of an odorant at a certain concentration in a probabilistic fashion, we apply the same sort of half-normalization for all MCs belonging to one HC, i.e. if the sum of output activities during one pattern in one HC is larger than one, it is set to one:

$$o\_j^{k'} = \begin{cases} o\_j^k / \sum\_j^q o\_j^k & \text{if: } \sum\_j^q o\_j^k > 1, \\ o\_j^k & \text{otherwise} \end{cases} \tag{14}$$

The indices *j* and *q* stand here for the MCs belonging to one HC. We will come back to this point of interpreting activity as the probability of perceiving a certain feature in the Discussion.

The recurrent weights between MCs situated in different HCs is then calculated in the same way as above in Equation (10) with the output activities *o<sup>k</sup> <sup>i</sup>* determining the probabilities of activation by replacing ξ *<sup>k</sup> <sup>i</sup>* in Equations (5, 6).

MCs belonging to the same HCs are not connected. The weights within a MC (from RSNP to PYR cells and between PYR cells) are set statically and not affected by this abstract learning algorithm. The same holds for the connectivity involving basket cells.

In order to be able to classify the distributed cortical representations after learning we train an additional layer of readout cells. For training the connectivity from PC to the readout layer we use the exact same formalism, but with only one single readout cell being active during a pattern. Hence, for readout cells we set *o<sup>k</sup> <sup>j</sup>* = 1 if *j* = *k* and 0 otherwise as a supervisor signal. This assumes that during learning the system is exposed to odorants in a pure form, in a sequential order (as separate patterns, i.e. responses are gained through separate simulations) and with the knowledge about the distinctness of odor patterns. This is also the condition for a correct recognition, when these abstract connection and bias values are transformed into the spiking network and "test patterns" are presented to the system, i.e. in the spiking context we regard a pattern as correctly classified if the corresponding readout cell has the maximal output firing rate. A readout neuron is not connected with other neurons and serves as a simple indicator if one pattern is perceived as present or not.

#### *2.5.3. Translating abstract learning results to biophysical model*

As described in the above section we obtain abstract connection matrices for feed-forward connections MT cells and PC, between MCs in PCs and from the PC to a readout layer which tries to identify input patterns with the presented patterns during the training. To transform the abstract connectivity obtained from Equation (10) we do a linear mapping from the abstract weights into biophysical weights, i.e. conductance values. If the resulting biophysical weight is below a threshold of 5 pS, the connection is discarded because it has no significant influence and to decrease computational costs. For OB to PC and the recurrent PC connections, negative values get linearly mapped to positive weights that target the inhibitory RSNP cells which in turn provide inhibition to the target MC. Positive values are linearly mapped to weights that target PYR cells. Based on the source and target cell type we use different linear transformation factors, e.g. we transform negative weights so that the most negative value corresponds to a conductance of 3 nS for MT to RSNP connections and 1.5 nS when the connection originates from a PYR neuron. When an MT cell excites a MC it targets 50% of all PYR in that MC, i.e. 15 cells. When an MT cell inhibits a MC it excites 75% of all RSNP in that MC, i.e. three RSNP neurons, which in turn inhibit the PYR cells in the that MC. For recurrent PC connections, positive weights are transformed into 45 excitatory long-range connections between the two respective MCs, which corresponds to 5% of all possible connections between the two MC. Negative weights are realized so that 10 out of 30 PYR cells from the source MC target 3 out of 4 RSNP cells in the target MC. Source and target cell pairs for recurrent PC connections are chosen randomly and multiple connections between the same source and target pair are replaced with a valid source-target pair. Connecting the readout layer takes into account all PYR cells in the source MC. After the linear transformation of the abstract weights into the cell type specific conductances, all conductances on the singlecell level are randomly changed by 10% in order to account for natural variability of neurons and synapses.

The full data on resulting number of synapses and neurons in the system is shown in **Table 2**.

#### **Table 2 | Neuron and connection numbers.**


## **3. RESULTS**

We will first show the response curves of ORNs and MT cells realizing the hypothesized fuzzy concentration interval code before we focus on the five functional tasks the system has been tested with (recognition, concentration invariance, noise robustness, pattern completion, pattern rivalry).

**Figure 3** shows the output rates of one family of ORNs to an odorant to which the OR has maximal affinity for different concentrations. Output rates are measured over one full simulation of 1600 ms in response to an odor puff (see **Figure 2**). The response curves are color coded depending on the target MT cell to which the ORN subgroup will project according to our hypothesized axon-sorting as described in sections 1.3.1. and 2.4.1. The MT response curves are averaged over ten trials with different random seeds modifying background noise and initial membrane potentials, error bars indicate the standard deviation. Through the projection patterns described in section 2.4.1 we achieve that individual MT cells code for only a certain concentration range.

#### **3.1. TASK 1: BASIC PATTERN RECOGNITION**

The fuzzy interval code realized by MT cells is the basis for our approach of interpreting the OB as a probabilistic sensor array which provides information about certain odor features to the PC. We have tested this coding scheme and the self-organized connection algorithm first in a simple pattern recognition task (in the following referred to as Task 1). The system expresses 40 ORs and has been trained by stimulating the ORNs and OB with 50 different patterns in sequence, i.e. separate simulations using odor puffs as input. **Figure 4A** shows the used set of random odor patterns, which correspond to artificial odor patterns at a medium concentration.

The OB response to these 50 pattern presentations was used to derive the connectivity to and within PC and to the readout layer as described above. As a basic proof of functionality, we then presented the exact same patterns to the system again and looked at the output rates of the readout cells for each pattern (see **Figure 4B**). The criterion for a correctly recognized pattern is that the readout cell responsible for the given pattern as defined by the supervisor signal (see section 2.5.2) must have the highest output rate measured over the whole simulation time of 1600 ms. According to this criterion all 50 patterns have been recognized correctly.

The activity of PYR cells averaged over all 50 patterns is very sparse and distributed. During each pattern 223 ± 34 neurons (∼2.0 ± 0.3%) of all neurons were active (being active measured as firing more than one spike per pattern). Still, firing rates of individual neurons could get as high as 150 Hz and mean firing rates averaged over all patterns and all cells that fired at least one spike are around 10 Hz. On average each neuron was active in only 1.0 ± 1.4 patterns (∼2.0 ± 2.7%). In total 68.5% of the PYR cells were active in at least one pattern, 19.5% in more than two and 2.7% showed spiking activity in more than three patterns.

## **3.2. TASK 2: CONCENTRATION INVARIANCE**

In order to test the system's capability of recognizing odors that appear at a different concentration, meaning that the effective activation for those OR that respond to the given odor is different, we selected the first 10 odor patterns from Task 1 and changed the affinity between an activated ORs and the odorant in five steps from −0.2 to +0.2 compared to the affinity in the training pattern (see **Figure 4C**). Changing the affinity is equivalent to changing the concentration as they are in our model dependent from each other *c* = OAV/(1 − OAV). This results in a set of 10 different odors with 5 different concentrations each and should be regarded as 50 test patterns. First, we tested the system as trained in Task 1 with this set of patterns and looked at the response of the readout cells. As the system was trained to distinguish 50 different odors at one single (medium) concentration pattern only (as shown in **Figure 4B**), the system did not recognize all patterns correctly, but three odors when presented at the lowest concentration were misclassified (data not shown). This could be interpreted as if the system would perceive these three odors at low concentration as being qualitatively different compared to higher concentrations.

Since odorants in real systems do occur at different concentrations and the perceived "effective" concentration varies during the sniffing or inhalation process, we trained the system with patterns representing odorants at different concentration. To achieve concentration invariance recognition we trained the system with the patterns representing 10 odors at 5 different concentration

train and test concentration invariance. Shown are the first 10 patterns from **(A)** with varying concentration (affinity). **(D)** Readout activity response to the patterns shown in **(B)** after training the system with these. Independent of the concentration, all patterns get recognized correctly after the training.

(**Figure 4C**) instead of single concentration odors only. The system then recognized the 10 different odorants correctly for all concentrations as shown in **Figure 4D**.

## **3.3. TASK 3: NOISE ROBUSTNESS**

To simulate a more realistic pattern recognition task, we presented noisy versions of the 50 "pure" patterns to which the system was trained in Task 1. As described by Equation (2) we modified each element in the affinity matrix by an increasing degree of noise σ and tested the system trained from Task 1 to recognize these noisy patterns. The blue curve in **Figure 5** shows the performance of the system for four different noise levels. For a degree of noise of abs(σ) ≤ 0.05 the system recognizes all patterns correctly, hence showing some noise robustness, but performance drops rapidly for larger σ. As we have chosen an extremely simple model for odorant-OR interaction without regarding possibly competitive interactions, it is not possible to relate these values to real systems in a meaningful way.

## **3.4. TASK 4: PATTERN COMPLETION WITH MODIFIED TEMPORAL INPUT**

A typical task to be mastered by a sensory system is to deal with incomplete patterns. We model incomplete patterns by taking the system from Task 1 and choosing a random number of ORs that get activated in the complete pattern (Task 1) to be inactive in the incomplete pattern. As an additional test for the dependency of the system to rely on precise spike timings we changed the input dynamics from the odor puff (with which the system has been trained) to the more variable sniffing input (see **Figure 2C**). The difference in stimulation dynamics is clearly visible on the ORN level, but is less pronounced on higher levels as shown in **Figure 6**.

This might be due to the strong influence of NMDA currents involved in feed-forward excitation, but also the self-excitation via excitatory autoreceptors on MT cell dendrites might attenuate the temporal structure imposed by the ORN layer. **Figure 6** shows the activity of MT, PYR and readout cells as raster plots to one example pattern in the training and test setup with half of the ORs being silenced. The complete training pattern is plotted with gray dots, whereas the response to the incomplete test pattern is marked with blue dots. Despite the difference in temporal input structure and the fact that activity in the OB and epithelium (not shown) is significantly less, the system is able to complete the pattern in the PC. This can be seen from the fact that cells being active during training overlap with the cells active during the test to a much higher extent than it is for MT cells, where a high number of gray dots indicate the incompleteness of the test pattern. Due to the recovered activity in the PC, the PYR cells drive the correct readout cell (the lowest readout cell, as it was pattern 0).

We have studied this pattern completion capability in a more systematic way by testing all patterns trained during Task 1 with different levels of completeness. Pattern completeness is defined by the fraction of ORs being active in the test pattern compared to the number of activated ORs in training patterns. Pattern completeness was varied from 80% to 30% and the number of correctly recognized patterns was counted as shown by the red trace in **Figure 5**. For all test patterns in Task 4 and 5 we used the sniffing input model in contrast to the training runs which uses an odor puff as input (see **Figure 2**).

The systems seems robust toward incomplete patterns for missing up to 40% of the odor components as the recognition performance stays above 90%. As shown by the example raster plots in **Figure 5**, the activity pattern in the PC seems very similar on the population level in the sense that the same MCs are active in comparison to the activity evoked during training, despite the missing odor information and the different temporal structure. This pattern completion capability is presumably due to the recurrent excitatory cortical connections which help to restore activity in MCs that receive less input from the OB during incomplete patterns. In order to prove this assumption we have tested a network with the exact same patterns but removed the long range connections between PYR and RSNP that have been trained during Task 1. The result is shown by the dotted red line in **Figure 5**. It shows a drastically impaired performance compared to an "intact" network with trained long-range connectivity between HCs.

## **3.5. TASK 5: PATTERN RIVALRY WITH MODIFIED TEMPORAL INPUT**

Perceptual bistability or rivalry occurs when stimulus patterns overlap so that two distinct perceptions are possible. This phenomenon can be explained by attractor dynamics in the networks involved in sensory integration and perception. We have investigated the system's response behavior to odor patterns that are constructed from varying subparts of distinct patterns from Task 1.

In order to study the system's responses to mixtures, we constructed new odor patterns by choosing a pair of two of the 50 distinct odors patterns, with which the system has been trained in Task 1, and generating a new set of patterns by varying the

number of components taken over from the respective two patterns. For example, a mixture of 0.4/0.6 between two arbitrary patterns B and R is built by choosing randomly 40% of the ORs active in pattern B and 60% or ORs activated in pattern R and combine them into a mixture pattern. This has been done for 50 different pattern pairs with a varying fraction of each pattern from 0.8/0.2 to 0.2/0.8 in steps of 0.1, taking over the previously chosen ORs into the next mixture pattern resulting in a sequence of mixture patterns which morphs from one to the other. This gave us seven different mixture patterns for each of the randomly chosen pure training pattern pair. We chose to pick 50 different odor pairs to generate in total 350 mixture patterns. This large set of mixture patterns has been presented to the system that has been trained with pure patterns as in Task 1 in sequential order.

We counted the number output spikes by the readout neurons corresponding to the two unmixed training patterns and averaged these over the 350 mixture patterns (see **Figure 7**). The average curve shows a smooth transition from one pattern to the other and rivalry behavior in between, meaning that the system recognizes both patterns at the same time (regarded over one pattern presentation). During the morphing process from one pattern into the other it often occurred that the readout layer recognized none of the two partial test patterns but interpreted the superposition as a different pattern.

When looking at the dynamics during a single example mixture as shown in the left two panels of **Figure 7**, the PC and the readout activity indicate two distinct odor percepts (as indicated by the color of the dots) at different times during the stimulation. Hence, the systems perception switches dynamically from one odor to the other which is characteristic for perceptual rivalry.

## **4. DISCUSSION**

In this study we have presented a generic architecture for selforganized pattern recognition and memory systems and implemented a spiking model thereof inspired by the first three stages of the mammalian olfactory system. We have proven the functionality of the system in different pattern recognition tasks involving concentration invariant recognition and pattern completion, and studied its robustness against noise and rivalry phenomena occurring with mixtures of odor patterns. Our approach is generic, because it can be used for other modalities as well (Lansner et al., 2009), as the format of the sensory array on which the learning algorithm operates is modality specific, but the cortical structure responsible for the integration and consolidation of sensory information is regarded to be modality independent.

#### **4.1. ORIGINAL HYPOTHESES**

One of our key hypotheses is that activity dependent axon sorting mechanisms contribute to the formation of a concentration interval code in the MT layer of the OB. The motivation behind the hypothesized interval coding is to use the OB as a probabilistic sensor array that serves as input for the BCPNN algorithm which allows for self-organization of the connectivity from OB to PC and within the PC based on the probabilistic interpretation of MT responses. Here, we explored the possibility of such a coding scheme in the context of odor concentration and showed that it is implementable in a spiking context.

Furthermore, we assume that the PC acts similar to other cortices as an attractor network and hence applied a modular network structure to simulate functions like pattern completion and rivalry. We are well aware of the fact that no columnar organization has been reported in olfactory cortices and we suggest that the computational structure is not necessarily visible from the spatial layout of cells as in other sensory systems, e.g. in V1 (Li et al., 2012), but rather implemented through the connectivity patterns, e.g. MCs could correspond to small, spatially dispersed populations with enhanced recurrent connections that connect to a common pool of inhibitory interneurons (corresponding to basket cells in our model). A softening of the rule

**FIGURE 7 | Task 5: Pattern rivalry with sniffing input. Left:** Raster plot showing PYR responses to a 0.6/0.4 mixture of two distinct patterns. The fraction of both patterns stays constant during the whole stimulation. Blue dots show spikes from cells being active during the "blue" odor in Task 1. Red dots show spikes from cells being active during the "red" odor in Task 1. Gray dots show spikes from cells that are active during the test pattern, but have not been active in either of the two mixture components. During the first 200 ms the red pattern evokes activity in both PYR and readout cells, but is then suppressed by the blue pattern becoming active after ∼500 ms of odor stimulation. A substantial part of PYR activity is related to none of the two patterns, exemplary for the

that basket cell inhibition targets only PYR belonging to the same HC was investigated in studies by Lundqvist et al. (2010, 2013). There the modular basket cell inhibition was replaced by a distance dependent inhibition and it was shown that the network dynamics change, but that the attractor behavior, and with that the computational capabilities of the system, are preserved without this modular inhibition. The computational capabilities of the presented model are rather based on the specific long-range excitation between MCs and the specific inhibition mediated through RSNP cells (Fransén and Lansner, 1998), whereas basket cell inhibition is required to regulate the network activity and balance the excitation. This is because RSNP cells in our model can not counterbalance the recurrent excitation within an attractor.

The strict columnar organization as used in our model was chosen to reflect the BCPNN algorithm more closely, but is likely softened in real systems. Hence, in this respect our network model should not be seen as a precise model of the biological counterpart but rather as a way to implement networks performing holistic computations and behaviorally relevant functions. One advantage of the modular structure and the assumed patchy connectivity is a shorter wiring length with the same pattern storage capacity when compared with a non-modular "pepper-and-salt" like organization (Meli and Lansner, 2013).

## **4.2. SUMMARY OF FINDINGS AND EXPLANATION**

We have shown that the self-organization algorithm previously used only in abstract models (Lansner et al., 2009; Persaud et al., 2013) can be translated into a spiking network context, and that pattern recognition can work on the time scale of a single sniff, comparable to results from behavioral studies (Uchida and Mainen, 2003). First of all, we have shown that a concentration interval code can be implemented with the help of known pathways in the OB with biophysically detailed neuron and synapse models. Furthermore, we have successfully

often occurring misclassifications during the recognition of odor mixtures. **Middle:** Raster plot corresponding to the pattern from left panel showing spikes emitted by readout cells that were active during the two respective training patterns. The stronger pattern is being recognized starting from ∼700 ms, i.e. approximately after 500 ms or two sniff cycles (simulated sniffing frequency is around 4 Hz. **Right:** Average curves showing the mean number of spikes emitted by the readout cells trained to recognize one of the two test patterns. Black curve shows the mean response from readout cells that code for none of the two test patterns. The blue and the red curve indicate a smooth transition from one pattern to the other depending on the relative strength in the mixture.

translated an abstract self-organization framework to a spiking network and shown its functionality in a simple pattern recognition task (Task 1). The key components to achieve this functionality is the projection from OB to PC and the connectivity within PC obtained from the BCPNN algorithm. The system has proven to be robust against changes in temporal dynamics and high levels of incompleteness at the same time. In a pattern completion task we have shown that the recurrent excitatory connectivity in the PC promotes the restoration of incomplete pattern activity and facilitates pattern recognition of incomplete patterns.

In addition, we have shown that concentration invariant recognition emerges after training the system with patterns at multiple concentrations. This brings us to the conclusion that concentration invariance could be learned through experience by exposure to odorants that effectively always vary during the sniffing or inhalation. A system without having been trained to perceive odors of different concentrations as belonging to the same odorant can lead to qualitative different percepts as we observed in our simulations, and has been reported for some odorants (Gross-Isseroff and Lancet, 1988; Johnson and Leon, 2000; Wright et al., 2005).

Surprisingly, little differences in bulbar and cortical activity were observed when different stimulation protocols were applied. One possible explanation for this could be that NMDA currents dominate the behavior more than fast excitatory currents do, as NMDA currents are found almost ubiquitously in the system, e.g. as source for self-excitation in MT cell dendrites. Thus, we conclude that the input dynamics including precisely timed spike patterns or sequences thereof do not play a crucial role for the pattern recognition capabilities of our model system. It remains subject to debate whether this finding can be seen as an argument against spike timing dependent codes, as different stages might use different ways of coding as suggested by Haddad et al. (2013). Similarly, other concentration coding schemes as the one used in our model could work equally well.

## **4.3. RESULTS IN CONTEXT TO OTHER EXISTING STUDIES**

In general our results are in qualitative agreement with recent experimental findings regarding odor representations in the PC and projections from OB to PC. The connectivity obtained by our self-organization method leads to PYR neurons that integrate information from distinct glomeruli as seen in Apicella et al. (2010). Furthermore, we observe sparse and distributed activity in PC in response to odor stimuli with activation levels in a comparable range to findings by Stettler and Axel (2009). In accordance with (Poo and Isaacson, 2011) we observed rather unspecific inhibition in PC, as connectivity involving basket cells is not dependent on the source or target cell's response properties. In addition, weights from OB to PC observed after training are often inhibitory and hence provide inhibition for a large number of odorants.

## **4.4. LIMITATIONS**

Despite the complexity of the presented model, there are a large number of limitations and aspects which have not been covered at all by our model. Regarding the general (structure), our model does not include any notion of the anterior olfactory nucleus (Brunjes et al., 2005), and other input sources into PC from other areas than the OB were not regarded in our model (see e.g. Luna and Morozov, 2012). Differences between the anterior and posterior PC have not been included in the model as well as learning in other structures (Morrison et al., 2013). Our model does also not include neurogenesis seen in the OB of rodents (Nissant et al., 2009; Sahay et al., 2011), but whether neurogenesis is crucial in the human olfactory system is still up for debate (Bergmann et al., 2012). Acetylcholine was not included in this model, but the role of cholinergic modulation might impact memory performance as shown in de Almeida et al. (2013). More specifically, our implementation of the concentration code is not easily extendable to larger neuron numbers, as this would require substantial retuning of various parameters to achieve the desired response curves.

One very important limitation of the model, as was presented here, is the lack of projections from the PC to the OB. The back-projections do play an important role in odor recognition, especially in tasks where attention or the expectation of an odor changes the signals represented in the system. This task-relevant information could be included in an extension of our model using external input into the PC and the inverted OB to PC weight matrix, which would make PYR neurons target granule cells, preferably connecting the respective glomerular module, so that task-relevant information acts like a template on the bulbar layer to filter or enforce certain patterns.

## **4.5. OUTLOOK**

In general, two broad directions could be taken starting from the presented model. One is making the model more realistic and trying to verify or falsify it, e.g. by using more realistic odor patterns, incorporating more experimental data specifying the circuits involved, adding cell types and structures that have been omitted in this model. The opposite direction is to simplify certain components even further (e.g. reducing the complexity of ORNs, and bulbar cell models) and test the model in different and more complex tasks, e.g. odor segmentation. The question on which scale the inhibition in cortical circuits acts in a computational meaningful manner, in our model represented by the size of a hypercolumnar module, and how the extent of this recurrent inhibition is sensitive to feed-forward excitation and the spread thereof is unknown and needs to be investigated in the future. As this study is only a first step in transforming abstract learning paradigms into the context of functional spiking network models and thereby trying to bridge the gap between system-level functions and biophysical detail, this model offers the possibility for versatile extensions and improvements, to be examined in future studies.

## **ACKNOWLEDGMENTS**

The authors thank the reviewers for their feedback, Pawel Herman for very useful discussions and comments on the manuscript, Simon Benjaminsson for the development of the machine learning and BCPNN technique, Benjamin Auffarth for fruitful discussions regarding the generation of odor patterns, Claudia Ramos Garcia for her work on the ORN model, Antoine Bergel for his work on the NEURON implementation of the ion channel modeling intrinsic excitability, and Malin Sandström, David Silverstein, and Pradeep Krishnamurthy for their help with the NEURON simulator environment. Code to reproduce figures and supplementary material is available on the author's github (Dabbish et al., 2012) page (Kaplan, 2013).

## **FUNDING**

This work was supported by projects FACETS-ITN (EU funding, grant number 237955), "BrainScaleS" (EU funding, grant number FP7-269921), and NEUROCHEM (EU funding, FP7- 216916). The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC Centre for High Performance Computing (PDC-HPC) at KTH, Stockholm.

## **REFERENCES**


cells from olfactory sensory neurons. *J. Neurosci.* 31, 8722–8729. doi: 10.1523/JNEUROSCI.0527-11.2011


bulb mitral cells. *J. Physiol.* 542, 355–367. doi: 10.1113/jphysiol.2001. 013491


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 01 November 2013; accepted: 09 January 2014; published online: February 2014. 07*

*Citation: Kaplan BA and Lansner A (2014) A spiking neural network model of selforganized pattern recognition in the early mammalian olfactory system. Front. Neural Circuits 8:5. doi: 10.3389/fncir.2014.00005*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Kaplan and Lansner. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# A methodology for fast assessments to the electrical activity of barrel fields *in vivo*: from population inputs to single unit outputs

## *Jorge J. Riera1,2\*, Takakuni Goto2 and Ryuta Kawashima2*

*<sup>1</sup> Department of Biomedical Engineering, Florida International University, Miami, FL, USA*

*<sup>2</sup> Department of Functional Brain Imaging, Institute of Development, Aging and Cancer, Tohoku University, Sendai, Japan*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Ehud Kaplan, Mount Sinai School of Medicine, USA Alex Proekt, Weill Cornell Medical College, USA*

#### *\*Correspondence:*

*Jorge J. Riera, Department of Biomedical Engineering, Florida International University, 1055 Flagler St., Miami, FL 33174, USA e-mail: jrieradi@fiu.edu*

Here we propose a methodology to analyze volumetric electrical activity of neuronal masses in the somatosensory barrel field of Wistar rats. The key elements of the proposed methodology are a three-dimensional microelectrode array, which was customized by our group to observe extracellular recordings from an extended area of the barrel field, and a novel method for the current source density analysis. By means of this methodology, we were able to localize single barrels from their event-related responses to single whisker deflection. It was also possible to assess the spatiotemporal dynamics of neuronal aggregates in several barrels at the same time with the resolution of single neurons. We used simulations to study the robustness of our methodology to unavoidable physiological noise and electrode configuration. We compared the accuracy to reconstruct neocortical current sources with that obtained with a previous method. This constitutes a type of electrophysiological microscopy with high spatial and temporal resolution, which could change the way we analyze the activity of cortical neurons in the future.

**Keywords: CSD, LFPs, brain current sources, neuronal activity, cerebral cortex, barrel field**

## **INTRODUCTION**

Currently many efforts are focused on decrypting canonical working principles of cortical microcircuits in mammalians. To this end, the barrel cortex of rats has been a very useful animal model. *In vivo* extracellular electric recording from these barrels provides information about the activity of large populations of neurons with an excellent temporal resolution. Although the extracellular electric recording technique was launched in the middle of the 19th century, it is now recapitulating its role with the rapid development of silicon-based microelectrode arrays (MEA). With the technological advances in the microelectromechanic systems (e.g., deposition, lithography, etching, die-preparation, Wise, 2005), MEAs with high spatial resolution are gradually being built with a variety of not only microelectrode local configurations (e.g., tetrodes, octodes, polytrodes) but also shank spatial arrangements (e.g., linear or "laminar," planar and three-dimensional) (Ulbert et al., 2001; Csicsvari et al., 2003; Buzsáki, 2004; Blanche et al., 2005; Kipke et al., 2008; Du et al., 2009; Ogawa et al., 2011; Riera et al., 2012). MEAs with threedimensional formats are ideal to obtain volumetric recordings from multiple barrels, a crucial step to understand trans-laminar and tangential interactions in the cortical microcircuits with an acceptable spatial and temporal resolution (Riera et al., 2012).

Unfortunately, the extracellular electric potentials do not represent directly the ionic flows generated by excitable membranes in active neuronal ensembles, i.e., the volumetric density of current sources *C*(*t*), but instead they are far-field external reflections of these electric currents through a highly conductive extracellular medium. Accurate biophysical models that included realistic profiles of the electric conductivity are required to properly characterize these external reflections at each particular cortical region. In order to have a good estimation of the current source density (CSD) *C*(*t*) inside a cortical region, extracellular electric potentials need to be observed, usually with respect to a common reference, from a large number of microelectrodes homogeneously distributed inside that region. This is named the CSD analysis. A priori information about the brain current sources is always required to uniquely solve the inverse problem underlying any CSD analysis. Evidence that brain current sources are actually smooth over extended regions within the barrel cortex has been accumulating over the last decade. Despite its clear value, this constraint has not been explicitly introduced in previous methods for CSD analysis.

In this study, we propose a new methodology for performing CSD analysis on volumetric extracellular recordings from the barrel cortex of Wistar rats that is based on:


We apply the proposed method to assess specific features of the current sources in the barrel cortex of adult Wistar rats undergoing whisker deflections. First, we determine the spatial extent of early thalamic inputs into layer 4 of the cortex and use it as a gold-standard to evaluate the performance of our method. In addition to obtaining volumetric current source patterns associated with local field potentials (LFP) during single whisker stimulation, we concurrently determine the spatiotemporal profiles of single cortical neurons by combining our method with those used for spike detection and single-unit classification (Quiroga et al., 2004; Sakata and Harris, 2009) from multiunit activity (MUA). Also, we use simulations to evaluate the stability of our method for different noise levels and electrode grid resolutions. For illustration purposes, we compare the performance of our method with that resulting from the use of an alternative method previously proposed in the literature to perform CSD analysis with three-dimensional MEAs (i.e., the iCSD3D method, Łe.ski et al., 2007). MATLAB scripts for the iCSD3D method are available at http://www.neuroinf.pl/Members/szleski/icsd.html.

## **MATERIALS AND METHODS**

## **ANIMAL PREPARATION**

All experiments were performed following the policies established by the Animal Care Committee at Tohoku University (Sendai, Japan). Adult Wistar rats (7–11 weeks of age, male) were used in the experiments. Animals were first anesthetized with intraperitoneal (IP) injections of urethane (1.2 g/kg), and immobilized with a stereotaxic system (Narishige, Japan) that comprises ear bars and a mouth/nose clamp. If necessary, an extra dose of urethane was administrated. Before surgery, all whiskers were trimmed to 1 cm. The right somatosensory barrel cortex was exposed through a craniotomy (5 mm in diameter, centered 2.4–2.5 mm posterior to the Bregma and 5.8–6.0 mm lateral from midline) and a small patch of dura matter was carefully removed. Non-conductive paraffin oil (Nacalai tesque) was applied over the exposed brain tissue to keep the cerebral cortex moistened. Two other craniotomies with 1 mm in diameter were made at the left posterior and right posterior parts to the lambdoidal suture to set the ground and reference screws, respectively. These screws were attached to the skull by dental cement and in direct contact with the brain's surface.

## **A THREE-DIMENSIONAL SILICON-BASED MEA**

In this study, we used a three-dimensional silicon-based microelectrode array (3D array) that was customized in collaboration with NeuroNexus Technologies, Inc. The 3D array is composed of multiple 2D planar probes (4 shanks each with 8 microelectrodes, 200μm inter-electrode distance) which are bound together (400μm inter-shank distance) using micro assembly technique (**Figure 1A**). **Figure 1B** shows an illustration of the 3D array after being inserted into a virtual barrel field of a rat. A picture with the probe in position to be inserted into an actual somatosensory barrel cortex is shown in **Figure 1C**. Each 3D probe has 128 microelectrodes in total covering a volumetric region of interest (ROI) of about 2 mm3, which means 4–9 adjacent barrels. By means of this 3D array, changes in the distribution of the extracellular electric potentials in such a ROI are observed with high temporal resolution.

## **THREE-DIMENSIONAL RECORDINGS OF EXTRACELLULAR ELECTRIC POTENTIALS**

photograph **(C)** of the 3D array right before its implantation into the

somatosensory cortex of a rat.

We implanted the customized 3D array into the exposed somatosensory barrel cortex in a way that the tip of each shank was at a depth of 1600μm. Due to the strong reactivity of the brain tissue compared to other probe formats (e.g., laminar and planar), the insertion of the 3D array constitutes one of the most difficult steps of the proposed methodology. Repelling forces make the tissue easily bent and recover upon attempted insertion of the probe. Therefore, we applied a gradual insertion method where the 3D array is iteratively inserted two steps forward (200μm) and one step backward (100μm) until a designated depth is reached. Each insertion was observed using a customized rotating digital microscope (KH-1300, Hirox; Narishige).The probe insertions were performed with a micromanipulator (Combi 25Z; Luigs and Neumann Feinmechanik, Ratingen, Germany) and the procedures were always monitored on the digital display of the micromanipulator's control system (SM5; Luigs and Neumann). Each shank of the 3D array was carefully painted with a lipophilic neuronal tracer carbocyanine (DiI, D282; Invitrogen) to reveal its actual position from histological images, which were obtained after each recording section. Experiments were early terminated for those rats with considerably cortical bleeding due to perforations of pial vessels. We also excluded from the analysis several rats whose histological images show signs of cortical swelling or abnormal lamination. It took us some years to master this insertion protocol." In most of the cases, deformations of the cortical tissues were observed neither during the experiment nor on the postmortem images.

For comparison with the iCSD3D method (Łe.ski et al., 2007), half of microelectrodes were excluded, resulting in an array of 64 microelectrodes, to mimic a 3D array with equidistant microelectrode arrangement (400μm inter-electrode distance and 400μm inter-shank distance). The 3D array was connected to the main amplifiers (PZ-2, Tucker-Davis Technologies, TDT) through a couple of 64 channel ZIF-Clip® headstages (ZC64; TDT). The PZ-2 amplifiers were connected to a signal processor unit (RZ-2; TDT) by optical fibers. The electric potentials at the microelectrodes were recorded with respect to the reference electrode, and with a sampling frequency of 25 kHz.

Individual whiskers were deflected by the piezoelectric bimorph actuator (TAYCA, Japan). The deflection angle, frequency and interval for each whisker deflection were set to 7.2◦, 1 Hz, and 100 ms, respectively. To that end, square pulses with these parameters were programmed in MATLAB and the resulting signals were used to energize a piezoelectric bimorph actuator through the D/A converter (PCI-6259, National instruments, USA) and the piezo driver (PCD-001, General Photonics, USA). For each condition, we recorded 100 trials.

## **HISTOLOGY**

After recordings, rats were perfused with 4% paraformaldehyde in 0.1 M sodium phosphate buffer saline solution, and their postmortem brains were kept in the same solution overnight. After that, the fixed brains were cut tangentially to the brain surface in 100μm thickness by a tissue sectioning equipment (Vibratome 1000-plus; Leica Microsystem). To reveal the barrels, the sections were treated with 3,3 diaminobenzidine (DAB, Sigma D8001) and cytochrome C oxidase from horse heart (Sigma, C2506) following the protocol by Civillico and Contreras (2006). Co-localized immunostaining images that reveal the shank positions and barrels were obtained by using an upright fluorescent microscope (SZX16, Olympus).

### **DATA PREPROCESSING**

The extracellular electric potential comprises two types of electrophysiological signals (Gray et al., 1995), i.e., the LFPs, which reflect spatiotemporal superposition of synaptic inputs to the neuronal populations, and the unit activity, which captures the action potentials produced by neurons in close proximity to the microelectrodes. To obtain LFPs from the raw data, we applied a Butterworth band-pass filter with cut-off frequency of 1 Hz and 500 Hz. Event-related potentials (ERPs, -(*t*), *t* = −50 − 100 ms) evoked by whisker deflections were calculated by averaging LFPs over 100 trials. Another band-pass filter with cut-off frequency of 500 Hz and 8 kHz was applied to the raw data. From the resulting high frequency components, we extracted MUA by negative edge detection *with a* threshold of 4 times the standard deviation and 1.5 ms dead time. Twenty samples (i.e., eight and twelve samples prior and posterior to the spike troughs, respectively) of the detected spikes were used for classification. Spikes at each microelectrode were divided into putative excitatory pyramidal cells (PCs) and interneurons (INs) by two-step clustering strategy (Ogawa et al., 2011). First, we represented the spikes using four-level Haar wavelets. From the resulting 20 wavelet coefficients, 10 representative coefficients were selected as the input for cluster analysis using the Kolmogorov–Smirnov test. The cluster analysis was performed using the superparamagnetic clustering method (Blatt et al., 1996) followed by a manual clustering strategy to avoid obvious outliers and misclassifications. The aforementioned data processing was carried out using the free-downloaded MATLAB toolbox, "Wave Clus" (Quiroga et al., 2004). Second, we extracted three features from the mean waveform of each classified spike cluster, i.e., the peak amplitude asymmetry, half width and trough peak. We applied kmeans clustering method to these features and we finally obtained two spike clusters (**Figure 2**). Based on the three features, we assumed that spikes whose waveforms show "wide" and "narrow" shapes were generated by putative PCs and INs, respectively (Sakata and Harris, 2009). The separability of these clusters was tested by the Hotelling's T-squared test (*P* = 0.022). It is well known that spiny stellate (SS) cells in Layer 4 are one of the INs in the neocortex. The spike's duration for SS cells is around 0.6 ms, which is within the range of that for the INs (i.e., 0.27– 0.65 ms) but different from that for the PCs, i.e., from 0.70 to 1.50 ms (Tierney et al., 2004). Therefore, based only on its duration it is difficult to distinguish a spike fired by a SS cell from one fired by a GABAergic INs. Meanwhile, a study using intracellular recording showed that SS cells in the stimulated barrel respond around 6–8 ms after the deflection (Armstrong-James et al., 1992). Based on this criterion, we selected the microelectrodes located around layer 4 of the barrel corresponding to the stimulated whisker. We picked up IN-like spikes observed at these microelectrodes in the post-stimulus period from 6 to 8 ms, and defined them as putative SS cells. The spiking times of PCs, INs and SS cells at each microelectrode were used as triggers to compute the spike-triggered average of the electric potentials (STAPs). The black cross in **Figure 2** corresponds to the features extracted for the SS cells. Clearly it is hard to distinguish SS cells from GABAergic INs in terms of spiking characteristics.

**FIGURE 2 | A classification of the detected spikes.** The **right panel** shows the classified mean spike waveforms of the excitatory pyramidal cells (PCs) and interneurons (INs). Black lines denote their mean spike waveforms. **Left panel** shows the spike waveforms as projected onto the three-dimensional feature space. The black cross indicates the mean spike waveform of the detected spiny stellate (SS) cells.

## **THE vCSD METHOD**

Neither the LFPs nor the unit activity independently represent the ionic flows across cell membranes, i.e., the volumetric CSD. Instead, they are external reflections of these electric currents through a highly conductive extracellular medium. The key component of our proposal is the vCSD method to reconstruct these trans-membrane ionic flows for both types of extracellular electric potentials. The main idea underlying the vCSD method is illustrated in **Figure 3**. Consider a 3D array of *N* = *nx* × *ny* × *nz* microelectrodes implanted in a neocortical ROI. The position of the probe inside the cortical regions is determined from the DiI traces left in the histological sections (**Figure 3A**). The symbols, *nx* and *ny* denote the numbers of shanks in the *x* and *y* directions, respectively, and *nz* represents the number of microelectrodes on each shank. The actual positions of these microelectrodes are *ri <sup>e</sup>* ∈ *R*3, (*i* = 1, 2, ··· , *N*) and the electric potentials observed at these microelectrodes are denoted by φ*i*. The ROI is divided into *M* = *mx* × *my* × *mz* cubic microscopic volumes. We called the resulting cubic mesh, with inter-node distance *d*, as the *"current source grid"* (**Figures 3B,C**). Discrete point current sources *Ij* (*j* = 1, 2, ··· , *M*) are defined at the grid points *r j <sup>s</sup>* ∈ *R*<sup>3</sup> of the current source grid. Note that the relationship between the actual vCSD value *Cj* and *Ij* at each grid point is represented by *Cj* = *Ij*/*d*3.

Under the validity of the quasi-static approach for the propagation of the electric field inside the brain tissue (Plonsey and Heppner, 1967), the Poisson equation is useful to relate the electric potentials and the current sources inside the brain

$$\nabla \cdot \left( \stackrel{\leftrightarrow}{\sigma} \, \nabla \phi \right) = -C,\tag{1}$$

where <sup>↔</sup> σ denotes the conductivity tensor. After solving the above partial differential equation independently for each time instant, the current sources defined on the discrete grid and the resulting electric potential φ*<sup>i</sup>* at the *i*th microelectrode can be related by the following biophysical model, known as the forward problem (Goto et al., 2010)

$$\Phi\_i = G(\vec{r}\_\varepsilon^i, \,\vec{r}\_s^j, \,\Theta)C\_j$$

$$= G\_{\vec{\imath}\vec{\jmath}}C\_{\vec{\jmath}},\tag{2}$$

where *G* is the generalized Green's function that is determined by the ROI's geometry, the boundary conditions, and the conductivity profile of the brain tissues, i.e., the volume conductor model. These physical properties are summarized in the parameter set in function *G*. Note that we are using Einstein symbolic sum notation. As a consequence of the superposition of the electric fields, the resulting electric potential at each microelectrode reflects contributions from all current sources. The relationship between electric potentials at all microelectrodes in the 3D array and the current sources at the grid points can be represented by the following algebraic equation

$$
\Phi = \mathbf{GC} \tag{3}
$$

where *-* = [φ<sup>1</sup> φ<sup>2</sup> ... φ*N*] *<sup>T</sup>* and *C* = [*C*<sup>1</sup> *C*<sup>2</sup> ... *CM*] *<sup>T</sup>* are vectors, and **G** is the discrete generalized Green's function matrix

$$\mathbf{G} = \begin{bmatrix} G\_{11} & G\_{12} & \cdots & G\_{1M} \\ G\_{21} & G\_{22} & \cdots & G\_{2M} \\ \vdots & \vdots & \ddots & \vdots \\ G\_{N1} & G\_{N2} & \cdots & G\_{NM} \end{bmatrix} . \tag{4}$$

For simplicity, we have ignored the time dependency in our definitions. The vCSD method consists of estimating **C** from measurements of *-*, which represents in fact an ill-posed inverse problem.

## **THE VOLUME CONDUCTOR MODEL**

The use of a realistic volume conductor model **G** constitutes one of the most significant differences between the vCSD method and other conventional methods for CSD analysis (e.g., Łe.ski et al., 2007; Potworowski et al., 2011). In some theoretical studies, inhomogeneity and/or anisotropy in the electric conductivity have been considered (Holt, 1998; Pettersen et al., 2006) on the basis of experimental evidence, e.g., in the cerebellum (Nicholson and Freeman, 1975; Okada et al., 1994) and in the neocortex (Hoeltzell and Dykes, 1979). However, most of CSD methods in the literature assumed an infinite, homogeneous and isotropic volume conductor model, denoted in this paper by the Green's function (*InfH*, **G**inf). It was demonstrated in the past that changes in the electric conductivity do not significantly affect the results obtained with the classic CSD method, which is based on the second-order spatial derivative of the electric potentials (Mitzdorf and Singer, 1980). Conversely, Goto et al. (2010) showed that misspecification of the volume conductor model in terms of both geometry and conductivity profile affect dramatically a more contemporary method, i.e., the iCSD3D method (Łe.ski et al., 2007). Goto et al. (2010) proved that the somatosensory cortex of rats can be locally approximated by six spherical shells, which were easily determined from fluorescent images of brain sections labeled by the fluorescent Nissl staining. Also, detailed measurements of the electric conductivity profile in this particular cortical region, revealed the existence of significant anisotropies (Goto et al., 2010). Based on this previous study, we used a spherical inhomogeneous and anisotropic (*SphIh*) volume conductor model, with corresponding Green's function **G**sph, for the somatosensory cortex of rats (**Figure 3D**), and used the mathematical strategy proposed by De Munck and Peters (1993) to calculate **G**sph numerically.

#### **SUPPRESSING THE EFFECT OF NOISE ON THE CSD RECONSTRUCTION**

Commonly, the number of point current sources is larger than the number of microelectrodes *M* >> *N*, and also, the linear operator based on function *G*(*r<sup>i</sup> <sup>e</sup>*, *r j <sup>s</sup>*, ) has a non-trivial null space; hence, the matrix **G** has an incomplete rank and is poorly conditioned. The use of a priori information about C has become a standard way to deal with this problem, giving rise to the well-known *"distributed inverse solution"* family. The low resolution electrical tomography (LORETA), which results

from a vector laplacian penalization to the optimization functional for the primary current density, constitutes, so far, one of the most acknowledged distributed inverse solutions for macroscopic EEG data (Pascual-Marqui et al., 1994). LORETA can be interpreted within the context of the *general smoothing splines* introduced by Wahba (1990) to solve noisy operator equations (Riera et al., 2006). LORETA inverse solution warrants not only smoothness of the reconstructed C but also forces it to be minimal on the boundary of the brain. Technically, the LORETA type of inverse solution of equation (3) results from minimizing the optimization functional *o* (**C**) = *-* − **GC**<sup>2</sup> + λ **LC**<sup>2</sup> respect to the CSD vector C. The matrix L is the discrete spatial Laplacian operator defined as

$$\begin{aligned} \mathbf{L} &= \frac{6}{d^2} (\mathbf{W} - \mathbf{E})\\ \mathbf{[W]}\_{\vec{\eta}} &= \begin{Bmatrix} \frac{1}{6}, \,\dot{\boldsymbol{y}} & \left\| \vec{r}\_s^i - \vec{r}\_s^j \right\| = d \\ 0 & \text{otherwise} \end{Bmatrix}, \quad \forall i, j = 1 \dots M \end{aligned} \quad (5)$$

where **E** is the *M* × *M* identity matrix.

Finally, the solution of the weighted linear regression problem is:

$$
\hat{\mathbf{C}} = \left(\mathbf{G'}\mathbf{G} + \lambda\mathbf{L'}\mathbf{L}\right)^{-1}\mathbf{G'}\Phi\tag{6}
$$

The estimation of the hyper-parameters λ is a problem of considerable importance, since it tells us about the accuracy of the electrophysiological instrument, the quality of the data in terms of the S/N ratio as well as the degree of smoothness to be introduced for the unknown vector C. In this paper, we used the generalized cross validation (GCV) method to estimate λ (Wahba, 1990). Therefore, the optimal λ minimizes the following evaluation

$$E(\lambda) = \frac{\|\mathbf{P}\Phi\|^2}{[tr(\mathbf{P})]^2} \tag{7}$$

where the projecting matrix *P* is defined as

function *E*(λ)

$$\mathbf{P} = \mathbf{E} - \mathbf{G} \left( \mathbf{G}' \mathbf{G} + \lambda \mathbf{L}' \mathbf{L} \right)^{-1} \mathbf{G}'.$$

The vCSD method was applied to grand-average ERPs and STAPs. To that end, immunostaining images were used to define the current source grid relative to the position of the microelectrodes. Based on the imprints of the shanks and insertion depth of the 3D array, we defined a rectangular current source grid, comprising *M* = 30 × 30 × 28 grid points with *d* = 50μm inter-grid distance. The CSD **C(t)** for each time instant was estimated by solving Equation (6) with **G** = **G**sph. The iCSD3D was also applied to the ERPs and the respective CSD in the current source grid was estimated. Note that in order to remove the dynamic effect of the signal observed at the reference electrode, we applied the average reference operator (Pascual-Marqui, 1999) to the Green's function matrices, ERPs and SRPs (Offner, 1950; Bertrand et al., 1985).

#### **EFFECT OF VOLUME CONDUCTOR MODEL ON THE CSD ANALYSIS**

Goto et al. (2010) have evaluated how misspecifications of the conductivity profile and the cortical geometry affect the CSD Riera et al. Volumetric CSD analysis

reconstruction using the iCSD3D method. Such a method is based on the assumptions of an infinite ROI with homogeneous and isotropic conductivity. Goto et al. (2010) found distortions in the CSD reconstructions, especially in the case of CSD distributions with charge-unbalances. In this study, we performed a simulation to ensure that such distortions are minimized by the proposed vCSD method when an appropriate volume conductor model is used. As in Goto et al. (2010), we employed a 3D array (*N* = 9 × 9 × 15 microelectrodes array, 100μm inter-electrode distance along the shank, 100μm inter-shank distances). We defined the rectangular current source grid which has *M* = 16 × 16 × 28 grid points with a resolution *d* = 50μm. Two types of CSDs were simulated. The first type was a sinusoidal function weighted by a Gaussian term, which represents charge balanced CSDs.

$$C\_{j} = \begin{cases} \sin\left[\frac{2\pi(z\_{j} - z\_{0})}{T}\right] \exp\left(-\frac{\sqrt{\mathbf{x}\_{j}^{2} + \mathbf{y}\_{j}^{2}}}{2l^{2}}\right) & \text{if } \left|z\_{j} - z\_{0}\right| < \frac{T}{2} \\ 0 & \text{otherwise} \end{cases} (8)$$

where *Cj* is the value of CSD at the *j*th grid point located in the tangential coordinates *xj*, *yj* and the radial depth *zj*, *l* is the full width at half-maximum (FWHM) in the *xy*-plane. The second type of CSD was a pure Gaussian function, which represents charge-unbalanced CSDs.

$$C\_{\dot{j}} = \exp\left(-\frac{\sqrt{\varkappa\_{\dot{j}}^2 + \wp\_{\dot{j}}^2 + \left(z\_{\dot{j}} - z\_0\right)^2}}{2l^2}\right) \tag{9}$$

where *l* is the FWHM in both the *xy*-plane and *z* direction. From these CSD distributions, we simulated electric potentials at the microelectrodes  by using equation (3) with the *SphIh* volume conductor model (i.e., **G**sph). After that, we performed both the vCSD analysis (i.e., **G** = **G**sph) and the iCSD3D method (i.e., **G** = **G**inf) to estimate *C*ˆ from the simulated data *-*. Finally, the reconstruction errors (RE) for both methods were evaluated from the estimated CSDs by the following criterion

$$\text{RE} = \sqrt{\sum\_{j=1}^{M} \left(\mathbf{C}\_{j} - \mathbf{\hat{C}}\_{j}\right)^{2}} \int \sqrt{\sum\_{j=1}^{M} \left(\mathbf{C}\_{j}\right)^{2}} \tag{10}$$

#### **CSD RECONSTRUCTION FROM NOISY DATA**

To assess the sensitivity of the proposed vCSD method to noise, we conducted another simulation study. We employed a cubic current source mesh which had *M* = 24 × 24 × 24 grid points with an inter-grid distance *d* = 50μm. The following Gaussian type of CSD distribution was used.

$$C\_{\vec{j}} = \exp\left(-\frac{\left\|\vec{r\_j} - \vec{r\_0}\right\|}{2l^2}\right) \tag{11}$$

where <sup>→</sup> *r*<sup>0</sup> is the center of the Gaussian function which for all trials was selected randomly within the ROI. The FWHM *l* was fixed at 400μm throughout this simulation study. The electric potentials at the microelectrodes  were calculated from this CSD distribution by using equation (3) with **G** = **G**inf. We computed the potentials β that included an additional noise term

$$
\Phi\_{\emptyset} = \Phi + \xi \tag{12}
$$

where **ξ** ∝ **N** 0, σ<sup>2</sup> is a Gaussian noise with zero mean and variance **σ**2. The variance **σ**<sup>2</sup> was determined from the sample variance of *-*

$$\sigma^2 = \beta \sum\_{i=1}^{N} \left( \phi\_i - \frac{1}{N} \sum\_{i=1}^{N} \phi\_i \right)^2 \tag{13}$$

where the parameter **β** ∈ [0.01, 0.05, 0.1, 0.5] determined the level of noise.

We estimated the CSD distribution *C*ˆ from simulated data β by the vCSD method, i.e., Equation (6) with **G** = **G**inf, and the iCSD3D method (**G** = **G**inf). The REs were calculated for each value of β. Additionally, we evaluated the impact of the resolution of the microelectrode array on the CSD reconstruction by calculating the respective RE for arrays with 200, 300, 400, and 600 μm inter-electrode distances.

#### **LOCALIZATION OF THE BARRELS**

To evaluate the accuracy of the vCSD and iCSD3D methods, we used estimated CSD distributions to detect the barrels corresponding to the particular deflected whiskers. We manually registered the anatomical barrels covered by the current source grid. First, we picked up one *xy*-plane **U** ⊂ *R*<sup>2</sup> in the current source grid at the approximated depth of layer 4. This plane (*ML*<sup>4</sup> = 24 × 24 grid points) was superimposed with the immunostaining image and the relative position of each grid point was defined as *rk* ∈ *U,* (*k* = 1, 2, ..., *ML*4). Note that grid points outside of the microelectrode grid, i.e., the outermost three grid points in both the *x* and *y* directions and the outermost two grid points in *z* direction in the 30 × 30 × 28 sized current source grid, were used to equivalently introduce a free boundary condition that allow us to accommodate outside current sources Łe.ski et al. (2007). CSD values at those grid points were ignored.

Second, we defined a binary value *ak* at each grid point in the two-dimensional grid plane *rk*, resulting in a vector **A** = (*a*1, *a*2, ..., *aML*<sup>4</sup> ). Third, we defined a barrel space **B** ⊂ **U** manually from the immunostaining images. And finally, the elements of **A** were set by the following criterion

$$a\_k = \begin{cases} 1 & \text{if } \vec{r}\_k \in \mathbf{B} \\ 0 & \text{otherwise} \end{cases} \tag{14}$$

We used the binary vector **A** (i.e., the anatomical barrel) as the Gold Standard for evaluating the accuracy with which barrel were detected by the CSD methods*.* The following thresholding method was used to detect a barrel:

1. We normalized the CSD distribution *C*¯(*rk*).

$$
\hat{C}\_n(\vec{r}\_k) = \hat{C}(\vec{r}\_k) \int \max\_{\vec{r}\_k \in U} \left( \hat{C}(\vec{r}\_k) \right) \tag{15}
$$

2. We defined another binary vector **F**(α) = *f*1(α), *f*2(α), . . . , *fML*<sup>4</sup> (α) as the functional barrel. A threshold α in the interval [0, 1] was used to define the elements of **F**(α) by the following criterion.

$$f\_k(\alpha) = \begin{cases} 1 & \text{if } \hat{\mathcal{C}}\_n(\vec{r}\_k) \ge \alpha \\ 0 & \text{otherwise.} \end{cases} \tag{16}$$

3. We determined the threshold α<sup>∗</sup> in a way that the functional barrel has same area as that of the corresponding anatomical barrel, i.e., difference of the total summation of the components in the binary vectors **A** and **F**(α) is minimized.

$$\alpha^\* = \underset{\alpha \in \{0, 1\}}{\arg\min} \left| \left\| \mathbf{A} \right\|^2 - \left\| \mathbf{F}(\alpha) \right\|^2 \right| \tag{17}$$

4. For evaluating the detection accuracy, i.e., the localization error, we used the normalized distance between the anatomical **A** and functional **F**(α∗) barrels.

$$\text{LocalizationError} = \frac{\left\|\mathbf{A} - \mathbf{F}(\boldsymbol{\alpha}^\*)\right\|^2}{\left\|\mathbf{A}\right\|^2} \tag{18}$$

In this formalism, the localization errors corresponding to the best and worst detected barrel are 0.0 and 1.0, respectively. We found no computational problems for all barrels analyzed with this method.

## **STATISTICAL ANALYSIS**

The Kolmogorov–Smirnov test was used to determine the wavelet coefficients that better represent the spikes. The Hotelling's Tsquared test was used to evaluate the separability of clusters in the spike's parameter space. Pair-based comparisons were performed using the one-tailed *t*-test and Mann-Whitney *U* test for the REs and *Localization Error*, respectively.

## **RESULTS**

**Figure 4** summarizes the methodology for fast assessments of the electrical activity of cortical networks in the barrel field of Wistar rats. Single whisker evoked-potentials were recorded from the somatosensory barrel cortex by using the 3D array. These potentials were separated into LFPs and unit activities by applying low and high range band-pass filters, respectively. We extracted LFPs for single trial responses, and also computed the ERPs. At the same time, neuronal spikes were detected at each microelectrode, and classified into excitatory PCs and INs. An additional criterion was applied to distinguish SS cells from INs (see Material and Methods). The spiking times of the classified cells were used to compute the STAPs. Cortical current sources associated with single trial LFPs, ERPs, and STAPs were processed through the vCSD method (**Figure 5**). The example movie of a single trial response can be seen in the Supplementary Video and an explanation for a particular time instant is in **Figure 6**.

**FIGURE 4 | The methodology for fast assessments to the electrical activity of cortical networks in the barrel field of Wistar rats.** Electric potentials recorded with the 3D array under single whisker deflections are divided into LFPs and unit activities via corresponding band-pass filters (1–500 Hz for LFPs and 0.5–8.0 kHz for unit activity). Event related and single trial potentials are computed from the LFPs. The spike triggered

average of the electric potentials (STAPs) for pyramidal cells (PCs) and interneurons (INs) are obtained through spike detection and clustering methods. Finally, the vCSD method is applied to the STAPs as well as to the averaged (ERPs) and single trial LFPs to estimate the spatiotemporal CSD maps associated with single unit activities and population inputs, respectively.

**FIGURE 5 | Example of the CSD distributions estimated for single unit (left panel) and population synaptic (right panel) activities.** The CSD distributions are represented in three dimensional contours. The contours denoted by meshes and patches represent the weak (30% of the maximum) and strong (70% of the maximum) intensity of the CSD, respectively. In the **left panel**, orange and magenta are used for the current sink of the excitatory [pyramidal (PC) and spiny stellate (SS) cells] and inhibitory (IN) neurons,

respectively. Blue and green are for the current sources generated by PC/SS and IN, respectively. In the **right panel**, red and blue are used to represent current sink and source, respectively. The CSD maps were estimated from instantaneous ERPs at 8 ms post-stimulus time of a single whisker deflection. On the right frame, the white circles in the histological image denotes the barrels and one of them, indicated by the arrow, corresponds to the barrel associated to the deflected whisker in this particular condition.

## **EFFECT OF VOLUME CONDUCTOR MODEL ON THE vCSD ANALYSIS**

We conducted computer simulations to evaluate the effect on the vCSD method of certain misspecifications in the volume conductor model. **Figure 7A** shows, for a single trial, the actual current sources (left panels) used to generate the electric potentials, as well as their reconstructions by means of the vCSD method in the cases of employing the *InfH* (center panels) and *SphIH* (right panels) volume conductor models, respectively. The current sources in the upper and lower panels were created using a balanced (*sinusoidal*) and an unbalanced (*Gaussian*) model,

respectively. When the *SphI*h volume conductor model was used, the CSD was accurately reconstructed for both small and large sized sources (reconstruction error, REs <2%). However, the CSD reconstructions obtained using the *InfH* volume conductor model showed significant distortions in the spatial configuration with larger REs. These distortions were more prominent for the case of charge-unbalanced models of the current sources. High REs for the charge-unbalanced CSD reconstructed by the iCSD3D method were also reported in Goto et al. (2010). The statistics for the REs reported in **Figure 7B** were obtained by performing ten single trial simulations with current sources centered at different depths along the cortical lamina.

## **CSD RECONSTRUCTION FROM NOISY DATA**

We have also performed a second simulation study to compare the noise sensitivity and the spatial resolution of the iCSD3D and vCSD methods. To this end, we simulated electric potentials which were contaminated with observational noise at different levels from 1 to 50%. In order to use the original iCSD3D method, we employed in this particular simulation study the *InfH* volume conductor model for both methods. Note that downloadable MATLAB code for the iCSD3D model is only available for *InfH* volume conductor. Additionally, these electric potentials were calculated for silicon-based probes with three dimensional microelectrodes arrays having different spatial resolutions, i.e., from 0.2 to 0.6 mm inter-electrode distances. **Figure 7C** shows the reconstructed current sources by both iCSD3D (upper panels) and vCSD (lower panels) methods for the particular case of a silicon-based probe with inter-electrode distance of 0.2 mm. In this figure, each column-wise panel shows the CSD reconstruction for different noise levels. Color maps with the REs for both types of CSD analysis methods as a function of the noise level and the inter-electrode distance are shown in **Figure 7D**. The iCSD3D method was able to correctly reconstruct current sources for low levels of observational noise. However, the REs, in this particular case, increased rapidly with the inter-electrode distance. The situation was dramatically inverted when large noise contamination in the observed electric potentials existed, with a very poor reconstruction for higher resolution MEA. However, substantial improvements were achieved when we reduced the electrode's resolution (i.e., increase inter-electrode distance). These improvements were observed in 50% of noise level only for inter-electrode distances longer than 0.4 mm. Even though they were smaller, significant differences in the REs were found (*P* < 0.001) between the vCSD and iCSD3D methods. This simulation study revealed an intrinsic tradeoff in the iCSD3D method, which results from the lack of a regularization term to stabilize an inverse operator defined from a highly ill-conditioned matrix. Such compromise between the noise level in the data and the spatial resolution of the microelectrode array was not observed in the case of using the vCSD method. The vCSD method kept acceptable performance even at 50% noise level and inter-electrode distance of 0.2 mm. In the current simulation study, we employed 50 trials for each noise level and microelectrode array's resolution.

## **LOCALIZING SINGLE BARRELS USING THE vCSD METHOD**

In this study, we used the actual anatomical barrels as the *"Gold standard"* to validate our methodology for single whisker deflection. The main reasons for using the anatomical barrels come from the structure of the barrels and their spatiotemporal synaptic responses to single whisker stimulation. First, the main inputs from the thalamus to the somatosensory barrel field arrive at layer 4 of the cortex, where the SS cells process them. The arriving times of these first sensory inputs are around 6–8 ms after the whisker deflection (Armstrong-James et al., 1992; Wilent and Contreras, 2004). After the SS cells receive these inputs, they increase their activity by self-feedback mechanisms within the corresponding barrel (Feldmeyer et al., 1999). During this period, synapses of the SS cells play the main role in producing post-synaptic potentials. The dendrites of SS cells located inside a particular barrel extend mainly to the center of the barrel, indicating most of these synapses are delimited to the single barrel (Woolsey et al., 1975; Petersen and Sakmann, 2000; Egger et al., 2008). Based on these facts, the synaptic activities of SS cells in response to the single whisker deflections are limited within the corresponding barrel, i.e., current sinks in the period of 6–8 ms after a single whisker deflection are confined within the barrel. This hypothesis has been supported by previous *in vitro* and *in vivo* studies. For instances, *in vitro* field EPSP recordings in the barrel field after the electric stimulation of its center showed that excitatory neuronal circuits within layer 4 are functionally confined to each barrel (Petersen and Sakmann, 2000). Additionally, *in vivo* VSDI imaging, and *in vivo* extracellular recordings by horizontal planar Utah intra-cortical microelectrode array (in combination with spike histogram analysis) showed that the barrels corresponding to the deflected whiskers could be well-localized (Petersen and Diamond, 2000; Petersen et al., 2003). Taking into account the anatomical and functional characteristics of the barrels, we used actual whisker ERPs recorded by our 3D array at this particular time instant to evaluate the performance of the vCSD method. By means of Dil staining, we were able to co-register the CSD distribution and the anatomical barrels, which were clearly determined from the *xy*-images of cytochrome C oxidase immunostaining at the level of layer 4 (**Figure 8A**). We estimated the CSD distributions in the period of 6–8 ms post-stimulus (**Figure 8B**). We defined the functional barrels from the CSDs based on a thresholding method. The thresholds were chosen in a way that the functional barrels have the same area as the corresponding anatomical barrels (**Figures 8C,D**). Both anatomical and functional barrels were represented by binary vectors whose lengths represented the total number of grid points on the *xy*-plane of the current source grid at the level of layer 4. **Figure 8E** shows localization errors of the functional barrels for reconstruction with the vCSD and iCSD3D methods. The results from single trial comparison are shown in **Figure 9**. We found that the localization error of the vCSD method was lower than 20% (19.0 ± 6.1%) and this method always produces more accurate reconstructions than the iCSD3D method (41.4 ± 10.1% localization error).

## **DISCUSSION**

We have demonstrated that, by the combination of mathematical methods and high technology, it is possible to image the activity of neuronal networks from extracellular electric recordings at resolutions unprecedented for electrophysiological methods. For the first time, to our knowledge, images with high spatial-resolution in both the horizontal plane (i.e., cortical columns) and translaminar axis (i.e., cortical layers) are obtained from electrophysiological recordings in similar fashions to those achieved via multi-photon fluorescent microscopes. From actual electrophysiological recordings, we reconstructed the CSD distributions at any depth of the barrel cortex separating cortical inputs from their outputs. Additionally, we were able to discriminate spike-related CSD distributions for different types of cells. Our methodology will be quite useful for a variety of applications in neuroscience,

**methods. (A)** CSD distribution at 6–8 ms post-stimulus with an illustration of the *xy*-plane at the level of layer 4. **(B)** Co-registration of the *xy*-plane of the current source grid at the depth of layer 4 and the anatomical image acquired from the cytochrome c oxidase stained tangential brain section of the barrel cortex. The position of the shanks are determined from Dil staining images. **(C)** Superposed pictures in the *xy*-plane of the CSD

distribution at the depth of layer 4 and the anatomical barrel denoted by cyan dots. **(D)** The functional barrel (yellow circle) obtained from the *xy*-slice of the CSD distribution and the anatomical barrels. **(E)** Localization errors between the anatomical barrels and the corresponding functional barrels estimated by iCSD3D and vCSD methods. The one-tailed Mann-Whitney *U* Test (*P* = 0.004, *n* = 5) was used to compare the performances of these two methods.

**FIGURE 9 | Five samples of anatomical barrels (cyan dots) and their corresponding functional barrels (yellow circles) estimated by the iCSD3D (top row) and vCSD (bottom row) methods.** It can be seen that the vCSD method provided better estimates of the functional barrels than those obtained by the iCSD3D method. We pointed out that among the main reasons for the inaccuracy of the iCSD3D method are the misspecification of the volume conductor model and/or the effect of systematic noise in the data.

from both a biophysical and an electrophysiological point of view. For example, from the cable theory for PCs, it is known that the spatial summation of the trans-membrane current must be zero. However, by means of the vCSD analysis, it was possible to evaluate this hypothesis for the case of STAPs (Riera et al., 2012). Also, in this latter study, the same analysis was applied to evaluate multipolar contributions to the LFP recordings. These issues are important in the light of recent interests in elucidating fundamental principles of the EEG and MEG genesis. Also, by the proposed methodology, it will be possible to identify layers, and determine detailed interactions between these layers but also columns (barrels) in behaving rats. Our group is currently using the proposed methodology to determine the spatial codifiers of the whisker velocity and direction (*unpublished data*). Our methodology could be extended in the future to study other cortical regions and species.

Methods to perform CSD analysis on data recorded with three-dimensional MEAs are still under development, with only different volumetric version of the inverse CSD (iCSD, Pettersen et al., 2006) method available in the literature (i.e., the iCSD3D method, Łe.ski et al., 2007; the kCSD method, Potworowski et al., 2012). The main idea behind these methods is to use interpolating splines to represent the extracellular electric potentials, and thus to indirectly introduce specific priors for the density of current sources *C*. The iCSD3D method was recently improved by formulating it in the context of reproducing kernel Hilbert spaces and introducing a Tikhonov regularization strategy (Potworowski et al., 2012). These authors used a cross-validation technique to determine the best value for the regularization parameter λ whenever the data are corrupted with noise. In contrast, the proposed method is the first introducing smoothing constraints directly to the brain current sources *C* over extended regions of the barrel cortex to solve the inverse problem underlying the CSD analysis. The performance of the proposed method was evaluated, and compared with that for the iCSD3D method, using simulated data with different noise levels and electrode grid resolutions.

Although the iCSD3D method can be trivially generalized to more complex volume conductor models, it originally assumed for simplification that the brain tissues are homogeneous and isotropic. In this study, we claim that more realistic volume conductor models for the brain tissues of interest must be used to considerably improve the accuracy of the three-dimensional CSD analysis. The iCSD3D method has been applied in the past to averaged extracellular electric potentials obtained from the deep forebrain of one adult male Wistar rat during whisker stimulation, with an insertion/recording strategy that allow to cover a volume of (2.8 × 3.5 × 4.9) mm with a total of 140 electrodes. However, we have evaluated its performance in this study using not only simulations but also an experimental paradigm for a gold standard.

The method developed in this work is directly applicable to perform CSD analysis whenever the following conditions are met: (a**)** recordings of extracellular potentials are performed with a tridimensional MEA, (b**)** the conductivity profile of the area of study is layer-wise inhomogeneous and anisotropic, and (c**)** the geometry is approximately spherical. As a consequence of its extensive use by the community, we have developed the entire methodology for the particular case of the barrel cortex of rats. Although the application to other cortical areas of the rats might be straightforward, its use in other species and brain regions must be carefully evaluated in accordance with the respective conductivity profiles and geometries (e.g., somatosensory cortex of cats, Hoeltzell and Dykes, 1979; CA1 of guinea pigs, Holsheimer, 1987; cerebellum of turtles, Okada et al., 1994; visual cortex of monkeys, Logothetis et al., 2007).

## **FUTURE DEVELOPMENTS**

The application of the multi-photon fluorescent imaging technique to study the brain constitutes one of the most remarkable achievements in the era of the colored revolution in neuroscience (Denk et al., 1990; Vonesch et al., 2006). By combining this technique with the bulk-loading method for membrane-permeable Ca2<sup>+</sup>-indicator dyes (Stosiek et al., 2003), both sensory-evoked and ongoing activity in neuronal populations have been observed *in vivo* from rodent/cat neocortex with the spatial resolution of single neurons. Recently, the technique has benefited from the latest technological and methodological advances in the evaluation of both neuronal spiking (Wallace et al., 2008) and volumetric activity (Göbel et al., 2007; Cheng et al., 2011). However, there are several limitations of the multi-photon fluorescent imaging technique, which make the methodology proposed in this study a better option for observing neuronal population activities in a variety of neuroscience problems. First, except when using voltage-sensitive fluorescent dyes, multi-photon microscopic imaging commonly constitutes an indirect measurement of the actual membrane potentials (i.e., it senses slow changes in the intracellular Ca2<sup>+</sup> concentrations). As a consequence, it is hard to distinguish subthreshold neuronal activity (i.e., post-synaptic inputs) from spiking (i.e., axonal outputs). In comparison to Ca2<sup>+</sup>-indicator dyes, the sensitivity of voltage-sensitive fluorescent dyes for imaging subthreshold electrical activity is excellent. Unfortunately, the latter lack single-cell spatial resolution *in vivo* (Kuhn et al., 2008) and are deficient in terms of the S/N ratio. Also, alterations in the cellular physiology have been associated with the use of voltage-sensitive fluorescent dyes (Mennerick et al., 2010). Second, multi-photon imaging still suffers from poor time resolution even though a lot of technical progresses have been made recently (Göbel et al., 2007; Planchon et al., 2011), Thus far, precisions of a few milliseconds have been achieved by combining some of these methods (Grewe et al., 2010), but actual reconstructions of spike dynamics, propagation and timings from fluorescence traces are just about to happen. Third, the neocortical tissues are high light-scattering media, which results in an imaging-depth limit (Theer and Denk, 2006). By combining regenerative amplifiers (Theer et al., 2003) with genetically encoded calcium indicators, even layer 5 (up to 800μm) have been recently imaged *in vivo* (Mittmann et al., 2011) although image resolution at that depth is poor. In principle, MEA could be combined with silicon photonics to take advantages of optical applications quickly developed in this decade. The methodology proposed in this study could also benefit from recent advances in MEA fabrication. For example, to improve interaction with neural cells, microelectrodes built from nanoscale bioactive coatings (e.g., polymers) have been proposed (Richardson-Burns et al., 2007). By means of multiplexing and telemetry techniques, miniaturized and wireless multi-channel systems are speedily developing for recording neural signals from behaving small animals (e.g., rats, Szuts et al., 2011).

The MATLAB code for the vCSD analysis is available at the following website http://web.eng.fiu.edu/jrieradi/.

## **ACKNOWLEDGMENTS**

Dr. Takeshi Ogawa participated in establishing the immunostaining protocols used in this study. We thank Neuronexus Tech. for working close to us during three years to achieve a suitable 3D array. Also, we would like to express our gratitude to Prof. Roberto Pascual-Marqui (University Hospital of Psychiatry, Zurich) for participating during the preparation of the present study and to Jose Matteo for revising the final manuscript. This work has been supported by the JSPS Grants-in-Aid (B) 18320062 and JSPS Grant-in-Aid for Young Scientists (B) 23700492.

## **SUPPLEMENTARY MATERIAL**

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/journal/10.3389/fncir. 2014.00004/abstract

## **REFERENCES**


by voltage-sensitive dye imaging combined with whole-cell voltage recordings and anatomical reconstructions. *J. Neurosci*. 23, 1298–1309. Available online at: http://www.jneurosci.org/content/23/4/1298.short


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 03 October 2013; accepted: 09 January 2014; published online: 05 February 2014.*

*Citation: Riera JJ, Goto T and Kawashima R (2014) A methodology for fast assessments to the electrical activity of barrel fields in vivo: from population inputs to single unit outputs. Front. Neural Circuits 8:4. doi: 10.3389/fncir.2014.00004*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Riera, Goto and Kawashima. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Modulatory effects of inhibition on persistent activity in a cortical microcircuit model

#### *Xanthippi Konstantoudaki 1,2, Athanasia Papoutsi 1,2, Kleanthi Chalkiadaki 1,2, Panayiota Poirazi <sup>2</sup> \* and Kyriaki Sidiropoulou1,2*

*<sup>1</sup> Department of Biology, University of Crete, Heraklion, Greece*

*<sup>2</sup> Institute of Molecular Biology and Biotechnology, Foundation for Research and Technology – Hellas, Heraklion, Greece*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Jason Sherfey, Boston University, USA Margarita Zachariou, University of Cyprus, Cyprus*

#### *\*Correspondence:*

*Panayiota Poirazi, Computational Biology Lab, Institute of Molecular Biology and Biotechnology, Foundation for Research and Technology – Hellas, 100, N Plastira str., GR 71110, Heraklion, Greece e-mail: poirazi@imbb.forth.gr*

Neocortical network activity is generated through a dynamic balance between excitation, provided by pyramidal neurons, and inhibition, provided by interneurons. Imbalance of the excitation/inhibition ratio has been identified in several neuropsychiatric diseases, such as schizophrenia, autism and epilepsy, which also present with other cognitive deficits and symptoms associated with prefrontal cortical (PFC) dysfunction. We undertook a computational approach to study how changes in the excitation/inhibition balance in a PFC microcircuit model affect the properties of persistent activity, considered the cellular correlate of working memory function in PFC. To this end, we constructed a PFC microcircuit, consisting of pyramidal neuron models and all three different interneuron types: fast-spiking (FS), regular-spiking (RS), and irregular-spiking (IS) interneurons. Persistent activity was induced in the microcircuit model with a stimulus to the proximal apical dendrites of the pyramidal neuron models, and its properties were analyzed, such as the induction profile, the interspike intervals (ISIs) and neuronal synchronicity. Our simulations showed that (a) the induction but not the firing frequency or neuronal synchronicity is modulated by changes in the NMDA-to-AMPA ratio on FS interneuron model, (b) removing or decreasing the FS model input to the pyramidal neuron models greatly limited the biophysical modulation of persistent activity induction, decreased the ISIs and neuronal synchronicity during persistent activity, (c) the induction and firing properties could not be altered by the addition of other inhibitory inputs to the soma (from RS or IS models), and (d) the synchronicity change could be reversed by the addition of other inhibitory inputs to the soma, but beyond the levels of the control network. Thus, generic somatic inhibition acts as a pacemaker of persistent activity and FS specific inhibition modulates the output of the pacemaker.

**Keywords: prefrontal cortex, NMDA, synchronicity, fast-spiking interneurons, connectivity, parvalbumin interneurons**

## **INTRODUCTION**

Neurons in the prefrontal cortex (PFC) have been shown to exhibit activity that often persists past the end of the stimulus, as recorded *in vivo* during the delay period of working memory tasks (Goldman-Rakic, 1995). This persistent activity corresponds to the on-line representation of a memory for a short period of time. Its emergence has been shown to depend on the balance of excitation, provided by glutamatergic neurons, and inhibition, provided by GABAergic interneurons (Goldman-Rakic, 1995; Compte, 2006) as well as on single neuron dynamics (Sidiropoulou et al., 2009; Yoshida and Hasselmo, 2009). Specifically, activation of NMDA glutamate receptors has been shown to have a highly significant role on supporting stable persistent activity, from computational studies (Wang, 1999; Compte et al., 2000), *in vitro* brain slice experiments (McCormick, 2003) and *in vivo* recordings in monkeys (Wang et al., 2013). Relatively few studies, however, have investigated how interneuron structure and physiology contributes to physiological prefrontal cortical (PFC) function (Rao et al., 1999; Wang et al., 2004).

Interneurons exhibit great diversity in their distribution, connectivity, neurochemistry, synaptic connections and electrophysiological properties. Three main classes have been identified based on their electrophysiological characteristics, namely the FS, the regular-spiking (RS) and the irregular-spiking (IS) interneurons (Markram et al., 2004).

FS interneurons exhibit fast, high-frequency and short duration action potentials. Morphologically, they have been identified as chandelier and basket neurons and they express the calciumbinding protein, parvalbumin (PV). They innervate the soma and the proximal dendritic compartments of pyramidal neurons in the PFC (Wang et al., 2002; Zaitsev, 2005). RS interneurons exhibit RS firing pattern, are mostly double bouquet and Martinotti-type cells andexpress the protein calbidin (CB) (Cauli et al., 1997). They have been shown to innervate the distal dendritic compartments of pyramidal neurons in the PFC. IS neurons exhibit IS firing pattern, are primarily bipolar cells and express the protein calretinin (CR). They project to the dendritic compartments of both PV- and CB- positive cells—suggesting that at least some types of CR-positive cells might be disinhibitory—as well as the distal dendritic compartments of pyramidal neurons in the PFC (Cauli et al., 1997).

Interneuron activity has been shown to contribute to cortical dynamics, network oscillations (Bartos et al., 2007), neuronal synchronization of pyramidal neurons (Guidotti et al., 2005), sensory processing (Börgers and Kopell, 2008), memory function (Jensen et al., 2007), goal-directed behavior (Kvitsani et al., 2013) and social behavior (Yizhar et al., 2011). Specific inactivation of PV interneurons was shown to lead to decreased gamma oscillations in the PFC (Sohal et al., 2009). With regards to working memory function and persistent activity, earlier experimental studies have suggested a role for GABAA-mediated inhibition in shaping the memory fields in PFC (Rao et al., 1999). Moreover, *in vitro* experiments suggest that GABAA activation prevents the generation of high frequency epileptiform bursts, while GABAB activation contributes to termination of up-and-down states, a physiological phenomenon related to persistent activity (Mann et al., 2009). Finally, computational studies have implicated the activity of PV/FS interneurons in persistent activity induction and of CB/RS interneurons in mediating the resistance to distractors which can prematurely terminate persistent activity (Wang et al., 2004).

However, there are still many unanswered questions with regards to which biophysical or connectivity properties of the different types of interneurons mediate persistent activity induction, firing frequency characteristics and neuronal synchronicity. In an effort to provide some answers to these questions, we extended a recently developed PFC microcircuit model (Papoutsi et al., 2013) to include the three main types of interneurons, i.e., the FS, regular-spiking (RS), and irregular-spiking (IS) interneurons. We used this modeling tool to dissect the role of different interneuron types in persistent activity and to determine whether the connectivity profile or the physiological properties of these interneuron subtypes mediate their roles in persistent activity.

Our simulations showed that (1) the NMDA current onto the FS interneuron can modulate the induction of persistent activity, but not its maintenance properties, (2) reducing FS model input to the pyramidal neuron models did not allow for NMDA or GABAB-dependent modulation of persistent activity induction, and significantly increased the firing frequency, ISI variability and neuronal synchronicity during persistent activity, (3) the firing frequency/ISI variability changes could not be altered by the addition of any other type of inhibitory input to the soma (RS or IS-mediated) and (4) the synchronicity change could be reversed, but beyond the levels of the control network by the addition of non-FS inhibitory input to the soma. Overall, our data suggest that somatic inhibition acts as a pacemaker of persistent activity with the FS interneuron modulating the output of the pacemaker.

## **MATERIALS AND METHODS**

Four different compartmental model cells were built, based on known electrophysiological data: one pyramidal neuron and three different interneurons, an FS model, an RS model and an IS model. They were connected in a network, which comprised 16 pyramidal models and 4 interneuron models (2 FS models, 1 RS and 1 IS model). Connectivity between the model neurons was based on experimental anatomical and electrophysiological data, as described below. All models are implemented in the Neuron simulation environment (Hines and Carnevale, 2001) and simulations were executed on a xeon cluster (8 core xeon processors).

## **PYRAMIDAL NEURON MODEL**

The pyramidal neuron model used was based on the one published in Papoutsi et al. (2013) and consists of a soma, a basal, a proximal and a distal dendritic compartment. It includes modeling equations for 14 types of ionic mechanisms, known to be present in these neurons, as well as modeling equations for the regulation of intracellular calcium (same equation as in Papoutsi et al., 2013). The passive and active properties of the pyramidal neuron model was validated according to experimental results of Nasif et al. (2004) (**Table 1** and **Figure 1**). The dimensions of the somatic, axonic, and dendritic compartments of the pyramidal model cell, as well as the passive and active parameters of the model neuron are listed in the supplemental text (Supplemental Tables 1, 2).

#### **INTERNEURON MODELS**

All three interneuron models included ionic mechanisms for the fast Na+, A-type K+, and delayed-rectifier K<sup>+</sup> currents, as well as modeling equations for the regulation of intracellular calcium buffering mechanism (same equations as in Papoutsi et al., 2013). In addition, each different interneuron model subtype included additional ionic mechanisms known to be present in each type (Toledo-Rodriguez et al., 2005), as detailed in the following paragraphs.

**Table 1 | Input resistance values of the model neurons and those obtained from electrophysiological data.**


**FIGURE 1 | Pyramidal neuron model validation. (A)** Model response to a current-step pulse at the soma (0.17 nA). **(B)** Experimental response of a PFC layer V pyramidal neuron to a current step-pulse (adapted from Nasif et al., 2004).

## **FS INTERNEURON MODEL**

The FS interneuron model consisted of three compartments: a somatic, a dendritic and an axonic compartment (Supplemental Table 1). The somatic compartment included mechanisms for the slow K<sup>+</sup> current (IKslow), the N-type high-threshold activated Ca++ current (N-type) and the hyperpolarization-activated cation current (Ih) (**Table 2**), in addition to the ones mentioned above. The membrane capacitance was set to 1.2μF/cm2 and axial resistance to 150 ohm/cm (**Table 2**). The resting membrane potential was adjusted to −73 mV and its resulting input resistance was 250 M- (Kawaguchi and Kubota, 1993) (**Table 1**). The APs of this FS model neuron had short duration and large afterhyperpolarization. It responded to a depolarizing current pulse (0.05 nA, 500 ms) with six spikes, as shown in **Figure 2A** (top), with an action potential threshold of −53 mV. A depolarizing current of 0.2 nA, 500 ms resulted in a (10 spikes 100 ms) 100 Hz response (**Figure 2A**, bottom).

## **RS INTERNEURON MODEL**

The RS interneuron model consisted of three compartments: a somatic, a dendritic and an axonic compartment (Supplemental Table 1) and included mechanisms for the low-threshold Ca++ current (T-type) and the Ih (**Table 3**). The membrane potential was adjusted to −64 mV (Kawaguchi and Kubota, 1993). The membrane capacitance was set to 1.2μF/cm2 and the axial resistance to 150 ohm/cm (**Table 3**). The resulting input resistance is 487 M- (**Table 1**). The model neuron responded to a depolarizing current pulse (0.05, 500 ms) with 15 spikes, with an action potential threshold of −51 mV (**Figure 2B**, bottom). A depolarizing current of 0.2 nA, 500 ms resulted in a 60 Hz response (**Figure 2B**, top).

## **IS INTERNEURON MODEL**

The IS interneuron model consisted of four compartments: a somatic, two dendritic and an axonal compartment, simulating a bipolar cell (Supplemental Table 1), and included mechanisms for slow K+ current, fast Ca++-activated K+ current and N-type Ca++ current (**Table 4**). The membrane potential was adjusted to -70 mV (Kawaguchi and Kubota, 1993), the membrane capacitance to 1.2μF/cm2, and axial resistance to


150 ohm/cm (**Table 4**). Its input resistance (∼545 M-) as indicated by electrophysiological data (Zaitsev, 2005) (**Table 1**). The typical discharge of this cell in response to depolarizing current pulses consisted of the emission of an initial cluster of two to six APs, depending on the level of depolarization, followed by APs emitted at an irregular frequency (Cauli et al., 1997). The discharge frequency increases as a function of the stimulation intensity according to electrophysiological results of (Cauli et al., 1997) (**Figure 2C**).

## **MICROCIRCUIT MODEL**

We constructed a microcircuit of 20 neuron models: 16 pyramidal models, based on Papoutsi et al. (2013), 2 FS interneuron models, 1 RS interneuron model and 1 IS interneuron model, so that the relative number of interneurons to pyramidal model neurons was 20% (Dombrowski et al., 2001) and the relative inhibitory input coming from FS interneurons was 50% (**Figure 4**). Connectivity properties including the location and number of synaptic contacts, the latencies between pairs of neurons, as well as the electrophysiological properties of their synaptic connections, were based on anatomical and electrophysiological data, similar to the values reported in Papoutsi et al. (2013). Specifically, pyramidal neuron models were fully connected recurrently (Wang et al., 2006) at their basal dendrites with latencies drawn from a Gaussian distribution with μ = 1.7 ms and σ = 0.9 (Thomson and Lamy, 2007). Autaptic contacts were also included and were adapted to 1/3 of excitatory connections (Lubke et al., 1996).

## **CONNECTIVITY**

The axon of each pyramidal neuron model projects to the basal dendrite of other pyramidal neuron models. Pyramidal neuron models also projected to the dendrites of FS models, IS model and RS model. However, specificity of synaptic innervations in the neocortex implies that the recurrent network is not randomly arranged (Yoshimura and Callaway, 2005). The axons of the FS interneuron models project to the soma of all pyramidal neuron models. The axon of the RS interneuron model projects to the distal apical dendrite of all pyramidal neuron models (Murayama et al., 2009). The axon of the IS interneuron model projects to the soma of the RS interneuron model, providing disinhibitory input to the micorcircuit, as well as the distal apical dendrite of all pyramidal neuron models. Furthermore, inhibitory autapses are present in the FS interneuron models (Bacci et al., 2003). A summary of synaptic connections present in the microcircuit is described in **Table 5**.

## **NUMBER OF SYNAPSES**

The total number of excitatory synapses to the three types of interneuron models and of inhibitory synapses on the pyramidal neuron model was based on the anatomical data (Tamás et al., 1997a,b; Markram et al., 2004). The total number of inhibitory synapses onto each pyramidal model neuron was 13% of the total excitatory synapses (Peters et al., 2008). A summary of the number of synapses introduced between each type of connection is described in **Table 5**.

## **VALIDATION OF THE SYNAPTIC MECHANISMS**

The conductances of excitatory and inhibitory synaptic mechanisms were adjusted according to electrophysiological recordings (Thomson and Deuchars, 1997; Angulo et al., 1999; Thomson and Destexhe, 1999; Xiang et al., 2002; Bacci et al., 2003; Woo et al., 2007; Wang et al., 2008; Wang and Gao, 2009). The conductance of a single AMPA-R synapse onto the pyramidal neuron model was adjusted so that it generated a voltage response of 0.1 mV at the soma (Nevian et al., 2007). The NMDA current was validated with a simulated voltage clamp protocol to replicate the results of Wang et al. (2008) (**Figure 3A**). AMPA- and NMDAmediated currents were recorded at −70 mV and +60 mV, respectively, in FS and RS neuron models, according to Wang and Gao (2009). Our results correspond to the experimental data,

profile of the FS model. RS neuron model response to increasing

as shown in **Figures 3B,C**. The relative proportion of NMDA and AMPA receptor mediated synaptic components of the FS models is standardized at 0.5 (Wang and Gao, 2009). The relative proportion of NMDA and AMPA receptor mediated synaptic components of RS models is standardized at 0.8 (Wang and Gao, 2009). In lack of experimental data for the IS neuron model, its AMPA- and NMDA- mediated currents were also simulated to match those of the FS and RS neuron models, whereas the NMDA-to-AMPA ratio was adapted so that the IS interneuron model could fire action potentials during the stimulus.

irregular spiking of the IS model.

Furthermore, GABAA receptor mediated currents (IPSCs), between the FS interneuron and the pyramidal neuron were validated, based on Woo et al. (2007) and the GABAB receptor mediated IPSC was validated against experimental data from Thomson

#### **Table 3 | Active and passive ionic properties of RS interneuron model.**


#### **Table 4 | Active and passive ionic properties of IS interneuron model.**


#### **Table 5 | Summary of synaptic connections in the microcircuit.**


et al. (1996) as in Papoutsi et al. (2013). According to Xiang et al. (2002), the amplitude of IPSCs for FS-Pyramidal pairs had a mean value significantly larger than RS-Pyramidal pairs. In particular, the GABAA mediated current between the RS-Pyramidal neuron pair should be 1/10 of the GABAA mediated current between FS-Pyramidal cell pair (Xiang et al., 2002). Due to lack of experimental data for the IPSCs of the IS-Pyramidal neuron

mediated by AMPA receptors; whereas at +60 mV, the currents were largely mediated by NMDA receptors (adapted from Wang et al., 2008, Copyright 2008 National Academy of Sciences, USA). **(A2)** Simulated voltage-clamp responses of the pyramidal model neuron at −70 mV (AMPA currents) and at +60 mV (NMDA currents). **(B1,B2)** Experimental **(B1)** and modeling **(B2)** data of AMPA and NMDA currents, in FS interneurons (**B1** is adapted from Wang and Gao, 2009). **(C1,C2)** Experimental **(C1)** and modeling **(C2)** data of AMPA and NMDA currents, in RS interneurons (**C1** is adapted from Wang and Gao, 2009).

pair, GABAA mediated current of this pair was estimated to be 1/10 of the GABAA of RS-Pyramidal pair. The autoinhibition of PV interneurons is much stronger than the inhibition between interneurons of different types, such as IS-RS pairs (Bacci et al., 2003). Autaptic inhibitory currents in FS interneurons evoked a relatively large transient current of 0.35 mA amplitude (Bacci et al., 2003). The aforementioned current was simulated as described in Papoutsi et al. (2013). Across different experiments, the NMDA-to-AMPA ratio of 1.25 and the GABAb-to-GABAa ratio of 0.2 were taken as control state.

#### **BACKGROUND NOISE**

In addition, for best simulation of membrane potential fluctuations as observed *in vitro* due to the stochastic ion channel noise (Linaro et al., 2011), an artificial current with Poisson characteristics (mean rate 0.02 Hz) was injected in the soma of all neuron models. Specifically, for the IS neuron model, the amplitude of this mechanism was larger (mean rate 0.035 Hz) (Golomb et al., 2007).

#### **STIMULATION PROTOCOL**

The proximal apical dendrites of the pyramidal neuron models were stimulated with 120 excitatory synapses (containing both AMPA and NMDA receptors), which were activated 10 times at 20 Hz (yellow arrows in **Figure 4**) (Kuroda et al., 1998). Since

neurons within a microcircuit share similar stimulus properties (Yoshimura and Callaway, 2005; Petreanu et al., 2009), the same initial stimulus was delivered to all pyramidal neurons.

## **ANALYSIS**

Data analysis was performed in Matlab (Mathworks, Inc). Inter-Spike-Intervals (ISIs) were calculated for the neuronal response of each neuron model of the microcircuit during the stimulus and during persistent activity. An average of the ISIs of each neuron of the network, as well as coefficient of variations, in 500 ms time bins was measured for each experimental state.

The Synchronization or de-synchronization of the neurons was measured using the SPIKE-distance measurement, which is sensitive to spike coincidences (Kreuz et al., 2011). For this measurement we obtained the spike trains simultaneously from the neuronal population of the microcircuit and then we calculated the time intervals between successive spikes occurring in any of the participating neurons. If there are no phase lags between the spike trains (neurons fire synchronously) the synchronization index will have values of zero. In general, small values of synchronization index indicate synchronicity, whereas large values indicate asynchronous spiking activity (as in Papoutsi et al., 2013).

As an additional estimation of the synchronization or desynchronization among spiking neurons in the microcircuit during each different condition, we measured the total number of spikes recorded in 1 ms time bins, and constructed plot with the discrete -time firing rate.

Power spectra were generated on the summed synaptic currents (AMPA, NMDA, and GABAA) generated by the pyramidal neurons in the network, averaged for 10 trials, over a 1-s period of steady-state persistent activity, 3 s after the end of the stimulus. The averaged synaptic currents were first decimated and then, the mean square power spectrum was calculated using the periodogram method.

## **MODEL AVAILABILITY**

The code of this model in the NEURON simulation environment will be available following contact with the corresponding author and will be posted on ModelDB database upon publication.

## **RESULTS**

We used a 20-neuron PFC microcircuit model that included 16 biophysically-detailed pyramidal cell models and 4 interneuron models: 2 FS, 1 RS, and 1 IS interneuron model, in order to study the role of these interneuron cell-types in persistent activity emergence and maintenance properties. All modeled neurons were validated against experimental data from intracellular recordings in brain slices (**Figure 2**—see Methods for details). In addition, the synaptic mechanisms were validated against experimental data (AMPA current, NMDA-to-AMPA ratio, GABA currents) (**Figure 3**—see Methods for details).

Persistent activity in the network was induced by an external excitatory stimulus to the apical dendrite (**Figure 4**). Similar to a smaller version of the microcircuit model (which included 7 pyramidal model neurons and 2 FS interneurons Papoutsi et al., 2013), persistent activity induction was dependent on the GABAB-to-GABAA and NMDA-to-AMPA ratio on the pyramidal neuron models (**Figure 5A**). Each neuron model had a different firing pattern during persistent activity, depending on its own electrophysiological characteristics (**Figure 5B**). The interspike intervals (ISIs) of the pyramidal neuron model during persistent activity were between 60 and 120 ms, i.e., firing frequency of 8-17 Hz (**Figure 5C**). The coefficient of variation (CV) of the ISIs, although not very high as observed *in vivo* (Compte, 2006), was greater during persistent activity compared to the CV during the stimulus (**Figure 5D**). Furthermore, we find that spiking activity of neurons in the network was synchronized both during the stimulus response and the persistent activity, although synchronicity during the stimulus was greater compared to that during persistent activity (**Figure 5E**). These properties are similar to the corresponding properties observed in persistent activity during working memory tasks (Constantinidis and Procyk, 2004).

Increasing the NMDA-to-AMPA ratio onto pyramidal neuron models decreased the ISIs, especially during the initial phases of persistent activity (**Figure 5F**), suggesting an increase in the firing frequency. On the other hand, modulating the NMDA-to-AMPA ratio onto FS models does modulate the % probability for induction of persistent activity (**Figure 5G**) but not the ISIs of the pyramidal neuron model (**Figure 5H**). This is in accordance with

pyramidal model neurons, as previously seen (Papoutsi et al., 2013). **(B)** Representative traces of all neuron models during the stimulus and during persistent activity. **(C)** Graph showing the ISIs during the stimulus and persistent activity in 500 ms bins, for all neuron models. **(D)** Graph showing the coefficient of variation during the stimulus and persistent activity in 500 ms bins, for all neuron models. The ISI is increased during the initial

This is not the case for the FS interneuron model. **(E)** Discrete-time firing rate stimulus and during persistent activity. **(F)** Changing NMDA-to-AMPA ratio on phases of persistent activity. **(G)** Changing NMDA-to-AMPA ratio on FS interneuron models modulated the probability for induction of persistent activity. **(H)** Changing NMDA-to-AMPA ratio on FS interneuron models did not modulated the ISIs.

the notion that regulation of NMDA receptors in FS interneurons modulates PFC function (Homayoun and Moghaddam, 2007), although the slow kinetics of the NMDA receptors may not allow for immediate change in network firing (Rotaru et al., 2011). Furthermore, modulating the NMDA-to-AMPA ratio either on pyramidal neuron model or the FS model doesn't have an important effect on the synchronicity among neurons in the microcircuit both during the stimulus and during persistent activity (Supplemental Tables 3, 4). For the rest of the study we used the following conditions: GABAB-to-GABAA ratio = 0.2, NMDAto-AMPA = 1.25 (pyramidal neuron model), and NMDA-to-AMPA = 0.5 (FS interneuron model).

In order to study the role of the different interneuron cell types in persistent activity, we next simulated "knock-out" networks for each interneuron subtype, Thus, we generated a PFC microcircuit without the FS models ("FS KO") (**Figure 6A1**), a microcircuit without a RS model ("RS KO" network) (**Figure 6A2**), and a microcircuit without an IS model ("IS KO" network) (**Figure 6A3**). We find that the probability for persistent activity induction is always 1, across all GABAB-to-GABAA and NMDAto-AMPA ratios in the "FS KO" network (**Figure 6B1**), while it is not significantly altered in the "RS KO" and "IS KO" network models (**Figures 6B2,3**). Therefore, the sensitivity to biophysical modulation is completely lost in the "FS KO" network. In

addition, the ISIs during the stimulus and during persistent activity are significantly decreased in the "FS KO" network to 15 ms (i.e., close to 80 Hz frequency), but not significantly altered in the "RS KO" and "IR KO" networks (**Figure 6C**). As well, the CV of the ISIs of pyramidal neurons during the stimulus and during persistent activity is significantly decreased in the "FS KO" network (**Figure 6D**). This indicates that the firing rate and its variability of pyramidal neuron models is tightly controlled by the activity of FS interneuron models, but not the RS and IR interneurons. Finally, neuronal synchronicity during persistent activity is also significantly decreased in the "FS KO" network, as evident by the desynchronization index measure and the discrete-time firing rate plot (**Figures 6E,F**). This is again in accordance with other studies suggesting a contribution of FS interneuron spiking on neuronal synchronization and oscillations (Sohal et al., 2009).

Since the "FS KO" net was the only one showing significant differences with regards to persistent activity properties, we wanted to further study the role of the FS interneuron model. Thus, we gradually decreased the number of GABAergic synapses (both GABAA and GABAB) from the FS interneuron model onto the pyramidal neuron model in order to simulate a less severe, and possibly more realistic, disruption in the FS neuronal functioning. We find that decreasing the FS model inputs onto the pyramidal neuron model increases the probability for persistent activity induction across the different GABAB-to-GABAA (NMDA-to-AMPA = 1.25), while when 40% or less of FS inputs remain, persistent activity is induced across all GABAB-to-GABAA ratios tested (**Figure 7A**). This suggests that once more than 50% of PV inputs are lost, then the PFC microcircuit behaves as if no FS model is present, with regards to induction of persistent activity. This large increase in persistent activity induction renders the microcircuit insensitive to modulation of GABAB.

Furthermore, as FS model inputs decrease, the ISIs of the pyramidal neuron model during persistent activity gradually decrease,

the number of synaptic inputs from the FS model neurons to the pyramidal model neurons increases the range of GABAB-to-GABAA ratios, in which persistent activity is induced. When less than 40% of FS inputs are present in the microcircuit, persistent activity is induced 100% across all GABAB-to-GABAA ratios. **(B)** As the number of the FS inputs

decreases, the ISIs of the pyramidal neuron model decreases. **(C)** As the number of the FS inputs decreases, the CV of the ISIs of the pyramidal neuron model decreases. **(D)** The synchronicity among all neuron models during persistent activity, is significantly reduced when no PV inputs are present in the microcircuit, while it increases by decreasing the number of synaptic inputs from the FS model neurons to the pyramidal model neurons.

hence the firing frequency gradually increases (**Figure 7B**). The variability of ISIs is also decreased when FS inputs decrease to 60% or more, making this index the most sensitive to FS inputs (**Figure 7C**). Finally, the desyncrhonization index among neuron models in the microcircuit gradually decreases while decreasing the number of FS inputs, but then increases when no FS inputs are present (i.e., "FS KO" net) (**Figure 7D**). This suggests that synchronicity actually increases when a percentage of FS inputs to the pyramidal neuron models are blocked but then decreases when no inputs are present.

Many of the roles of FS neurons on cortical network functions have been attributed to its specific connectivity, specifically the projection of FS neurons to the soma of the pyramidal neurons (Lovett-Barron et al., 2012; Royer et al., 2012). However, by design experimental manipulations cannot differentiate between the target location of an interneuron and its physiological characteristics. So, the next step was to study in detail the role of this specific connectivity by changing the projection site of the FS neuron model to different dendritic locations of the pyramidal neurons other than the soma; on the basal dendrites (D0 net) (**Figure 8A1**), on the proximal dendrites (D1 net) (**Figure 8A2**), on the distal dendrites (D2 net) (**Figure 8A3**). When the FS input is located anywhere else but the soma, then, the probability for induction of persistent activity increases to 100% (**Figure 8D**), while the ISIs and ISI variability during persistent activity significantly decrease (**Figures 8B,C**). Furthermore, when the FS model input is located to dendritic locations and not the soma the desynchronization index decreases, suggesting an increase in synchronicity (**Figure 8E**). Therefore, if FS neuron models do not project to the soma, network activity during persistent activity resembles the state where 50% of less FS inputs to the soma are active (**Figure 7**).

Somatic inhibition provided by the FS interneuron seems to be necessary for the induction of proper firing frequencies during persistent activity. In order to eliminate the possibility that the same perisomatic inhibitory effect could be achieved by the other types of interneurons, we modified the network by reversing the projection and number of FS interneuron with RS interneuron (Reverse RS net) (**Figure 9A1**) and with IS interneuron (Reverse IS net) (**Figure 9A2**). When one FS interneuron is projecting in distal dendrites of pyramidal neurons, but two RS or two IS interneurons provide somatic inhibition on pyramidal neurons, the probability for induction of persistent activity increases (**Figure 9E**), while the ISIs and ISI variability during persistent activity significantly decrease (**Figures 9B,C**). Finally, the desynchronization index decreases in both of the reverse networks (**Figure 9D**). The above results were resistant to changes in the kinetics of the excitatory synaptic mechanisms of RS and IS models and to changes in the conductance values of the inhibitory synaptic mechanism that the RS and IS models provide to their connected neuron models (Supplemental Figures 1, 2). Our results were also resistant to increasing the frequency (40 Hz) of the stimulation used to initiate persistent activity (Supplemental Figure 3). These results suggest that somatic inhibition provided

specifically by the FS neuron is necessary for the firing frequency during persistent activity and allowing for modulation of persistent activity induction. On the other hand, synchronicity of the PFC microcircuit could be increased by changing either the projection of the FS model or changing the physiological profile of the interneuron models projecting to the soma.

In an effort to compare our results to the available literature with regards to changes in network oscillations in the presence of defects in inhibition, we analyzed the power spectra of the summed synaptic currents in the different model networks reported above. In our control network, we observe the presence of a peak in the power spectrum at 20 Hz and a smaller peak at 40 Hz (**Figure 10A**, dark blue trace). Both peaks are absent in the FS KO network, suggesting a significant role of the FS model neuron in maintaining these oscillations. In addition, only the 40 Hz peak is decreased in the RS KO network, while the power spectrum of the IS KO network is the same as the control (**Figure 10A**). Both peaks are absent when 50% or less of the FS input to the pyramidal model neuron remain, however, the peak at 40 Hz is already decreased when 80% of the FS inputs remain (**Figure 10B**). Finally, when the somatic inhibition is provided by the RS or IS neuron, the peaks at 20 and 40 Hz are even larger (**Figure 10C**). Therefore, our results suggest that the FS input is critical for maintaining network oscillations, and also reveals a novel role of the RS neuron model in maintaining primarily the 40 Hz oscillation.

## **DISCUSSION**

In this study, we have specifically delineated distinct and specific roles of the FS interneurons in persistent activity properties. First, we identified that NMDA current input onto interneurons only modulates persistent activity induction but not its spiking properties during persistent activity. Second, we find that the FS neuronal inputs to the pyramidal neurons modulate the induction of persistent activity, in an all-or-none way, while the properties of spiking during persistent activity in a gradient manner. Third, moving the FS inputs away from the soma and onto other dendritic compartments has similar effects to completely removing the FS neurons from the network, indicating the significant role of the projecting site of the FS neuron. Finally, we show that replacing somatic inhibition by either the RS or IS neuron models does not reverse the induction or firing frequency changes

**FIGURE 9 | Different variations of the PFC microcircuit were constructed in order to study the effect of the spiking profile of the neuron model that provides somatic inhibition to the pyramidal neuron. (A)** Graphical representations of the microcircuit in which 2 RS interneuron models are projecting to the soma, while 1 FS interneuron model projects to the distal dendritic compartment of pyramidal neuron models (Reverse RS) **(A1)**, and of another microcircuit in which 2 IS interneuron models are projecting to the soma, while 1 FS interneuron model projects to the distal dendritic compartment of pyramidal neuron models (Reverse IS) **(A2)**. **(B)** Graph showing ISIs before and during persistent in 500 ms bins for the control and the two "reverse" states of the network. The ISIs of pyramidal neuron models are decreased in all

but does alter neuronal synchronization (increases beyond the control condition).

### **NMDA RECEPTORS IN PYRAMIDAL NEURON vs. INTERNEURONS**

The role of pyramidal neuron NMDA currents in persistent activity and working memory tasks is well established (Wang, 2001). Our simulation results reinforce the significance of pyramidal neuron NMDA currents in persistent activity induction and modulation of both induction and the firing properties during persistent activity. However, the role of NMDA receptors on interneurons has received much less attention, particularly from modeling studies of persistent activity. Our modeling results show that an increase in NMDA currents onto FS interneurons decreases the probability for induction of persistent activity, while decreasing the NMDA currents onto FS interneurons increases the probability for induction of persistent activity (**Figure 5G**). This bidirectional modulation suggests that NMDA receptors at FS interneurons have a critical role in persistent activity induction, and subsequently working memory performance. Our results partly agree with a more generic cortical model (Spencer, 2009), showing the effects of NMDA on FS interneurons on network activity and synchronization. Furthermore, since NMDA on FS interneurons modulates persistent activity induction in our model, we predict that this could impair working memory microcircuits in which the FS model neuron project to the distal dendritic compartments of the pyramidal neuron models while either of the other two neuron models are projecting to the soma of pyramidal neurons. **(C)** Graph showing CVs of ISIs before and during persistent in 500 ms bins for the control and the two "reverse" states of the network. The CVs of ISIs of pyramidal neuron models are decreased in all microcircuits in which the FS model neuron project to the distal dendritic compartments of the pyramidal neuron models while either of the other two neuron models are projecting to the soma of pyramidal neurons. **(D)** Graph showing the synchronicity index in the control and the two "reverse" states of the network. **(E)** Graph showing persistent activity induction in the control and the the two "reverse" states of the network.

and other PFC functions. Indeed, removing functional NMDA receptors from FS interneurons has been shown to result in such behavioral defects (Belforte et al., 2009).

### **THE ROLE OF INTERNEURONS IN NEURONAL SYNCHRONICITY AND GAMMA-FREQUENCY OSCILLATIONS**

Cortical oscillations, particularly in the gamma-frequency, have been suggested to significantly contribute to several cognitive functions, such as selective attention, perception. These oscillations are thought to reflect synchronous activity of rhythmically firing neurons (Jensen et al., 2007). Activity of PV/FS interneurons has been found in several studies, both experimental and computational, to have a significant role in maintaining the above oscillations and neuronal synchronization (Borgers et al., 2008; Cardin et al., 2009; Sohal et al., 2009; Vierling-Claassen et al., 2010).

Gamma oscillations have been shown to increase the mutual information between incoming synaptic frequency and output of action potentials (Sohal et al., 2009).

In our model, there is a bidirectional modulation of neuronal synchronicity by the FS interneuron. Decreasing the FS input results in increased synchronicity, while a "KO" simulated condition results in decreased synchronicity. Replacing the FS input with either the RS or IR neurons increased synchronicity, but past

the levels of the control network. This suggests that the effects of FS input on neuronal synchronicity are complex. Small reductions of FS inputs (20%) result in both increased synchronicity and a small deviation in the firing rate and ISI variability, suggesting that this could be beneficial for the network activity and could possibly result in working memory enhancements. However, 40% or greater reductions in FS-mediated synaptic inputs result not only in increased synchronicity but also in increased firing frequency and decreased ISI variability, indicating a possible defect that could move the network activity toward epileptiform behavior. Therefore, as mentioned in Yu et al. (2004), "it is not the weaker or stronger but an appropriate synchronous state may be of more functional significance in sensory encoding."

#### **CHANGES IN INTERNEURONS AND DISEASE**

Converging experimental and clinical evidence suggests that dysfunction in the GABAergic system and the consequent imbalance between excitation and inhibition in the cerebral cortex underlies at least part of the pathophysiology of several neuropsychiatric disorders, such as schizophrenia, epilepsy and autism (Marín, 2012).

In particular, interneuron defects have been associated very strongly with schizophrenia (Lewis et al., 2005). Schizophrenic patients have been shown to express decreased levels of the GABA-synthesizing enzyme, GAD67, and PV (Akbarian, 1995; Volk et al., 2001). In addition, GAD67 and PV are also decreased in several animal models of schizophrenia (Braun et al., 2007; Lodge et al., 2009). Furthermore, both reduced working memory load and reduced power of gamma oscillatory activity in the dorsolateral PFC have been found in schizophrenia (Gonzalez-Burgos and Lewis, 2008), which is also observed when inactivating the PV interneurons with light (Sohal et al., 2009).

Decreased markers of inhibitory transmission, such as number of differentiated PV and CB interneurons is found in animal models of autism (Eagleson et al., 2010; Fu et al., 2012) while a decrease in GAD67 (a marker for inhibitory transmission) is also decreased in human autistic patients (Fatemi et al., 2002). Furthermore, decreased gamma response in the occipital cortex was also found in a human study (Wright et al., 2012).

Epilepsy is another condition that is associated with decreased interneuron populations, as evident mostly from the emergence of epileptic behavior in animal models with reduced number of interneurons (Cobos et al., 2005; Butt et al., 2008; Gant et al., 2009; Peñagarikano et al., 2011). Moreover, epilepsy is characterized by excessive neuronal synchrony (Traub and Wong, 1982), suggesting that decreased interneuron function can also be associated with increased synchrony.

Our results predict that decreases greater than 50% in the number of interneurons or GABAergic synapses lead to disruption in stimulus-specific persistent activity induction. Therefore, any type of stimulus irrespective of neuromodulation could results in persistent activity, a condition that should greatly impair performance in working memory tasks and other PFCdependent cognitive functions, such as attention and behavioral flexibility.

#### **CONNECTIVITY vs. PHYSIOLOGICAL PROPERTIES**

All three distinct interneuron subtypes differ in their physiological characteristics as well as the location of their synaptic targets. Specifically for the FS interneuron, it has a FS, high frequency physiological profile and it primarily targets the somatic region of the pyramidal neurons (Markram et al., 2004). It has been suggested that the observed function of the specific interneuron subtypes is mostly attributed to the location of their synaptic targets (Wang et al., 2004). The significance of the projection site in the pyramidal neurons has also been shown in our study, since the differences in persistent activity induction, firing frequency and ISI variability are seen when the FS interneuron projects to any dendritic compartment and not the soma. However, our simulations also show that the FS physiological profile is also necessary, since replacing the FS neuron model with either the RS or the IR neuron models at the soma does not reverse the induction, firing frequency and ISI variability changes. Instead, the synchronicity changes are reversed, but not to baseline levels. Specifically, when somatic inhibition is provided by the RS or IS neurons, both the synchronicity and network oscillations are stronger. This suggests that somatic inhibition provided by interneurons with FS activity is absolutely crucial for the different properties of persistent activity. However, any type of inhibition, mediated by either the RS or IR neuron models, can maintain and even further increase neuronal synchronicity. Thus, generic somatic inhibition can serve as a pacemaker of persistent activity, but FS-mediated somatic inhibition is necessary for proper expression of persistent activity properties.

### **MODEL LIMITATIONS**

Our model microcircuit includes different types of model neurons (a pyramidal neuron and 3 different types of interneurons). As seen in the methods, the model network used in this work is heavily constrained with available experimental data. However, sources of inaccuracy can be introduced by the variability of preparations used to produce the experimental data, as well as by the limited availability of data with regards to the specific brain region of our study (prefrontal cortex) and specific layer (layer V). Whenever possible, the data used to validate the models were taken from studies of layer V pyramidal neurons or the different types of interneurons in the prefrontal cortex (Zaitsev, 2005; Peters et al., 2008; Wang et al., 2008; Wang and Gao, 2009). However, in cases there were no available data from the specific region and specific layer, data from non-specified frontal cortex (Kawaguchi and Kubota, 1993; Lubke et al., 1996; Thomson and Lamy, 2007; Woo et al., 2007) or specific primary sensory areas for either pyramidal neurons or interneurons (Cauli et al., 1997; Tamás et al., 1997a,b; Xiang et al., 2002; Toledo-Rodriguez et al., 2005) and were used. Another issue with regards to the available data used to constrain the model is the age of the animals used in the experimental studies. Most of the above studies used to validate our models come from rodents of very young age (up to a month old), although there are some in adult animals, for example (Wang and Gao, 2009). Since it is becoming evident that age plays a very significant role in the cellular physiology and underlying cellular mechanisms (McCutcheon and Marinelli, 2009), our conclusions are limited by the use of the specific available data. Should more specific data from the prefrontal cortex, preferably from adults, become available the specific or future models can be constrained in a more strict way.

Furthermore, there is also variability in the available data with regards to the specific layer examined. Hence, studies using primates suggest that recurrent networks mostly in layer III and to a lesser extent in layer V PFC mediate the persistent activity observed during working memory tasks (Wang et al., 2013). Several models of working memory in the literature (Compte et al., 2000; Wang et al., 2004) simulate layer III recurrent networks (Kritzer and Goldman-Rakic, 1995), in a larger scale compared to the model reported here. The main results, however, with regards to the contribution of NMDA currents, for example, are similar. Even the layer III models constrain their biophysical parameters using layer V electrophysiological recordings (Seamans et al., 1997) or recordings not confined to a specific layer (Connors and Gutnick, 1990; Hammond and Crepel, 1992). In light of recent evidence with regards to differences in biophysical properties of layer III and layer V pyramidal neurons (Bremaud et al., 2007; de Kock, 2013), future modeling studies could be constrained even further with layer specific recordings, possibly revealing layer-specific information coding or persistent activity properties.

#### **MODEL PREDICTIONS AND FUTURE USES**

Our model generates two important predictions that could be tested experimentally. First, it predicts that NMDA current modulation on interneurons only modulates induction of persistent activity and not neuronal excitability or synchronicity during persistent activity. Thus, NMDA receptor blockade specifically in interneurons could increase the emergence of up-and-down states in *in vitro* experiments or increase persistent activity to non-selective stimuli in *in vivo* tasks. Second, it predicts that gradual decrease in the percent of FS-mediated GABAergic synapses will significantly increase the firing rate during persistent activity and decrease the variability of ISIs. For example, in animal models with decreased GABAergic neurons (Peñagarikano et al., 2011; Vidaki et al., 2012), one would expect a very significant increase in the emergence of up-and-down states, during which the neuronal firing rate will be increased and ISI variability decreased.

Furthermore, because of its level of the biophysical detail and extensive validations, the model network can be used as a tool to further delineate the role of interneurons in persistent activity and stimulus-dependent activity. Some examples of studies that could use and/or extend the model include, but are not limited to, (a) identifying the role of the other two interneuron types (RS and IS models) under different stimulation protocols, (b) studying the role of specific biophysical mechanisms on interneurons either on persistent activity or stimulus-dependent activity, and (c) extending the model network to incorporate plasticity rules specific to the different types of interneurons.

## **ACKNOWLEDGMENTS**

This work was supported by a Marie-Curie IOF grant (FEAR MEMORY TRACE-253388) and a NARSAD young investigator award to Kyriaki Sidiropoulou and an ERC Starting Grant to Panayiota Poirazi ('dEMORY', ERC-2012-StG-311435).

## **SUPPLEMENTARY MATERIAL**

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/journal/10.3389/fncir. 2014.00007/abstract

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 01 November 2013; accepted: 14 January 2014; published online: 31 January 2014.*

*Citation: Konstantoudaki X, Papoutsi A, Chalkiadaki K, Poirazi P and Sidiropoulou K (2014) Modulatory effects of inhibition on persistent activity in a cortical microcircuit model. Front. Neural Circuits 8:7. doi: 10.3389/fncir.2014.00007*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Konstantoudaki, Papoutsi, Chalkiadaki, Poirazi and Sidiropoulou. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## Genesis of interictal spikes in the CA1: a computational investigation

## *Shivakeshavan Ratnadurai-Giridharan1, Roxana A. Stefanescu2, Pramod P. Khargonekar 3, Paul R. Carney1,4 and Sachin S. Talathi 1,4,5\**

*<sup>1</sup> J Crayton Pruitt Family Department of Biomedical Engineering, University of Florida, Gainesville, FL, USA*

*<sup>2</sup> Department of Otolaryngology, Kresge Hearing Research Institute, University of Michigan, Ann Arbor, MI, USA*

*<sup>3</sup> Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA*

*<sup>4</sup> Department of Pediatrics, University of Florida, Gainesville, FL, USA*

*<sup>5</sup> Qualcomm Corp R&D, San Diego, CA, USA*

#### *Edited by:*

*A. Ravishankar Rao, IBM Research, USA*

#### *Reviewed by:*

*Alessandro Viganò, Sapienza University of Rome, Italy Roberto Latorre, Universidad Autónoma de Madrid, Spain*

#### *\*Correspondence:*

*Sachin S. Talathi, Qualcomm Corp R&D, 5828 Pacific Center Blvd, 5775 Morehouse Dr, San Diego, CA 92121, USA e-mail: talathi@gmail.com*

Interictal spikes (IISs) are spontaneous high amplitude, short time duration <400 ms events often observed in electroencephalographs (EEG) of epileptic patients. *In vitro* analysis of resected mesial temporal lobe tissue from patients with refractory temporal lobe epilepsy has revealed the presence of IIS in the CA1 subfield. In this paper, we develop a biophysically relevant network model of the CA1 subfield and investigate how changes in the network properties influence the susceptibility of CA1 to exhibit an IIS. We present a novel template based approach to identify conditions under which synchronization of paroxysmal depolarization shift (PDS) events evoked in CA1 pyramidal (Py) cells can trigger an IIS. The results from this analysis are used to identify the synaptic parameters of a minimal network model that is capable of generating PDS in response to afferent synaptic input. The minimal network model parameters are then incorporated into a detailed network model of the CA1 subfield in order to address the following questions: (1) How does the formation of an IIS in the CA1 depend on the degree of sprouting (recurrent connections) between the CA1 Py cells and the fraction of CA3 Shaffer collateral (SC) connections onto the CA1 Py cells? and (2) Is synchronous afferent input from the SC essential for the CA1 to exhibit IIS? Our results suggest that the CA1 subfield with low recurrent connectivity (absence of sprouting), mimicking the topology of a normal brain, has a very low probability of producing an IIS except when a large fraction of CA1 neurons (>80%) receives a barrage of quasi-synchronous afferent input (input occurring within a temporal window of ≤24 ms) via the SC. However, as we increase the recurrent connectivity of the CA1 (*P*sprout > 40); mimicking sprouting in a pathological CA1 network, the CA1 can exhibit IIS even in the absence of a barrage of quasi-synchronous afferents from the SC (input occurring within temporal window >80 ms) and a low fraction of CA1 Py cells (≈30%) receiving SC input. Furthermore, we find that in the presence of Poisson distributed random input via SC, the CA1 network is able to generate spontaneous periodic IISs (≈3 Hz) for high degrees of recurrent Py connectivity (*P*sprout > 70). We investigate the conditions necessary for this phenomenon and find that spontaneous IISs closely depend on the degree of the network's intrinsic excitability.

#### **Keywords: interictal spikes, hippocampal CA1 region, computational models, paroxysmal depolarization shift, temporal lobe epilepsy**

## **1. INTRODUCTION**

Mesial temporal lobe epilepsy (MTLE) is a chronic neurological disease that affects the hippocampus and the inner regions of the temporal lobe. MTLE is characterized by recurrent seizures (ictal activity) and interictal spikes (IISs), which typically occur in between seizure epochs in the form of transient discharge events which are clearly discernible from background EEG activity. Studies involving long-term EEG monitoring in animal models of MTLE show that IISs also occur prior to the first instance of spontaneous ictal activity (Buzsáki et al., 1991). In chronic *in vivo* animal models of MTLE, it has been observed that IISs start within a few weeks after initial brain injury and steadily increase in frequency of occurrence (Buzsáki et al., 1991). Despite an overwhelming evidence for an IIS as a characteristic observable feature in EEG of MTLE patients (Engel, 1996), the role of IISs and its clinical manifestation in MTLE remain unclear. For example, while there is evidence to suggest that IISs interfere with normal cognition and learning (Holmes and Lenck-Santini, 2006; Kleen et al., 2010) and may facilitate the development of spontaneous seizure activity (Staley et al., 2005), recent *in vitro* experiments suggest that an increase in interictal spiking activity may serve as an anti-epileptogenic agent (Avoli et al., 2006). In order to completely understand the role of IISs in MTLE, we need to study the effects of selectively invoking or suppressing IISs on demand. Progress in this direction will most certainly first require a fundamental understanding of the network mechanisms underlying the generation of an IIS in an epileptic brain.

In MTLE, IISs are thought to originate from the CA3/2 region of the hippocampus involving a group of pacemaker pyramidal (Py) cells (Jefferys, 1990; Wittner and Miles, 2007). IISs propagate as population bursts throughout the CA3 subfield and on to the CA1 subfield via the Schaffer collaterals (SC) (Stoop and Pralong, 2000). A number of *in vivo* and *in vitro* studies have demonstrated that when the SC fibers are cut or the CA3 removed, CA1 loses its ability to generate IISs (Lewis et al., 1990; Stoop and Pralong, 2000). While the CA3 may be necessary for the initiation of IISs in the hippocampus, the CA1 subfield is critical for propagating the IIS to subcortical brain structures outside the hippocampus via the subiculum and the entorhinal cortex (Lopes da Silva et al., 1990; van Groen and Wyss, 1990; Dvorak-Carbone and Schuman, 1999). Furthermore, in MTLE, the CA1 is one of the first hippocampal subfields that undergoes rapid morphological and structural changes, such as recurrent pyramidal axonal sprouting and neuronal cell death (Lehmann et al., 2000). It is therefore essential to understand how the morphological and structural changes implicated in the CA1 subfield of an MTLE brain influence the subfields ability to exhibit IISs in response to afferent input from the SC.

The cellular correlate for an IIS is the epileptiform bursting activity of Py cells commonly referred to as the paroxysmal depolarization shift (PDS) (McCormick and Contreras, 2001; Staley and Dudek, 2006). The PDS represents a large (20–40 mV), long lasting (50–200 ms) neuronal depolarization which results in the initiation of high frequency burst of action potentials (200–300 Hz) (Kandel et al., 2000). The depolarization wave is usually followed by a slow afterhyperpolarization (AHP). An example of PDS recorded from resected hippocampal tissue of a TLE patient is shown in **Figure 1B**. The PDS phenomenon is attributed to a number of factors including increased extracellular *K*<sup>+</sup> concentration, reduced extracellular *Ca*2<sup>+</sup> concentration (Yaari et al., 1983; Formenti et al., 2001; Burgo et al., 2003; Smith et al., 2004; Golomb et al., 2006), increased synaptic drive (Jefferys, 1990) and channelopathies (McNamara, 1994). In the pathological CA1 Py cell population, the duration of a PDS burst its AHP can have variable durations. Furthermore, the PDSs themselves can occur with varying degree of synchronization (Netoff and Schiff, 2002). Identifying the correspondence between the features of these cellular events and the extent of their synchronization is critical for exploring their role in the formation of IISs in the CA1.

The primary goal of this study is to develop a biophysically relevant computational network model of the CA1 subfield in order to investigate the network mechanisms implicated in the formation of IISs within the subfield. Using experimental data on IISs recorded from an *in vivo* animal model of chronic limbic epilepsy, we first ask the following question: what are the characteristics of PDS events that are implicated in the generation of an experimentally observable IIS? We develop a method for analyzing recorded IISs in order to empirically estimate the underlying PDS characteristics. These include the depolarization time interval, the hyperpolarization duration of a typical

PDS event and the degree of synchronization between these PDS events. This data is used to tune a synaptically reduced neuronal network model in order to enable the model to generate PDSs with features matching those obtained using the empirical estimation procedure. The tuning procedure allows us to estimate the relative strengths of the excitatory and inhibitory neuronal populations implicated in the generation of PDS events in the minimal network model. This information in addition to data from literature (Kandel et al., 2000; Demont-Guignard et al., 2009) is used to build a bio-physically relevant network model for the CA1 subfield. We then use this model to investigate the following specific questions: (1) How does the formation of an IIS in the CA1 depend on the degree of sprouting (recurrent connections) between the CA1 Py cells and the fraction of CA3 Shaffer collateral (SC) connections onto the CA1 Py cells? and (2) Is synchronous afferent input from the SC essential for the CA1 to exhibit IIS? Our results suggest that the CA1 network is capable of eliciting IIS activity primarily through two mechanisms of network synchronization: (1) input-induced synchronization, where the CA1 network with low intrinsic excitability characterized by low degree of recurrent connections between the CA1 Py cells can elicit IIS in response to synchronized barrage of afferent input from the SC with high degree of SC to Py connectivity and (2) emergent synchronization, where the CA1 network with high degree of recurrent connections between the excitatory Py cells can elicit spontaneous IIS activity even in response to asynchronous afferent input from SC.

A major motivation for identifying the various conditions under which the CA1 can exhibit IIS, is to eventually develop a control strategy to disrupt IIS events. Any control strategy should take into consideration the fact that different network conditions may require different control schemes to disrupt IIS. This affects not only the control approach but also possibly the choice of actuation used in the implementation of a control system. We anticipate that the CA1 network model presented here and our findings of the general conditions under which the CA1 can elicit an IIS response could serve as a computational tool to effectively investigate and develop various control paradigms for the ultimate purpose of controlling IIS.

## **2. MATERIALS AND METHODS**

## **2.1. EXPERIMENTAL SETUP**

Adult male Sprague Dawley rats (*n* = 3) of age 63 days and weighing between 200 and 265 g were used for the experiments. Thirtytwo microwire recording electrodes were bilaterally implanted into the CA1 region of each rat's hippocampus. Chronic limbic epilepsy was induced in the rats using the *in vivo* selfsustaining electrical status epilepticus animal model (Lothman et al., 1989). The Institutional Animal Care and Use Committee of the University of Florida approved all protocols and procedures (IACUC protocol D710). The rats were housed in a controlled environment and monitored with continuous video and CA1 local field potential recordings. At the end of the recording session, the rats were sacrificed and the intact brains were excised. The isolated intact brains were imaged with high strength magnetic resonance microscopy to confirm the location of the electrode placement within the CA1 region of the hippocampus (Talathi et al., 2009). Data from a single electrode implanted in the contral-lateral CA1 subfield was analyzed to extract IIS shape profiles (Talathi et al., 2009), which were detected using a modified spike clustering algorithm (Fee et al., 1996) that sorted spikes into separate clusters.

Data for reference PDSs were obtained from Dr. K. Srinivasa Babu (Christian Medical College, Vellore, See Acknowledgments). Ventral hippocampal horizontal slices of 400 μM thickness were dissected out from a 4 week old Wistar rat. The slices were transferred into an interface chamber containing artificial CSF with 118 mM NaCl, 2.5 mM KCl, 2 mM CaCl2, 2 mM MgCl2, 25 mM NaHCO3, 1.24 mM NaH2PO4, and 10 mM glucose equilibrated with dissolved Carbogen (95% O2 and 5% Co2) at room temperature and at a pH of 7.4. After 1 h of incubation, the slices were perfused with 20μM Bicuculline to induce epileptic bursting. Recordings were performed with Glass electrodes (4–8 mOhm) fabricated from borosilicate glass capillaries, filled with pipette solution containing (in mM) 135 Kmeso4, 8 NaCl, 10 HEPES, 2 Mg2ATP, and 0.3 Na3GTP. Signals were digitized at 10 KHz and recorded using Clampex software (Molecular devices, USA).

## **2.2. COMPUTATIONAL SETUP**

All computational networks were built using a custom C++ framework. The code was compiled using g++ (ver 4.4.6) and run using a RHEL 6 cluster. Up to 64 simulations could be run in parallel using the OpenMPI framework. The computational cluster consisted of 2 Intel Xeon dual-quad core dual-rackmounts. This provided 32 cpu cores. The total RAM available to the system was 48 GB. Networks of 300 neurons for a simulation time of 1 s, typically took between 10 and 30 min depending on the number of synapses implemented for a given network. Analysis of experimental data, simulation data, and template based studies were done on an Intel Xeon quad core PC using Matlab. Numerical integration was performed using the fourth order Runge-Kutta(RK4) algorithm with a time step of 0.01 ms. The RK4 technique was used with delay-differential equations (See Methods: Synaptic model) after verifying that there were no noticeable deviations in simulation results with the Euler method of numerical integration. We compare the CA1 model's output for different integration methods and different time steps in the Supplemental data section. The codes used for producing some of the results will be made available on modelDB.

## **2.3. NEURON MODELS**

In this section we introduce the neuronal models that we employ for the construction of our network. We use single compartment, standard Hodgkin-Huxley type neuronal models for both the pyramidal cells (Py) and interneurons. For interneurons, we model the predominant populations found in the CA1 which are the basket (B) and orien/alveus (OA) cells. The Py cells are modeled using the Golomb neuron model (Golomb et al., 2006), the B cells are modeled using the Wang-Buzsaki model (Wang and Buzsáki, 1996), and the OA cells are modeled using the Wang neuron model (Wang, 2002). The neuron models are mathematically structured as follows:

$$\begin{aligned} \boldsymbol{c}\dot{V} &= I\_{DC} - I\_{\boldsymbol{\xi}}(V, \mathbf{n}) - I\_{\text{KCa}}(V, \mathbf{C}a^{2+}) + I\_{\text{syn}}\\ \dot{\mathbf{n}} &= \mathbf{G}(V, \mathbf{n})\\ \mathbf{m}\_{\infty} &= \mathbf{U}(V) \end{aligned} \tag{1}$$

where, *c* is the membrane capacitance, *V* is the membrane voltage, *Ig* represents the sum of the currents flowing due to voltage gated intrinsic membrane ion channels (Na, K, Ca etc.), *IKCa* is the membrane current due to calcium gated potassium channels, *I*syn is the total synaptic current and *IDC* is an intrinsic current that sets the cell's excitability. **n** ∈ [0, 1] and **m**<sup>∞</sup> ∈ [0, 1] are the gating variables vectors for ion channels present on the neuron membrane with finite rise time constants and instantaneous risetime kinetics, respectively. We provide the details on the neuron models channel currents and kinetics in the Supplemental data section.

## **2.4. SYNAPTIC MODEL**

The synaptic current contribution for all the neuron models is modeled as: *I*syn(*t*) = *gsS*(*t*)(*V* − *Es*), where *gs* can be the α-Amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) or γ -Aminobutyric acid (GABA) synaptic conductance, *Es* is the reversal potential of the synapse (approximately 0 mV (Andrasfalvy and Magee, 2001) for AMPA and −72 mV for GABA Cohen et al., 2002), and *S(t)* represents the fraction of bound synaptic receptors and has the following form:

$$\dot{S}(t) = \frac{S\_0(V\_{\rho\text{re}}(t-\tau\_\text{x})) - S(t)}{\hat{\mathfrak{t}}(S\_1 - S\_0(V\_{\rho\text{re}}(t-\tau\_\text{x})))} \tag{2}$$

where

$$S\_0(V\_{\rm pre}) = 0.5(1 + \tanh(120(V\_{\rm pre} - 0.1)))\tag{3}$$

where *Vpre*(*t*) is the pre-synaptic neuronal spike, τ*<sup>x</sup>* is the synaptic delay for synapse type *x* ∈ {Py,B,OA}. τ*Py* was assumed to be near instantaneous at 0.5 ms. Basket cells were assumed to have a larger delay of 5 ms due to their distance from Py cells. OA cells that were the furthest away from Py cells had synaptic delays of 10 ms with Py cells. The delay between OA and B was 5 ms. The delay from synapses of interneurons was calculated assuming a conduction velocity of 0.1 m/s (Salin and Prince, 1996) and considering the average distance between neuronal populations. The rise and decay time constants of the synaptic current is expressed as τ*<sup>R</sup>* = τˆ(*S* − 1) and τ*<sup>D</sup>* = τˆ*S*, respectively. τ*<sup>R</sup>* for all cells was assumed to be near instantaneous at 0.1 ms. τ*<sup>D</sup>* for Py, fast firing B and slower OA synapses were set at 1 ms, 3 ms, and 5 ms, respectively (Geiger et al., 1995; Bartos et al., 2002, 2007; Taxidis et al., 2012).

In **Table 1** we list the values of the synaptic strength conductances used for the CA1 model. This was done by matching the post-synaptic potentials (PSPs) of various neurons with physiological recordings (Cobb et al., 1997; Ali et al., 1998; Taxidis et al., 2012). As we were unable to obtain physiological data for PSPs of the synaptic connections for Py-Py and SC-B, we assumed SC-B strengths to be the same as other inhibitory synapses and the Py-Py excitatory strength was assumed to be the same as SC-Py's. We estimated the strength of SC-Py using empirical techniques (see Results: Estimating the synaptic parameters of the CA1 network).

## **2.5. DETAILED CA1 NETWORK MODEL**

We construct a detailed representation of a section of the CA1 using information interpreted from literature (Kandel et al., 2000; Andersen et al., 2006; Demont-Guignard et al., 2009). Following from Demont-Guignard et al. (2009), we construct the CA1 network with 80%−20% excitatory-inhibitory neuron ratio using 225 Py cells, ∼22 B cells, and ∼22 OA cells that are distributed in a 3D cuboid of 0.21 × 0.21 × 0.21 mm3 as shown in **Figure 2A**. Each of the neuron types are distributed uniformly within their individual cuboid layers. The connection between any given pair of cells depends on the euclidean distance between the pairs and follows a Gaussian distribution with standard deviation σ*x*,*y*, where x and y are the pair of connected cells. We initially



*The resting membrane potential of the neurons in the CA1 network model from which the PSPs were measured were:* −*65.73 mV for Py cells,* −*64.02 for B cells, and* −*61.54 mV for OA cells.*

choose σ*Py*,*Py* = 20μm such that the probability of the connection between any two Py cells is low. The other parameters for the network are taken from Demont-Guignard et al. (2009) as: σ*Py*,*<sup>B</sup>* = 166.6μm, σ*B*,*Py* = 233.3μm, σ*Py*,*OA* = 166.6μm, σ*OA*,*Py* = 280μm, σ*B*,*<sup>B</sup>* = 233.3μm, σ*OA*,*<sup>B</sup>* = 280μm. OA cells do not exhibit recurrent inhibition and B cells do not synapse on to the OA cells. The CA3's input to the CA1 via SC is simulated as a barrage of action potentials through AMPA synapses that synapse on to Py and B cells. 100% of the B cells each receive an SC synapse, while the percentage of Py cells each receiving SC input can be varied (Typically 70%). We illustrate the connectivity of the CA1 network in **Figure 2B**.

Since IISs are a hyper-excitable phenomenon, it is necessary to ensure that the CA1 network is in a hyper-excitable state. In the Results section, we have provided details on estimating synaptic conductances that are conducive for this purpose. We also increase the Py cell's excitability by setting the *IDC* value of Pyramidal cells to 0.3 nA/cm2. We additionally set *IDC* = −0.3 nA/cm<sup>2</sup> for OA cells to prevent them from firing spontaneously and inhibiting the Py cells. Bio-physically this can be interpreted as due to decreased interneuronal cell activity. The effects of sprouting in the network model was a critical component in this paper. We define the degree of sprouting (*P*sprout) by the average number of incoming synapses from other Py cells that each Py cell receives. For example, *P*sprout = 40 corresponds to an average of 40 input synapses from other Py cells to every Py cell. We define *P*sprout = 100 to be the upper-limit in our simulations due to the observations in our results that IIS frequency saturates at this value of sprouting (see Results: IIS formation as an induced or emergent synchronization phenomenon).

## **2.6. LOCAL FIELD POTENTIAL MEASUREMENT**

The local field potential (LFP) refers to extracellular voltage signals (≤500 Hz) recorded sub-durally from the brain(Dzhala and Staley, 2003). EEG level signals are usually found in the range of ≤100 Hz while signals from multiunit recordings are found at around ≥300 Hz. Both these components are usually present in various combinations in LFP depending on the type of electrode and site of the electrode placement (Dzhala and Staley, 2003). For our computational model we focus on the lower frequency EEG-range signals as IISs fall in the EEG-range signal category. To model this, we must be able to capture transmembrane potentials that might contribute to an IIS (Buzsáki et al., 2012). Researchers seeking to reproduce LFP signals in a computational environment commonly consider summed synaptic current contributions. This however may not be sufficient to capture detailed aspects of highly synchronized events such as IIS. Transmembrane potentials can build up significantly under synchronous conditions to affect extracellular measurements (Buzsáki et al., 2012). These include fast spikes(in bursts) and AHP. When there is an inflow of cations from the extracellular medium into the neuron, a current source is created extracellularly. To maintain electro-neutrality, there will be an outflow of ions into the extracellular field creating a sink. A dipole created in such a manner obeys Kirchoff's current law. An LFP's main component is from the dipoles (Buzsáki et al., 2012) whose measured potential varies with the inverse square of distance between the current source-sink site and the recording

electrode position. Recent studies have shown that there may also be significant contributions from monopoles whose creation does not follow Kirchoff's law (Destexhe and Bedard, 2012; Riera et al., 2012). However, the effects of monopoles on recorded LFP from the brain are not completely understood. Hence, we choose not to model our LFP based on monopoles. With these factors in mind, we choose to model the LFP as a direct function of the membrane voltage of individual Py neurons. Similar approaches have also been used by other researchers such as Ursino and La Cara (2006).

$$LFP(t) = \sum\_{i}^{N} \frac{V\_{si}(t)}{r\_i^2} \tag{4}$$

*N* is the total number of pyramidal cells, *Vsi* is the *i*th neuron's soma membrane voltage,and *ri* is the distance of the *i*th neuron from the measuring electrode. In our network model, the LFP electrode is placed in the vicinity of the stratum pyramidale layer. This arrangement indicates that the major LFP component will be from the pyramidal soma.

## **3. RESULTS**

## **3.1. TEMPLATE BASED ANALYSIS OF THE TEMPORAL PROFILE OF PDS EVENTS**

We begin by identifying the temporal characteristics of PDS events and the degree of PDS synchronization that is implicated in the generation of an experimentally observed IIS. In **Figure 1A**, we show an example of a typical IIS (normalized with peak value of 1) recorded from the CA1 subfield of an epileptic rat (Talathi et al., 2009). In agreement with data from earlier works (Jayakar et al., 1989; Adjouadi et al., 2005), the IIS profile exhibits the following empirical characteristics: (a) The total duration *t* of the IIS, *t* = -*RP* + -*PF* + -*FQ*, is between 50 ms and 400 ms. (b) The two half waves (R to P and P to F) satisfy the condition that their absolute difference is less than or equal to their calculated average (Jayakar et al., 1989): |-*RP* − -*PF*| ≤ 0.5(-*RP* + -*PF*) (c) The downward deflection voltage *A*2, is larger than the upward deflecting voltage *A*1, satisfying the condition: 0.25 ≤ *R* ≤ 2, where *R* = *A*1/*A*2.

An example of a typical PDS, obtained from a rat hippocampal slice perfused with Bicuculline (see Methods) is shown in **Figure 1B**. In general, there is a large variability in the shape profile of PDSs (Hwa et al., 1991), however the following empirical characteristics are commonly observed in majority of PDS shape profiles: (a) The amplitude of the depolarization peak (*ADP*) is usually greater than the amplitude of the afterhyperpolarization peak (*A*AHP) (Kandel et al., 2000) and (b) The duration of PDS burst τ*<sup>B</sup>* is less than or equal to the duration of after-hyperpolarization τAHP, τ*<sup>B</sup>* ≤ τAHP.

The transformation : (τ*B*, τAHP) → -*RP*, -*PF*, -*FQ* from individual PDS events into a mean field IIS event is mediated via synchronization of PDS events, which can be quantified by estimating the distribution in the timing τ*s*, of the occurrences of PDS in the CA1 Py cells. In order to characterize this transformation, we use a template based method to first construct an artificial PDS template (ρ) parameterized by τ*<sup>B</sup>* and τAHP as follows:

$$\rho(t) = \begin{cases} -\frac{\mathcal{E}t}{\mathsf{T}\_B} e^{\frac{\mathsf{T}}{\mathsf{T}\_B}} & \text{if } t \le 0 \\\ -\frac{\beta et}{\mathsf{T}\_{\mathsf{AHP}}} e^{-\frac{\mathsf{T}}{\mathsf{T}\_{\mathsf{AHP}}}} & \text{if } t > 0 \end{cases} \tag{5}$$

This empirical function produces a PDS envelope with the PDS depolarization peak normalized to 1 as illustrated in **Figure 1D**. These artificial PDS constructs are then used to generate a template IIS event as follows:

$$\text{IIS}\_{\text{temperature}}(t) = \sum\_{k} \rho(t - t\_0 + t\_k) \tag{6}$$

where *tk* ∈ <sup>−</sup>τ*<sup>s</sup>* <sup>2</sup> , <sup>τ</sup>*<sup>s</sup>* 2 represents the time of occurrence of *k*th PDS event relative to a reference time *t*0. We note that the parameter τ*<sup>s</sup>* controls the effective degree of synchronization of the PDS events that are implicated in the generation of the IIS. Using the IIS template in equation 6, we estimate the set of parameter values {τ<sup>∗</sup> *<sup>B</sup>*, <sup>τ</sup><sup>∗</sup> AHP, <sup>τ</sup><sup>∗</sup> *<sup>s</sup>* } that produce an IIS template with a "closest-fit" match [root mean squared error (RMSE) <0.1] to the experimentally recorded mean field IIS (**Figure 1A**). The key results of our template based analysis are summarized in **Figure 3**. In **Figures 3A–C**, we plot the RMSE as function of τ*<sup>B</sup>* and τAHP for three specific values of τ*<sup>s</sup>* = {20, 40, 80} ms ,respectively. In **Figure 3D**, we present an example of a valid IIS template event with the closest match (lowest RMSE) to the experimental IIS. In **Figures 3E,F**, we show examples of invalid IIS template events generated using PDSs with τ*<sup>B</sup>* << τAHP and τ*<sup>B</sup>* >> τAHP, respectively. From **Figure 3A** we identify PDS parameters {τ*B*, τAHP, τ*s*} = {50 ms, 350 ms, 20 ms} that produces a template IIS with low RMSE against experimental IIS(≈ 0.1). We use these parameters as initial conditions and minimize the template IIS RMSE using the Nelder-Mead simplex method (Lagarias et al., 1998) to obtain τ<sup>∗</sup> *<sup>B</sup>* <sup>=</sup> 37 ms, <sup>τ</sup><sup>∗</sup> AHP <sup>=</sup> 300 ms, and <sup>τ</sup><sup>∗</sup> *<sup>s</sup>* = 24 ms.

## **3.2. ESTIMATING THE SYNAPTIC PARAMETERS FOR THE SCHAFFER COLLATERAL AFFERENTS**

In this section we present results of our analysis of a synaptically reduced model for the CA1 subfield that is used to estimate the excitatory synaptic strength of SC input onto the epileptic CA1 Py cells. The SC-Py synaptic strength is chosen such that the Py cell elicits PDS bursts with measured temporal features (τ*B*, τAHP) matching those obtained from the template PDS analysis. We define the synaptically-reduced model of the CA1 subfield by incorporating just the essential connectivity patterns between a single CA1 Py neuron and the two prominent interneuron types; the basket cell (B) and the orien-alveus cell (OA). The synaptically-reduced network's architecture is illustrated in **Figure 4A**. The conductance values for all synapses (Py − B = Py − OA = *gex* and SC − B = B − Py = OA − Py = *gin*) except for SC − Py = *g*SC, are estimated by matching the PSP magnitudes from each cell type (see Methods: Synaptic models). We then systematically varied *gSC*, corresponding to the strength of excitatory synaptic input from the SC onto the Py, in order to trigger a template matched PDS response from the CA1 Py cell. The results of these calculations are reported in **Figure 4B**. We notice that the error in τ*<sup>B</sup>* and τAHP values decreases and saturates between 1.5 ≤ *gSC* ≤ 2.5 such that the synaptically reduced network elicits a PDS with temporal features {τ<sup>∗</sup> *<sup>B</sup>*, <sup>τ</sup><sup>∗</sup> AHP} ≈{37, 370} ms, which conform with the parameters identified for the template based PDS event. An example of this model generated PDS is shown in **Figure 1C**. These synaptic parameters are next used in the construction of a biophysically relevant CA1 network model that is capable of generating IISs which matches the features of experimentally recorded IISs.

**FIGURE 4 | (A)** The synaptically reduced CA1 network consisting of the basic neuron types with synaptic interconnections (except autapses). The network is used to determine synaptic parameters (*gi* and *gsc* ) that produces a PDS matching that predicted by the template studies. **(B)** Shows the variation of burst width (τ*B*) in green, AHP width (τAHP) in blue, and the RMSD error (red curve which is scaled by a factor of 30 to match the green Y-axis on the right) of these values from the template-analysis PDS parameters, as a function of *gsc* .

**(D)** Shows a valid IIS profile while **(E,F)** show cases when τ*<sup>B</sup>* << τAHP and τAHP << τ*B*, respectively.

## **3.3. ELICITING IIS FROM THE CA1 NETWORK MODEL**

The synaptic parameters obtained from the synaptically-reduced network model of CA1 neurons were incorporated into a biophysically relevant network model of the CA1 subfield comprising of excitatory Py neurons (≈230) and two major interneuronal subtypes, the basket (B) cells (≈30) and the orien-alveus (OA) cells (≈30). All the neurons were modeled as single compartment model neurons following the conductance based Hodgkin-Huxley framework. To address the issue of variability in biological neurons and its effect on system dynamics (Marder and Taylor, 2011), in particular the formation of IIS, we investigated two different single compartment CA1 Py neuron models in our construction of the CA1 network. Interneuronal dynamics are implemented using well-established single-compartmental neuron models, the Wang-Buzsaki model for fast spiking basket cells (Wang and Buzsáki, 1996) and the Wang bursting neuron model for the OA cells (Wang, 2002). Further details on the neuron model types and the distribution of synaptic connectivity in the network are provided in the Methods Section. Using this model we investigate how pathological changes in the CA1 subfield including sprouting of CA1 Py neurons and the variability in the afferent input from the SC affects the CA1 subfield's ability to elicit IISs. Toward this end, we systematically investigated the likelihood for CA1 to elicit an IIS as a function of (a) the degree of synchronization in the volley of afferent input from SC onto the CA1 neurons. (b) the percent of CA1 Py cells that receive direct afferent input from the SC and (c) the degree of CA1 Py neuronal sprouting.

The degree of synchronization in the afferent spike volley was varied as a function of a temporal window τ*SC* Syn in which a given fraction *fSC* of CA1 neurons receive afferent input from SC with uniform probability. The degree of sprouting was quantified in terms of percent sprouting *P*sprout, corresponding to the average number of synapses a given CA1 Py cell receives from other Py cells in the network. For example, *P*sprout = 40 implies any given CA1 Py neuron in the network receives synaptic input from 40 other CA1 Py neurons in the network. In **Figures 5A–E**, we summarize the key results of our simulation studies to identify

input and output rasters and the LFP measured from the activity of the CA1 during low sprouting. The Py cells receive low frequency Poisson input via SC but at 500 ms, there is a quasi-synchronous barrage of input that causes the Py in the CA1 to fire in a synchronous manner. This activity is an example snapshot of the multiple simulations done to create **(A–E)**.

the set of parameter values α = {τ*SC* Syn, *fSC*, *P*sprout} for the CA1 network that can trigger an IIS that satisfies the empirical criteria for an experimentally observed IIS. The color code corresponds to the RMSE between the simulated IIS and the mean profile of the experimentally recorded IIS. The white space in the color plots correspond to the regions in the parameter space α, where the CA1 network failed to trigger an IIS that satisfied the empirical criterion. In **Figure 5G**, we show a spike raster plot of a typical input received by CA1 Py cells from the SC and in **Figure 5H**, we show the spike raster of the response of CA1 Py neurons to the SC input. Finally, in **Figure 5I**, we show the corresponding LFP activity including the presence of an IIS that is generated by the CA1 network model.

As can be seen from **Figures 5A–E**, the CA1 network with very low sprouting, mimicking the topology of a normal brain, will not elicit an IIS except when a large fraction of CA1 neurons (*fSC* ≥ 80%) receives a quasi-synchronous barrage of afferent input (τ*SC* Syn ≤ 20 ms) via SC. Increased recurrent connectivity of CA1 (*P*sprout > 40) can evoke IIS in the CA1 network even in the presence of low synchrony SC afferent input (τ*SC* Syn > 80 ms) and a low fraction of SC input to CA1 pyramidal cells (*fSC* ≈ 30%). This suggests that the ability of CA1 to trigger an IIS increases with increasing recurrent connections in the network and is less dependent on the variability in the afferent input from the SC. In turn, this indicates that IISs may become more frequent with increased axonal recurrent sprouting as a wider range of SC input is now sufficient to elicit an IIS. Considering a previous hypothesis, that the epileptic CA1 pyramidal axonal sprouting increases over time and enhances the CA1's ability for local recruitment during population bursts (Smith and Dudek, 2001), our computational observations suggests that the frequency of IIS events may increase over time during epileptogenesis. Indeed, this phenomenon has been reported in animal models of epilepsy (Buzsáki et al., 1991).

We next investigated how the results presented above vary with the choice of the CA1 Py model neuron used in the implementation of the CA1 network. We employed a CA1 pyramidal cell model, recently developed and validated by Nowacki et al. (2011). **Figures 6A–E** show that our general conclusion for the region in the parameter space of α where the CA1 network can trigger an IIS remains unaltered. Minor differences in the degree of Py sprouting and the percent of CA1 neurons receiving direct synaptic input from SC is attributed to the difference in the intrinsic excitability of the model Py neurons. Following from these findings we observe that the ability to trigger an IIS in an excitable CA1 network is primarily dependent on the ability of an individual CA1 neuron to generate a PDS like burst in response to synaptic input from SC and is less dependent on the exact details of the mechanism of the PDS generation itself. **Figure 6F** compares an IIS generated using the Nowacki model for PY in the CA1, against the mean experimental IIS.

Finally, we investigated the degree of PDS synchronization implicated in the generation of an IIS. From the template based analysis, we predicted that an IIS will result when individual PDS events occur within a temporal window of τ<sup>∗</sup> *<sup>s</sup>* = 24 ms. In order to verify the significance of this prediction, we analyzed the distribution of PDS events generated in the CA1 network in response to afferent drive from the SC. Specifically, for each IIS event triggered in the CA1 network, we look for the fraction of PDS events (*fPDS*) that fall within a temporal window of ±12 ms around the peak of the triggered IIS event. In **Figure 7**, we plot the distribution of this fraction. We find that when an IIS occurs,

synchronization (jitter). In each of the colormaps for a fixed input window (t*SC* Syn), the input percentage from SC to pyramidal cells (f*SC*), and the degree of sprouting (Psprout) was varied. **(F)** Shows an example model generated IIS in comparison with the mean experimental IIS.

on average ≥50% of the PDS events generated in the network lie within a 24 ms temporal window, in agreement with the PDS empirical estimation results.

## **3.4. SPONTANEOUS GENERATION OF IIS FROM HIGH RECURRENCE AND SPARSE RANDOM INPUT**

Motivated by our finding that the CA1 network with a high degree of recurrent connections is able to elicit an IIS even in the absence of a synchronous barrage of afferent input from the SC, we investigated the response of the CA1 network to an asynchronous afferent input from the SC modeled as a sequence of identically independently Poisson distributed spikes occurring at 5 Hz. The results are summarized in **Figures 8A–C**. We observe that in response to Poisson distributed random SC input, the CA1 network, with sufficiently high degree of sprouting (*P*sprout > 65)

responds by emitting a periodic sequence of spontaneously triggered IIS occurring at ≈3 Hz. This type of quasi-periodic synchronization in response to random input have been observed in generic networks of coupled excitatory and inhibitory neuronal populations(van Vreeswijk and Hansel, 2001; Kudela et al., 2003). On further investigation, we noticed that the spread of network activity during a population burst corresponding to these spontaneously triggered IIS events was not fixed. This is illustrated in **Figure 9**. In the two selected IISs from the same simulation, the spread of synchronized bursting can be seen occurring in opposite directions. This observation suggests that specialized neurons (like hub neurons) may not be necessary for this kind of burst synchronization. We also decreased the percentage of recurrent excitatory sprouting and as suggested from the reported findings in the previous section, all IIS activity in the network was abolished (**Figures 8D–F**).

We also investigated the effect of scaling on the network's ability to exhibit spontaneous periodic IIS activity by increasing the size of the CA1 network 30 fold (by scaling the network in X and Z dimensions shown in **Figure 2B**) while maintaining the network connectivity, the level of neuronal excitability and the synaptic strengths. The scaled network consisted of 8000 pyramidal cells, 1000 basket, and 1000 oriens/alveus cells. The sprouting was maintained at approximately *P*sprout = 65. The network was stimulated with 5 Hz Poisson distributed random input via SC. The scaled network's rasters and LFP responses are shown in **Figures 10A–C**. We observed that the scaled model exhibited identical behavior to that of the original model in that IIS were generated at a similar rate of ≈3 Hz in the scaled network as well.

## **3.5. IIS FORMATION AS AN INDUCED OR EMERGENT SYNCHRONIZATION PHENOMENON**

From results presented in the previous Sections, we see that the CA1 network can exhibit IIS via (a) induced synchronization

LFP of network activity when sprouting is low (normal case).

of the PDS events elicited from CA1 Py neurons in response to synchronous barrage of afferent input from the SC and (b) emergent synchronization of PDS events elicited from CA1 Py neurons with high degree of recurrent connections in response to asynchronous Poisson distributed SC input. In **Figures 11A,B**, we show a schematic diagram depicting the scenarios that can trigger the emergence of IIS in the CA1 network. The mechanism of induced synchronization leading the emergence of IIS is relatively straightforward and does not require recurrent connections between CA1 Py cells as a necessary condition. In the absence of

recurrent sprouting (*P*sprout ≈ 70).

any recurrent CA1 Py neuronal connections, if a large fraction of CA1 Py cells receive sufficiently synchronized SC input; the CA1 Py cells will respond with a high probability of eliciting a PDS more or less simultaneously, resulting in the observation of an IIS in the mean field CA1 network activity.

On the other hand, the spontaneously occurring IIS in the CA1 network in response to asynchronous Poisson distributed SC input requires recurrent CA1 Py connections as a necessary condition. We refer to this phenomenon as emergent synchrony of the PDS events because the synchrony emerges from within the network as opposed to primarily from the input. In addition to recurrent connections between the CA1 Py neurons, a number of additional network parameters that regulate the network excitability are implicated in the emergence of spontaneous IIS within the CA1 network. In particular, the strength of inhibitory synaptic connections and the frequency of Poisson distributed SC input play major roles in the emergence of spontaneous IIS.

We begin by analyzing the role of sprouting in the emergence of spontaneous IIS. We systematically varied the degree sprouting (recurrent connections between the CA1 Py neurons) from *P*sprout = 0 to *P*sprout = 100, while keeping all other CA1 network parameters in the default state (see Methods). In **Figure 11C**, we plot the result of these calculations (shown in blue trace) by measuring the rate of spontaneous IIS as function of the degree of sprouting in the CA1 network. We observe that spontaneous IIS emerge in the network when *P*sprout > 25, with the rate of spontaneous IIS saturating to a maximum value of ≈3 Hz for *P*sprout = 100 sprouting in the network. Further investigations led us to the observation that the slow activation time constant τ*<sup>Z</sup>* of the potassium M-current of Py cells is primarily implicated in determining the maximum rate of spontaneous IIS activity in the CA1 network. This is illustrated in **Figure 11C**, where we plot the rate of spontaneous IIS in the CA1 network as a function of *P*sprout for different values of τ*Z*. We observe that the maximum rate of spontaneous IISs monotonically decreases with increasing value of τ*Z*. This is because for smaller value of τ*Z*, the M-current activates faster resulting in truncating the number of spikes per burst of PDS activity, while faster deactivation of M-currents result in an increased rate of PDS bursting activity, which in turn triggers the emergence of spontaneous IIS at an increased rate.

We next analyzed the dependence of the rate of spontaneous IIS on the frequency of Poisson distributed random SC input. For a default network configuration with *P*sprout = 70, we

**FIGURE 11 | (A)** This figure schematically represents the concept of induced synchrony: When an already synchronous barrage of input arrives at the CA1 network, the output also tends to be synchronous in nature. This has been observed in networks with both low and high degree of sprouting. **(B)** Another case when IIS is possible but without synchronous input. In this case we observe that the CA1 network must have a high degree of sprouting and receive sparse random input continuously. **(C)** IIS frequency vs. sprouting trends. We see that all IIS frequencies tend to saturate by 100% sprouting. Lower values of the M-current decay constant (τ*<sup>Z</sup>* ) significantly changes the maximum rate of spontaneous IIS exhibited in the network. The mean trend is shown in dark colors while the lighter bands indicate the standard error.

**(D)** The nature of IIS rate vs. Poisson input frequency is non- monotonic. Networks that are less excitable (*gsc* = 1.0 mS/cm2, *IDC* = 0.1μA/cm2, *P*sprout = 30) show an increase in IIS rate up to a certain point (30 Hz) beyond which the nature of the poisson input goes from synchrony-conducive excitability to synchrony-disruptive excitability. The mean trend is shown in dark blue and the standard error is depicted in the light blue band. **(E)** Increasing GABA synaptic strengths can interfere with synchronization and hence decrease the rate of spontaneous IISs. The mean trend is shown in dark blue and the standard error is depicted in the light blue band. **(F–H)** Input and output rasters and the LFP of a network with high GABA synaptic strength (*gin* = 25 ms/cm2).

systematically increased the frequency of Poisson input from 5 Hz to 30 Hz. The results of this analysis are presented in **Figure 11D**. We see that, with default network parameters, the network initially responds by generating spontaneous IIS at ≈3 Hz, but as the frequency of Poisson input increases, the network exhibits high frequency-low amplitude non-IIS like spiking activity. By estimating the degree of synchronization amongst the CA1 Py neurons using a well-established synchronization measure Hansel and Sompolinsky (1992); Kudela et al. (2003), we find that the network receiving Poisson distributed random SC input at 30 Hz exhibits 70% less synchronization than is the case when the network exhibits spontaneous IIS in response to Poisson distributed random SC input at 5 Hz. This suggested that increasing the rate of random spikes to the CA1 via SC disrupts network synchronization making the network less susceptible to exhibit spontaneous IIS. It should be noted that this finding was observed in a network that was already in an excitable state, i.e. high degree of recurrent connections and Py cells are highly input sensitive (*IDC* = 0.3μA/cm2). We next investigated if this finding varied with a network of lower intrinsic network excitability. We reduced the degree of sprouting to 30 and at the same time reduced τ*<sup>Z</sup>* to 25 ms, such that the network does not exhibit spontaneous IIS at the saturating rate of 6 Hz observable for network with *P*sprout = 100. We also reduced the Py cells intrinsic excitability (*IDC* = 0.1μA/cm2, *gsc* = 1.0 mS/cm2). In this case, as we increased the frequency of Poisson distributed random SC input, the rate of spontaneous IIS activity is increased reaching a peak of 6 Hz for Poisson input frequency of 30 Hz. Thus, depending on the level of network excitability the rate of random SC input can be either conducive or disruptive to emergent synchronization in the network. In **Figure 12** we show examples of LFPs generated by the CA1 network model for different values of SC input rates and excitability levels of Py cells (*IDC*). We notice that in general as the rate of SC input increases the network goes from producing IIS-like spikes to higher frequency oscillations.

Finally, we analyzed how the rate of spontaneous IIS depends on the strength of inhibitory synaptic connections in the network. For different values of *P*sprout = {35, 50, 65}, we gradually increased the strength of inhibitory synaptic couplings in the network. From **Figure 11E** we see that the frequency of spontaneous IIS decreases with increase in network inhibition and eventually saturates. Furthermore, we also observe that the rate at which the IIS frequency saturates depends upon the degree of sprouting of pyramidal cells in the CA1 network. We see that for *P*sprout = 65, the frequency of spontaneous IIS takes longer to saturate than in the case of *P*sprout = 35 where the frequency of IISs saturates to a non-zero minimum (<1 Hz) much faster. Finally, we make the observation that when *P*sprout is sufficiently high to induce spontaneous IISs, even high values of GABAergic conductances


**FIGURE 12 | The figure shows the LFP of the CA1 network for different SC input rates (Poisson frequency varied) and different levels of neuronal excitability (***Idc* **).** In general as the rate of SC input increases, the network goes from producing IIS-like spikes to higher frequency oscillatory activity.

are unable to completely eliminate IISs. This possibly suggests that when a CA1 network reaches a sufficiently hyperexcitable state through recurrent pyramidal cell sprouting, even enhanced interneuronal activity may be insufficient in completely suppressing epileptiform activity (Franck and Schwartzkroin, 1985; Franck et al., 1988; Bausch, 2005). In **Figures 11F–H**, we show an example of raster and LFP plots when GABA strength (*gin*) values were set at 25 mS/cm2. The Figures clearly show that spontaneous IISs can still occur with high network inhibition but many of the synchronization attempts of the Py cells are thwarted by interneuronal interferences. This results in much weaker islands of synchronization (**Figure 11G** at 1500 and 2000 ms) or no synchronization at all (at 900 ms).

## **4. DISCUSSION**

Our primary goal was to first develop a biologically relevant platform for modeling IISs in the CA1, and then to analyze the network conditions necessary for IISs. In order to build the CA1 network model such that it possesses the capability to exhibit IISs, we had to consider various cellular and network level parameters possibly implicated in the role of IIS genesis. In order to avoid the impracticality of a computationally expensive parametric exploration of a high dimensional system for IIS generation, we chose to dissociate the cellular level parameters from network level parameters and analyze them separately in a hierarchic fashion.

We introduced a template based approach for estimating the empirical features of the cellular correlate of IIS, i.e., the PDS burst width (τ*B*) and the PDS after-hyperpolarization duration (τAHP), from experimentally recorded IIS. These features were used to estimate the synaptic parameters of a reduced CA1 network, which in turn allowed us to develop a biologically relevant model of the CA1 network capable of generating an IIS event with empirical characteristics matching those obtained from experimental recordings from the CA1 of an *in vivo* animal model of epilepsy. We believe that the proposed approach, leveraging experimental data to estimate network parameters, may be used for the analysis and development of models for other LFP features besides IIS. We however note that the template based approach proposed here provides a ballpark estimate for some of the critical network parameters. For example, PDSs that occur during an IIS would most likely have a distribution of burst width and AHP durations with the mean values presumably close to our estimated values. To our knowledge not many other methods are currently available for estimating computational network parameters solely from experimental recordings of LFP data such as IIS. This step is important for the biological validation of computational models that attempt to capture experimental features of interest.

We next developed a biophysically relevant model of the CA1 network in order to identify the network conditions under which the CA1 network can elicit an IIS. We found that the CA1 can trigger an IIS event under a variety of conditions. For network configurations characterized by a low degree of sprouting, IISs can be evoked by a synchronous barrage of afferent input from the SC with high percentage of SC to Py connectivity. The simulations also indicated that many pyramidal cells (>80%) are recruited via the SC when an IIS is triggered. For higher degrees of sprouting, we found that the formation of IIS is less dependent on the degree of input synchronization and the percentage of SC to Py connections. Indeed, even in the presence of low input synchronization (τ*SC* Syn = 240 ms) and low *fSC*, we noticed that IISs could still form for a sufficiently large degree of CA1 sprouting (*P*sprout ≥ 40). These findings suggest that sprouting may play a significant role in synchronization of PDSs, resulting in the manifestation of an IIS. We also observed that in the presence of sufficiently high degree of Py cell sprouting, sequence of asynchronous afferent input onto the CA1 Py cells from the SC can trigger spontaneously generated IIS events that occur in periodic fashion. These results indicate that the CA1 network is less influenced by the nature of SC input as the degree of Py sprouting increases. If we consider the hypothesis suggested by Staley et al. (2005) that sprouting is a phenomenon that progresses over a period of weeks in the TLE brain, then in conjunction with our results this indicates that IISs will gradually increase in their frequency over a timespan. This phenomenon was in fact observed experimentally by Buzsáki et al. (1991).

The CA1 network is capable of producing IISs primarily through two mechanisms of synchronization, (a) input-induced synchronization and (b) emergent synchronization. We found that while induced synchronization is a straightforward manifestation of SC input on Py output synchronization, emergent synchronization was a more complex phenomenon. IIS formation by emergent synchronization seems to depend on the CA1 network's intrinsic excitability. The level of the CA1 network excitability depends on many parameters. We observe that when the CA1 network has achieved a sufficient degree of Py cell recurrent sprouting, even enhanced GABAergic strengths cannot completely suppress epileptiform activity. This observation is in agreement with prior experimental studies (Franck and Schwartzkroin, 1985; Franck et al., 1988) where enhanced GABAergic activity alone was insufficient in controlling epileptic hyperexcitability. We also see that if the network is already in an excitable state (primarily high recurrent Py sprouting and sufficiently excitatory synaptic strengths) and is capable of exhibiting IISs, increasing the SC drive significantly (30 Hz) to the CA1 may actually disrupt IISs and instead result in high amplitude oscillatory activity. Hence there is a domain of network excitability in terms of SC input and network parameters for which IIS formation is possible.

We note that NMDA synapses were not incorporated in our implementation of the CA1 model primarily because it has been demonstrated that NMDA synapses are not critical for burst initiation in the hippocampus when the brain is already in an epileptic state (Stoop and Pralong, 2000; Stoop et al., 2003). It may, however, be worth investigating the effects of NMDA on long-term changes in the CA1, such as sprouting and synaptic plasticity between recurrent pyramidal cells.

While IISs have been the focus of this work, the CA1 network model may be capable of exhibiting other significant LFP patterns. For instance, we observe that in the event of significantly increased random input drive from the SC (@ 30 Hz), the CA1 network with a high degree of sprouting produces non-IIS oscillatory activity in the theta range (4–8 Hz). This LFP activity is

the frequency of SC input drive to the CA1 (@ 30 Hz). **(C)** Sequence of IIS events observed in an EEG recording of a rat induced with TLE. **(D)** EEG recordings of the tonic phase of an ictal event.

extremely similar to the tonic phase of an ictal event (Quiroga et al., 1997). We illustrate comparisons between our model's LFPs and experimentally recorded data for the IISs and the tonic phase of an ictal event in **Figure 13**. **Figures 13A,C** show IISs generated from the model CA1 network and from experimentally recorded EEG, respectively. **Figure 13D** shows the EEG recording of the tonic phase of an ictal event. In comparison, the model is able to generate a similar LFP waveform as shown in **Figure 13B** when the rate of Poisson SC input is increased to 30 Hz. We anticipate that our modeling paradigm may serve as a framework for future investigators interested in incorporating further details in the CA1 model in order to better understand the mechanisms of epileptogenesis.

## **ACKNOWLEDGMENTS**

We would like to thank Dr. K. Srinivasa Babu and Deepak Subramanian from the Neurophysiology Unit, Department of Neurological Science, Christian Medical College, Vellore, India for information and data on PDSs. We also thank Deepak Subramanian for his constructive suggestions during the development of this work.

## **FUNDING**

This research was funded by startup funds to Sachin S. Talathi from the Department of Pediatrics at the University of Florida; the intramural grant on Computational Biology at the University of Florida; and the Wilder Center of Excellence for Epilepsy Research and the Children's Miracle Network. Pramod P. Khargonekar was partially supported by the Eckis Professor Endowment at the University of Florida.

## **REFERENCES**


of persistent bursting and associated long-lasting changes in CA3 recurrent connections. *J. Neurosci.* 23, 5634–5644.


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 02 August 2013; accepted: 07 January 2014; published online: 27 January 2014.*

*Citation: Ratnadurai-Giridharan S, Stefanescu RA, Khargonekar PP, Carney PR and Talathi SS (2014) Genesis of interictal spikes in the CA1: a computational investigation. Front. Neural Circuits 8:2. doi: 10.3389/fncir.2014.00002*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Ratnadurai-Giridharan, Stefanescu, Khargonekar, Carney and Talathi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## **APPENDIX**

## **NEURON MODEL DETAILS**

In this section we provide details on the channel currents and conduction parameters for the three neuron models described in the methods section. The channel currents for the Golomb model (Golomb et al., 2006) are: *Ig* = {*INa*, *IKDR*, *INaP*, *IL*, *IA*, *IM*} and *IKCa* = 0 where,

$$\begin{aligned} I\_{\rm Na} &= \lg\_{\rm Na} m\_{\infty}^{3} h (V - E\_{\rm Na}) \\ I\_{L} &= \lg\_{L} (V - E\_{L}) \\ I\_{\rm KDR} &= \lg\_{\rm KDR} n^{4} (V - E\_{K}) \\ I\_{\rm NaP} &= \lg\_{\rm NaP} p\_{\infty}^{3} h (V - E\_{\rm Na}) \\ I\_{A} &= \lg\_{\infty} b (V - E\_{K}) \\ I\_{M} &= \lg\_{\rm M} z (V - E\_{K}) \end{aligned} \tag{7}$$

The gating variables are {*n*, *h*, *b*,*z*, *m*∞, *p*∞, *a*∞}, where

$$\begin{aligned} \dot{h} &= \phi(\Gamma(V, \theta\_h, \sigma\_h) - h) / (1 + 7.5 \ast \Gamma(V, \tau\_{th}, -6.0)) \\\\ \dot{b} &= (\Gamma(V, \theta\_b, \sigma\_b) - b) / \tau\_b \\\\ \dot{n} &= \phi(\Gamma(V, \theta\_n, \sigma\_n) - n) / (1 + 7.5 \ast \Gamma(V, \tau\_{th}, -15.0)) \\\\ \dot{z} &= (\Gamma(V, \theta\_z, \sigma\_z) - z) / \tau\_z \\\\ m\_{\infty} &= \Gamma(V, \theta\_m, \sigma\_m) \\\\ \rho\_{\infty} &= \Gamma(V, \theta\_p, \sigma\_p) \\\\ a\_{\infty} &= \Gamma(V, \theta\_a, \sigma\_a) \end{aligned} \tag{8}$$

where, φ = 1 and (*V*, θ, σ) = <sup>1</sup> 1+exp( <sup>−</sup>(*V*−θ) <sup>σ</sup> ) and the parameters are: θ*<sup>m</sup>* = −30 mV; σ*<sup>m</sup>* = 9.5 mV; θ*<sup>h</sup>* = −45 mV; σ*<sup>h</sup>* = −7 mV; θ*<sup>n</sup>* = −35 mV; σ*<sup>n</sup>* = 10 mV; θ*<sup>a</sup>* = −50 mV; σ*<sup>a</sup>* = 20 mV; θ*<sup>b</sup>* = −80 mV; σ*<sup>b</sup>* = −6 mV; θ*<sup>z</sup>* = −39 mV; σ*<sup>z</sup>* = 5 mV; τ*<sup>b</sup>* = 15 ms; τ*<sup>z</sup>* = 75 ms; τ*th* = −40.5 ms; τ*tn* = −27 ms.The channels reversal potentials and conductances are as follows: *ENa* = 55 mV; *EK* = −90 mV; *EL* = −70 mV; *gNa* = 35 mS/cm2; *gNaP* = 0 mS/cm2; *gKDR* = 6 mS/cm2; *gL* = 0.05 mS/cm2; *gA* = 1.4 mS/cm2; *gM* = 1 mS/cm2. We set *IDC* = 0.3μA/cm2.

For the Wang-Buzsaki model (Wang and Buzsáki, 1996) the channel currents are: *Ig* = {*INa*, *IK*, *IL*} and *IKCa* = 0,where

$$I\_{Na} = \text{g}\_{Na} m\_{\infty}^{3} h (V - E\_{Na})$$

$$I\_{L} = \text{g}\_{L} (V - E\_{L})\tag{9}$$

$$I\_{K} = \text{g}\_{K} n^{4} (V - E\_{K})$$

The gating variables are {*n*, *h*, *m*∞},where

$$\begin{aligned} \dot{h} &= \phi \left( \alpha\_h (1 - h) - \beta\_h h \right) \\\\ \dot{n} &= \phi \left( \alpha\_n (1 - n) - \beta\_n n \right) \\\\ m\_{\infty} &= \frac{\alpha\_m}{\alpha\_m + \beta\_m} \end{aligned} \tag{10}$$

The gate variables rate constants are given as:

$$\begin{aligned} \alpha\_m(V) &= \frac{-0.1(V+35)}{\exp(-0.1(V+35)) - 1} \\ \beta\_m(V) &= 4 \exp(-(V+60)/18) \\ \alpha\_h(V) &= 0.07(\exp(-(V+58))/20) \\ \beta\_h(V) &= 1/\exp(-0.1(V+28) + 1) \\ \alpha\_n(V) &= \frac{-0.01(V+34)}{\exp(-0.1(V+34)) - 1} \\ \beta\_n(V) &= 0.125 \exp(-(V+44)/80) \end{aligned} \tag{11}$$

The model parameters are: *ENa* = 55 mV; *EK* = −90 mV; *EL* = −65 mV; *gNa* = 35 mS/cm2; *gK* = 9 mS/cm2; *gL* = 0.1 mS/cm2; *c* = 1μF/cm2; φ = 5. *IDC* was set to 0 mV.

The Wang model's (Wang, 2002) channel currents are: *Ig* = {*INa*, *IK*, *Ih*, *ICa*, *IL*} and *IKCa* = *gKCa*[*Ca*<sup>2</sup>+]/([*Ca*<sup>2</sup>+] + *KD*) (*V* − *EK*), where

$$\begin{aligned} I\_{Na} &= \text{g}\_{Na} m\_{\infty}^{3} h (V - E\_{Na}) \\ I\_{K} &= \text{g}\_{K} n^{4} (V - E\_{K}) \\ I\_{h} &= \text{g}\_{h} H (V - E\_{h}) \\ I\_{Ca} &= \text{g}\_{Ca} m\_{\infty}^{2} (V - E\_{Ca}) \\ I\_{L} &= \text{g}\_{L} (V - E\_{L}) \end{aligned} \tag{12}$$

The gating variables {*n*, *h*, *m*∞} follow the same dynamics as described in equations 10 and11. The additional gating variables are {*H*˙ , ˙ [*Ca*<sup>2</sup>+]} which have the following dynamics:

$$\begin{aligned} \dot{H} &= \frac{H\_{\infty} - H}{\mathfrak{r}\_{H}} \\ H\_{\infty}(V) &= 1/(1 + \exp((V + 80)/10)) \\ \mathfrak{r}\_{H}(V) &= \frac{20}{(\exp((V + 70)/20) + \exp(-(V + 70)/20))} + 5 \\ \dot{Ca}^{2+} &= -\alpha I\_{Ca} - [Ca^{2+}]/\mathfrak{r}\_{Ca} \end{aligned} \tag{13}$$

The parameters for {*INa*, *IK*, *IL*} are the same as those described in the Wang-Buzsaki model. The additional parameters are: *gKCa* = 10 mS/cm2; *KD* = 30μM; *ECa* = 120 mV; *gCa* = 1 mS/cm2; *Eh* = −40 mV; τ*Ca* = 80 ms; α = 0.002.

#### **NUMERICAL INTEGRATION DETAILS**

In this section we show that the network behavior remains invariant for different numerical integration methods. In paricular we compare the effects of different time steps (0.01 and 0.05 ms) on the CA1 network's model simulation. We also compare the Euler integration method against the fourth order Runge-Kutta method. The results are illustrated in **Figure A1**. For a fixed value of *P*sprout = 65 and keeping other model parameters at

their default values, we ran simulations to compare the numerical integration time steps and methods. In the case of the Euler integration method, we see that a time step of 0.01 ms (**Figure A1A**) and a time step 0.05 ms (**Figure A1B**) both produce very similar spontaneous IIS LFP traces. Even in the case of the fourth order Runge-Kutta integration method, we continue to see no noticable deviation between the LFP traces for simulation time steps of 0.01 ms (**Figure A1C**) and 0.05 ms (**Figure A1D**). Furthermore, we also do not notice any significant differences between the LFP traces obtained from the Euler and fourth order Runge-Kutta integration methods. These observations suggest that the choice of time step (0.01 ms) for our simulations is valid.

# N2A: a computational tool for modeling from neurons to algorithms

## *Fredrick Rothganger\*, Christina E. Warrender , Derek Trumbo and James B. Aimone\**

*Cognitive Modeling Department, Sandia National Laboratories, Albuquerque, NM, USA*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA A. Ravishankar Rao, IBM Research, USA*

#### *\*Correspondence:*

*Fredrick Rothganger and James B. Aimone, Cognitive Modeling Department, Sandia National Laboratories, 1515 Eubank Blvd MS-1327, Albuquerque, NM 87185, USA e-mail: frothga@sandia.gov; jbaimon@sandia.gov*

The exponential increase in available neural data has combined with the exponential growth in computing ("Moore's law") to create new opportunities to understand neural systems at large scale and high detail. The ability to produce large and sophisticated simulations has introduced unique challenges to neuroscientists. Computational models in neuroscience are increasingly broad efforts, often involving the collaboration of experts in different domains. Furthermore, the size and detail of models have grown to levels for which understanding the implications of variability and assumptions is no longer trivial. Here, we introduce the model design platform *N2A* which aims to facilitate the design and validation of biologically realistic models. N2A uses a hierarchical representation of neural information to enable the integration of models from different users. N2A streamlines computational validation of a model by natively implementing standard tools in sensitivity analysis and uncertainty quantification. The part-relationship representation allows both network-level analysis and dynamical simulations. We will demonstrate how N2A can be used in a range of examples, including a simple Hodgkin-Huxley cable model, basic parameter sensitivity of an 80/20 network, and the expression of the structural plasticity of a growing dendrite and stem cell proliferation and differentiation.

**Keywords: neuroinformatics, computational modeling, computational neuroscience, structural plasticity, biologically realistic modeling**

## **INTRODUCTION**

Computational neuroscience methods for constructing and simulating biologically realistic models have increasingly been recognized as important for understanding the function of complex neural circuits. The role for computational tools will continue to grow in the near future, with significant policy efforts such as the EU Human Brain Project (Markram, 2012) and the proposed Brain Activity Map (Alivisatos et al., 2013). These programs emphasize the high-throughput collection of neural data through both connectomics research and large scale physiology measurements of neuronal behavior in circuits. While the role of computational tools for modeling and simulation is increasingly recognized, the path from this raw data to interpretable model results is unclear.

Constructing neural simulations typically involves several distinct stages once a conceptual approach has been established (**Figure 1A**). **(1)** Relevant data from the biological world must be identified, filtered, and represented in a computationally tractable form. This is often a challenge because a substantial portion of neurobiological data is qualitative in nature. **(2)** A model must be assembled from this raw data, which involves critical decisions on the appropriate level of abstraction and desired scope. **(3)** The model is typically simulated, either directly in the model construction tool or in a separate environment. **(4)** Finally, the simulation data must be analyzed, which is often non-trivial due to the potential scale of models today. Each of these four stages is unique, often requiring distinct forms of insight and benefiting from different aspects of expertise on the part of the user.

There are numerous software applications available for parts of one or several of these stages, some of which have been optimized over decades (**Table 1**). In particular, the simulation of neural systems **(step 3)** has benefited greatly from tools such as NEURON and GENESIS/MOOSE which facilitate the representation and simulation of complex neuronal dynamics and morphologies (Hines and Carnevale, 1997; Bower and Beeman, 1998; Dudani et al., 2009). Recently introduced simulators such as Brian and NEST have focused more on network simulations, and similar capabilities have been added to NEURON and GENESIS (Gewaltig and Diesmann, 2007; Goodman and Brette, 2008). Many of these network simulators have been parallelized to run on supercomputers. In general, simulators require the user to describe models in a programming language. Notably, some simulators, such as NEURON and GENESIS, also provide an integrated modeling environment that facilitates the user's work at various steps in the process, such as editing models and managing simulations. Having a programming language such as Python or C at the foundation of a neural modeling tool is greatly enabling for its functionality, as in theory these languages are both agnostic to scale or complexity.

Nevertheless, despite this plethora of tools, modeling neural systems is becoming ever more challenging, particularly as available computing resources and available biological data approach previously unimaginable heights. This trend toward incorporating more biological detail into models and integrative community efforts has led to the development of XML based descriptions such as NeuroML and NineML (Gleeson et al., 2010, 2011; Raikov, 2010) and model generation tools such as PyNN and NeuroConstruct (Gleeson et al., 2007; Davison et al., 2008) that are moving the community beyond stand-alone platforms toward model-sharing. Nonetheless, the use of standard parts, which is useful for model interchange, can be limiting when building models with complex features, such as structural plasticity or non-standard dynamics. Relying on a formal coding interface to go beyond pre-packaged modeling components often presents a challenge to the typical neuroscientist user. We expect this need to be particularly notable when a prospective modeler faces challenges such as structural plasticity (often important for clinical models), uncertainty quantification (necessary for any model with numerous free parameters), and parallelization of large-scale simulations. These are general problems with solutions that are often specific to a given network. For instance,

while some network architectures map well onto GPUs (Richert et al., 2011), other networks map better to different system architectures.

Here, we present a new tool, *Neurons to Algorithms*, or *N2A,* which complements these existing approaches. Rather than focus on the simulation aspects, which are often specialized to the type of model being computed, we focus on the first two stages of modeling, the computational representation of neurobiological data (e.g., describing the projection pattern from DG to CA3 as a narrow Gaussian with sparse connection probability) and the descriptions of models themselves (**Figure 1B**). In this respect, it is most similar to PyNN, though with several important differences. First, N2A represents information in a flexible computable format that permits almost any neurological dynamics; whereas PyNN is more specialized to use canonical standards or native models represented within lower-level simulators. Second, N2A's hierarchical and relational design is inherently scale agnostic, forming a computable database for neural data. Finally, the part-relationship representation is suitable for both standard dynamical simulations as well as higher level network analysis.

We have designed N2A to be general in how it represents models, so we expect that it will be suitable for a wide range of neural modeling approaches. However, we recognize that some tools are exceptional in certain application areas (i.e., biophysical single neuron multi-compartment models in NEURON), and we expect

*Primary scope is illustrated by bold arrows, with limited capabilities shown by shaded arrows.*

those to remain the tools of choice in those domains and will seek to integrate N2A with their existing functionality. Rather, we believe that N2A will provide differentiating capabilities in high fidelity, large scale network models. These models have several key characteristics, including many distinct neuron and connection types, non-trivial connectivity patterns and part-to-part variability, and large parameter spaces with at times poor biological constraints that will require considerable sensitivity analysis and parameter exploration. This type of high-detail modeling is relatively new to neuroscience and is an increasingly common approach, enabled in large part by modern computing resources and the advances in high density physiology and anatomical data acquisition (Izhikevich and Edelman, 2008; Aimone et al., 2009; Richert et al., 2011; Markram, 2012) and by the recognition that therapeutic models will require consideration of the complexity of neural dynamics (Aimone and Weick, 2013).

## **OVERVIEW OF N2A FRAMEWORK**

The translation from raw biological information into a model suitable for simulation is a non-trivial process. We recognized that a systematic approach capable of model development would require a structured language, a dedicated software platform, and use of community resources. Along these lines, the overall N2A framework we describe here has three significant components: the N2A language, the N2A software, and integration into the broader community. First, we will introduce the N2A language, which is our approach for describing neural models that enables the description of neural data in a computable format from which models can be constructed. Second, we will describe the current N2A software application, which includes both a user interface and a custom database. Third, we will discuss our vision for how N2A fits into the broader neuroscience community, which includes both the integration of N2A into existing neuroinformatics frameworks and collaborative N2A peer-to-peer networks.

The N2A tool is open source and is available at http://code. google.com/p/n2a.

## **MODEL DESCRIPTION LANGUAGE**

The N2A language was designed with the primary goal of being capable of representing as much neural data as possible in a simple computable format. In this context, computable refers to the ability for an observer, whether a human or a machine, to read the description and integrate it into a simulation. A simple ruleof-thumb is that for a model to use neural information, it either has to be represented by an equation or in the structure of the model. For some classes of neural data, such as the behavior of ion channels and membrane voltage dynamics often characterized in electrophysiology studies, representation in a computable format is as simple as writing the canonical differential equations (see HH example below). For other types of data, computability is less straightforward; for instance describing the dynamics of dendrite growth will likely be a non-trivial pursuit involving approaches such as L-Neuron (Ascoli and Krichmar, 2000). N2A refers to units with largely self-contained dynamics (e.g., a neuron or a dendritic spine) as *parts* and the equations governing its dynamics as its *equation set*.

The conversion of neural anatomy information into a model's structure is a major goal of N2A which is best illustrated by an example. **Figure 2** illustrates a few cell types in the hippocampus from one common point of view. Ontologies, such as those at NeuroLex and Open Source Brain (Gleeson et al., 2012; Larson and Martone, 2013), describe the parts and relationships of a system. Each object in the ontology can have any number of attributes, and an important job of the ontology is to provide consistent naming of those attributes across the entire community. Attributes may contain any kind of data, from a single number to text to an entire data series captured by a physiological experiment.

Examples of attributes might be:

Name = Hippocampus CA3 pyramidal call Organism = Vertebrata Neurotransmitter released = Glutamate Dendrite Length = 12481.9 ± 2998.9 um

Most attributes can be thought of as a simple pair: attribute = value. N2A takes this one step further by representing the dynamics of a part as a set of equations. The attributes are the variables, and the values describe how those variables evolve over time. Equations describe how attributes interact with each other in an explicit computable manner. Such a mathematical representation can be incorporated into the metadata of any part in the ontology. The Examples section below shows what several models look like in practice.

## **PART INHERITANCE AND INCLUSION**

The N2A language specifies rules for how equation sets are combined which are motivated by object oriented principles from programming. When part **C** also *is a* part **P** (e.g., a granule cell *is* a neuron), part **C** *inherits* all the equations and metadata contained in **P**. **C** can inherit from any number of parents. A named value (equation or metadata) that is defined directly in **C** hides any value with the same name in a parent.

When a part **M** *has a* part **P** (e.g., the dentate gyrus *has* granule cells), a prefix is added to each equation from **P** as it is *included* into **M**. This allows the user to reference equations within included parts. N2A uses the full-stop character (.) to delimit prefixes. A value with **P**'s prefix that is defined directly in **M** hides any value in **P** with the same name, in much the same way that names in **M** hide names in **M**'s parents. **P** may in turn include a part **Q**, whose equations are all prefixed and placed in **P**. **M** can then hide any name in **Q** by using both prefixes. This can continue any number levels deep. For example, the brain model includes a hippocampus which includes a granule cell model. The brain model could contain an equation that specifically sets the number of granule cells in the population.

## **CONNECTIONS**

To understand connections in N2A, it is important to recognize the difference between a part and an *instance* of that part. N2A distinguishes these notions in much the same way that an objectoriented language such as Java distinguishes between a class and an object. When a simulation runs, each part in the model can generate an entire population of instances, and each instance has its own distinct set of values for the state variables defined in the part. An equation set should be thought of as a template for stamping out instances.

A part **C** that *connects* two parts **X** and **Y** (e.g., the mossy terminal in **Figure 2**) is able to access their equation sets and make statements about how they couple to each other. **C** associates a prefix with each of **X** and **Y**, and uses those prefixes to access the respective variables. During a simulation, an instance of **C** may add to or otherwise modify values in the connected instances. **C** specifies rules about which members of population *A* to connect with which members of population *B*. Instances of **C** are created or destroyed automatically as the populations grow and shrink.

#### **STRUCTURAL DYNAMICS**

When an instance of part **P** *becomes* an instance of part **Q** (e.g., the progenitor cell *becomes* a granule cell in **Figure 2**), all values with matching names are copied into the new instance. **P** can split into any number of types, allowing one to model development and population dynamics. The N2A language commits to the notion that all morphology and connectivity are the consequence of the dynamics governing individual parts. These include rules for creating and destroying parts, splitting and changing type, and moving in space. The language provides a way to express all of these as equations.

## **SCALE INDEPENDENCE**

The N2A language is designed to model a system at a wide range of scales. Gene regulatory networks can be represented either as coupled parts or as a collection of state variables within a given part. Common protein interaction sequences, such as the MAPK pathway, can be represented as a part that is included in many other structures.

The interaction of neuron populations is illustrated in **Figure 2**. Entire brain regions can be wrapped into parts and

connected with each other. Each level of model can be represented by either a simple or a detailed part, allowing successive abstraction as one studies a system (**Figure 3**).

## **SOFTWARE**

The N2A software attempts to ease many of the obstacles that researchers face while developing, executing and fine-tuning physiological models. To this end the software embodies these basic principles: transparency, traceability, repeatability, and sharing.

The system is a Java-based desktop application (**Figure 4**) with an embedded database (**Figure 5**). The interface provides the user with a method to locate models and other supplemental records, modify models, and create new sets of simulations ("run ensembles") against a given model. Supplemental records could be references to papers, associated lab results, input data, or other related information that you want to track alongside the models. The user interface provides context sensitive help. It shows part hierarchies along with associated equation sets, metadata and references.

To support repeatability, the N2A software stores all model runs. A planned part of the design is to keep a version history for models (see below), so researchers can make changes without affecting the equations used in a previous simulation. Currently, run ensembles and individual runs maintain all parameter information in addition to their associated model. Run results are stored separately for analysis. Post-run/analysis products can potentially be tracked by the software given the right plug-in support. By recording every aspect of the model creation and execution, including system-generated random numbers and seeds, we enable repeatability for quality assurance and double-checking purposes.

No single tool can serve all purposes, so N2A is built from the ground up with extensibility in mind. The software uses a plug-in

**FIGURE 3 | Illustration of application of part-connection-model framework to different scales of neural simulations.** The structure of the N2A language allows it to be applied in scales ranging from molecular kinetics models of cell signaling to neural network models comprised of complex neurons and synapses.

infrastructure to allow others to extend the product to meet their needs. A key class of extension is the handler ("backend") for a given simulator, and a simple interface is provided for creating new ones. Additionally, new types of model and supplemental records can be added and visualized in the user interface according to the plug-in designer's wishes.

within a model, and incorporate metadata and references regarding literature

## **SIMULATION CAPABILITIES**

N2A is a model description language, but to make it useful in practice the tool is able to translate models into inputs to several different simulators. Each simulator is handled by a separate "backend" module. Currently, N2A has backend modules for two simulators: C++ and Xyce. To be fully useful it will need additional backends to support commonly used simulators such as NEURON, GENESIS, Brian, or NEST, and common middleware such as PyNN. As N2A becomes integrated with evolving neuroinformatics standards such as NeuroML, we hope notation and constants defined directly.

to leverage multiple additional simulators. This is a key part of future work.

## **C++**

The C-backend is the reference implementation of the N2A language. It is capable of simulating any construct expressible in the language, including structural dynamics. The price for such generality is a loss of efficiency in specialized cases. For example, the C-backend is primarily designed for a general dynamical system, so it is less efficient on large spiking networks. The C-backend works by translating the model into a set of C++ classes, which are then coupled with a runtime library that handles object management and numerical integration. The entire simulation is a self-contained executable program.

## **XYCE**

Xyce is a parallelized version of the electrical circuit simulator SPICE that is capable of natively simulating large scale circuits

on supercomputers. Recently, we have extended its capabilities for very large scale simulations of neural networks (Schiek et al., 2012). In addition to its traditional devices (transistors, capacitors, etc.), Xyce now also has neuron and synapse "devices." Xyce parses and solves a broad range of explicit mathematical expressions, so model dynamics not covered by built-in devices can also be included. Currently, N2A is capable of translating most of its neural models into Xyce simulations, through a combination of direct equations and specialized neural devices.

## **FUTURE CAPABILITIES: MODEL SHARING AND INTEGRATION INTO THE NEUROINFORMATICS COMMUNITY**

The N2A tool is still under development, and the methods of sharing described in this section are aspirational, but high-priority future work. We summarize existing and projected capabilities in **Table 2**.

Ideally, all models associated with a given part should be stored in a central repository accessible to everyone, such as the Neuroscience Information Framework (NIF) or Open Source Brain (Gleeson et al., 2012). NIF is particularly compatible with our vision because they organize all data according to the NeuroLex ontology and they offer curation for small quantities of data. Since N2A models are very concise they fit into this category.

**Figure 6** illustrates a second means of sharing. A user asks the N2A tool to act as a server online and allow peers to access data and compute resources. This *Peer-to-Peer (P2P)* arrangement brings up two closely related issues: versioning of models and the repeatability of simulations. The problem is this: if a researcher configures a model a certain way, simulates it, and later some part that the model depends on is changed, it is no longer possible to produce exactly the same simulation again.

We propose to keep all parts/models under *version control*. Examples of version control systems in the software-development world include Subversion, Git, Mercurial, etc. An N2A data-store would not directly use these tools, but instead implement similar

#### **Table 2 | Status of current and future features of N2A.**


collaborators or with the broader community through opening their database to the broader neuroinformatics community.

concepts. Any time a model is transmitted between two peers or simulated, a version is permanently recorded in the database. All parts it depends on are also permanently versioned. Ongoing development of a part goes into a subsequent version, and does not have any influence on the content of a model. To ensure repeatability of simulations, it is necessary to record a number of details beyond the model itself, such as the simulator used, random number seeds, platform, etc. It may not be possible to capture every detail and make a simulation perfectly repeatable, but a record of the key variables will help in interpreting the results of the experiment.

To further drive integration into the neuroinformatics landscape, we envision that the N2A tool will be compatible with existing tools by leveraging the increasingly common standards for model definition, such as NeuroML, LEMS, and NineML. As other simulation frameworks and environments specializing in other classes of neural simulations develop support these growing standards, we expect that linking the models defined within N2A into those simulation environments to be relatively straightforward.

## **EXAMPLES**

Here, we show three different examples of the neuroscience systems implemented within the N2A tool to illustrate how it represents progressively more sophisticated neural circuits. These are not a complete sample of N2A's applicability; rather these examples are intended to highlight the scope of N2A and its eventual vision.

## **HH MODEL**

The Hodgkin-Huxley (HH) model of spike generation and propagation underlies many computational modeling studies and is well suited to illustrate how N2A represents neural dynamics (Hodgkin and Huxley, 1952). Briefly, the Na+/K+ ion channel version of the HH model is a system of four differential equations with two state variables governing the dynamics of Na+ ion channels (*m* and *h*), one state variable governing dynamics of K+ ion channels (*n*) and a state variable (*V*) representing the internal voltage of the neuron or axon. *V* is often represented by the equation

$$CV' = \text{g}\_{\text{Na}} m^3 h (E\_{\text{Na}} - V) + \text{g}\_K n^4 (E\_K - V)$$

$$+ \text{g}\_{\text{leak}} (E\_{\text{leak}} - V) + I$$

where *C* is membrane capacitance; *gNa*, *gK*, and *g*leak are maximum conductances for Na+, K+ and leak currents, respectively; *ENa*, *EK*, and *E*leak are the reversal potentials for those respective currents; and *I* is input current. The state variables *m*, *n*, and *h* typically take the form

$$\mathbf{x}' = \alpha\_\mathbf{x}(V) \left( 1 - \mathbf{x} \right) - \beta\_\mathbf{x}(V) \ge 0$$

where α*x(V)* and β*x(V)* are functions of voltage specific to each state variable.

## **N2A REPRESENTATION**

Within N2A, we represented the HH model using the equations outlined in (Koch, 2004) in a simple 3-segment cable configuration (**Figure 7**). While N2A can represent the HH dynamics of individual compartments using a part that contains all of the equations for the sodium, potassium, and leak currents, we chose to construct the demonstration model as a part with only passive membrane dynamics that "includes" the appropriate ion channels, in this case Na+ and K+. This separation of ion channels from host compartments facilitates the reuse of well-tuned ion channels in multiple independent neuron models as well as the rapid interchange of one ion channel to another within a given model. Each of the three HH compartments are coupled by a simple connection part that implements the cable equation

$$A.V' = \mathcal{g}\_r (B.V - A.V)$$

$$B.V' = \mathcal{g}\_r (A.V - B.V)$$

where *A.V* and *B.V* are the voltages of the two connected HH compartments and *gr* is the lateral membrane conductance.

Below is the complete set of equations expressed in the N2A language (**Figure 7A**). This example contains seven parts: the abstract ion channel, two ion channels that inherit from it, the abstract passive compartment, the HH compartment that inherits from it and includes the two ion channels, the HH connection, and finally the model that incorporates the HH compartment and HH connection into a cable. For a more thorough explanation of how the language expresses this model, see the "N2A Language Overview" in the supplementary material.

We illustrated simple HH dynamics and propagation of action potentials by injecting 10pA into the left compartment (**Figures 7B,C**) and, in an effective current clamp condition, observed voltage deflection representing the 100 mV spiking event in the compartment (**Figure 7D**). The spike propagates to the right-most compartment with a short delay (**Figure 7E**). A longer current injection yields a series of spikes in the leftmost compartment (**Figure 7F**) that again is manifested two compartments away (**Figure 7G**), albeit at a short delay and with a notable failure to propagate of one spike.

## **SENSITIVITY ANALYSIS OF BALANCED EXCITATION/INHIBITION NETWORKS**

Balanced excitation/inhibition (E-I) networks have attracted attention as a coarse model of cortical dynamics (Vogels and Abbott, 2005; Brette et al., 2007). Often containing a mixture of 80% excitatory and 20% inhibitory spiking neurons (though studied with both other ratios and in non-spiking systems), E-I networks can show a range of non-trivial "phases" of dynamical network activity, including oscillatory and chaotic (or near-chaotic) behaviors. Balanced E-I models are interesting for a number of reasons, among which is their increasing relevance in understanding motor and prefrontal cortex dynamics and their relationship to the reservoir computing research area in machine learning. Specifically, it appears that the chaotic dynamics observed under certain conditions are computationally uniquely powerful (Laje and Buonomano, 2013).

Clearly, not all configurations will produce complex chaotic or near-chaotic behavior; indeed understanding the effects of design and parameters on these dynamics is an active area of research (Litwin-Kumar and Doiron, 2012). Here, we illustrate the parameter exploration capabilities of the N2A tool by systematically varying two basic parameters that affect the behavior phase: strength of recurrent excitation (E) and strength of recurrent inhibition (I).

## **N2A REPRESENTATION**

We implemented the model described in benchmark 3 of (Brette et al., 2007) in N2A, then used N2A to define and execute a "run ensemble" of 121 simulations with different values of synaptic conductance (**Figures 8**, **9**). Building the 80-20 network model consisted of creating the necessary parts, defining cell populations ("Layers"), and defining connections between cells ("Bridges") both within and across populations. The N2A parts used for this model were: (1) a variant of a Hodgkin-Huxley neuron described in the Brette paper (**Figure 8A**) (2) a conductance-based synapse also described in the Brette paper (**Figure 8B**), (3) an artificial "Spiker" neuron to provide input into the network, and (4) another exponential synapse to connect the "Spiker" cells to selected cells in the main population. Xyce has built-in implementations of the Brette neuron and synapse models, so the N2A parts included metadata indicating that those implementations should be used. **Figures 8C,D** shows how populations and connections are identified in N2A. All neurons in the 80-20 network have the same dynamics, so we created a single population of neurons using the same N2A part, but made excitatory connections only to the first 80% by index. Excitatory and inhibitory connections used the same "Brette synapse" part. We used connection equations both to override part parameter values as appropriate for excitatory or inhibitory connections and to specify which neurons can be connected.

The "Runs" tab shown in **Figure 9A** allows the user to create and run one or more simulations of the model. Any parameter defined in the model can be dragged from a pre-populated list into the run ensemble definition, with search strategies ranging from simple step protocols to Monte Carlo and Latin Hypercube sampling. The figure below shows selection of the two synaptic conductance coefficients varied to produce **Figures 9B–D**, the number of values for each and how they varied. In this case we simply stepped through a range of values at fixed intervals. Certain simulation parameters such as seed or integration method can be chosen or varied in the same way.

Unsurprisingly, for low E the network exhibits very low average firing rates, whereas high E with low I yields very high average firing rates (**Figure 9B**). For roughly balanced E and I levels, the overall firing rates appear to be comparable in spite of absolute magnitude. However, a simple measure of activity distribution (**Figure 9C**) shows that even for E, I combinations with comparable firing rates, the dynamical state of the network can differ considerably; suggesting that there are at least four clear states of network activity observable in our small search space (it should be noted that it is not surprising that high-dimensional networks such as these can exhibit many different phases of behavior). **Figure 9D** shows representative examples of network activity at different positions in the parameter space.

**FIGURE 8 | 80/20 E-I network definition in N2A. (A)** Screenshot of part equation set for spiking neurons used in 80/20 model. **(B)** Screenshot of synapse equation set for connections between neurons in 80/20 model. **(C)** Simple network illustration of 80/20 model from a block perspective (left) and instantiation perspective (right). **(D)** Screenshot of model definition equation set for 80/20 network in N2A; differences in excitatory and inhibitory connections (namely conductance and reversal potential) as well as the sparse inputs are defined at this level.

It is interesting to note that even this simple illustration of the parameter searching capabilities of the N2A tool provides results that merit more detailed exploration. It was not surprising that these networks not only exhibit silent (**1**) and hyperactive (**4**) states in addition to the originally published asynchronous state (**2**), but we did not expect this simple parameter exploration exercise to show a state where the network activity is preferentially localized to a subset of highly active neurons (**3**). What is not clear from this study (or indeed many other studies of these abstract networks) is how these dynamics relate to real *in vivo* cortical function. For instance, it has been suggested that working memory in the pre-frontal cortex (and other cortical areas) involves a switch from asynchronous activity to a more persistent activity of a subset of neurons holding a trace (Durstewitz et al., 2000; Wang, 2001). These illustrative parameter search results are far too preliminary to make any strong links to this neurobiological phenomenon, however it would be interesting to expand the search to include the both more realistic network connectivity (Litwin-Kumar and Doiron, 2012)and extrinsic neuromodulatory influences such as dopamine(Brunel and Wang, 2001) that may effectively alter the excitation/inhibition balance dynamically.

## **STRUCTURAL DYNAMICS**

In addition to challenges in understanding parameter sensitivity of models, many neural systems involve dynamics or structures that are not well suited to existing tools. One such example is structural plasticity of neural systems. While most modeling studies treat neural circuits as effectively fixed, at most implementing plasticity in synaptic weights, there are many neural processes that necessitate changing the network itself over extended time scales. These include neurological and psychiatric disorders, development, and even structural plasticity in the healthy adult brain through neurogenesis and dendritic spine dynamics.

#### **N2A REPRESENTATION**

The structure of parts within N2A allows for the representation of the regulated birth, transitions, and death of instances. Here, we show two examples of how structural plasticity would be represented within the N2A language. There are two key language commands: assigning **type** to an instance of a part will transition it to a different type of part (i.e., differentiation), and assigning multiple types to an instantiated part will replicate the instance. This division can either be symmetric (where both children are equivalent to one another, regardless of whether the parent's type

can be varied either randomly or systematically and in isolation or in conjunction with other parameters. **(B)** Firing rates in response to parameter sweep of differing E and I synaptic conductance levels. **(C)** Distribution of activity in response to parameter sweep. **(D)** Representative raster plots of spiking from E-I network at different positions in parameter state space; 1—all neurons effectively silent; 2—asynchronous firing dynamics; 3—skewed firing dynamics, with subset of neurons exhibiting persistent activity; 4—hyperactive network with most neurons persistently active.

is maintained) or asymmetric (where the children are different parts, with one perhaps retaining the parent's type).

**Figure 10** shows two examples of what the N2A language will accommodate. **Figure 10A** shows a growing dendrite, with a dynamic growth cone (purple) at the end. This growth cone is capable of linear growth (basically splitting into an ordinary compartment and the growth cone), branching (splitting into multiple growth cones and a stable compartment), differentiation (growth cone becomes a compartment), or death (growth cone simply disappears).

**Figure 10B** illustrates a different form of structural dynamics, the proliferation and differentiation of a dentate gyrus stem cell into an eventual granule cell population during the adult neurogenesis process. In this example, a radial glial cell (RGC, green), which is considered the primordial stem cell type in the adult dentate gyrus, is capable of asymmetric division, producing a

neural progenitor cell (NPC, red) as well as a "copy" of itself. This NPC subsequently exhibits several rounds of symmetric cell division, amplifying the number of children. Finally, the NPCs will either die (black) or differentiate into granule cells (blue).

## **SUMMARY**

N2A has been designed to enable the general neuroscientist to achieve the scope and depth of models that heretofore have been mostly limited to those with considerable programming expertise. Concepts such as structural plasticity and parameter searching that are illustrated can all be achieved using other tools or conventional languages, but they often require considerable work on the modeler's part. We believe that the trends in neuroscience toward more detailed characterization of systems and increased emphasis on clinical conditions (such as diseases and therapeutic mechanisms) will further amplify the importance of having a tool to effectively capture neurobiological complexity in a straightforward manner.

The increase in high throughput data acquisition, improved neuroinformatics tools, and growing availability of computing resources all will facilitate the trend toward more biologically detailed approaches to modeling neural systems. An important consideration is that the rationale behind biologically realistic models is quite different than that of other modeling approaches, such as large scale simple models and abstract models of neural system functions. Briefly, in contrast with models that illustrate how a neural circuit can map to a known function, "bottom up" detail oriented models can suggest novel computational functions for neural processes that otherwise would not have been considered. Such work has in the past been useful in identifying the functions of complex neural processes; for instance a high resolution model of neurogenesis was able to suggest that new neurons may provide a previously unknown function of encoding time into episodic memories (Aimone et al., 2009), a function that has subsequently been measured in rats (Morris et al., 2013).

Notably, the potential value of "bottom up" models in providing novel functional insight into a brain region is dependent on a design of the model that is not biased toward desired results. The brain is, of course, considerably more complex than any single model is capable of representing, and abstraction is thus always necessary to some extent. However, abstraction should be performed with careful consideration to minimize disruption to behavior, not simply guided by the ease of implementation or the ready availability of data. The N2A tool is well suited for this challenge; as its representation of neuronal dynamics enables the incorporation of complex processes that are often neglected into models, such as adult neurogenesis or cellular protein kinetics. Furthermore, we believe that as N2A is further integrated into the broader neuroinformatics community, modeling biases due to the local availability of information will be minimized (e.g., someone may include the oft-ignored CA2 region in a hippocampal model if N2A can pre-populate the relevant details).

We recognize that N2A's data-centric, dynamical representation of neural information makes it less well suited for other modeling approaches, for which we expect many existing tools to be preferable. This includes Monte Carlo type simulations of molecular dynamics (e.g., MCell) and morphologically defined models of dendritic dynamics (e.g., NEURON). Notably, the original motivation of N2A was to automatically extract computational structures (the "algorithms" in the name) from data about neurons and their interconnections. Although the goal of automatic model reduction for algorithm discovery is now considered remote, its influence lingers in the design of the language and tools. For example, an increasing fraction of data in neuroscience exists in databases and is machine readable and computable, incorporating both graphical structures and dynamics.

The long-term goal of understanding the computation of the entire brain appears in the community sharing and neuroinformatic aspects of the tool. It is necessary, after all, to have a *computational* framework capable of representing the entire nervous system. In that sense N2A shares aspirations with cognitive frameworks such as ACT-R and SOAR, but it makes far fewer commitments to specific structure. Rather the expectation is that a large community of experts will jointly assemble what they know onto the scaffolding to create a digital mind. Undoubtedly the current incarnation of the language will evolve many times and perhaps even go extinct before the community reaches that goal.

#### **ACKNOWLEDGMENTS**

This work was supported by the Laboratory Directed Research and Development (LDRD) program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. We would like to thank Rich Schiek, Corinne Teeter, and Alex Duda for helpful discussions and comments.

## **SUPPLEMENTARY MATERIAL**

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/Neural\_Circuits/10.3389/ fncir.2014.00001/abstract

## **REFERENCES**


Wang, X.-J. (2001). Synaptic reverberation underlying mnemonic persistent activity. *Trends Neurosci.* 24, 455–463. doi: 10.1016/S0166-2236(00) 01868-3

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 30 September 2013; accepted: 06 January 2014; published online: 24 January 2014.*

*Citation: Rothganger F, Warrender CE, Trumbo D and Aimone JB (2014) N2A: a computational tool for modeling from neurons to algorithms. Front. Neural Circuits 8:1. doi: 10.3389/fncir.2014.00001*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2014 Rothganger, Warrender, Trumbo and Aimone. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Identification of neuronal network properties from the spectral analysis of calcium imaging signals in neuronal cultures

#### *Elisenda Tibau1, Miguel Valencia2 and Jordi Soriano1 \**

*<sup>1</sup> Neurophysics Laboratory, Departament d'Estructura i Constituents de la Matèria, Universitat de Barcelona, Barcelona, Spain <sup>2</sup> Neurophysiology Laboratory, Division of Neurosciences, CIMA, Universidad de Navarra, Pamplona, Spain*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA Silvina P. Dawson, Universidad de Buenos Aires, Argentina*

#### *\*Correspondence:*

*Jordi Soriano, Neurophysics Laboratory, Departament d'Estructura i Constituents de la Matèria, Universitat de Barcelona, Av. Diagonal 645, Barcelona E-08028, Spain e-mail: jordi.soriano@ub.edu*

Neuronal networks *in vitro* are prominent systems to study the development of connections in living neuronal networks and the interplay between connectivity, activity and function. These cultured networks show a rich spontaneous activity that evolves concurrently with the connectivity of the underlying network. In this work we monitor the development of neuronal cultures, and record their activity using calcium fluorescence imaging. We use spectral analysis to characterize global dynamical and structural traits of the neuronal cultures. We first observe that the power spectrum can be used as a signature of the state of the network, for instance when inhibition is active or silent, as well as a measure of the network's connectivity strength. Second, the power spectrum identifies prominent developmental changes in the network such as GABAA switch. And third, the analysis of the spatial distribution of the spectral density, in experiments with a controlled disintegration of the network through CNQX, an AMPA-glutamate receptor antagonist in excitatory neurons, reveals the existence of communities of strongly connected, highly active neurons that display synchronous oscillations. Our work illustrates the interest of spectral analysis for the study of *in vitro* networks, and its potential use as a network-state indicator, for instance to compare healthy and diseased neuronal networks.

#### **Keywords: neuronal cultures, multineuron calcium imaging, spectral analysis, network development, excitationinhibition balance, GABA switch, synchronous oscillations**

## **1. INTRODUCTION**

Living neuronal networks, from the smallest neuronal assembly up to the human brain, are one of the most fascinating yet intricate structures in Nature. The subtle interplay between the architecture of the neuronal network and the dynamics of the neurons give rise to a vast mosaic of complex phenomena that are still a major paradigm in neuroscience (Bassett and Gazzaniga, 2011), including spontaneous activity patterns (Blankenship and Feller, 2009; Deco et al., 2010; Luczak and MacLean, 2012), information processing and routing (Bullmore and Sporns, 2012), synchronization (Salinas and Sejnowski, 2001), plasticity and adaptability (Destexhe and Marder, 2004), together with remarkable self-organizing properties and critical behavior that suggest an efficient yet flexible *modus operandi* (Chialvo, 2010; Bullmore and Sporns, 2012).

The interplay between single cell dynamics and network topology is tremendously complex, particularly when applied to the comprehension of the human brain (Chicurel, 2000; Alivisatos et al., 2012; Abbott, 2013). However, in the last two decades we have attended to an outbreak in the development of techniques to investigate the brain *in vivo*. Advances in brain functional and mapping techniques such as fMRI, EEG, MEG, or DTI, together with resources from graph theory and signal processing (Bullmore and Sporns, 2009; Feldt et al., 2011), have provided unprecedented detail on brain functional interactions and their dependence with the underlying circuitry. They have also opened new perspectives in our comprehension of dysfunctional circuits. Indeed, severe neurological disorders and behavioral deficits are associated to alterations of the neuronal circuitry (Seeley et al., 2009), abnormal neuronal activity coordination (Uhlhaas and Singer, 2012), or deficient neuronal machinery (Maccioni et al., 2001). Autism, for instance, has been ascribed to an underconnectivity or overconnectivity of local brain circuits combined with long-distance disconnection. Schizophrenia has been associated with an imbalance of the excitatory and inhibitory circuits, among other factors (Lynall et al., 2010; Yizhar et al., 2011b). Epileptic brains, compared to those of healthy subjects, display a richer functional connectivity with a clear modular structure (Chavez et al., 2010), while brain networks in Alzheimer's disease patients are characterized by a loss of the small-world network feature (Stam et al., 2007).

These advances have provided novel clinical prognosis tools by linking specific functional failures to topological traits of the anatomical network. They have evidenced that the information obtained from functional and anatomical techniques contain several signatures that reveal the properties of brain functions, both in normal and disease states. Nevertheless, a major difficulty in analyzing this information has been the sheer size and complexity of the human brain. The activity recorded from the intact brain results from the occurrence of several, simultaneous processes involving a huge number of interacting cells, thus complicating the understanding of the ultimate mechanisms that regulate neural activity. These difficulties have called for more controlled, accessible and simplified systems that would allow to investigate the basis of brain operation. Neuronal cultures have emerged as one of those systems. These *in vitro* preparations are typically derived from dissociated rat cortical or hippocampal tissues, can be maintained for several months, and their activity monitored by a number of recording techniques that are able to track single cell behavior (Eckmann et al., 2007). The flexibility of neuronal cultures to fit diverse experimental platforms, as well as the ability to *act* on them by chemical, electrical or other means, have made them very attractive for a large number of investigations, most notably the emergence and richness of spontaneous activity patterns (Wagenaar et al., 2006a; Orlandi et al., 2013), the interplay activity-connectivity (Volman et al., 2005), the network's self-organizing potential (Pasquale et al., 2008), and criticality (Tetzlaff et al., 2010).

Here we propose to use analytical tools based on spectral analysis to investigate the functional and structural topology of neural cultures. We use fluorescence calcium imaging to monitor the spontaneous activity of the neuronal network with single cell resolution. In a first set of experiments, we investigate the development of the network along the first 3 weeks of maturation, a period in which the average neuronal connectivity, circuitry topology, and the excitatory-inhibitory balance change significantly. In a second set of measurements, we perturb the topology of a mature culture by gradually weakening the excitatory connections. This action results in a gradual decay of collective spontaneous activity until it is fully disrupted. The analysis of the power spectrum in these two scenarios evidences that spectral data can capture dynamical features of the neuronal network. Our study is a preliminary investigation that, although it requires a thorough exploration and modeling, may help understanding the use of statistical descriptors to detect and quantify distinct topological and dynamical traits in neuronal networks.

## **2. MATERIALS AND METHODS**

#### **2.1. NEURONAL CULTURES**

Rat cortical neurons from 18 to 19-day-old Sprague-Dawley embryos were used in the experiments. All procedures were approved by the Ethical Committee for Animal Experimentation of the University of Barcelona, under order DMAH-5461. Following standard procedures described in previous studies (Soriano et al., 2008; Orlandi et al., 2013), dissection was carried out in ice-cold L-15 medium (Life) enriched with 0.6% glucose and 0.5% gentamicin (Sigma-Aldrich). Embryonic cortices were isolated from the rest of the brain and neurons dissociated by pipetting.

Cortical neurons were plated on 13 mm glass coverslips (#1 Marienfeld-Superior). Prior to plating, glasses were washed in 70% nitric acid for 2 h, rinsed with double-distilled water (DDW), sonicated in ethanol and flamed. To facilitate a homogeneous distribution of neurons in the cultures, glasses were coated overnight with 0.01% Poly-l-lysine (PLL, Sigma). Cultures were incubated at 37◦C, 95% humidity, and 5% CO2 for 4 days in plating medium [90% Eagle's MEM—supplemented with 0.6% glucose, 1% 100X glutamax (Gibco), and 20μg/ml gentamicin—with 5% heat-inactivated horse serum, 5% heatinactivated fetal calf serum, and 1μl/ml B27]. The medium was next switched to changing medium [90% supplemented MEM, 9.5% heat-inactivated horse serum, and 0.5% FUDR (5-fluorodeoxy-uridine)] for 3 days to limit glia growth, and thereafter to final medium [90% supplemented MEM and 10% heatinactivated horse serum]. The final medium was refreshed every 3 days by replacing half of the culture well volume. Plating was carried out with a nominal density of 1 million cells/well (5000 neurons/mm2), providing a final density in the range 200–400 neurons/mm2.

Cultures prepared in these conditions contain both excitatory and inhibitory neurons, whose strength can be controlled by the application of 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX, Sigma), an AMPA-glutamate receptor antagonists in excitatory neurons; or through bicuculine-methbromide (Sigma), a GABAA receptor antagonist in inhibitory neurons.

#### **2.2. PREPARATION OF THE EXPERIMENTS**

Our study encompassed two groups of experiments. In a first one we monitored neuronal activity along the maturation of the network; in a second one we studied the disintegration of the network by gradually blocking AMPA-excitatory connections through CNQX.

The study of the evolution of the network as a function of the culture age (days *in vitro*, DIV) started with the preparation of 2–3 batches that contained 24 identical cultures each. One of the batches was next selected for analysis, which was carefully inspected before the beginning of the series of measurements. We used only those batches whose cultures contained a similar number of neurons, and homogeneously distributed over the substrate. Measurements then consisted in the systematic recording of spontaneous activity in the cultures of the batch, in 24 h intervals along 3 weeks.

We verified that the culture medium changes did not biased the results presented here, particularly those related with the maturation of the network. This verification was carried out by measuring neuronal activity along 2 weeks in batches where we either replaced completely the mediums in each change, or in batches where we replaced only 1/3 of the culture well volume. All development experiments showed the same trend within experimental error, independently of the medium change protocol.

The disintegration experiments were also carried out in cultures that were prepared and inspected as the above. As described later, we considered cultures in the range 8–16 DIV, which were sufficiently mature to show rich spontaneous activity during the different stages of disintegration.

## **2.3. EXPERIMENTAL SETUP**

Measurements consisted in the recording of spontaneous activity through calcium imaging, which allows the monitoring of neuronal firing by the binding of Ca2<sup>+</sup> ions to a fluorescent indicator (Grienberger and Konnerth, 2012). Prior to imaging, cultures were incubated for 40 min in External Medium (EM, consisting of 128 mM NaCl, 1 mM CaCl2, 1 mM MgCl2, 45 mM sucrose, 10 mM glucose, and 0.01 M Hepes; treated to pH 7.4) in the presence of the cell-permeant calcium sensitive dye Fluo-4-AM (Gee et al., 2000), with 4 μl Fluo-4 per ml of EM. The culture was washed with fresh EM after incubation and finally placed in a recording chamber containing 4 ml of EM.

The recording chamber was mounted on a Zeiss inverted microscope equipped with a 5X objective and a 0.32X optical zoom. Spontaneous neuronal activity was monitored through a Hamamatsu Orca Flash 2.8 CMOS camera attached to the microscope, in combination with a light source for fluorescence. Images were acquired with a speed of 20 or 33 frames per second (respectively, 50 or 30 ms interval between two consecutive frames) and a spatial resolution of 4.40μm/pixel. Images had a size of 960 × 720 pixels with 256 gray-scale levels. This settings provided a final field of view of 4.2 × 3.2 mm2 that contained on the order of 3000 neurons. Camera, microscope and light source settings were optimized to minimize photo-bleaching and photodamage while providing the best signal to noise ratio throughout the measurements.

#### **2.4. EXPERIMENTAL PROCEDURE AND PHARMACOLOGY**

For the experiments where we investigated the development of the network, we proceeded as follows. We first recorded spontaneous activity as a long sequence of images with a total duration of 30 min, with both excitation and inhibition active (*"E* + *I" network*). We next fully blocked inhibitory synapses with 40μM bicuculline, a GABAA antagonist, so that activity was solely driven by excitatory neurons (*"E-only" network*). We then left the culture in darkness for 10 min for the drug to take effect, and finally measured again for 30 min with identical experimental settings.

For the experiments where we monitored the disintegration of the network, we first completely blocked inhibition with 40μM bicuculline as well as NMDA receptors with 20 μM APV. We then waited 10 min and measured spontaneous activity for 20 min ("Eonly" activity). Next, we started a sequence of gradual application of CNQX, and explored concentrations of 50, 100, 200, 400, 800, and 2000 nM. After each application we waited 5 min for the drug to take effect, and measured spontaneous activity for 15 min. The total duration of the experiment was about 2 h. We verified by washing off the drug and measuring again "E-only" network activity that the culture health was not compromised by the long duration of the experiment. Other studies that used almost identical disintegration protocols confirmed the good health of the culture throughout the experiment (Soriano et al., 2008; Jacobi et al., 2009).

In all experiments we also quantified the background signal of the recording system to assess our ability in resolving neuronal firings from actual noise. To do this, we removed the culture from the recording chamber and measured the noise of the camera as well as possible additional artifacts, such as fluctuations in the light of the fluorescence lamp or contamination from indirect light sources in the laboratory. We finally verified that the results presented here were not influenced by any artifact from the experimental system.

## **2.5. DATA ANALYSIS**

At the end of each experiment we took bright-field images for a better identification of the neuronal cell bodies (see **Figure 1**). We then manually marked each neuron as a squared region of interest (ROI) with a typical lateral size of 10 pixels (about 40μm). Each experiment typically contained about 2000 ROIs, i.e., individual neurons. The analysis of the average gray level in each ROI along the entire acquired image sequence finally provided the fluorescence intensity *F* for each neuron as a function of time.

Long trains of neuronal activity may contain a small drift of the baseline signal due to photo-bleaching. Although we observed such an effect only in about 5% of the neurons, we automatically corrected this artifact by applying a moving median filter of width 2000 points. We verified that such a correction did not modify the shape of neuronal signal during firing events.

Finally, the fluorescence trace *F*(*t*) was normalized for each neuron to correct for its background brightness level by computing *F*˜(*t*) = (*F*(*t*) − *F*0)/*F*<sup>0</sup> ≡ -*F*/*F*0, where *F*<sup>0</sup> is the average amplitude of the background fluorescence signal at rest. The illustrative traces of **Figure 1**, as well as all the data shown in this work, correspond to such a corrected data.

Neuronal activity in our cultures is characterized by episodes of intense, network-spanning activity events (bursts) combined with quiescent interval of erratic individual firing. The interval between bursting episodes was calculated over the average signal of the neuronal network to take advantage of the almost synchronous bursting episodes. We first determined the onset time of neuronal activation, which was achieved by detecting those events in the fluorescence signal that were at least four times above the standard deviation of the signal. Second, we computed the difference between consecutive onset times, to finally provide the interburst interval distributions.

## **2.6. SPECTRAL ANALYSIS**

To analyze the spectral content of the fluorescence signals, we computed the power spectral density of the normalized traces *F*˜(*t*) = -*F*/*F*<sup>0</sup> by using the Welch periodogram method (Welch, 1967; Halliday et al., 1995) implemented in Matlab 7.12.0. Signal is divided into Hamming windows of 256 points (approximately 10 s), 50% overlapped. To estimate the FFT, 1024 points are used, applying zero-padding. Because we use a sample frequency of 20 Hz for young cultures and 33 Hz for mature culture, the frequency resolution is of 0.019 Hz and 0.032 Hz, respectively. The corresponding frequency ranges are (0.078–10) Hz and (0.128–16.5) Hz. Finally, the averaged spectrum for the whole set of neurons was computed when required, for instance to compare global network characteristics during the maturation of the cultures.

For the studies where we investigated the spatial distribution of the *local* energy across the different frequencies we calculated—for each neuron—the average signal of the selected neuron and its *n* = 100 closest neighbors. Then, the resulting time-series were analyzed following the same procedure described above. By plotting the spectral energy of each neuron at a frequency of interest we obtained a two-dimensional representation of spectral energy that revealed those neurons or

entire neuronal culture 13 mm in diameter at day *in vitro* 9. The rectangular box shows the actual field of view (FOV) of the camera and illustrates the size of the monitored area. **(B)** Detail of a small region of the monitored area containing about 100 neurons. Images correspond to bright field (left) and fluorescence (center), together with an example of the selection of the regions of interest for monitoring single neuronal activity (right). The fluorescence image has been integrated over 100 frames, and bright spots correspond to firing neurons. **(C)** Illustrative fluorescence traces of neuronal activity in three regions of interest (labeled and marked with arrowheads in the above images) along 2 min. Fluorescence traces are expressed as -*F*/*F*<sup>0</sup> (background corrected fluorescence divided by the resting fluorescence). Both excitation and inhibition were active during the recording ("E+I" networks). Note the variability in fluorescence amplitude from neuron to neuron. The bottom trace in blue shows the fluorescence signal averaged over all the neurons in the field of view.

groups of neurons with the strongest power at that frequency of interest.

The smoothing of the fluorescence signal by averaging with neighboring cells significantly reduced the noise of the PSD data. We tested different *n* values and observed that 100 was the appropriate value to balance a neat PSD signal and low overlap, particularly in the studies of spatial distribution of spectral energy. For the latter, we indeed verified that the results did not change significantly up to *n* 500.

## **3. RESULTS**

## **3.1. NEURONAL CULTURES AND NETWORK ACTIVITY**

The neuronal networks that we study are constituted by an ensemble of thousands of neurons that have been dissociated from rat cortical tissue and homogeneously plated on glass cover slips 13 mm in diameter, as shown in **Figure 1A** and described in detailed in the Materials and Methods section. Neurons grown in these conditions have a remarkable self-organizing potential, connecting to one another within hours and showing spontaneous activity as early as day *in vitro* (DIV) 4–6 (Chiappalone et al., 2006; Pasquale et al., 2008; Soriano et al., 2008). Although neurons develop in a relatively large area, with our imaging instrumentation we observe a small but representative region of 13.4 mm<sup>2</sup> that contains few thousand neurons. A detailed inspection of our cultures reveal their spatial distribution which, despite some clustering, is compatible with a homogeneous distribution of neurons (**Figure 1B**). We monitor neuronal activity with fluorescence calcium imaging. As shown in the panels of **Figure 1B**, the spatial resolution of our measuring device is sufficient to trace the behavior of all the neurons in the field of view, with single-cell resolution, and along several hours.

**Figure 1C** provides examples of fluorescence traces in our cultures, for measurements with both excitation and inhibition active ("E+I" networks). The traces correspond to a developing culture at DIV 9. Fluorescence displays a fast onset due to neuronal activation, followed by a slow decay back to the baseline and that corresponds to the slow unbinding rate of calcium ions from the fluorescent probe.

Neuronal network activity in cultures is characterized by episodes of collective neuronal activation termed *bursts* where the neurons fire in a quasi-synchronous manner in a short time window of ∼200 ms. Almost the entire population of neurons participate in a bursting episode, which is observed in the traces of **Figure 1C** by the quasi-simultaneous occurrence of firing across the neurons. The timing of the bursts themselves is in general regular, with average interburst intervals on the order of 10 s in the provided example. In between bursts, neuronal activity is characterized by sparse, asynchronous firings across the network.

The properties of spontaneous activity, and in particular the structure of the bursting episodes, depends both on the excitability of the neurons, i.e., their ability to spontaneously fire, and the connectivity of the network, i.e., the ability to recruit, amplify and propagate activity from other neurons. The latter is particularly important since connectivity significantly changes during the maturation of the network.

### **3.2. NETWORK DEVELOPMENT**

To investigate distinct features of spontaneous activity due to varying neuronal connectivity, we first treat the scenario in which the network grows and matures along several days *in vitro* (DIV).

Neurons in our preparations are plated homogeneously on the glass substrate and lack any initial connectivity. However, development occurred rapidly. We already observed connections as early as 24 h after plating and, consistently with other studies (Soriano et al., 2008), neurons were electrically excitable by DIV 2–3 (data not shown). Spontaneous activity appeared by DIV 5– 6, subsequently changing in strength and structure as the culture matured and evolved further. **Figure 2** illustrates this behavior for a given culture batch, and with both excitation and inhibition active ("E+I" network). Representative fluorescence traces of average network activity in a period of 15 days of development are provided in **Figure 2A**. For this batch we observed the first occurrence of bursting at DIV 6. At earlier days, the bursting dynamics was either absent or too sparse to be detected. Although the presence of bursts is clear at DIV 6, their interburst timing is irregular and the firing amplitudes low. By DIV 8 the fluorescence amplitude has substantially increased and bursting has become more regular, reaching a stage of high periodicity by 2 weeks after plating. At later stages of development we observed different trends from batch to batch, with firing amplitudes and interburst intervals stabilizing or decreasing.

**Figure 2B** depicts the shape and strength of a burst among different evolutionary stages. Bursts are time-shifted for the onset of network activation to coincide. The plot reveals the gradual increase in bursting amplitude during the early stages of development, and the sudden jump at DIV 9, which hints at strong changes in both neuronal excitability and network connectivity.

The example of **Figure 2B** highlights the dominance of the burst shape (amplitude and width) on the structure of the recorded signal. This is further evidenced in **Figure 2C**, which shows the distribution of fluorescence amplitudes for the population-averaged signal along maturation. The distribution at DIV 5 is close to a Gaussian distribution, indicating the absence of firing events sufficiently strong to be detected by the camera. As development continued, the histogram of amplitudes became distinctly right-skewed, with progressively higher values of fluorescence. A detailed statistical analysis of the changes in fluorescence is provided in **Figure 2D**, and illustrates the strong asymmetry of the fluorescence distributions. Interestingly, the major changes in firing amplitude occur by the end of the first and second weeks *in vitro*. The average firing amplitudes (denoted by a black square) as well as the maximum measured amplitudes (up triangles) abruptly jump at these stages.

**FIGURE 2 | Network development. (A)** Examples of fluorescence traces of spontaneous activity along 15 days of development. All measurements correspond to cultures from the same batch. Traces are the average over the monitored network population (2000 neurons). The peaks of fluorescence signal identify bursting episodes. The time elapsed between two consecutive bursts define the interburst interval (IBI). The measurement at day *in vitro* (DIV) 6 corresponds to the first observation of spontaneous bursting activity in the batch. **(B)** Detail of a bursting event (averaged over the monitored network population) during the early stages of development to illustrate the substantial increase in fluorescence amplitude after DIV 8. **(C)** Histogram of the network-averaged fluorescence signal for representative stages of

development. Bursting activity is absent at DIV 5, giving rise to a fluorescence histogram that is close to a Gaussian distribution. The distributions broaden as bursts emerge and increase in amplitude. **(D)** Box plots of the statistical analysis of the fluorescence distributions. Note the logarithmic scale in the vertical axis. The mean of the distribution (-) and its maximum value () substantially increase by DIV 8 (pink) and after DIV 15 (yellow), suggesting major evolutionary switches of the network. In the figure, whiskers represent 25 and 75% confidence intervals, and crosses (x) 1 and 99%, respectively. **(E)** IBIs box plot analyses. The broad IBI distribution observed for young cultures significantly changes to a narrow distribution with stable IBI timing after DIV 8, to change again toward a higher variability by DIV 15.

These changes in network dynamic behavior are also captured by the distribution of interburst intervals (IBIs), which show a tendency to become well timed as the cultures mature (**Figure 2D**). The average IBI reduces from high, broadly distributed values in the range 100–200 s at DIV 5–8 to narrowly distributed values around 10–20 s after DIV 8. By DIV 16 the network dynamics changes again toward a more erratic behavior and larger IBIs.

## **3.3. EMERGENCE OF INHIBITION DURING DEVELOPMENT**

The role of inhibition during development is depicted in **Figure 3**. A first interesting feature is the observation that the blockade of inhibition ("E-only" recordings, see Materials and Methods) at early stages of development silences the network or strongly disrupts its activity, as shown in the network-averaged traces at DIV 5 and 8 in **Figure 3A**. Such a disruption is a consequence of the depolarizing action of GABA at early developmental stages and that confers it an excitatory role (Ben-Ari, 2002). Therefore, the blockade of GABAA effectively reduces excitation and, in turn, the mechanisms for the network to spontaneously fire.

GABA changes to its normal inhibitory action by DIV 7, an event known as GABA switch (Ganguly et al., 2001; Soriano et al., 2008). The blockade of inhibition at this and subsequent stages results in strong bursting due to the excess in excitation, which is revealed by the high fluorescence amplitudes at DIV 15 (**Figure 3A**).

The distribution of fluorescence amplitudes of **Figure 3B** also illustrates the changing role of inhibition during development. "E+I" networks show bursting activity already at DIV 5, with broad fluorescence distributions that gradually increase in width as bursts strengthen in maturer stages. "E-only" networks, however, show at DIV 5 a distribution of fluorescences close to a Gaussian distribution, although the slight deviation at high fluorescences hint at some sporadic, individual neuronal activity. Bursting is observed by DIV 7–8, though very erratic due to GABA switch. At the other extreme of development (DIV 15) network behavior completely changes, and the bursting amplitudes in the "E-only" condition are much higher than in the "E+I" one.

In general, the blockade of inhibition in cultures older than 1 week leads to a substantial increase of the fluorescence amplitudes, larger interburst intervals and a higher regularity of bursting episodes. These distinct traits of "E-only" networks are a consequence of the absent firing-regulatory role of inhibition, which causes the neurons to fire until the excitatory neurotransmitter's pool is exhausted (Cohen and Segal, 2011).

We observed that GABA switch could be well identified by analyzing the network average fluorescence signal in terms of the power spectrum density (PSD), and comparing the two network conditions along development. As shown in **Figure 3C**, at DIV 5 and 6 the "E-only" signal is below the "E+I" one. The spectra for the "E-only" case also scales with lower slopes, indicating a much different behavior of the network, which is either silent or very

**FIGURE 3 | Influence of inhibitory action during development and GABA switch. (A)** Illustrative population-averaged traces of spontaneous activity during development, and comparing "E+I" (top traces) and "E-only" signals (bottom ones) on the same culture. GABA has an excitatory role at early developmental stages and therefore its blockade effectively reduces excitation and silences the network. GABA switches to its normal inhibitory role by DIV 6 − 7. In maturer cultures, the blockade of inhibition increases excitation and the strength of the bursting episodes. **(B)** Corresponding fluorescence amplitude distributions, depicting the gradual increase in values as maturation progresses. At DIV 8 the blockade of inhibition neither silences the network nor strengthens firing, signaling the GABA switch event. **(C)** Power spectrum densities (PSD) of the spontaneous activity signals, averaged over the monitored population, and along representative stages of development. The gray curve shows the PSD associated to the noise of the camera. The PSD for "E+I" (blue) and "E-only" recordings (red) are markedly different except during GABA switch, at DIV 7 − 8, signaling its occurrence. The lines and their slopes are a guide to illustrate the markedly different behavior of the PSD between noise and actual measurements.

weak in activity. By DIV 7–8 the spectral curves cross one another. Most likely inhibition has here a mixed role across the culture during the GABA switch event, leading to a similar spectral trend in the two network conditions. GABA is completely inhibitory at DIV 9 and maturer cultures, and the "E-only" curves are now the ones with the highest energy compared to the "E+I" case.

We also show in **Figure 4** the evolution of the PSD for three different batches and covering different ranges of the maturation process. We show only the "E+I" data to emphasize developmental traits. The plots depict the general trend that the power spectra moves upwards and with progressively higher slopes as the cultures mature and the bursts strengthen. At DIV 5, which corresponds to the first occurrence of bursting activity for this batch, the corresponding PSD curve is distinctly above the noise level. The shape of the PSD curves and their relative shift substantially change during evolution, signaling the progressive increase in bursting amplitudes and frequency. After the second week *in vitro*, however, the cultures seem to reach a stable phase, with all spectra showing similar amplitudes and effectively collapsing into one another. The PSD here fits well a power law behavior *P* ∼ *f* <sup>−</sup>α, with 2.3 α -2.8.

#### **3.4. NETWORK DISINTEGRATION**

Here we investigate the deterioration in spontaneous activity when the excitatory connectivity of the network is progressively weakened by CNQX, an AMPA-glutamate receptor antagonist in excitatory neurons (see Materials and Methods). In these experiments we fully blocked NMDA and GABAA receptors to restrict ourselves to the simplest scenario. **Figure 5A** illustrates, for a mature culture at DIV 16, the evolution of the average "E-only" spontaneous activity for increasing concentrations of CNQX. We also provide the activity data for the unperturbed, "E+I" network for comparison. For [CNQX] = 0 (full connectivity strength), the network spontaneous activity shows the usual high-amplitude bursting behavior together with the large interburst intervals characteristic of the dynamics solely driven by excitation. Small additions of CNQX mainly disrupt the average interburst interval, which increases remarkably compared to the initial case. As the disintegration progresses, concentrations of [CNQX] 200 nM modify both the fluorescence amplitude and the interburst intervals. At extreme values of weakening, [CNQX] 2000 nM, global network activity is very rare or has stopped completely.

While high concentrations of CNQX completely disrupted bursting, i.e., population-spanning coherent activity, we should note that uncorrelated, neuron-to-neuron activity was still present. Although these events were scarce, we systematically detected their presence in the studied cultures.

To investigate variability in culture age, we carried out the same disintegration protocol for cultures at different stages of maturation. As depicted in **Figure 5B**, the bursting amplitudes in all these cases show a similar trend. Initially, the blockade of inhibition in the transition from "E+I" to "E-only" connectivity

tendency of the PSD to scale as a power law *P* ∼ *f* <sup>−</sup>α. Mature cultures at DIV 13 appear close to one another, suggesting that approximately after 2 weeks in *in vitro* cultures have reached a stable stage. The dotted black lines are a guide to the eye to illustrate the increasing values of α along maturation.

ramps up the bursting amplitude to a maximum, but the subsequent gradual network disintegration leads to a progressive decay in amplitudes until bursts disappear altogether.

This general trend in the decay of bursting amplitudes does not hold for the bursting activity of the network, which is quantified as 1/IBI. As shown in **Figure 5C**, although most of the cultures at DIV - 13 exhibit a gradual decay in activity upon CNQX application, those cultures at DIV 14 display an increase in activity at specific concentrations of CNQX. This erratic behavior seems indeed a distinct feature of mature cultures, and hints at the existence of network mechanisms in these cultures that promote activity, possibly to compensate the weakening in connectivity. Moreover, the fact that the increase in activity upon CNQX application occurs at different concentrations from one culture to another may indicate that development drives each culture to slightly different circuit architectures and connectivity strengths.

The study of the disintegration process in terms of the PSD is shown in **Figure 6A** for a culture at DIV 13. This figure portrays the general trend observed in most of the experiments. The PSD initially increases from the "E+I" condition to the "Eonly" one due to the large amplitude of the bursts in the absence of inhibition. Next, the gradual addition of CNQX decreases the overall power as well as the PSD slope, concurrently with the progressive decay in bursts amplitudes. However, for large concentrations of CNQX—and rare or inexistent bursting—the PSD exhibits a scaling trend that is distinctly different from both the bursting behavior and the background noise. This scaling suggests that the PSD is capturing temporal correlations between neurons' individual firing events. We note that these neuron-to-neuron interactions could not be detected in measurements with strongest connectivity strengths ([CNQX] - 400 nM) due to the dominance of bursting behavior in network activity.

This general trend actually showed some interesting variations, illustrated in **Figures 6B,C**. For the example at DIV 14 (**Figure 6B**) we observed evidences of peaks in the PSD at frequencies *f* 7 − 8 Hz. These peaks were particularly strong at CNQX concentrations of 100 and 200 nM. Remarkably, these concentrations also correspond to the ones in which network activity increases upon disintegration. Indeed, we systematically observed a correlation between those experiments in which activity increased at specific values of CNQX and the presence of peaks in the PSD. Another example is provided for a culture at DIV 16 (**Figure 6C**). In this case we observed two peaks (at around 5 and 7 Hz) for [CNQX] = 200 nM, the concentration at which network activity increases for this culture.

## **3.5. NETWORK SPATIAL TRAITS**

To further explore the PSD potential in characterizing neuronal network features, we analyzed the spatial distribution of spectral energy across the culture. We first considered the average energy, i.e., the mean value of the PSD distribution. **Figure 7A** shows the map of spectral energy for the PSD data of the culture at DIV 16 depicted in **Figure 6C**. Spectral energy is shown for the "E-only" condition along different stages of disintegration. The "E+I" data is also provided for reference.

shows a detail of the peaks.

We note that, by considering the entire spectral energy, the PSD values are dominated by the low frequency contributions, i.e., those associated with the amplitude of the bursts. Hence, the map of spectral energy in these conditions effectively shows the distribution of bursting amplitudes across the network.

and the PSD reveals temporal correlations arising from individual neuronal

An interesting feature of the map shown in **Figure 7A** is that the distribution of energy is inhomogeneous. Neurons with high bursting amplitudes are concentrated in the top-right corner of the field of view, and constitute by themselves a group of spatially close neurons that fire together with similar amplitudes, a quality that is maintained even at high levels of disintegration. We also note that in the transition from "E+I" to "E-only" connectivity, the spatial location of the "highly energetic" neurons substantially changes, evidencing that the balance between excitation and inhibition plays an important role in shaping network's local dynamical features.

The physical closeness of these "highly energetic" neurons is emphasized in **Figure 7B**, which shows the spectral energy as a function of the neuron index, with neurons ordered by spatial proximity. The plot marks two particularly relevant communities, labeled R0 and R1, whose containing neurons maintain a high spectral energy up to complete disintegration of the network. The location of these two groups in the monitored region of the culture is shown in **Figure 7C**. We remark that we monitor only a small region of the culture. Therefore, these groups of neurons may also share some traits with (or their dynamics influenced by) other neurons outside the field of view. For sake of discussion, we also provide in **Figure 7D** the neuronal density map, which highlights those regions in the field of view that are more densely populated. A direct comparison with **Figure 7C** shows that the two communities R0 and R1 of energetic neurons do not correlate with particularly dense areas, revealing the importance of nonlocal phenomena (both in circuitry and dynamics) in shaping specific neuronal activity traits.

We carried out this spatial analysis with all the monitored cultures, and covering from very young (DIV 5–6) to mature (DIV 20) cultures. In general we observed that young cultures up to DIV 10 displayed a rather homogeneous spatial disintegration, with no identifiable "highly energetic" communities. However, for cultures at DIV 14 and older we systematically observed an inhomogeneous disintegration combined with the existence of communities. The location of these communities varied from culture to culture, and confirmed that mature cultures break the initial network isotropy and develop slightly different connectivity layouts.

#### **3.6. COHERENT NEURONAL OSCILLATIONS**

**Figures 6B,C** introduced the observation that some cultures had a PSD characterized by the presence of peaks at frequencies *f* in the range 5–10 Hz. These peaks were stronger at

specific concentrations of CNQX, suggesting the emergence—or reinforcement—of collective oscillatory modes in the network for a precise coupling strength between neurons.

To further investigate these oscillatory modes, we considered again the experiment at DIV 16 whose PSD is shown in **Figures 6C**, **7**. Here, however, we analyze the PSD properties at the frequency *f* = 5.54 Hz, where a peak was well identifiable at [CNQX] = 200 nM. **Figure 8A** shows the spatial distribution of energy at this frequency for the two network conditions, "E+I" and "E-only", as well as along gradual disintegration through CNQX.

We first note the remarkable contrast in the spatial distribution of energy at *f* = 5.54 Hz between the "E+I" and "E-only" conditions. The former shows a compact spot of energetically similar neurons, while the latter displays an almost symmetric coverage, with a low energy region on the left that contrasts with a high energy one on the right. Again, these distinct maps reveal the importance of inhibition in shaping network dynamics.

Second, the study also reveals the evolution of this highly energetic spot throughout weakening. Indeed, for the "E+I" condition, the difference in energies between this spot and the neighboring areas is relatively small, by 10%, which made difficult its detection in the PSD of **Figure 6C**. As the connectivity of the network shifts to the "E-only" condition and CNQX is applied, we observe that the difference between the energy in this spot and its neighborhood ramps to about 45% at [CNQX] = 200 nM, a difference that progressively decreases as the disintegration progresses, although the compactness of the spot is well maintained.

We additionally investigated in more detail the differences in the PSD between the observed compact spot and the neighboring areas. For simplicity, we restricted the analysis to the "E-only" connectivity condition at [CNQX] = 200 nM weakening. **Figure 8B** depicts four investigated communities. In each community we selected a central neuron and averaged its PSD with the 100 closest neighbors (white dots within a circle in

**FIGURE 8 | Emergence of synchronous oscillations during network disintegration through CNQX.** Data corresponds to the experiment at DIV 16 shown in **Figure 6C**. **(A)** Spatial distribution of the PSD at a frequency of 5.54 Hz and for different connectivity conditions, "E+I" and "E-only" with gradual weakening. The presence of a compact spot at the center-right of the PSD map highlights a neuronal community (termed Z0) that synchronously oscillates at this frequency. Oscillations with strong amplitude also appear along the right edge. For [CNQX] = 200 nM the Z0

**Figure 8B**). We label as Z0 the community that corresponds to the "spot" mentioned above, and by R1–R3 the rest of communities. The corresponding PSD distributions are shown in **Figure 8C** together with the average over the entire network for clarity. We first note that the Z0 and R1 communities have a much higher energy than the others, and that both are markedly characterized by a peak in the PSD at 5.54 Hz. This peak is difficult to observe in the other communities. By comparing these results with the network-averaged PSD, we conclude that both Z0 and R1 are the main contributors to the observed peak at 5.54 Hz, and that Z0 is the community that remains highly coupled throughout disintegration.

To gain insight into the origin of these synchronous oscillations, we also carried out an analysis in which we investigated the link between the oscillations and the bursts themselves. As shown in **Figure 9A**, we first separated the original fluorescence signal into two contributions, one containing the low–frequency modulation associated to the shape of the bursts, and another one containing the rest of the signal. The corresponding PSD analysis (**Figure 9B**) revealed that the shape of the bursts dominates the behavior of the spectral curves and therefore masks the dynamics

community displays the highest difference in energy compared to the neighboring regions. **(B)** Location of 4 different communities. For each community, the central dot marks the position of a selected neuron whose power spectrum is averaged over all the 100 closest neurons (white dots within a circle). **(C)** PSD of the four communities for the "E-only" connectivity at 200 nM. The community Z0 and R1 show a clear peak in the PSD at *f* = 5.54 Hz. The PSD at this frequency is also higher in these two communities compared to the others.

of the network. On the contrary, the PSD of the filtered data retains both the dynamical traits of the network and completely captures the oscillatory behavior. We also investigated the properties of the signal in between bursting episodes, and excluded any contribution of the background signal to the presence of the oscillations. We therefore confirmed that the oscillations occurred concurrently with the bursts themselves. This is highlighted in **Figure 9C**, which compares the traces of the filtered signal along the different bursts. In all cases, the onset of the oscillatory behavior practically coincides with the beginning of bursting (*t* = 0s in the plots). The frequency analysis of these traces (averaged over all the bursting episodes) is shown in **Figure 9D**, revealing a peak at 5–7 Hz, i.e., the range of the initially described characteristic frequencies.

We extended all the above analyses to other cultures characterized by peaks in the PSD. We observed qualitatively similar traits, i.e., the existence of communities with markedly strong synchronous oscillations, the presence of specific CNQX concentrations at which the strength of the oscillatory mode was maximum, and the link between oscillations and bursts. The frequencies of the oscillatory modes as well as their spatial

**FIGURE 9 | Oscillations originate in the bursts.** Data corresponds to the experiment at DIV 16 for [CNQX] = 200 nM. **(A)** The top trace depicts the average fluorescence time series of a group of 100 bursting neurons that constitute the *Z*0 community. The first burst of this series and its manipulation is shown in detail in the bottom panel. The blue trace corresponds to the original fluorescence signal, and reveals a well pronounced oscillatory behavior at the peak of bursting. The original signal is separated into two contributions: the burst shape (red) and the oscillatory signal (black). Burst shape is estimated by applying a median filter with length *L* = 33 frames (1 sec); the resulting trace follows the slow dynamics of the burst while the oscillatory signal keeps the higher frequency components. Activity out of the busting episodes (background signal, gray trace) is computed by connecting the periods in between bursting events. **(B)** The spectral analysis of the resulting signals reveals that the burst trace

distribution significantly varied among cultures and developmental ages, emphasizing again the formation of specific network features during maturation.

## **3.7. UNHEALTHY CULTURES**

**Figure 6** showed that the PSD could capture, in a regime of suppressed bursting, temporal correlations between individual neuronal firings. Such a burst elimination was achieved by significantly reducing neuronal coupling through CNQX. Based on this observation, we hypothesized that such a network-spanning affectation could also occur in conditions where the health of the culture was compromised. To test such a possibility, we carried out a simple test in which we left the cultures to degrade, at the end of a normal experiment, by leaving them in the recording system for several hours.

Photo-damage in such an experiment induced neuronal death and severe disruption in the normal neuronal network behavior, which was evidenced by the extinction of bursting episodes. However, close inspection of the recordings showed that local activity, in the form of individual firing or groups of persistently active neurons, was still identifiable. **Figure 10A** shows traces of network-averaged fluorescence to compare the healthy and unhealthy states. We also show the fluorescence signal corresponding to the noise of the camera.

The corresponding PSDs of these measurements are shown in **Figure 10B**. Remarkably, the PSD for the deteriorated, unhealthy culture displays a neat scaling that is not masked by the bursts' structure. Also, the PSD is qualitatively similar in shape as the one for healthy cultures and [CNQX] = 800 nM. Interestingly, we measured clearly different exponents α. For the healthy and dominates the shape of the power spectrum, and actually keeps the greatest fraction of energy from the original signal. The oscillatory component has a much lower energy but retains network activity correlations (in the range 1 − 5 Hz approximately) as well as the oscillations at 5.5 and 7.5 Hz. The signal out from the bursting episodes does not exhibit any oscillatory components. Curves are vertically shifted a factor 50 from one another (and using the background signal as reference) to better highlight the different shapes of the power spectrum. **(C)** Fluorescence trace of the oscillatory signal for all the 11 bursting episodes of the recording, locked to the initiation of each episode (dashed white line), showing that oscillations originate within the bursts themselves. **(D)** Their corresponding averaged Gabor transform, picturing the presence of an oscillation in the 5–7 Hz range that only appears once the bursts have reached their maximum amplitude and start the decaying phase.

CNQX-drugged networks we consistently measured exponents of α 2.0, while for the unhealthy experiments we obtained α 3.0. Such a different values reveal different temporal correlations or dynamical modes in the network, and hints at the potential of PSD analysis to quantify the state of neuronal networks.

## **4. DISCUSSION**

Our experiments fall within the context of *functional multineuron calcium imaging* (fMCI), a technique based in the ability to examine network activity in large neuronal populations and with single-cell resolution (Stosiek et al., 2003; Ohki et al., 2005; Bonifazi et al., 2009; Takahashi et al., 2010a,b). fMCI has received substantial attention in the last years driven by the spectacular development of optogenetic tools and genetically encoded calcium indicators, which allow to monitor and probe neuronal circuits *in vivo* without the need of electrodes or other invasive measuring techniques (Yizhar et al., 2011a).

Given the challenge in fMCI to link the measured calcium fluorescence signal with the structural and dynamical traits of the underlying network, *in vitro* preparations have emerged as valuable platforms to probe neuronal circuitry and investigate the properties of the measured fluorescence signal. In this work we have utilized spontaneous activity in cortical cultures as the main measure to investigate the relation between activity, fluorescence signal and network connectivity. We have used two major approaches to access different neuronal circuitries, namely the monitoring of network development along 3 weeks and its controlled disintegration through application of CNQX. In both cases we observed distinct features in the shape of the fluorescence signal and its associated power spectrum density (PSD). The

PSD could capture relevant events during development, revealed locality features in the neuronal network, and highlighted the presence of synchronous oscillatory modes within neuronal communities.

#### **4.1. FLUORESCENCE SIGNAL AND POWER SPECTRUM**

The recorded fluorescence signal displayed different traits depending on both the age of the neuronal culture and its connectivity strength. First, young cultures under DIV - 5 did not display bursts, and the networks dynamics was characterized by sparse individual neuronal firings of very low amplitude. We detected the presence of these events in the histograms of fluorescence amplitude (**Figure 3B**), which deviate from Gaussian distributions at high fluorescence values. However, the PSD curves corresponding to these "young" traces were similar to the ones obtained by measuring the noise of the camera. Hence, in very young cultures and with the experimental settings that we used in the present work, we could not use the power spectrum to quantify temporal correlations between neurons or other dynamical features.

Second, cultures at DIV 6 did show bursts, with a structure (amplitude, width and interburst timing) that depended on maturation. The corresponding PSDs reflected such variations, and we could detect GABA switch as well as the relative strength between excitation and inhibition by comparing the PSD curves of the "E+I" and "E-only" conditions (**Figure 3**). Also, the rise in bursting amplitudes during development was reflected in the PSD by a gradual increase in the average power (**Figure 4**). The PSD curves for mature cultures showed a rather good collapse with a slope α 2.5, indicating the advent of a more stable network state. Despite the variations from culture to culture, such a trend was systematic. Hence, in principle we could "guess" the developmental stage of a culture, and even some coarse properties, based in the average energy and slope of the PSD.

We must note, however, that the shape of the PSD arises from a complex combination of factors, including the fast jump in fluorescence at the beginning of bursting, the width of the bursts, the slow decay of fluorescence back to the resting state, as well as the time between burst. One would therefore need a detailed exploration of these different parameters to fully understand the information that the PSD can provide. Given the variety of bursting regimes that a neuronal culture can convey (Van Pelt et al., 2004b; Wagenaar et al., 2006a,b), such a exploration is a considerable endeavor.

As a third major remark, we observed distinct features in the PSD between the development of the network and its disintegration through CNQX. The former includes the growth and strengthening of connections, both locally and globally, and thus the overall network dynamics constantly evolve. The latter weakens homogeneously the excitatory connectivity in the network, leading to essentially a similar network dynamics with progressively reduced bursting. Hence, young cultures are not equivalent to fully disintegrated mature cultures. The two experimental approaches are therefore complementary and reveal distinct features. Indeed, a remarkable observation in the experiments with CNQX is that, for concentrations that led to almost no bursting at [CNQX] 800 − 2000 nM, we observed significant individual neuronal firing across the culture. Given the maturation of the network, these firings were of sufficient strength to exceed the noise of the system. Only in these conditions the PSD followed a scaling that we believe was capturing correlations between neurons (**Figure 6**).

The investigation of temporal correlations from PSD analyses is indeed a powerful concept since it may unveil dynamical traits of the network, e.g., in the form of synaptic inputs or intrinsic neuronal interactions (Thurner et al., 2003; Destexhe and Rudolph, 2004; El Boustani et al., 2009). The significance of the scaling by itself in our data, as well as the information that these correlations provide about the interplay activity-connectivity in the network, needs detailed investigation. Notably, the observation that healthy and unhealthy cultures exhibit different scaling exponents suggest that such studies could provide a basis to describe pathological or deteriorated cultures from the analysis of the PSD. In this context, an additional experimental tool that would provide valuable insight is the incorporation of connectivity guidance in the culture substrate, for instance in the form of biochemical fixation or physical trapping (Eckmann et al., 2007; Wheeler and Brewer, 2010). Dynamics in such "patterned cultures" substantially differ from standard ones due to the dictated connectivity (Shein Idelson et al., 2010; Tibau et al., 2013), and would possibly give rise to different temporal correlations.

## **4.2. DEVELOPMENT AND NETWORK TRAITS**

Several works in the literature have investigated the emergence of network-spanning bursting episodes during development. Consistently with our work, bursts were reported to appear by DIV 5–6 (Kamioka et al., 1996; Opitz et al., 2002; Wagenaar et al., 2006a), showing a low amplitude and irregular timing. These studies used micro-electrode arrays (MEAs) as activity-measuring technique, and also revealed that the activity contained both individual firing events and bursts. As said before, this individual spiking was also present in maturer networks (DIV 10 and older), and we actually used the valuable information that they provide to reconstruct neuronal connectivity in the context of Transfer Entropy (Stetter et al., 2012). Mature cultures exhibited stronger and more regular bursting as a consequence of the progressive maturation of synapses and the increase in their number (Muramoto et al., 1993; Kamioka et al., 1996; Opitz et al., 2002). Interestingly, we observed a stabilization in bursting amplitudes as well as a decrease in bursting firing frequency by DIV 18–20 (**Figures 2D,E**). These results are consistent with the studies of Van Pelt et al. (2004b,a) who reported that, in cortical cultures similar to ours, burst duration and firing amplitudes reached maximum values by DIV 18, to later stabilize or decrease as network evolved further.

The different spatial analysis of the PSD (**Figures 7**, **8**) for mature cultures during network disintegration revealed strong inhomogeneities in the distribution of spectral energies, with compact spots of high energy. Spectral energy is directly linked to the amplitude of the bursts which, in turn, is related to the number of the elicited action potentials (Sasaki et al., 2008). If we assume that neurons firing with large bursting amplitudes have a higher input connectivity, then the combination of strong firing and spatial closeness identifies neuronal communities that are highly interconnected. The cohesion within a community is maintained up to complete disintegration of the network. Chiappalone et al. (2006) showed that spatially close neurons are progressively more functionally connected as the network matures; and Soriano et al. (2008) showed that, in CNQX disintegration experiments similar to ours, groups of neurons spatially close maintained their interconnectivity and collective firing when stimulated electrically.

Hence, we ascribe this spatial inhomogeneities in the PSD to the formation of highly conserved topological communities that maintain unique local features despite changes in global network dynamics. We indeed hypothesize that the communities observed by Chiappalone et al. (2006) are the same as our groups of "highly energetic neurons."

## **4.3. HIGH FREQUENCY SYNCHRONOUS OSCILLATIONS**

The PSD curves upon CNQX disintegration revealed the existence of high-frequency oscillations in the range 5–10 Hz, which were remarkably strong and spatially localized at particular concentrations of CNQX. These oscillations were observed solely in mature cultures and, in general, we detected them both in the "E+I" and "E-only" conditions. A detailed study of the fluorescence traces revealed that the oscillatory modes originated from activity within the bursts themselves. Interestingly, Shein Idelson et al. (2010) reported oscillations in small neuronal circuits formed by compact cell aggregates. They observed collective oscillatory modes within network bursts in the range 25–100 Hz, and the authors associated them to synchronous oscillations during the decaying phase of the network burst.

Our observed oscillations are markedly strong in localized communities, suggesting that the oscillations emerge as a result of recurrent activity within these communities. We found puzzling, however, the observation that the CNQX concentrations at which the oscillations had the highest amplitude coincided with sudden increases in global network dynamics. We suggest that the network may activate correction mechanisms at a critical connectivity weakening to prevent the deterioration of activity. These mechanisms may arise from local alterations in synaptic strength or connectivity, as well as from changes in the excitability of the neurons themselves.

It also may occur that these communities of oscillatory activity play a role in the network, for instance as centers for the initiation of activity. Orlandi et al. (2013) recently introduced the concept of "noise focusing", the amplification and propagation of network background activity toward specific foci or basins of attraction where bursts ultimately initiate. It would be enlightening to investigate if there is a relation between these foci of burst initiation and our oscillatory communities.

Finally, we remark that these oscillations seem to be inexistent in young cultures (or too weak to be detected), which strengthens the argument that strong coupling within the cell community is required for their generation. An aspect that requires investigation, however, is what parameters tune the frequency of the oscillations, for instance by exploring the relative weight between AMPA, NMDA and GABA receptors. Shein Idelson et al. (2010) indeed showed that the oscillations disappeared altogether when GABA was fully blocked, which does not occur in our case.

## **ACKNOWLEDGMENTS**

The authors acknowledge Javier G. Orlandi for fruitful discussions and insight. Research was supported by the Ministerio de Ciencia e Innovación (Spain) under projects FIS2010-21924- C02-02 and FIS2011-28820-C02-01. We also acknowledge the Generalitat de Catalunya under project 2009-SGR-00014. Miguel Valencia and Jordi Soriano acknowledge financial support from the Gobierno de Navarra, Education Department, Jerónimo de Ayanz Programme. Miguel Valencia acknowledges financial support from the Spanish Ministry of Science and Innovation, Juan de la Cierva Programme Ref. JCI-2010-07876.

## **REFERENCES**


in cultured neural networks. *IEEE Trans. Biomed. Eng.* 51, 2051–2062. doi: 10.1109/TBME.2004.827936


Yizhar, O., Fenno, L. E., Prigge, M., Schneider, F., Davidson, T. J., O'Shea, D. J., et al. (2011b). Neocortical excitation/inhibition balance in information processing and social dysfunction. *Nature* 477, 171–178. doi: 10.1038/nature10360

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 14 October 2013; accepted: 01 December 2013; published online: 18 December 2013.*

*Citation: Tibau E, Valencia M and Soriano J (2013) Identification of neuronal network properties from the spectral analysis of calcium imaging signals in neuronal cultures. Front. Neural Circuits 7:199. doi: 10.3389/fncir.2013.00199*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2013 Tibau, Valencia and Soriano. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# Effect of phase response curve skew on synchronization with and without conduction delays

## *Carmen C. Canavier1,2 \*, ShuoguoWang1 † and Lakshmi Chandrasekaran1 †*

<sup>1</sup> Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, Louisiana State University Health Sciences Center, New Orleans, LA, USA

<sup>2</sup> Neuroscience Center, Louisiana State University Health Sciences Center, New Orleans, LA, USA

#### *Edited by:*

A. Ravishankar Rao, IBM Research, USA

#### *Reviewed by:*

Timothy J. Buschman, Princeton University, USA Sachin S. Talathi, University of Florida, USA

#### *\*Correspondence:*

Carmen C. Canavier, Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, Louisiana State University Health Sciences Center, 1901 Perdido Street, New Orleans, LA 70112, USA e-mail: ccanav@lsuhsc.edu

#### *†Present address:*

Shuoguo Wang, Department of Genetics and the Human Genetics Institute of New Jersey, Rutgers, The State University of New Jersey, Piscataway, NJ, USA; Lakshmi Chandrasekaran, Stowers Institute for Medical Research, Kansas City, MO, USA

A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.

**Keywords: synchrony, synchronization, pulsatile coupling, phase locking, phase resetting**

## **INTRODUCTION**

A role has been proposed for synchronous oscillations in binding of sensory experiences (Singer, 1993) and attention (Fries et al., 2001). Synchronization that occurs between distal brain regions is almost always associated with oscillatory activity (Konig et al., 1995). This synchrony is achieved rapidly (Singer, 1999) and persists only transiently. The role of reciprocal coupling in synchronizing neural oscillators is supported by the observation that strong inter-hemispheric phase locking in the gamma frequency band with zero phase lag occurred in cat visual cortex could be disrupted by severing the corpus callosum (Engel et al., 1991). The inter-hemispheric conduction delays were on the order of 4–6 ms, which is about a sixth to a third of a gamma cycle. A role for altered synchronization tendencies in disease states (Uhlhaas and Singer, 2006) is supported by the observations that long distance synchronization is reduced in schizophrenia and epilepsy, whereas local synchronization in epilepsy is enhanced. Phase resetting theory (Glass and Mackey, 1988; Winfree, 1990; Ermentrout and Terman, 2010) is often used to study the synchronization tendencies of regularly spiking neurons. A phase response curve (PRC) shows how much an input advances or

delays the next spike as a function of where in the cycle the input is applied. Type 1 PRCs (Hansel et al., 1995) are comprised of either all advances (for excitation) or all delays (for inhibition), whereas type 2 PRCs exhibit both advances and delays.

Neurons with type 1 PRCs tend not to synchronize via weak mutual excitation (Hansel et al., 1995; Ermentrout, 1996). Nonetheless, the ability of pulse-coupled leaky integrate and fire (LIF) and other oscillators with type 1 PRCs to synchronize due to strong mutual excitation is well known (Peskin, 1975; Mirollo and Strogatz, 1990). The PRC of this model at late phases has a strongly stabilizing slope due to the ability of an input to trigger a spike immediately on arrival at very late phases, which creates a linear "causal limit" region in the PRC. This region accounts for synchrony at zero delay (Canavier and Achuthan, 2010), and as we show in this study, also accounts for the existence of a gradual transition between synchrony and antiphase as the conduction delay is increased, regardless of PRC skew. In contrast, a critical role for PRC skew in networks of type 1 neurons connected by mutual synaptic excitation was demonstrated by Ermentrout et al. (2001). If the maximum resetting (of either sign) occurs

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in the first half of the cycle, the PRC is left skewed; on the other hand if it occurs in the right half, it is right skewed. If the right skew is increased, the tendency to approximately synchronize with small time lags is increased for pairs of type 1 neurons coupled via mutual synaptic excitation or electrical coupling (Pfeuty et al., 2003; Zahid and Skinner, 2009), and skewing the PRC toward the left stabilizes the antiphase mode. In contrast, the antiphase mode is stabilized by skewing the PRC to the right (Ladenbauer et al., 2012) for pair with type 1 mutual inhibition. There are several ways in which altering the conductances (Ermentrout et al., 2001, 2012; Pfeuty et al., 2003; Gutkin et al., 2005; Stiefel et al., 2009) can change the shape of a type 1 PRC for a regularly spiking neuron. Therefore, one way to quickly reverse the synchronization tendencies of neurons is to modulate the intrinsic ion channels that alter the PRC shape (Ermentrout et al., 2012), which could provide a switch to turn synchrony on and off rapidly.

Here we examine the effect of changing the skew of a type 1 PRC on the ability of pairs of neurons characterized by these PRCs to synchronize in the presence of conduction delays. We quantify the tendency of a network to synchronize using a global method that requires the identification of the unstable solutions comprising the boundaries between the attractive basins of the stable solutions, and compares the size of the sets of initial conditions, or basins of attraction, that lead to synchrony versus any other competing stable modes. In some cases we also use a local measure that infers the rate of convergence to synchrony in the neighborhood of a stable solution using the slopes of the PRC at the phases at which inputs are received with a possibly non-zero delay after spikes in the presynaptic neuron(s). The solution structure for pairs of coupled neurons with type 1 PRCs in the presence of conduction delays is highly dependent upon the skew of the PRC. In particular, right skew enhances the ability of mutually excitatory pairs to preserve synchrony in the presence of small delays, but diminishes that of inhibitory pairs. Overall, inhibitory synchrony (Van Vreeswijk et al., 1994; Wang et al., 2012) is much more robust to conduction delays. These results have implications for synchronization in larger networks as well (see Implications of Generic Modes for Larger Networks).

## **MATERIALS AND METHODS**

#### **WANG–BUZSAKI MODEL**

The Wang and Buzsaki (1996) conductance-based model neuron has the following parameters unless otherwise noted. The reversal potentials *E*Na, *E*K, and *E*<sup>L</sup> were set to 55, −90, and −65 mV, respectively and the capacitance was set to 1 μF/cm2. The maximal sodium (*g*Na), potassium (*g*K), and leak (*g*L) conductances were set to 35, 9, and 0.1 mS/cm2, respectively. *I*stim is the applied current and was set at 1.0 μA/cm2. The synaptic current is given by *I*syn = *g*syn*s*(*V* − *E*syn), where *g*syn is the maximum synaptic conductance and *E*syn is equal to −75 mV for inhibitory synaptic connectivity and equal to 0 mV for excitatory synaptic connectivity. The rate of change of the gating variable s in units of ms−<sup>1</sup> is d*s*/d*t* = 6.25(1 − *s*)/[1 + exp(−*V*pre/2)] − *s*/τsyn, where *V*pre is the voltage of the presynaptic cell, and τsyn is the synaptic decay time constant of 1.0 ms.

## **MEASUREMENT OF PRC IN ISOLATED WANG–BUZSAKI NEURONS**

**Figure 1A1** shows the measurement of the PRC for a Wang– Buzsaki model neuron where the input is the synaptic conductance waveform (**Figure 1A1**, bottom trace) that results from a spike in the presynaptic neuron. The phase φ is 0 at an upward crossing of a predetermined threshold (here −14 mV), and the phase φ at which a stimulus is received is *t*s/*P*0, where *P*<sup>0</sup> is the intrinsic period and *t*s = φ*P*<sup>0</sup> is the stimulus interval, defined as the interval between the time of the action potential and the receipt of an input. The recovery interval *t*r is defined as the interval between the receipt of an input by a neuron and the next action potential in the same neuron: *t*r = *P*<sup>0</sup> − *t*s + *P*0*f*(φ), where the phase resetting *f*(φ) is given by the normalized change in the cycle length that contains the perturbation *f*(φ) = (*P*<sup>1</sup> − *P*0)/*P*0. In this study we do not focus on second order resetting that is evidenced by changes in length of the second cycle following the perturbation, but in some cases the second order phase resetting must be considered (Oprisan et al., 2004; Maran and Canavier, 2008; Woodman and Canavier, 2011). A positive resetting signifies a phase delay and a negative resetting signifies a phase advance. **Figure 1A2** shows a typical PRC for the Wang–Buzsaki model, consisting of all delays in response to an inhibitory synaptic input.

### **LEAKY INTEGRATE AND FIRE NEURON MODEL PRC**

The LIF model is given by d*V*/d*t* = −γ*V*(*t*) + *S*0, where *V*(*t*) is the membrane potential, γ is the magnitude of the leak, and *S*<sup>0</sup> is the applied current. When *V*(*t*) = 1 the neuron is presumed to fire and *V*(*t*) is reset to 0 (**Figure 1B1**). Following the methods of Peskin (1975) and Mirollo and Strogatz (1990), the neurons are instantaneously pulse-coupled such that an input depolarizes the membrane by a fixed amount ε or brings the membrane potential to threshold, whichever among the two values is less. For two coupled neurons i and j, when *Vi*(*t*) = 1, meaning one neuron reaches spike threshold, then the potential in the partner is set to *Vj*(*t*) = min [1,*Vj*(*t*)+ε], j -= i, meaning that inputs that occur late within the cycle can immediately trigger a spike. At an initial condition of *V*(0) = 0, we can explicitly solve for the voltage such that *V*(*t*) = (*S*0/γ)(1 − e−γ*<sup>t</sup>* ). From this expression, solving for the elapsed time (*t*s) to reach a given value of voltage *V*(*t*), we obtain *t*s = (1/γ) ln{*S*0/[*S*<sup>0</sup> − γ*V*(*t*)]}. The intrinsic period of the oscillator is the elapsed time required to reach *V*(*t*) = 1, which is *C*/γ, where *C* = ln[*S*0/(*S*<sup>0</sup> − γ)]. We can solve for the phase advance due an instantaneous jump from *Vj*(*t*) to*Vj*(*t*)+ε by taking the difference between the elapsed time required to reach *Vj*(*t*) corresponding to a given phase φ = *t*s γ/*C* and the elapsed time required to reach *Vj*(*t*) + ε in the absence of a perturbation. For *Vj*(*t*) + ε < 1, this difference is equal to (−1/γ){ln[(*S*<sup>0</sup> − γ)/(*S*<sup>0</sup> − γεe*C*φ)] − 1}, which is then normalized by the intrinsic period in Eq. 1. For *Vj*(*t*) + ε ≥ 1, the resetting is limited by the fact that an input cannot advance the next spike time to a time before the neuron receives an input, so the phase is advanced by exactly the normalized time remaining until the next input (1 − φ), with a sign reversal due to our definition of phase resetting in which advances are negative. The resetting at 0 and 1 are not the same because the effect of an input is assumed to end when a spike is produced; a more physiological model for coupling would assume any excess charge beyond that required

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cycle length f(φ) = (P<sup>1</sup> − P0)/P0. Alternatively, the perturbed cycle length P1 is equal to the sum of the stimulus (ts) and recovery (tr) intervals. **(A2)** Typical PRC for an inhibitory input to a Wang–Buzsaki neuron with <sup>g</sup>syn <sup>=</sup> 0.35 mS/cm2, <sup>τ</sup>syn <sup>=</sup> 1 ms, <sup>I</sup>stim <sup>=</sup> 2.0 <sup>μ</sup>A/cm2, and otherwise as the Methods. **(B1)** PRC measurement in a leaky integrate and fire model neuron. An instantaneous increment in membrane potential (black arrow) either advances the phase or immediately causes the neuron to reach threshold. **(B2)** Typical PRC for leaky integrate and fire neuron. Beyond a phase of about 0.8, a spike is triggered immediately by the input. In this "causal limit" region, interval in neuron 2 plus twice the delay δ, and the pink shaded area illustrates a similar constraint for the stimulus interval in neuron 2. The time lags, or firing intervals between neurons, can be inferred from the stimulus and recovery intervals. **(D)**. Predicting closed loop modes with open loop data. Plotting the algebraic combination of intervals with quantities that must be equal in a phase-locked mode on the same axis ensures that the intersections represent the stimulus and recovery intervals in phase-locked modes. The delay was 20% of the intrinsic period P0. The axes are all normalized by the intrinsic period of the component oscillators and therefore dimensionless.

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to evoke a spike is applied in the next cycle, causing an advance in that cycle, which at a phase of 1 would equal the resetting at a phase of 0.

Given the constraints on how much a spike can be advanced, the phase resetting (see **Figure 1B2**) is given by

$$f(\phi) = -\min(|(-1/\mathcal{C})\{\ln[(S\_0 \cdot \gamma)/(S\_0 \cdot \gamma \text{sc}^{C\phi})] - 1\}|, |\phi - 1|) \tag{1}$$

The negative sign in Eq. 1 was necessary to make the sign of the PRC consistent with the convention used in this work. The PRC in **Figure 1B2** was calculated directly from Eq. 1. This nonphysiological feature by which an input instantaneously triggers a spike introduces a linear region in the PRC at late phases in which the phase advance is φ − 1, exactly equal in magnitude and opposite in sign to the fraction of the cycle remaining when the input is applied. Since this limit is imposed by causality, we call this linear region the causal limit region of the PRC.

### **PREDICTION OF NETWORK ACTIVITY USING PHASE RESETTING**

The assumptions required to apply the stimulus and recovery intervals measured in isolated neurons with no feedback to the closed loop circuit are simply that the spikes remain essentially the same in the presence of feedback, and that the effect of each perturbation dies out within a single network period after it is received. Detailed stability calculations are given inWoodman and Canavier (2011). We use a method (Woodman and Canavier, 2011; Wang et al., 2012) very similar to the spike time difference method (Acker et al., 2003) with the advantage that it is easily extendable to longer conduction delays.

Our method takes advantage of the algebraic relationship shown in **Figure 1C** between the stimulus and recovery intervals in one to one phase-locked periodic modes. The stimulus interval (in the absence of any second order resetting) is simply *P*0φ as described above, and the dependence of the recovery interval on the phase was determined using the phase resetting protocol also described above. The key idea (Woodman and Canavier, 2011) is that there is a feedback loop through which a spike in one neuron influences, after a conduction delay, the timing of a spike in its partner, and this spike in turn, after another conduction delay, affects the timing of a spike in the original neuron. The duration of this feedback loop is always the sum of the two delays plus the recovery interval in the partner. For short equal conduction delays, the duration of this feedback loop is exactly the stimulus interval in the original neuron, as illustrated in **Figure 1C**. This condition must be met with respect to both the stimulus interval in neuron 1 (**Figure 1C**, gray shaded area) and the stimulus interval in neuron 2 (**Figure 1C**, red shaded area), so there are two symmetric criteria that must both be satisfied in order to establish a periodic one to one phase locking. However, longer feedback loops are also possible, in which the duration of the feedback loop is still equal to twice the conduction delay plus the recovery interval in the partner, but one or more spikes occur in the original neuron before the feedback from a given spike is received. The duration of the feedback loop in the original neuron is then equal to the stimulus interval in the original neuron plus *k* − 1 network periods *P*N, where the parameter *k* − 1 is the number of spikes that occur before the feedback loop is closed, and the network period is the sum of the stimulus and recovery intervals associated with any given input phase.

The stimulus and recovery intervals measured using the PRC protocol can be plotted for each isolated neuron with the axes arranged as in **Figure 1D** so that the intersection points meet both criteria for the duration of the feedback loop described above that must be satisfied in a periodic one to one locking by the stimulus and recovery intervals in each neuron. The observable time lags between neural firings can be calculated using the algebraic relationships shown in **Figure 1C** (Woodman and Canavier, 2011). In addition to the phasic relationships within a periodic mode, we also need to know the stability of each mode. The stability can also be read from the graph in **Figure 1D** (Wang et al., 2012), at least for *k* = 1. The stability criterion for the *k* = 1 mode mandates that if the absolute value of the slope of the black curve is greater than the slope of the red curve at an intersection, then that intersection is stable, hence a steeper black curve at the intersection point guarantees stability. The derivation follows from the stability criterion for modes with *k* = 1, which is −1 < [1 − *f* (φ1)][1 − *f* (φ2)] < 1 where *f* (φ1) and *f* (φ2) are slopes of the PRC evaluated at the phase locking points of φ<sup>1</sup> and φ2. Stability is guaranteed if the slope of the PRC at both locking points is positive and <2. Since ts depends only on phase, and *t*r depends on both the phase and the phase resetting, algebraic manipulation reveals that the slope of the black curve for neuron 1 for *k* = 1 is [*f* (φ1) − 1]−<sup>1</sup> and the slope of the red curve for neuron 2 for *k* = 1 is [*f* (φ2) − 1]. Dividing all terms in the stability criterion by [1 − *f* (φ1)] and considering the cases for which [1−*f* (φ1)] is positive or negative gives the stability criterion in terms of the relative steepness of the slopes. For *k* = 2, the stability criterion is −1 < [1 − *f* (φ1) − *f* (φ2)] < 1. For higher values of *k*, the appropriate stability criterion must be applied (Woodman and Canavier, 2011).

## **RESULTS**

## **TWO LIF NEURONS PULSE COUPLED BY EXCITATION TRANSITION GRADUALLY BETWEEN SYNCHRONY AND ANTIPHASE AS THE CONDUCTION DELAY IS INCREASED**

Solutions that were obtained as the conduction delay was varied in pairs of LIF model neurons coupled via excitatory pulses are shown in **Figure 2**. With no delay, all initial conditions converged to synchrony (**Figure 2A**), as expected (Peskin, 1975; Mirollo and Strogatz, 1990). For delays >0 but up to about 40% of the intrinsic period, a "leader–follower" mode was obtained in which the smaller time lag between the firing of the two neurons was equal to the delay (second blue bar in **Figure 2B**). This mode is observed because the follower fires exactly when the delayed input from its partner arrives, but the leader does not fire immediately upon receiving an input from the follower. Convergence occurs within a single cycle for reasons explained below. The lack of robustness of synchrony mediated by excitatory pulse coupling to delays was also expected (Ernst et al., 1995). For delays equal to about 40–50% of the intrinsic period (see **Figure 4C**), an exact antiphase mode was obtained (**Figure 2C**) in which the time lags are each equal to half the network period, and because each neuron fired immediately upon receiving an input, the delays (horizontal blue bars) were exactly equal to the time lags. For delays equal to about 50– 85% of the network period, we again obtained a leader–follower

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mode in which one neuron, but not the other, fired immediately after the delayed input from its partner arrived (second blue bar in **Figure 2D**). In this case the longer of the two time lags is equal to the delay, but convergence generally does not occur within one cycle. Finally, for delays longer than 90%, a nearly synchronous mode emerged in which the firing order of the two neurons switched on every cycle (leapfrog mode in Maran and Canavier, 2008; Oh and Matveev, 2008).

midrange delays antiphase is observed in which each time lag (asterisks) is

We can understand how the modes in **Figure 2** arise by examining how the delays alter the generic periodic solutions for two identical, identically coupled oscillators in which the receipt of an input at late phases can immediately trigger a spike. We consider as generic only 1:1 modes, in which no oscillator fires twice in a row before the other oscillator fires. The inset in **Figure 3** shows two oscillators coupled with equal conduction delays. **Figure 3** shows a schematic representation of the generic modes: the two oscillators can fire together in exact synchrony (**Figures 3A,E**), they can alternate in exact antiphase with the same time lags (**Figure 3C**), or they can fire alternately with different intervals between spikes (**Figures 3B,D**). The meaning of the integer *k* can

S<sup>0</sup> = 1, ε = 0.05.

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be better understood by observing the paths marked by dashed lines in **Figures 3A,B**. The path begins with a spike in the top neuron and shows whether the timing of that spike affects the timing of the next spike in the spike neuron via the feedback loop through the other neuron. **Figure 3A** shows that after one delay, an input is received by the other neuron, then one recovery interval later the other neuron spikes, then after one more delay an input is received by the first neuron. This input arrives too late to affect the timing of the very next spike in the first neuron, but will affect the timing of the second, so *k* = 2. On the other hand, the dashed lines in **Figure 3B** show that the first spike in the top neuron does affect the timing of the very next spike in the same neuron via the feedback loop though the other neuron, so *k* = 1 for this case.

The pink shaded areas in **Figure 3** show the relationship of the stimulus intervals to the conduction delay, and the infinity symbol represents the steady value of the intervals in a periodic mode after all transients have decayed. This relationship is important because it allows us to predict the phase that at which inputs will be receive in a given model directly from the value of the conduction delay. For synchrony at both early (**Figure 3A**) and late phases (causal limit synchrony, **Figure 3E**), the conduction delay is equal to the stimulus interval in each neuron (δ = *t*s = φ*P*0). Therefore the

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phase at which an input is received in the synchronous modes is always equal to the normalized delay (φ = δ/*P*0). We refer to **Figure 3E** as causal limit synchrony because a spike is triggered immediately when the delayed input is received; this is not the case in **Figure 3A**. The pink shaded area in **Figure 3B** shows that the delay is half the stimulus interval (*t*sL) for the leader, so the phase at which an input is received is half the normalized delay. The pink shaded area in **Figure 3C** shows that in the antiphase mode the stimulus interval in one neuron is equal to twice the delay plus the recovery interval in the other neuron. There is no pink shaded area in **Figure 3D** because there is no integral relationship between one stimulus interval and the conduction delay; instead the delay is half the sum of the two stimulus intervals. This result is obtained by noting *t*sF + *t*lFL + *t*sL = *t*lFL + 2δ and canceling the time lag term tlFL.

The blue shaded areas in **Figure 3** show the relationship of the time lags observed in each mode to the conduction delay. **Figures 3B,D** show that for leader–follower modes with *k* = 1 and 2 respectively, the delay is equal to the time lag between the leader and the follower (*t*lLF), meaning that a spike is triggered in one neuron, but not the other, immediately when a delayed input is received. The blue shaded area in **Figure 3E** shows that in the causal synchrony mode, a spike is triggered in both neurons immediately upon receipt of the delayed input, and consequently both stimulus intervals as well as the network period are equal to the conduction delay. These relationships are a direct consequence of the ability of a delayed input to immediately trigger a spike upon its arrival.

Each of the generic modes in **Figure 3** corresponds to the panel with the same letter in **Figure 2**. However, in order to complete our analysis on the coupled LIF system, we need to apply the insights gained in **Figure 3** to the PRC for the individual LIF neurons given the assumed form of pulse coupling described in Section "Materials and Methods." **Figure 4** uses only the information in the PRC (shown in both **Figures 4A,B**) to predict the two time lags (defined in **Figure 1C**) that comprise one to one periodic locked modes (shown in **Figure 4C**) associated with each mode obtained in **Figure 2** by integrating the differential equations for the pair of pulse-coupled LIF oscillators. The time lags are the intervals between a spike in one neuron and the next spike in its partner (**Figure 1C**). For synchrony, one lag is arbitrarily set to 0 and the other to the network period. For antiphase, both time lags are equal (indicated by filled circles) so only one is visible. Only stable modes (black symbols) can be observed as a result of simulations, but the prediction method also identified the unstable (red symbols) and neutrally stable (blue circles) modes. Both the axes with the time lags (intervals) and delay are normalized with respect to the intrinsic period; the phase is the stimulus interval normalized by the intrinsic period. The lowercase letters along the middle of **Figure 4C** indicate the delays corresponding to the solutions in the corresponding panels in **Figure 2** and the schematic representation in **Figure 3**.

The overall picture given in **Figure 4** with respect to the generic modes is as follows. There are three solution branches, corresponding to synchrony, leader–follower and antiphase. Synchrony is stable (a) at zero delay (open black circles) but in region (b) splits into an unstable synchronous branch (pairs of open red

**FIGURE 4 | Predicting the solution structure for pulse-coupled LIF pairs. (A)** Phase response curve for a leaky integrate and fire neuron with parameters as in **Figure 2**. The unstable branch is to the left of φCL. The stable, causal limit branch of the PRC is to the right of φCL, and neurons receiving an input on this branch fire immediately upon receipt of the input. The input phases in leader–follower mode φ<sup>L</sup> and φ<sup>F</sup> lie on the left and right branches, respectively, and must have equal phase resetting f(φL) = f(φ<sup>F</sup> ) as indicated by horizontal dotted line. Open circle denotes the average of φ<sup>L</sup> and φ<sup>F</sup> . The vertical dotted lines from **(A)** to **(C)** give the boundaries in **(C)** of the leader–follower mode for k = 2 and the causal limit synchrony region. The dashed line labeled y = 2φ − 1 give the input phase for the antiphase mode φAP with zero delay. If the center of the PRC (open circles) falls to the right of this line, the leader–follower k = 1 branch exists. **(B)** The PRC is replotted at half scale to show the generic relationships between the normalized stimulus interval, phase (φ = ts/P0) and the normalized delay (δ/P0) in the leader–follower mode for k = 1 and the antiphase mode. The phase φ<sup>L</sup> at which the leader receives an input for the k = 1 leader–follower mode is twice the normalized delay (φ<sup>L</sup> = tsL/P0; tsL = 2δ, see **Figure 3B**, so φ<sup>L</sup> = 2δ/P0). The follower receives an input at phase φ<sup>F</sup> and fires immediately. This leader–follower mode ceases to exist when twice the normalized delay value reaches φCL; beyond that point, the antiphase mode gains stability. In this antiphase mode, both neurons receive an input at the same phase (see **Figure 3C**) on the right stable branch indicated in **(B)**, and fire immediately upon receiving the input. On this branch, the normalized stimulus interval for each neuron is equal to twice the normalized delay. **(C)** Predicted solution structure as delays are varied for two neurons coupled via the PRC in **(B1)**. The two time lags (Continued)

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#### **FIGURE 4 | Continued**

between the firings of the two neurons are represented by a pair of red symbols (unstable mode), a pair of black symbols (stable mode) or a pair of blue circles (neutrally stable mode). In an antiphase mode, both time lags are the same, which is indicated by a filled symbol. For synchrony, one time lag of each pair must be 0 and the other is equal to the network period. The leader–follower mode for k = 1 persists for normalized delays up to φCL/2 indicated by the leftmost vertical dotted line from **(B)** to **(C)**. In the stable antiphase mode for normalized delays from φCL/2 to 0.5, both time lags are exactly equal to the delay (see **Figure 3C**). The leader–follower mode reestablishes itself for normalized delays >0.5 and persists until the normalized delay reaches φCL. This leader–follower mode has a k = 2 and the normalized delay is equal to (φ<sup>L</sup> + φ<sup>F</sup> )/2 (see **Figure 3D**). Neutrally stable causal limit synchrony (see **Figure 3E**) is observed starting at normalized delays greater than φCL indicated by rightmost vertical dashed line from **(A)** to **(C)**. Note the diagonal line formed by the black and blue symbols indicates that in every stable or neutrally stable mode, at least one time lag is equal to the delay. The lower a, b, c, d, and e in **(C)** correspond to the labels of the same letter in **Figures 2** and **3**.

circles) and a stable leader–follower branch (open black squares). In the same regions, antiphase is unstable (solid red circles). Unlike the weak coupling approach, the network period is not equal to the intrinsic period, because the network period includes a nonnegligible contribution from the phase resetting in the circuit. Therefore, the normalized time lag is not 0.5 for antiphase because the normalization is by the intrinsic period and not the network period. At the start of the region labeled (c), the stable leader– follower branch and the unstable antiphase branch coalesce into a stable antiphase branch. Therefore, the stable leader–follower branch allows for a gradual transition between synchrony and antiphase as the delay is lengthened. At the start of the region (d) these two branches again diverge with antiphase losing stability and leader–follower regaining existence. At the start of region (e) the stable leader–follower and unstable synchronous branches merge into neutrally stable causal limit synchrony (open blue circles). Neutrally stable synchrony implies that near synchronous solutions, like the one shown in **Figure 2E**, do not converge to synchrony.

The diagonal line of symbols in **Figure 4C** indicates that for every predicted one to one stable or neutrally stable phase-locked mode, one or both time lags are equal to the delay. We will show that the PRCs in **Figures 4A,B**, along with the understanding of the generic modes presented in **Figure 3**, can explain the relationship of these time lags to the delay as well as why solution branches coalesce, diverge, or change stability. The key characteristic of the PRCs in **Figures 4A,B** is that they have two branches, a left branch with a negative destabilizing slope and a right branch with a maximally stabilizing slope to the right of the phase marked φCL, which is the causal limit (CL) region described in Section "Materials and Methods." Inputs received at phases in the causal limit region immediately trigger a spike.

## **WHY IS EXACT SYNCHRONY STABILIZED BY THE CAUSAL LIMIT REGION OF THE PRC AND DISRUPTED BY CONDUCTION DELAYS?**

At zero delay, indicated by the point labeled "a," there is a stable synchronous solution (black circles in **Figure 4C**) and an unstable antiphase solution (solid red circle). For the synchronous solution, **Figure 3A** shows that both neurons receive an input at a phase equal to the normalized delay. Synchrony at zero delay is a special case because *k* = 1 for that case, and the relevant stability criterion for synchrony with no delay depends upon the slope of the PRC at the two ends, *f* (0+) and *f* (1−). Specifically for synchrony stability requires that−1<[1−*f* (0+)][1−*f* (1−)]<1 where the+and− superscripts indicate the limit from the right and left, respectively (Oprisan and Canavier, 2001; Achuthan and Canavier, 2009). The quantity [1 − *f* (0+)][1 − *f* (1−)] is a scaling factor that operates in the vicinity of synchrony and multiplies the phasic deviation from synchrony on one cycle to give the deviation on the next cycle.

If infinitesimally small delays are introduced, each spike no longer affects the timing of the very next spike in the same neuron via the feedback loop through the partner (**Figure 3A**). Instead, the effect is felt on the second spike after the spike that triggered the input, so *k* = 2 and the stability criterion becomes −1 < [1 − *f* (0+) − *f* (0+)] < 1 (Woodman and Canavier, 2011). For the negative slopes just to the right of zero, the scaling factor 1 − *f* (0+) − *f* (0+) is >1, resulting in deviations from synchrony that grow and render synchrony unstable. The major effect is not the change in the form of the stability criterion, but rather the loss of the stabilizing slope at a phase just to the left of one (1−), where the slope is nearly 1 so the scaling factor is nearly 0. The bottom line is that the slope of the left branch of PRC for excitation does not favor synchrony at short delays; therefore, zero time lag synchrony with mutual excitation is not robust to delays for this PRC shape. Since the stimulus interval is equal to the delay, the normalized delays and input phases on the PRC are numerically equal and synchrony remains unstable along the left branch of the PRC in **Figure 4A** until the normalized delay exceeds φCL (blue circles in region including the label e in **Figure 4C**). The neutrally stable causal limit branch emerges at that point with one time lag equal to the delay and the network period as shown in **Figure 3E**. Recall that the scaling factor that determines stability is 1 − *f* (φ1) − *f* (φ2) for *k* = 2. Both input phases are the same (φ<sup>1</sup> = φ2) and fall on the causal limit line with a slope of 1 [*f* (φ1) = *f* (φ2) = 1]. Therefore, the scaling factor that determines whether perturbations from synchrony grow or decay is equal to −1. This implies that synchrony is neutrally stable, which means that perturbations do not decay; also the negative sign of the scaling factor guarantees that the firing order switches on every cycle preventing convergence to exact synchrony as shown in **Figure 2E**.

## **WHEN DO YOU GET UNEQUAL TIME LAGS THAT TRANSITION BETWEEN SYNCHRONY AND ANTIPHASE?**

In the leader–follower mode shown in **Figure 2B**, the follower neuron (red trace) but not the leader (black trace) fires immediately upon the delayed receipt of an input (see **Figure 3B**), thus its phase locking point lies in the causal limit region of the PRC. This particular mode has a conduction delay of 20% of the period, and is indicated by the open circles in the predictive plot for *k* = 1 in **Figure 1D** as well as by the black squares above and below the letter b in **Figure 4C**. As illustrated schematically in **Figure 3B**, one time lag (*t*lLF) is equal to the delay δ, and the stimulus interval for the leader (*t*sL) is exactly twice the delay. Therefore the PRC in **Figure 4B** is plotted so that the normalized stimulus intervals for the leader (the phase φL) line up with the

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corresponding normalized delay (half the stimulus interval). For each φ<sup>L</sup> on the left branch, the leader–follower mode for *k* = 1 exists if there is a corresponding φ*<sup>F</sup>* point on the right, causal limit branch with the same resetting value, as illustrated by the horizontal dashed line. This leader–follower solution branch ends at φCL and coincides with the stabilization of the antiphase mode. The ability of a delayed input to immediately trigger a spike guarantees stable solutions for which the time lag is equal to twice the delay (see caption of **Figure 3D**) and enables near synchrony with short time lags. For *k* = 1, the scaling factor for deviations from the phase-locked mode is [1 − *f* (φ*<sup>F</sup>* )][1 − *f* (φL)], which is 0 given that *f* (φ*<sup>F</sup>* ) is 1, so convergence is rapid.

The defining characteristic of the antiphase mode (**Figures 2C** and **3C**) is that the two time lags are equal, the two stimulus intervals are equal and the two recovery intervals are equal. Each stimulus interval is also equal to the recovery interval plus twice the delay: *P*φAP = *P* − *P*φAP + *Pf*(φAP) + 2δ. This implies that the phase at which an input is received in the antiphase mode is φAP = [1 + *f*(φAP)]/2 + δ/*P*. For zero delay, the intersection of the line *y* = 2φ − 1 with the PRC occurs at φAP, because on that line φ = [1 + *f*(φ)]/2. In **Figure 4**, for normalized delays less than φCL, the phase corresponding to the antiphase mode falls on the left, unstable branch of the PRC. The stability of antiphase at zero delay is critical: the stability of this mode at 0 usually implies the absence or lack of stability of the near-synchrony modes (squares with short delays) and competes with synchrony if it exists. Beyond φCL the recovery intervals become 0, so the stimulus intervals become equal to twice the delay and the region between the vertical dashed lines in **Figure 4B** forms the boundaries for the stable antiphase mode (black circles in the vicinity of c in **Figure 4C**), as the time lags become exactly equal to the delay, and the phase at which an input is received in the stable antiphase model falls on the stable causal limit branch in **Figure 4B**, where the phase is twice the normalized delay. The scaling factor for *k* = 1 antiphase is given by [1 − *f* (φAP)][1 − *f* (φAP)], which is 0 on the causal limit line and implies convergence within a single cycle in the neighborhood of the fixed point.

The antiphase mode loses stability because φAP "wraps around" and falls on the destabilizing left branch of the PRC for delays greater than half the intrinsic period (Woodman and Canavier, 2011). For normalized delays between 0.5 and φCL, the leader– follower mode reappears (**Figure 2D** and region near d in **Figure 4C**). For the *k* = 2 leader–follower mode, the sum of the stimulus intervals equal to twice the delay (**Figure 3D**). Using the definition of the stimulus intervals, we obtain that the normalized delay is equal to (φ<sup>L</sup> + φ*<sup>F</sup>* )/2, marked as open circles in the PRC in **Figure 4A**. The horizontal dashed lines show a minimum normalized delay of about 0.5 is required for φ<sup>L</sup> = 0 and φ*<sup>F</sup>* = 1 + *f*(0), and a maximum normalized delay of φCL, beyond which causal limit synchrony emerges as described above. The scaling factor for the *k* = 2 leader–follower mode is 1 − *f* (φL) − *f* (φ*<sup>F</sup>* ). Since *f* (φ*<sup>F</sup>* ) = 1, the scaling factor reduces to −*f* (φL), which is positive. If the latter slope is <1, which it generally is, stability is guaranteed.

In order to confirm that our graphical analysis of the PRC yields the correct predictions for modes with unequal time lags (specifically leader–follower modes for the LIF model) regardless of whether the PRC is right or left skewed, as well as to confirm

that the stable synchronous modes results from the steep slope at 1<sup>−</sup> and not directly from right skew, we constructed the counterexample in **Figure 5**. The pulse coupling was made to be very strong in order to extend the causal limit region of the PRC in **Figure 5B** leftward. **Figure 5A** illustrates with one example set of initial conditions that for zero delay, all initial conditions converge to synchrony. Synchrony at zero delay remains stable, and the intersection of the line *y* = 2φ − 1 with the PRC that gives the stability of the antiphase mode at zero delay still falls on the unstable branch, and the leader–follower modes with unequal time lag still mediate a gradual transition from synchrony to antiphase as the conduction delays are lengthened. At a delay corresponding to the value φCL, the antiphase mode is stabilized. This extreme, artificial example that shows that right skew is not required for synchrony at zero delay nor the gradual transition with near synchronous modes at small delays. However, in the more realistic examples given in the next section, increasing right skew does promote synchrony and near synchrony for excitatory coupling.

## **LEFT SKEW STABILIZES ANTIPHASE AT SHORT DELAYS AND PROMOTES BISTABILITY FOR CONDUCTANCE-BASED MODEL WITH EXCITATORY COUPLING, UNLIKE THE LIF RESULTS**

The pulse-coupled LIF is not very physiological, especially with respect to the instantaneous pulse coupling in the voltage waveform. The generic modes observed in the LIF are modified in networks of real neurons, and their closer analogs, conductancebased models, because a spike in one neuron cannot immediately trigger a spike in another – there must be a finite delay. **Figure 6A1** shows a typical left skewed type 1 PRC for a Wang–Buzsaki model neuron receiving excitatory synaptic input. The PRC has a left branch with a destabilizing slope and a right branch with a stabilizing slope. The vertical dotted line separates the branches. Unlike the extreme example of left skew for a pulse-coupled LIF neuron given in **Figure 5**, the left skew in a more realistic model does not give rise to synchrony with zero delay, nor to the leader– follower branch with near synchrony at small delays. Instead, the synchronous mode is unstable for zero delay because the destabilizing slope at 0<sup>+</sup> dominates the less steep stabilizing slope at 1−. Synchrony remains unstable for normalized delays to the left of the vertical dotted line (red circles to the left of the dotted line in **Figure 6A2**). The leader–follower branch does not emerge at small delays because of the left skew as explained below. One important consequence of the non-zero recovery intervals in realistic models (and real neurons) is that synchrony with normalized delays greater than φCL is stabilized, as opposed to neutrally stable and unobservable as for the case of the pulse-coupled LIF. The slope on the right branch is less steep ensuring convergence because the scaling factor 1 − 2*f* (φ) is guaranteed to have an absolute value <1 for positive slopes <1. Optimal convergence occurs when the slope at the locking point equals 0.5.

The most critical result of this paper, which is the effect of skew on the existence of unequal modes, can be explained as follows. The key idea is that the same line that determines the location and hence the stability of the antiphase mode also determines whether a positive value of the conduction delay can support the near synchrony that is part of the leader–follower solution branch. For *k* = 1, for identical oscillators with identical delays, we obtain

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**FIGURE 5 | Right skew is not rigorously required for zero lag synchrony or leader–follower modes in excitatory pairs.** An extreme example was constructed to show that right skew is not required for these phenomena. Parameters for the coupled LIF pair are γ = 0.9, S<sup>0</sup> = 1, ε = 0.05. **(A)** Convergence to synchrony with no conduction delay. **(B)** Left skewed phase resetting curve enters the causal limit region beyond the phase φCL at about 0.4. Again, the dashed line labeled y = 2φ − 1 give the input phase for the unstable antiphase mode φAP with zero delay. **(C)** Solution structure for delays less than half the intrinsic period shows a stable leader–follower mode with time lags proportional to the delay emanating from synchrony at zero delay. The symbols have the same meaning as in **Figure 4**, and all quantities are normalized by the intrinsic period.

2δ = *t*sL − *t*r*<sup>R</sup>* (see **Figure 7A**), where *t*r*<sup>R</sup>* is recovery interval for the phase locking point on the right branch and *t*sL, the stimulus interval for the phase locking point on the left branch of the PRC. Therefore the recovery interval has to be less than the stimulus interval. [Note that if *f*(φR) = φ<sup>R</sup> − 1, which occurs when φ<sup>R</sup> falls in the causal limit region and *t*r*<sup>R</sup>* = 0, this condition is automatically satisfied, as in **Figures 4C** and **5C**.] Substituting for the recovery and stimulus intervals yields a normalized delay δ/*P*<sup>0</sup> =(φL+φR)/2−[1+*f*(φR)]/2, where *f*(φR)=*f*(φL). Since the normalized delay has to be non-negative, unequal interval modes with *k* = 1 only exist if the average (φ<sup>L</sup> + φR)/2, indicated by the open circles in **Figure 6A1**, is greater than or equal to [1 + *f*(φ)]/2. Since the phase resetting corresponding to each open circle is given by the *y*-axis value, this condition is satisfied for phases that lie to the right of the dashed line *y* = 2φ − 1, the same line that determines the phase φAP for antiphase at zero conduction delay, because along this line φ = [1 + *f*(φ)]/2. Since all possible circles lie to the left of this line in **Figure 6A2**, no *k* = 1 branch of solutions with unequal time lags emerges. However, the vertical dotted line shows that a *k* = 2 solution branch (red squares to the right of the line in **Figure 6A2** with unequal time lags does emerge at delays equal to (φL+φR)/2. In fact, it is easy to see that the *k* = 2 unequal times lags (including leader–follower) mode always exists, because it is not possible for the delay to be negative in this scheme. However, this mode is not guaranteed to be stable. **Figure 7B** shows that for unequal time lag modes with *k* = 2, δ − *t*sL = *t*s*<sup>R</sup>* − δ which implies that the sum of the stimulus *t*sL and the recovery intervals *t*r*<sup>R</sup>* equals twice the delay: 2δ = *t*sL + *t*s*R*. In this case, the normalized delay δ/*P*<sup>0</sup> is the average (φ<sup>L</sup> + φR)/2, which is the same expression as that of the leader–follower mode for *k* = 2. In contrast to the LIF example, this branch of solutions with unequal time lags is unstable because the slope on the destabilizing left branch dominates due to the shallower slope of a right branch that does not fall on the causal limit.

The same line representing *y* = 2φ − 1 gives the phase φAP antiphase mode for zero delay at the intersection with the PRC (**Figure 6A1**). The left skew favors the stability of the antiphase mode for zero delay because it extends the stabilizing right branch of the PRC to smaller phases, and this stability persists for a range of delay values (black circles marked *k* = 1 in **Figure 6A2**). Since φAP = [1 + *f*(φAP)]/2 + δ/*P*, increasing the delay shifts the antiphase mode rightward. The synchronous solution (*k* = 2) with φ = δ/*P* is stabilized by its arrival on the right branch before the antiphase solution reaches the end of the right branch and loses stability as it jumps to the left branch. This overlap enables bistability for some delays. As delays are further increased, the *k* increases to 3 and the generic solutions recur (Woodman and Canavier, 2011). Left skew promotes bistability by increasing the length of the stabilizing branch compared to the destabilizing branch, increasing the likelihood that solutions for different *k* values at the same delay can be concurrently stable.

## **RIGHT SKEW FAVORS A GRADUAL TRANSITION FROM NEAR SYNCHRONY TO ANTIPHASE IN CONDUCTANCE-BASED MODELS WITH EXCITATORY COUPLING, SIMILAR TO LIF RESULTS**

A right skewed PRC (**Figure 6B1**) was obtained by increasing the potassium conductance. The antiphase mode again emerges

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symbols (unstable mode), or a pair of black symbols (stable mode). Only one symbol is visible for antiphase because the two time lags are equal, indicated by a filled symbol. For synchrony one time lag is 0. **(B1)** Right skewed type 1 Wang–Buzsaki PRC with <sup>g</sup><sup>K</sup> <sup>=</sup> 40 mS/cm2. The antiphase mode for zero delay falls on the unstable branch. The open circles that indicate the center between the two branches fall to the right of the dashed line, so there is an unequal time lag branch at short delays (black squares for k = 1 in **B2**). The blue bar shows a delay that falls on this branch. **(B2)** Predicted solution structure as delays are varied for two neurons coupled via the PRC in **(B1)**. k Values are given for the stable (black) branches.

for zero delay at the intersection of the line *y* = 2φ − 1 (gray line in **Figure 6B1**) with the PRC, but the right skew destabilizes the antiphase mode by causing it to fall on the destabilizing left branch, and the destabilization persists for short delays. The slope on the right branch at 1<sup>−</sup> is not in the causal limit region, and is insufficiently steep to stabilize synchrony with zero delay. The synchronous solution branch is qualitatively similar to that for left skew. However, the right skew enables the existence of the modes with unequal time lags by the same mechanism that it stabilizes antiphase; shifting the PRC with respect to the line *y* = 2φ − 1. In contrast to the open circles representing the average phase (φ<sup>L</sup> + φR)/2 of a pair with the same resetting, there are open circles in **Figure 6B1** that lie to the right of this line. The blue bar indicating the phase gap between the line and the

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open circle gives the magnitude of the normalized delay for that mode. The pairs of black squares in **Figure 6B2** at short delays show that the time lag can be quite short for small delays, so near synchrony can potentially be enabled by right skew in oscillators with type 1 PRCs. These modes are stable because the right skew tends to make the slope on the stabilizing right branch steeper than that on the left, favoring stability by keeping the scaling factor [1 − *f* (φL)][1 − *f* (φR)] below 1. A positive slope (≤1) decreases the magnitude of [1−*f* (φR)], which compensates for [1−*f* (φL)] being >1. The unequal modes lose existence at a delay equal to δ/*P*<sup>0</sup> = (φ<sup>L</sup> + φR)/2 − [1 + *f*(φR)]/2, where φ<sup>L</sup> = φR, exactly the same delay δ/*P*<sup>0</sup> = φAP − [1 + *f*(φAP)]/2 at which the antiphase mode gains stability because φAP also shifts to the stable right branch at the point on the PRC at which φ<sup>L</sup> = φ<sup>R</sup> in a bifurcation that is generic for pairs of oscillators coupled by excitation with right skewed type 1 PRCs. This is evidenced by the pairs of black squares coalescing to a region with only one filled circle visible in **Figure 6B2**.

A *k* = 2 branch of unequal time lag solutions emerges before antiphase loses stability. As in the case of left skew, the average phase (φ<sup>L</sup> + φR)/2 of a pair with the same resetting is equal to the normalized delay (see also schematic in **Figure 7B**) as indicated by the vertical dotted line emanating from the open circle in **Figure 6B1** and demarcating the end of the leader–follower *k* = 2 branch in **Figure 6B2**. Unlike the analogous mode for left skew, this mode is stabilized by the steeper slope of the PRC on the right branch compared to the left, again caused by the rightward skew, because the scaling factor 1 − *f* (φL) − *f* (φR) is <1.

## **LEFT SKEW FAVORS SYNCHRONY THAT IS ROBUST TO SUBSTANTIAL DELAYS IN PAIRS COUPLED WITH INHIBITION**

The effect of potassium conductance on the skew of PRCs measured in response to synaptic inhibition is opposite the effect for excitation. Therefore the potassium conductance was reduced to obtain a left skewed PRC (**Figure 8A1**) for a Wang–Buzsaki model neuron receiving an inhibitory synaptic input. For type 1 PRCs in response to inhibition, the slope of the left branch is stabilizing and the slope of the right branch is destabilizing, which is the opposite of the situation for excitation. Synchrony with zero delay is stable for this example with left skew because the stabilizing slope at 0<sup>+</sup> is steeper than the destabilizing slope at 1−. The robustness of the synchronous solution to delays is striking, as the synchronous solution (pairs of black circles with one time lag equal to 0 in **Figure 8A2**) persists for delay values nearly half the intrinsic period. The scaling factor for early synchrony with *k* = 2 is 1 − 2*f* (φ), where the phase corresponds to the normalized delay, so for small positive PRC slopes, the synchronous mode remains stable in the presence of conduction delays. Intuitively and in contrast to the case for excitation in **Figures 3C** and **5C**, the slope at 1<sup>−</sup> is not required for stability, and the loss of the effect of this slope when conduction delays are introduced does not affect the stability of synchrony. Stability of synchrony is lost only when the normalized delay value exceeds the phase that marks the beginning of the right branch of the PRC with a negative, destabilizing slope.

Since the antiphase mode for zero delay occurs at a phase determined by the intersection of the line 2φ − 1 = *f*(φ) with the PRC, left skew destabilizes the antiphase mode by extending the unstable right branch to earlier phases such that this intersection occurs on the unstable branch as in **Figure 8A1**. The antiphase mode is unstable for delays up to about half the intrinsic period (indicated by red filled circles in **Figure 8A2** for short delays for *k* = 1) because that is the length of the unstable branch. The same mechanism that destabilizes antiphase prevents the existence of modes with unequal time lags for short delays, and also for the most part destabilizes the *k* = 2 leader–follower branch of unequal time lags at delays near 0.5 in **Figure 8A2**. The open circles in **Figure 8A1** marking the average phase for pairs of phases with equal phase resetting on the left and right branches of the PRC fall to the left of the 2φ − 1 = *f*(φ), so they correspond to unrealizable negative delay values and the *k* = 1 unequal time lags mode (**Figure 7A**) does not exist. The location of the squares in **Figure 8A2** indicates

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**FIGURE 8 | Synchrony is robust to conduction delays for skewed type 1 PRCs in response to inhibition, although right skew favors antiphase for small conduction delays.** k Values are given for the stable (black) branches. **(A1)** Typical type 1 left skewed Wang–Buzsaki PRC with inhibitory coupling, <sup>g</sup>syn <sup>=</sup> 0.06 mS/cm<sup>2</sup> and <sup>I</sup>stim <sup>=</sup> <sup>1</sup> <sup>μ</sup>A/ms, <sup>g</sup><sup>K</sup> <sup>=</sup> 5 mS/cm2. The intersection of the dashed line y = 2φ − 1 with the PRC gives the phase for the unstable antiphase mode φAP with zero delay. Open circles are the average (φ<sup>L</sup> + φR)/2 for pairs of phases on the left and right branches with equal phase resetting, and since they fall to the left of the dashed line, there is no unequal time lag mode at short delays. **(A2)** Predicted solution structure as delays are varied for two neurons coupled via the PRC in **(A1)**. The two time lags between the firings of the two neurons are represented by a pair of red symbols (unstable mode) or a pair of black symbols (stable mode). Only one symbol is visible for

antiphase because the two time lags are equal, indicated by a filled symbol. Synchrony is stable for delays less than about half the intrinsic period, and antiphase is stable for delays greater than half the intrinsic period. **(B1)** Type 1 Wang–Buzsaki model right skewed PRC with gsyn = 0.06 mS/cm<sup>2</sup> and <sup>I</sup>stim <sup>=</sup> <sup>1</sup> <sup>μ</sup>A/ms, <sup>g</sup><sup>K</sup> <sup>=</sup> 9 mS/cm2. The line <sup>y</sup> <sup>=</sup> <sup>2</sup><sup>φ</sup> <sup>−</sup> 1 intersects the PRC on the stable left branch, so antiphase with zero delay is stable. The open circles that indicate the center between the two branches fall to the right of the dashed line, so there is an unequal time lag branch at short delays (red squares for k = 1 in **B2**), but it is unstable. The blue bar shows a delay that falls on this branch. **(B2)** Predicted solution structure as delays are varied for two neurons coupled via the PRC in **(B1)**. At the shortest delays, synchrony and antiphase are bistable. The basin of attraction for antiphase is large at zero delay but shrinks with increasing delay until antiphase loses stability.

the delay values for the unequal time lags near a delay of 0.5, but since the destabilizing slope on the right branch is in general less steep than that on the stabilizing left branch, these modes are mostly unstable because the scaling factor 1 − *f* (φL) − *f* (φR) is usually >1. For longer delays, at *k* = 3 the generic solutions recur.

## **RIGHT SKEW FAVORS ANTIPHASE AND BISTABILITY FOR SHORT DELAYS IN PAIRS COUPLED WITH INHIBITION**

**Figure 8B1** shows a right skewed PRC of the Wang–Buzsaki model neuron for the same parameter values as in **Figure 6A1** except for the reversal potential of the synaptic conductance, which is inhibitory for the PRC in **Figure 8B1**. The synchronous branch is

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qualitatively the same as for the left skewed PRC in **Figure 8A1**, and it is stable for conduction delays up to half the intrinsic period for the same reasons. However, the antiphase solution branch is qualitatively different. Since the antiphase mode for zero delay occurs at a phase determined by the intersection of the line 2φ − 1 = *f*(φ) with the PRC, right skew stabilizes the antiphase mode by extending the stable left branch to later phases such that this intersection occurs on the stable branch as in **Figure 8B1**. The antiphase mode is stable for delays up to about a 10th of the intrinsic period (indicated by black circles in **Figure 8B2** for time lags of about 0.5 at short delays for *k* = 1) because that is the length of stable branch at phases greater than φAP. In contrast to **Figure 6B** for inhibition, here the same mechanism that stabilizes antiphase also enables the existence of modes with unequal time lags for short delays. The open circles in **Figure 8B1** marking the average phase for pairs of phases with equal phase resetting on the left and right branches of the PRC fall to the right of the 2φ − 1 = *f*(φ), so they correspond to the delay values for the *k* = 1 unequal time lags mode indicated by the red squares in **Figure 8B2** that fall between stable synchrony and stable antiphase at short delays. The blue bar in **Figure 8B1** indicating the phase gap between the line and the open circle gives the magnitude of the delay for that mode. Because the destabilizing slope on the right branch is steeper than that on the stabilizing left branch, the scaling factor [1 − *f* (φL)][1 − *f* (φR)] is < −1, so these modes are unstable. Significantly, the structure of the unequal modes solution with small time lags at early delays transitioning into delays equal to half the network period causes the basin of attraction for synchrony to be quite small for very short delays such that most initial conditions lead to antiphase. However, this effect quickly dissipates with increasing delay and synchrony quickly becomes robust over a significant region of delays as in the case for left skew.

The location of the open circles in **Figure 8B1** indicates the delay values for the unequal time lags for *k* = 2 near a delay of 0.5, but because the destabilizing slope on the right branch is steeper than that on the stabilizing left branch, the scaling factor 1 − *f* (φL) − *f* (φR) exceeds 1, destabilizing these modes as indicated by the second set of red squares in **Figure 8B2**. For type 1 inhibition, right skew rather than left skew promotes bistability, because bistability depends upon lengthening the stable branch and the slopes and synchronization tendencies of the left and right branches of the PRC are inverted compared to excitation. For longer delays, at *k* = 3 stable antiphase recurs. The bottom line is that for type 1 PRCs in response to inhibition, left skew destabilizes and right skew stabilizes the antiphase mode, therefore left but not right skew favors synchrony at short conduction delays.

## **DISCUSSION**

## **SUMMARY**

The major result of this paper is to understand how the shape of the PRC determines the generic modes that are observed in pairs of neurons (or other oscillators) with no delays, and how conduction delays affect the tendency of pairs of neurons to synchronize. Specifically, a gradual transition from synchrony to antiphase with increasing conduction delay exists only if the center of the two

branches lies to the right of the invariant line whose intersection with the PRC determines the intrinsic phase at which each neuron receives an input in the antiphase mode with no delay. For type 1 PRCs and mutual excitation, right but not left skew enables near synchrony at short delays by shifting the center of the two branches to the right of this invariant line. In contrast, for type 1 PRCs and mutual inhibition, left but not right skew favors synchrony at short delays by destabilizing the competing antiphase mode by causing the intersection with the invariant line to occur on the unstable right branch. We show that exact synchrony with no delay for type 1 inhibitory but not excitatory PRCs is robust to conduction delays, because only the PRC for excitation relies on the stabilizing slope of the PRC at late phases to stabilize synchrony with no delay. A recent experimental study (Wang et al., 2012) confirmed the fragility of the synchronous mode for excitatory synaptic coupling in the presence of conduction delays and the robustness of this mode for inhibition. Generic solution structures are given herein for type 1 PRCs; however, the existence and stability criteria for all generic modes are general and apply to any shape PRC. Consistent with previous work, the effect of skew also manifests itself via differential effects on the slopes of the two PRC branches. Several stability features of the generic solutions for excitatory coupling depend critically on the increase in the steepness of the slope of the PRC at late phases mandated by causality.

## **EXTENSION TO OTHER PRC SHAPES**

For PRCs with more than two branches, any two branches could in principle give rise to the solutions with unequal time lags that provide a gradual transition between synchrony and antiphase. For example, a type 2 PRC in response to excitation typically has two lobes (Ermentrout, 1996): at early phases the first lobe consists of delays and the second lobe of advances. If the center of the second lobe lies to the right of the invariant line, then modes with unequal time lags and relatively short delays could be enabled and stabilized, so right shift of the extremum of the second lobe would favor such modes. Furthermore, the branch between the maximum advance and the maximum delay is unstable, so shifting the peaks so that the intersection with the invariant line does not fall on this branch removes bistability of antiphase with synchrony at zero delay, favoring synchrony. As the frequency is increased, the first lobe of the type 2 PRC shrinks (Fink et al., 2011). In principle the effect of any PRC shape can be understood by applying the methods described in this study. The only critical assumptions are that each neuron emits one spike for every spike emitted by the partner, that the PRC of each isolated neuron in response to an input from the partner is known, that the PRC still characterizes the response of the neuron to an input received within the coupled network, and that the effect of each input does not persist after the next spike in the same neuron that received the input.

## **GENERIC NATURE OF OUR RESULTS COMPARED WITH SPECIFIC MODEL APPROACHES**

The three major approaches (Ermentrout and Chow, 2002) to studying coupled oscillators are (1) to study specific model such as the LIF model or the Hindmarsh–Rose model, (2) to use a weak coupling assumption, or (3) to use a pulsatile coupling

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assumption. We chose to use the latter. Previously, Dhamala et al. (2004) showed that time delays can enhance neural synchrony by calculating the largest Lyapunov exponent for time delayed networks of diffusively coupled Hindmarsh–Rose model neurons. Clearly our methods also illustrate how delays can enhance neural synchrony. For example, in **Figure 6A**, synchrony is unstable for delays less than about a third of the intrinsic period, but is stable for delays from a third of an intrinsic period to an intrinsic period. For a 40-Hz gamma oscillation, regions separated by 8– 25 ms would synchronize optimally. Our approach does not rely on knowledge of the differential equations that describe particular neurons, only of the relevant PRC, therefore it is quite general.

Our work on the LIF oscillator was motivated by studies of pulsatile coupling (Ernst et al., 1995, 1998) that extended the results of Peskin (1975) and Mirollo and Strogatz (1990) to the case of two pulse-coupled oscillators reciprocally coupled with delays up to half the intrinsic period. Their assumptions implicitly defined a PRC and allowed the construction of return maps. The stable fixed points of these maps revealed that for small delays and strong excitatory coupling, at all coupling strengths synchronization with a phase lag equal to the delay was found to be always stable, analogous to our leader–follower mode. For inhibition, bistability between synchrony and antiphase was observed but as the coupling strength was increased only synchrony remained. Zeitler et al. (2009) studied the bifurcation structure of pairs of similar oscillators also coupled via conduction delays and noted that for these systems, only antiphase and synchrony could be stable for identical, identically coupled oscillators, but modes with unequal time lags could acquire stability in pairs coupled by excitation, similar to what we have found. The solution structure of simplified models such as the pulse-coupled LIF is not always representative of that obtained for conductance-based models. Real neurons (and conductance-based models) can exhibit much more nuanced PRCs, and the theoretical framework presented here includes and expands previous work on pulse-coupled oscillators with delay.

## **DIVERGENT PREDICTIONS OF PULSATILE COUPLING THEORY VERSUS WEAK COUPLING THEORY**

Weak coupling theory cannot be used to analyze pulsatile coupling of the type proposed by Peskin (1975) and Mirollo and Strogatz (1990), in which a finite and constant perturbation in voltage results from a presynaptic threshold event, with the caveat that an increase in the postsynaptic membrane potential beyond threshold has no additional effect. The PRC for an infinitesimal perturbation in membrane potential (equivalent to an infinitesimal perturbation in membrane current) has been derived for the LIF oscillator (Brown et al., 2004), and the PRC given in Eq. 1 for strong coupling cannot be derived from the infinitesimal PRC. Furthermore, the stability results for weakly coupled LIF oscillators and pulsecoupled oscillators are not in agreement. Weakly coupled type 1 oscillators do not synchronize with excitation (Hansel et al., 1995), but pulse-coupled oscillators synchronize both for the two oscillator circuit (Peskin, 1975) and all-to-all coupled circuits of *N* oscillators (Mirollo and Strogatz, 1990). Weak coupling does not account for the increase in the slope on the right branch imposed by causality as the conductance is increased, but instead assumes the PRC scales with increasing coupling the same way at all phases. This is an important limitation of weak coupling theory that has not been previously documented, and applies to synchrony in all circuits of oscillators that have PRCs with a destabilizing slope at a phase of 0 but a stabilizing slope at a phase of 1. Another disagreement between weak coupling theory and strongly pulse-coupled theory is that weak coupling (for example, Ladenbauer et al., 2012) assumes that in the presence of delays, synchrony always exists with both neurons receiving an input at zero phase, but clearly for oscillators coupled by strong excitation, at sufficiently long delays, the synchronizing input actually occurs at a late phase on or near the causal limit region of the PRC. Finally, weak coupling does not recognize how the stability criterion changes with the duration of the feedback loop.

## **FUNCTIONAL SIGNIFICANCE: PRC SKEW AND THUS SYNCHRONIZATION PROPERTIES CAN BE MODULATED**

Ermentrout et al. (2012) proposed that modulation of intrinsic ion channels could quickly reverse the synchronization tendencies of neurons by altering the PRC shape, providing a switch to turn synchrony on and off rapidly. There are several ways in which altering the conductances (Ermentrout et al., 2001, 2012; Pfeuty et al., 2003; Gutkin et al., 2005; Stiefel et al., 2009) can change the shape of a type 1 PRC for a regularly spiking neuron. Reducing restorative potassium currents or increasing regenerative sodium currents favors left skew if these currents are active at rest, whereas manipulations in the opposite direction favor right skew. This principle was used to manipulate the skew of the PRC in the Wang–Buzsaki model neuron used in this study. For the baseline potassium conductance value *g*<sup>K</sup> = 9 mS/cm2, the PRC for the Wang–Buzsaki model neuron in response to excitation was left skewed but the PRC in response to inhibition was right skewed. For excitation, *g*<sup>K</sup> was increased to change the PRC skew from left to right, and for inhibition, the *g*<sup>K</sup> was decreased to change the skew from right to left. Taken to the extreme, manipulations of currents active at rest that favor right skew can change the underlying bifurcation and PRC type from 1 to 2 (Ermentrout et al., 2001; Prescott et al., 2008; Stiefel et al., 2008), which often changes the stability by changing the sign of the slope at zero phase. On the other hand, manipulations of currents that are only activated by spikes cannot in general change the PRC type, but they can alter its shape (Ermentrout et al., 2001, 2012; Gutkin et al., 2005). Ermentrout et al. (2001) also showed that adding either recurrent inhibition or adaptation with a sharp, depolarized threshold such that it was only evoked by spikes, preserved the type 1 character of the PRC but shifted the skew to the right as expected for increases in outward current. However, an exception to this general pattern was found in which increasing an outward current that contributes to the afterhyperpolarization following a spike promoted left rather than right skew, because the primary effect of the change was to increase sodium channel availability (Ermentrout et al., 2012). Thus, there are many plausible modulatory targets available for changing the synchronization tendencies of biological networks.

## **PREVIOUS STUDIES EXAMINING SKEW IN THE CONTEXT OF WEAK COUPLING WITH NO DELAY**

Weak coupling (Ermentrout and Kopell, 1990, 1991; Ermentrout, 2002) identifies one to one phase-locked modes

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in identical coupled pairs by finding the zero crossings of *H*(φ) − *H*(1 − φ), in which the *H* function is equivalent to our phase resetting *f*(φ) except opposite in sign. Instead of neglecting changes in frequency caused by the coupling, our method finds the equivalent of the zero crossings of *H*(φ) − *H*[1 − φ − *H*(φ)], which contains an extra *H* function within the argument of another *H* function in order to update the elapsed time by the non-negligible resetting in the partner neuron. One consequence of neglecting the contribution of phase resetting to the network period is that for weak coupling, antiphase is always assumed to occur at an intrinsic phase of 0.5 instead of 2φAP − 1 = *f*(φAP). A critical role for PRC skew in networks of type 1 neurons connected by mutual synaptic excitation was demonstrated by Ermentrout et al. (2001), who showed that for certain model neuron pairs with slightly right skewed type 1 PRCs in response to excitation, both synchrony and antiphase were unstable, and near antiphase was the only stable solution. They used weak coupling to explain their results, and plotted *H*(1 − φ) − *H*(φ) to get the phase-locked modes from the zero crossings and the stability from the slope *H* (1 − φ) − *H* (φ), which must be >0 for stability. The stability criterion is slightly different than the ones we utilize because changes in the network period due to resetting are neglected as explained above. Nonetheless, the stability analysis is usually quite similar, for example, the stability of synchrony is determined entirely by whether the slope of the PRC is steeper before or after the spike, with the former case implying stability for the case of pairs of neuron coupled via type 1 PRCs in response to excitation. Ermentrout et al. (2001) then skewed type 1 PRCs farther to the right by flattening the slope at early phases while increasing the steepness at late phases. The increased skew caused the stable zero crossings to shift toward near synchrony, in which one neuron of the pair fires just before the other, as shown in our **Figure 6B**. Our extension of their work is that we explain directly in terms of the shape of the PRC how the stabilization occurs by bringing the unequal time lags solution branch into existence.

Ermentrout et al. (2012) also give an example of a different stabilization mechanism for a pair of Golomb and Amitai (1997) model neurons with reciprocal synaptic excitation and type 1 PRCs in which exact synchrony is stable for the baseline parameters given because of the right skew of the PRC. In this mechanism the right skew preferentially steepens the stabilizing slope at 1<sup>−</sup> compared to the nearly flat destabilizing PRC slope at 0+. Pfeuty et al. (2003) had complementary results showing that skewing the PRC toward the left stabilized the antiphase mode for two mutually electrically coupled neurons by causing the antiphase mode near a phase of 0.5 to fall in the region of stable slope. Similarly, Zahid and Skinner (2009) showed that for pairs of electrically coupled neurons, right skewfavors small phase lags because both synchrony and antiphase were unstable for the type 1 PRCs they observed, but sufficient left skew can stabilize antiphase and cause it to be globally attracting. In that study, skew was quantified by the fraction of the area under the PRC that fell to the left of a phase of 0.5, and weak coupling theory was invoked to show how destabilization of the antiphase mode by right skew led to the emergence of nearly synchronous modes with one small time lag. Electrical coupling is more analogous to excitation than inhibition in spiking neurons if the effect of the depolarizing effect of the suprathreshold spike dominates (Chow and Kopell, 2000), so these results are consistent with the framework presented in this paper. The advance in theory presented in this paper is that we do not make the weak coupling assumption, but instead show graphically that the destabilization of antiphase mode and the emergence of near synchrony depends on the location of the peak of the PRC relative to the location of the line that gives the phase of the antiphase mode at zero delay in terms of the intrinsic period.

## **PREVIOUS STUDIES EXAMINING SKEW IN THE CONTEXT OF WEAK COUPLING WITH CONDUCTION DELAY**

Remme et al. (2009) examined oscillatory dendritic compartments separated by passive cylindrical dendritic compartment of different electrotonic lengths, somewhat analogous to introducing a delay. Under weak coupling assumptions, they found that a left skewed PRC, or interaction function *H*(φ), yields bistability between synchrony and antiphase, whereas a right skewed interaction function yields gradual transitions between the two modes as the delay was increased. Again, results for electrical coupling parallel our results for synaptic excitation in **Figure 6**. Ladenbauer et al. (2012) also showed that increasing right skew in pairs of type 1 neurons coupled by synaptic excitation favored smaller phase lags decreasing to 0 at no delay, and favored the leader–follower mode by destabilizing the antiphase mode in the presence of conduction delays. For inhibition, increasing right skew stabilized the antiphase mode and promoted bistability with synchrony that persisted with short conduction delays. The weak coupling analysis of an adaptive exponential integrate and fire neuron (aEIF) in **Figure 6** of that paper is consistent with our **Figures 6** and **8**.

## **EFFECT OF DISCONTINUITIES**

The criteria for exact synchrony given in this paper are only strictly valid if there is no resetting in the cycle following the perturbation (Oprisan et al., 2004; Achuthan and Canavier, 2009), called second order resetting. Second order resetting is most prominent for inputs given just before a spike, so adding conduction delays for the most part precludes receipt of an input just before a spike and minimizes the importance of second order resetting. A complete treatment of stability with discontinuities must take into account that the first order phase resetting at a phase of 1 is 0 because an input applied after the cycle is over cannot affect that cycle. Effects of discontinuities are treated in Ladenbauer et al. (2012), Dodla and Wilson (2013), and Wang et al. (2012, supplementary material).

#### **IMPLICATIONS OF GENERIC MODES FOR LARGER NETWORKS**

Some of the results presented herein may also be extendable to networks of all to all connected neurons. For type 1 PRCs in response to excitation, the right branch of the PRC tends to stabilize synchrony, since if a neuron spikes later than the group, it receives an input at a late phase (1−, just to the left of 1) that advances it more than the group on the next cycle bringing it closer to synchrony. On the other hand, the left branch tends to destabilize, since a neuron that spikes before the group receives an input

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at an early phase (0+, or just to the right of 0) that advances it more than the group, taking it farther from synchrony. Simulations of pulse-coupled LIF neurons (Ernst et al., 1995; Coombes and Lord, 1997) have previously shown that the globally attracting synchronization of *N* pulse-coupled oscillators (Peskin, 1975; Mirollo and Strogatz, 1990) with type 1 PRCs comprised of all advances (excitation) is easily disrupted by conduction delays. Therefore the population activity is predicted by the activity of a single pair, in which synchrony is also disrupted by conduction delays.

Another possible extension is to clustering in larger networks. Bistability between synchrony and antiphase supports clustering (Gutkin et al., 2005; Jeong and Gutkin, 2007; Achuthan and Canavier, 2009). Chandrasekaran et al. (2011) have shown how two clusters that fire even slightly out of phase with each other can enforce synchrony within each cluster, even if exact synchrony within the isolated cluster in unstable, so this mechanism should generalize to enforce near synchrony in larger networks with one cluster firing slightly before the other. The leader–follower mode has been shown to stabilize clusters to some degree in networks with delay (Ernst et al., 1995). The strongly stabilizing effect of the causal limit region of excitatory PRCs is only adequately considered using the methods for strong coupling described herein.

The most important extension of these results is to synchronization between distal brain regions. Previously it was thought that long projections connecting brain regions were excitatory, but recently long distance inhibitory connections have also been identified (Melzer et al., 2012). For two mutually coupled populations in two different brain regions, the results from this study and our previous study (Wang et al., 2012) show that inhibitory projections may more reliably synchronize these populations in the presence of conduction delays between distal regions, and that some heterogeneity and noise can be tolerated. Alternatively, if the connections are excitatory and the PRCs type 1, then right skew in the PRC is likely required during episodes of near synchrony. If the unit oscillator is not a single neuron, but rather a network oscillator, the relevant PRCs for the network oscillation can be measured and analyzed for synchronization tendencies in a similar fashion to that for a single neuron (Akam et al., 2012).

A final possible extension relates to the dynamic relay hypothesis which suggests that synchronization among distal neurons can be achieved via symmetric coupling through a hub neuron. Viriyopase et al. (2012) studied the simplest such system with two outer neural oscillators each reciprocally connected to a third neuron, the relay neuron via identical reciprocal delays. They identified a "pacemaker" regime in which all three neurons fired simultaneously in the causal limit synchrony mode, that is, all neurons fired immediately upon receiving delayed input from the neuron or neurons to which it is connected. They also identified two other modes, "slave synchrony" in which the outer neurons were leaders and the relay neuron was a follower, and a "driven synchrony" mode in which the converse was true. Therefore the concepts developed herein for two neurons are directly extendable to *N* neurons each reciprocally connected to a hub (but not directly to each other).

## **ACKNOWLEDGMENT**

This work was funded by NIH grants 5R01MH085387-03 and 5R01NS054281-08 to Carmen C. Canavier under the CRCNS program.

## **REFERENCES**


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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 17 June 2013; accepted: 23 November 2013; published online: 11 December 2013.*

*Citation: Canavier CC, Wang S and Chandrasekaran L (2013) Effect of phase response curve skew on synchronization with and without conduction delays. Front. Neural Circuits 7:194. doi: 10.3389/fncir.2013.00194*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2013 Canavier, Wang and Chandrasekaran. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited andthatthe original publication inthis journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

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# Sudden synchrony leaps accompanied by frequency multiplications in neuronal activity

#### *Roni Vardi <sup>1</sup> \*†, Amir Goldental 2†, Shoshana Guberman1,2†, Alexander Kalmanovich1, Hagar Marmari 1† and Ido Kanter 1,2\**

*<sup>1</sup> Gonda Interdisciplinary Brain Research Center and the Goodman Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan, Israel*

*<sup>2</sup> Department of Physics, Bar-Ilan University, Ramat-Gan, Israel*

#### *Edited by:*

*Ehud Kaplan, The Fishberg Department of Neuroscience and Friedman Brain Institute, The Mount Sinai School of Medicine, USA*

#### *Reviewed by:*

*Ehud Kaplan, The Fishberg Department of Neuroscience and Friedman Brain Institute, The Mount Sinai School of Medicine, USA Thomas Kreuz, CNR, Italy*

#### *\*Correspondence:*

*Roni Vardi and Ido Kanter, Gonda Interdisciplinary Brain Research Center, Bar-Ilan University, Ramat-Gan 52900, Israel e-mail: ronivardi@gmail.com; ido.kanter@biu.ac.il †These authors have contributed equally to this work.*

A classical view of neural coding relies on temporal firing synchrony among functional groups of neurons, however, the underlying mechanism remains an enigma. Here we experimentally demonstrate a mechanism where time-lags among neuronal spiking leap from several tens of milliseconds to nearly zero-lag synchrony. It also allows sudden leaps out of synchrony, hence forming short epochs of synchrony. Our results are based on an experimental procedure where conditioned stimulations were enforced on circuits of neurons embedded within a large-scale network of cortical cells *in vitro* and are corroborated by simulations of neuronal populations. The underlying biological mechanisms are the unavoidable increase of the neuronal response latency to ongoing stimulations and temporal or spatial summation required to generate evoked spikes. These sudden leaps in and out of synchrony may be accompanied by multiplications of the neuronal firing frequency, hence offering reliable information-bearing indicators which may bridge between the two principal neuronal coding paradigms.

**Keywords: network, topology, firing synchrony,** *in vitro* **modular networks, neuronal circuit**

## **INTRODUCTION**

One of the major challenges of modern neuroscience is to elucidate the brain mechanisms that underlie firing synchrony among neurons. Such spike correlations with differing degrees of temporal precision have been observed in various sensory cortical areas, in particular in the visual (Eckhorn et al., 1988; Gray et al., 1989), auditory (Ahissar et al., 1992; Nicolelis et al., 1995), somatosensory (Nicolelis et al., 1995), and frontal (Vaadia et al., 1995) areas. Several mechanisms have been suggested, including the slow and limited increase in neuronal response latency per evoked spike (Vardi et al., 2013b). On a neuronal circuit level its accumulative effect serves as a non-uniform gradual stretching of the effective neuronal circuit delay loops. Consequently, small mismatches of only a few milliseconds among firing times of neurons can vanish in a very slow gradual process consisting of hundreds of evoked spikes per neuron.

The phenomenon of sudden leaps from firing mismatches of several tens of milliseconds to nearly zero-lag synchronization, below a millisecond, is counterintuitive. Since the dynamical variations in neuronal features, e.g., the increase in neuronal response latencies per evoked spike, are extremely small, one might expect only very slow variations in firing timings. Moreover, relative changes among firing times of neurons require dynamic relaxation of the entire neuronal circuit to achieve synchronization. Hence, sudden leaps, in and out of synchrony, seem unexpected.

In the present study, we propose a new experimentally corroborated mechanism allowing leaps in and out of synchrony. The procedure is based on conditioned stimulations enforced on neuronal circuits embedded within a large-scale network of cortical cells *in vitro* (Marom and Shahaf, 2002; Morin et al., 2005; Wagenaar et al., 2006; Vardi et al., 2012). These stimulations varied in strength, so that the evoked spikes of selected neurons required temporal summation. We demonstrate that the underlying biological mechanism to sudden leaps in and out of synchrony is the unavoidable increase of the neuronal response latency (Aston-Jones et al., 1980; De Col et al., 2008; Ballo and Bucher, 2009; Gal et al., 2010) to ongoing stimulations, which imposes a non-uniform stretching of the neuronal circuit delay loops.

## **MATERIALS AND METHODS**

## **CULTURE PREPARATION**

Cortical neurons were obtained from newborn rats (Sprague– Dawley) within 48 h after birth using mechanical and enzymatic procedures (Marom and Shahaf, 2002; Vardi et al., 2012, 2013b). All procedures were in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals and Bar-Ilan University Guidelines for the Use and Care of Laboratory Animals in Research and were approved and supervised by the Institutional Animal Care and Use Committee.

The cortex tissue was digested enzymatically with 0.05% trypsin solution in phosphate-buffered saline (Dulbecco's PBS) free of calcium and magnesium, supplemented with 20 mM glucose, at 37◦C. Enzyme treatment was terminated using heat-inactivated horse serum, and cells were then mechanically dissociated. The neurons were plated directly onto substrateintegrated multi-electrode arrays (MEAs) and allowed to develop functionally and structurally mature networks over a time period of 2–3 weeks *in vitro*, prior to the experiments. Variability in the number of cultured days in this range had no effect on the observed results. The number of plated neurons in a typical network is in the order of 1,300,000, covering an area of about 380 mm2. The preparations were bathed in minimal essential medium (MEM-Earle, Earle's Salt Base without L-Glutamine) supplemented with heat-inactivated horse serum (5%), glutamine (0.5 mM), glucose (20 mM), and gentamicin (10 g/ml), and maintained in an atmosphere of 37◦C, 5% CO2 and 95% air in an incubator as well as during the electrophysiological measurements. All experiments were conducted on cultured cortical neurons that were functionally isolated from their network by a pharmacological block of glutamatergic and GABAergic synapses. For each plate, 12–20μl of a cocktail of synaptic blockers was used, consisting of 10μM CNQX (6-cyano-7-nitroquinoxaline-2,3-dione), 80μM APV (amino-5-phosphonovaleric acid), and 5μM Bicuculline. This cocktail did not block the spontaneous network activity completely, but rather made it sparse. At least 1 h was allowed for stabilization of the effect.

## **MEASUREMENTS AND STIMULATION**

An array of 60 Ti/Au/TiN extracellular electrodes, 30μm in diameter and spaced either 200 or 500μm from each other (Multi-Channel Systems, Reutlingen, Germany) was used. The insulation layer (silicon nitride) was pre-treated with polyethyleneimine (Sigma, 0.01% in 0.1 M Borate buffer solution). A commercial setup (MEA2100-2x60-headstage, MEA2100-interface board, MCS, Reutlingen, Germany) for recording and analyzing data from two 60-electrode MEAs was used, with integrated data acquisition from 120 MEA electrodes and 8 additional analog channels, integrated filter amplifier and 6-channel current or voltage stimulus generator (for both MEAs). Mono-phasic square voltage pulses (−900 to −100 mV, 100–500μs) were applied through extracellular electrodes. Each channel was sampled at a frequency of 50 k sample/s. Action potentials were detected online by threshold crossing. For each of the recording channels a threshold for spike detection was defined separately, prior to the beginning of the experiment.

## **CELL SELECTION**

Each circuit node was represented by a stimulation source (source electrode) and a target for the stimulation—the recording electrode (target electrode). These electrodes (source and target) were selected as the ones that evoked well-isolated, well-formed spikes and reliable responses with high signal-to-noise ratio. This examination was done with stimulus intensity of −800 mV using 30 repetitions at a rate of 5 Hz followed by 1200 repetitions at a rate of 10 Hz.

## **STIMULATION CONTROL**

A node response was defined as a spike occurring within a typical time window of 2–10 ms following the electrical stimulation. The activity of all source and target electrodes was collected, and entailed stimuli were delivered in accordance to the circuit connectivity.

## *Circuit connectivity,* **τ**

Conditioned stimulations were enforced on the circuit neurons embedded within a large-scale network of cortical cells *in vitro*, according to the circuit connectivity. Initially, each delay was defined as the expected time between the evoked spikes of two linked neurons; e.g., conditioned to a spike recorded in the target electrode assigned to neuron A, a spike will be detected in the target electrode of neuron B after τ*AB* ms. For this end, conditioned to a spike recorded in the target electrode of neuron A, a stimulus will be applied after τ*AB*-*LB*(0) ms to the source electrode of neuron B, where *LB*(0) is the initial latency of neuron B.

In cases where missed evoked spikes caused a termination of the neuronal circuit activity, stimulation was given to neuron A after a period of 100 ms, to restart the circuit's activity.

All neurons were stimulated at a rate of 10 Hz (**Figures 1**, **3**) or 8 Hz (**Figure 2**), before the leap to synchronization.

Strong stimulations, (−800 mV, 200μs), resulting in a reliable neural response, were given to all circuit neurons excluding neuron C (**Figures 1**, **2**) and E (**Figure 3**). Weak stimulations (**Figure 1**: −450 mV, 40μs. **Figure 2**: −600 mV, 60μs. **Figure 3**: −700 mV, 60μs) were given to neuron C (**Figures 1**, **2**) or E (**Figure 3**), so that an evoked spike is expected only if the time-lag between two consecutive weak stimulations is short enough. In cases where the time-lag between two consecutive stimulations was shorter than 20μs (from the end of the first stimulation to the beginning of the consecutive one), a unified strong stimulation was applied, to overcome technical limitations. The weak stimulations were defined for each neuron separately, due to differences in their threshold.

*TTS* (TS stands for temporal summation) is the maximal timelag between two weak stimulations which typically results in an evoked spike. This quantity was empirically estimated by gradually changing the time-lag between two weak stimulations, and found to differ between neurons.

## **DATA ANALYSIS**

Analyses were performed in a Matlab environment (MathWorks, Natwick, MA, USA). Action potentials were detected by threshold crossing. In the context of this study, no significant difference was observed in the results under threshold crossing or voltage minima for spike detection. Reported results were confirmed based on at least ten experiments each, using different sets of neurons and several tissue cultures.

## **RESULTS**

## **LEAP TO SYNCHRONY ACCOMPANIED BY A DOUBLED FIRING FREQUENCY**

#### *Experimental results*

We first demonstrate leaps to synchrony using a neuronal circuit consisting of four neurons and conditioned stimulations split into weak/strong stimulations (**Figure 1A**). A strong stimulation consists of a relatively high amplitude and/or relatively long pulse duration such that an evoked spike is generated reliably, whereas a weak stimulation consists of a lower amplitude and/or pulse duration, such that an evoked spike is expected only if the time-lag between two consecutive weak stimulations is short enough. All delays (denoted on connecting lines between neurons in **Figure 1A**) were selected to initially include the response latency of the target neuron, e.g., the time-lag from neuron A to B, τ*AB*, was initially set to τ-*LB*(0) where *LB*(0) stands for

Experimental measurements of -(Stim*C*) as a function of the spikes of neuron A. -(Stim*C*) is initially set to ε ≈ 0.8 ms (green line) with τ = 50 ms and *TTS* ≈ 0.24 ms (presented by the dashed horizontal green line). A unified longer stimulation was given in events where the time-lag between the weak stimulations <20μs [presented by -(Stim*C*) = 0]. Sync*AB* is presented by the blue line, indicating a sudden leap from τ = 50 ms to nearly zero-lag synchronization. **(C)** Spike trains of the four neurons. A sudden leap to Sync*AB* ≈ 0 occurs at time/2τ = 122.5 (at spike 121 of neuron A) immediately following a single evoked spike of neuron C. It is accompanied by a doubled

**(A)** is now represented by a population comprised of 40 Hodgkin-Huxley neurons, each one innervated by four randomly chosen neurons from each of its driving clusters. The delays between neurons are taken from a Gaussian distribution centered at the delays of the single neuron case with a variance of 0.2 ms. For simplicity, each time a neuron fires all of its outgoing delays are increased by 0.04 ms. The simulation parameters were ε = 2 ms and *TTS* ≈ 1.3 ms. **(G)** Raster plot of the 120 neurons comprising nodes A, B, and C. A leap to synchrony occurs at time/2τ ≈ 20, accompanied by a doubling of the firing frequency.

the initial response latency of neuron B. For τ = 50 ms, neurons A and B initially fire alternately, in and out of phase, at a frequency of ∼10 Hz (**Figure 1B**). Neuron D fires ∼τ/2 ms laggard to neuron A (**Figure 1C**) and the time-gap between two weak stimulations arriving at neuron C, -(Stim*C*), is initially ε (**Figures 1A,B**). The experimentally estimated maximal time-gap between stimulations of neuron C which generates an evoked spike (temporal summation) is denoted by *TTS*, thus for

**FIGURE 2 | A sudden leap to synchrony accompanied by tripled frequency.** Notations used: Sync*AB*, the absolute time-lag between the spikes of neurons A and B; -(Stim*C*), the absolute time difference between two weak stimulations to neuron C; -*LD*, the increase in response latency of neuron D after n evoked spikes. **(A)** Schematic of a neuronal circuit as in **Figure 1A**, however, the delay from neuron B to A is now 2τ. **(B)** Experimental measurements of -(Stim*C*), similar to **Figure 1B**, with ε ≈ 0.5 ms, 3τ = 125 ms and *TTS* ≈ 0.2 (presented by the dashed horizontal green line). Sync*AB*, (blue line) indicating a sudden leap from τ ≈ 125/3 ms to nearly zero-lag synchronization. **(C)** Spike trains of the four neurons. A sudden leap to synchronization, Sync*AB* ≈ 0, occurs at time/3τ = 44 (at spike 44 of neuron A) consecutive to three evoked spikes of neuron C. This is accompanied by tripled firing frequency of neurons A and B, from ∼8 to ∼24 Hz. Sync*AB* ≈ 0 is robust to response

failures of neuron C, e.g., time/3τ = 46.33. **(D)** Sync*AB* as a function of the spikes of neuron A for various ε, where the number of spikes to the leap to synchrony increases with ε. The data for ε = 0.5 (blue) is the same as in **(B,C)**. The observed oscillations in Sync*AB* before a leap to synchrony originate from response failures of neuron C, and similarly oscillations in a leap out of synchrony originate from response failure of either neuron A or B. **(E)** -*LD*, for repeated stimulations at 8 Hz. -*LD* at the leap for different ε are indicated and colored following **(D)**, approximately verifying Equation (1), e.g., for ε = 0.8 ms and *TTS* ≈ 0.2 ms, -*LD*(197) gives ∼0.6 ms. Note that Spike*<sup>D</sup>* is equal to Spike*<sup>A</sup>* in **(B,D)**. **(F)** Results of population dynamic simulations similar to **Figures 1F,G** with ε = 2 ms, *TTS* ≈ 1.3 ms and 3τ = 125 ms. **(G)** Raster plot of the 120 neurons comprising nodes A, B, and C. A leap to synchrony occurs at time/3τ ≈ 20, accompanied by tripled firing frequency.


**in frequency.** Notations used: Sync*AE* , the absolute time-lag between the spikes of neurons A and E; -(Stim*<sup>E</sup>* ), the absolute time difference between two weak stimulations to neuron E; -*L*, defined as -*LB* + -*LC* + -*LD* − -*LF* . **(A)** Schematic of a neuronal circuit consisting of six neurons and weak/strong stimulations represented by dashed (green)/full (black) lines. **(B)** Experimental measurements of -(Stim*<sup>E</sup>* ), similar to -(Stim*C*) in **Figure 1B**, with ε ≈ 1.7 ms, τ = 50 ms and *TTS* ≈ 0.5 ms (presented by the dashed horizontal green line). The time delay between neurons A and E, ∼2τ, is denoted by the dashed horizontal black line. The firing region of neuron E (blue dots bounded by dashed vertical guidelines), which is at nearly zero-lag synchronization with the firing of neuron A, Sync*AE* ≈ 0, starts after 77 spikes of neuron A. The

(**Figure 1C**). Note that the sudden multiplication in frequency, by itself, shortens Sync*AB* from 100 to 50 ms, however, it cannot lead to zero-lag synchrony. The sudden emergence of Sync*AB* ≈ 0 ms requires only a single firing of neuron C, and is then maintained by the mutual firing of neurons A and B, independently of the firing of neuron C (**Figure 1C**). For a given *TTS*, the number of evoked spikes of neuron D until the leap to synchrony, *n*, increases with ε (**Figure 1D**). Quantitatively, using the experimental response latency profile of neuron D, *LD*, one can find n fulfilling the equality:

$$
\Delta L\_D(\mathfrak{n}) \approx \mathfrak{e} - T\_{TS} \tag{1}
$$

where -*LD*(*n*) stands for the increase in response latency of neuron D after n evoked spikes (**Figure 1E**). Note that neuron D is laggard to neuron A, thus the number of evoked spikes of neuron A until the leap to synchrony increases with ε as well, in accordance with Equation (1) (**Figure 1D**). Since *TTS* varies between neurons and even within the same neuron over different trials, temporary firing of E terminates after ∼200 spikes of neuron A. **(C)** Spike trains of neurons A, F, and E, indicating a steady firing frequency (∼10 Hz) of the neuronal circuit independent of the firing of neuron E, where an epoch of synchrony, Sync*AE* ≈ 0, begins at time/2τ = 77 (at spike 77 of neuron A). **(D)** The number of spikes prior to the firing of neuron E increases with ε. The mild increase in the firing mismatch, Sync*AE* , is attributed to the additional increase by ε of the initial 2τ delay loop (E fires ∼2τ + ε laggard to A, however, the time-gap between consecutive firings of A is ∼2τ + 2ε). The data for ε = 1.7 (blue) is the same as in **(B,C)**. **(E)** -*L* for repeated stimulations at 10 Hz. -*L* at the synchrony leap for different ε are colored following **(D)**. The number of spikes per neuron (e.g., Spike*A*), *n*, until the leap to synchrony increases with ε and can be obtained from Equation (1), where -*LD* is substituted by -*L*.

deviations from this equation are expected (e.g., -*LD* for ε = 0.8 ms and ε = 1 ms are almost the same, **Figures 1D,E**). A slow gradual increase in Sync*AB* after a leap to synchrony (**Figure 1D**) is theoretically attributed to the difference in the increase of neuronal response latencies |-*LA*(*n*)--*LB*(*n*)| and the leap out of synchrony (**Figure 1D**) is a consequence of a response failure of neurons A and/or B (see Section "Slow Divergence out of Synchrony" in Appendix). Similar results were obtained and exemplified for spatial summation (not shown), where weak stimulations were given to a neuron *through two different source electrodes*. An evoked spike is expected only if the time-lag between two consecutive weak stimulations, controlled by the relative stimulation timings of the source electrodes, is short enough.

## *Simulations of population dynamics*

The sudden leap to synchrony was experimentally verified under the limitation where each circuit node is represented by a single neuron, and is demonstrated to be robust under simulations of population dynamics (**Figures 1F,G**). Each one of the four nodes (**Figure 1A**) now represents a population comprised of 40 Hodgkin-Huxley sparsely connected neurons (for simulation details, see Vardi et al., 2013a). For the parameters used, *TTS* ≈ 1.3 ms, ε = 2 ms and 0.2 ms variance for the Gaussian distribution of the delays, a leap to synchrony is expected following Equation (1) after ∼20 spikes of cluster A (**Figure 1F**). The simulated Sync*AB* is defined as the absolute difference between the average spiking times of the neurons comprising clusters A and B, where at least 50% of the neurons in a cluster fired (**Figure 1G**). Initially, several neurons in cluster C fire as a result of relatively close stimulations from either cluster A or D. This sporadic firing is a consequence of the Gaussian distribution of the delays between populations, however, their impact on the firing activity of cluster B is negligible. As neurons of cluster D fire repeatedly, -(Stim*C*) decreases and more neurons from cluster C fire. Consequently, the activity of cluster C is enhanced such that a leap to synchrony is observed, accompanied by frequency doubling from ∼10 to ∼20 Hz (**Figures 1F,G**). A leap out of synchrony was not observed in the simulations, since population dynamics are more robust to a single neuron's response failure in comparison to a neuronal circuit where each node is represented by a single neuron (**Figures 1A,D**). Low connectivity, as well as a wider Gaussian distribution of delays between populations are expected to enhance fluctuations and response failures, and will eventually lead to a leap out of synchrony.

Population dynamics exhibit consistency with most of the experimental results, hence minimizing the possibility of these results as being only an artifact of the tissue culture. Nevertheless, the verification of our results in more realistic scenarios is required, including shorter delays and their interplay with the neuronal refractory period, the morphology of the neurons instead of considering neurons as points (Doiron et al., 2006), as well as possible adaptation mechanisms in the form of short and long term synaptic plasticity (Abbott and Regehr, 2004; Izhikevich, 2006). In addition, more accurate and systematic statistical measures of synchrony (Kreuz et al., 2009; Shimokawa and Shinomoto, 2009) can be adopted to describe the transition to synchrony in the case of population dynamics.

### **LEAP TO SYNCHRONY ACCOMPANIED BY TRIPLED FIRING FREQUENCY**

More general features of a sudden leap to synchrony are exemplified by increasing the delay from neuron B to A, τ*BA*, from τ (**Figure 1A**) to 2τ (**Figure 2A**). The circuit now consists of two delay loops, ∼3τ (A-B-A) and ∼4τ (A-C-B-A) (**Figure 2A**). Since GCD(4,3) = 1, zero-lag synchronization is theoretically expected, conditioned to the firing of neuron C. Initially, neurons A and B fire at a frequency of ∼8 Hz (3τ = 125 ms) (**Figure 2C**) and Sync*AB* ≈ τ (**Figure 2B**). Neuron C starts to fire as -(Stim*C*) ≤ *TTS* ≈ 0.2 ms, resulting in Sync*AB* ≈ 0 which is accompanied by tripled firing frequency (**Figure 2C**). The number of evoked spikes by neuron D (or its leader neuron A) to the leap increases with ε in a non-linear manner following -*LD*(*n*), in accordance with Equation (1) (**Figures 2D,E**).

Typically, several leaps in and out of synchrony between neurons A and B occur before arriving at a stable nearly zero-lag synchronization (**Figure 2D**). These oscillations are attributed to unreliable responses of neuron C, and increase the duration of the relaxation to synchrony (**Figure 2D**). Similar oscillations on the way out of synchrony (**Figure 2D**) are attributed to the first response failure of either neuron A or B. Consequently, neurons A and B fire alternately in time-lags τ and 2τ. The final exit out of synchrony occurs in the second response failure of neurons A or B.

Simulation results (**Figures 2F,G**) confirmed the robustness of the experimentally observed leap to synchrony in population dynamics. The oscillations in the relaxation to synchrony are attributed to response failures of cluster C. These failures are a consequence of fluctuations in the firing timings of clusters A and D and the Gaussian distribution of their delays to cluster C.

## **EPOCHS OF SYNCHRONY NOT ACCOMPANIED BY A CHANGE IN FREQUENCY**

A mechanism to leap out of synchrony as well as the interrelation between the sudden leap to synchrony and the firing frequency are at the center of the next examined neuronal circuit (**Figure 3A**). This circuit consists solely of a 2τ-delay loop, hence neurons A and F fire alternately in ∼τ ms time-lags. Nevertheless, neuron A affects neuron E by weak stimulations arriving from two comparable initial delay routes; ∼2τ ms (A-F-E) and ∼2τε ms (A-B-C-D-E) (**Figure 3A**). Initially, neuron E does not fire since ε ≈ 1.7 ms > *TTS* ≈ 0.5 ms. Since the overall increase in the neuronal response latency of a chain is accumulative, proportional to the number of neurons it comprises, -(Stim*E*) gradually decreases below *TTS* (**Figure 3B**) and neuron E suddenly starts to fire. Consequently, since neuron A fires every ∼2τ ms and neuron E fires ∼2τ ms laggard to A, Sync*AE* ≈ 0 (**Figures 3B,C**). As -(Stim*E*) decreases, the response of neuron E becomes more reliable (**Figures 3B,C**) and a leap out of synchrony is observed when -(Stim*E*) again exceeds ∼*TTS* (**Figure 3B**). Since neuron E's firing does not close a new neuronal loop, the leaps in and out of synchrony do not affect the firing frequency of the neuronal circuit (**Figure 3C**). The number of spikes to synchrony increases with ε as well as the time-gap between neurons during synchronization, Sync*AE* (**Figures 3D,E**). Simulation results (not shown) confirmed the robustness of the experimentally observed leap in and out of synchrony without a frequency change in population dynamics.

## **DISCUSSION**

Understanding the brain mechanisms that underlie firing synchrony is one of the great challenges of neuroscience. There are many variants of population codes, where a set of neurons in a population acts together to perform a specific computational task (Palm, 1990; Eichenbaum, 1993; Ainsworth et al., 2012). There is much discussion over whether *rate* coding or *temporal* coding is used to represent perceptual entities in populations of neurons in the cortex. A number of reports suggest that almost all the information in a stimulus is embedded in the rate code of active neurons (Aggelopoulos et al., 2005), while others suggest that synchrony among spiking of neuronal populations carry the information (deCharms and Merzenich, 1996). Experimental support for changes solely in firing rate when the perceptual task is modified (e.g., Lamme and Spekreijse, 1998; Roelfsema et al., 2004) is as compelling as those works that show changes in synchrony in the absence of firing rate changes (e.g., Womelsdorf et al., 2005), whereas in other experiments changes in both rate and spike correlations are observed concurrently (e.g., Biederlack et al., 2006). In any case, the usefulness of rate coding and temporal coding as information carriers of brain activity is a function of the decoding complexity, which is tightly correlated with their accuracy.

Rate and temporal coding are typically inaccurate in brain activities, although there are several well-known exceptions where neurons fire with high temporal accuracy (Bullock, 1970; Bullock et al., 1972; Moortgat et al., 2000). Rate precision, measured by inter-spike interval (ISI) distributions, typically follows a broad distribution, deviating from a Poissonian one (Amarasingham et al., 2006). In addition, relative spike timings between coactive neurons are usually within the precision of several milliseconds (Kayser et al., 2010; Wang, 2010). In the case of a broad distribution of ISIs, the mission to grasp gradual changes in temporal and/or rate coding (e.g., changes from an average firing rate of 5–6 Hz), on a timescale of a few ISIs, is a heavy computational mission which might not be satisfactorily resolved. The underlying cause of this computational difficulty is the broad distribution of the ISIs which is overlapped between gradually changed temporal codes or gradually changed rate codes.

To overcome this difficulty we proposed a mechanism which enables the emergence of a sudden leap to synchrony together with or independent of a leap in the firing frequency. This mechanism results in leaps from firing mismatches of several dozens of milliseconds to nearly zero-lag synchronization, and can be accompanied by a sudden frequency multiplication of the neuronal firing rate. These sudden changes occur on a time scale of extremely few ISIs, and are easily detectable as the distributions of the ISIs before and after the leaps are non-overlapping. Hence, one ISI is sufficient to detect the transition without accumulatively estimating the ISI distribution. These fast and robust indicators might be used as reliable information carriers of time-dependent brain activity.

The proposed mechanism also allows for the simultaneous emergence of sudden leaps in rate and temporal synchrony, hence bridging between these two major schools of thought in neuroscience (Eckhorn et al., 1988; Gray et al., 1989; Ahissar et al., 1992; Nicolelis et al., 1995). This mechanism requires *recurrent* neuronal circuits, and synchrony appears even among neurons which do not share a common drive. Sub-threshold stimulations (e.g., the stimulations to neuron C in **Figures 1**, **2** and to neuron E in **Figure 3**) serve as a switch that momentarily closes or opens a loop in the neuronal circuit. The state of the switch changes a global quantity of the network, the GCD of the entire circuit's loops, which determines the state of synchrony (e.g., zero-lag synchrony, cluster synchrony, shifted zerolag synchrony) (Kanter et al., 2011; Nixon et al., 2012). These demonstrated prototypical examples call for a theoretical examination of more structured scenarios, including multiple leaps in and out of synchrony. In addition, a more realistic biological environment has to be examined containing synaptic noise and adaptation.

## **ACKNOWLEDGMENTS**

We would like to thank Moshe Abeles and Evi Kopelowitz for stimulating discussions. Fruitful computational assistance by Yair Sahar and technical assistance by Hana Arnon are acknowledged. This research was supported by the Ministry of Science and Technology, Israel.

## **REFERENCES**


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 15 August 2013; accepted: 13 October 2013; published online: 30 October 2013.*

*Citation: Vardi R, Goldental A, Guberman S, Kalmanovich A, Marmari H and Kanter I (2013) Sudden synchrony leaps accompanied by frequency multiplications in neuronal activity. Front. Neural Circuits 7:176. doi: 10.3389/fncir.2013.00176 This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2013 Vardi, Goldental, Guberman, Kalmanovich, Marmari and Kanter. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

## **APPENDIX**

## **SLOW DIVERGENCE OUT OF SYNCHRONY**

The slow increase in Sync*AB* (**Figure 1D**) is analytically examined below for a case of two phase-to-phase neurons, A and B, as depicted in **Figure A1**. The derivation below is in the spirit of Ermentrout's analysis of coupled type I membranes (Ermentrout, 1996). We first define the following quantities and assumptions:

*ti*(*q*) ≡ the timing of the *q*th spike of neuron i, e.g., *tA*(0) is the timing of the first spike of neuron A, where the count starts at 0.

*Li*(*q*) ≡ neuronal latency of neuron i at its *q*th spike.

The initial time delays are τ*AB* = τ*BA* ≡ τ.

Assuming initial conditions, *t* = 0, where both neurons fire simultaneously, i.e., *tA*(0) ≡ 0, *tB*(0) = 0.

The spiking times of neurons A and B are given by

$$\begin{cases} \text{(i)}\ t\_B(q) = t\_A(q-1) + \mathfrak{r} + L\_B(q) \\ \text{(ii)}\ t\_A(q) = t\_B(q-1) + \mathfrak{r} + L\_A(q) \end{cases}$$

**FIGURE A1 | Slow divergence out of synchrony between two phase-to-phase neurons.** Notation used: Sync*AB*, the time-lag between the spikes of neurons A and B. **(A)** Schematic of two bidirectional interconnected spiking neurons. The initial delays between the neurons are equal, τ*AB* = τ*BA* = τ. **(B)** Response latency of both neurons as a function of spike number. The latencies were taken to be *LA* = 0.5 × ln(*q* + 5) + 2, *LB* = 0.3 × sqrt(*q* + 2) + 3, qualitatively similar to latency profiles observed in experiments. **(C)** Sync*AB* as a function of spike number for the latencies depicted in **(B)**, assuming Sync*AB*(0) = 0. The calculation (brown line) was done using Equation (A1) and is in a good agreement with straightforward simulations of exact spike times (black dots). For simplicity, the simulated Sync*AB* is only displayed for even numbers of spikes.

Substituting (ii) into (i) and vice versa:

$$\begin{cases} t\_B(q) = t\_B(q-2) + \mathfrak{r} + L\_A(q-1) + \mathfrak{r} + L\_B(q) \\ t\_A(q) = t\_A(q-2) + \mathfrak{r} + L\_B(q-1) + \mathfrak{r} + L\_A(q) \end{cases}$$

one can find that the solution of these coupled recursive equations is given by:

$$\begin{cases} t\_B(q) = \sum\_{q'=1}^{\frac{q}{2}} L\_B\left(2q'\right) + L\_A\left(2q'-1\right) + 2\pi \\\ t\_A(q) = \sum\_{q'=1}^{\frac{q}{2}} L\_A\left(2q'\right) + L\_B\left(2q'-1\right) + 2\pi \end{cases}$$

Consequently, the firing time-gap between the two neurons is given by

$$\begin{aligned} \text{Sync}\_{AB}(q) & \equiv \left| t\_B(q) - t\_A(q) \right| \\ \text{Sync}\_{AB}(q) &= \left| \sum\_{q'=1}^{\frac{q}{2}} L\_B \left( 2q' \right) + L\_A \left( 2q' - 1 \right) \right| \\ & - L\_A \left( 2q' \right) - L\_B \left( 2q' - 1 \right) \Bigg| \\ \text{Sync}\_{AB}(q) &= \left| \sum\_{q'=1}^{\frac{q}{2}} \left( L\_A \left( 2q' - 1 \right) - L\_A \left( 2q' \right) \right) \right| \\ & - \sum\_{q'=1}^{\frac{q}{2}} \left( L\_B \left( 2q' - 1 \right) - L\_B \left( 2q' \right) \right) \Bigg| \end{aligned}$$

Under the assumption of continuous increase in latency and large *q*

$$\frac{\mathrm{dL}\_{i}\left(2q'\right)}{\mathrm{d}\left(2\mathrm{q'}\right)} \approx -\left(L\_{i}\left(2q'-1\right) - L\_{i}\left(2q'\right)\right) > 0$$

$$\mathrm{Sync}\_{AB}(q) = \left| \int\_{0}^{\frac{4}{2}} \frac{\mathrm{dL}\_{A}\left(2q'\right)}{\mathrm{d}\left(2\mathrm{q'}\right)} \frac{\mathrm{dq'}}{\mathrm{d}\left(2\mathrm{q'}\right)} \mathrm{d}\left(2\mathrm{q'}\right)$$

$$-\int\_{0}^{\frac{4}{2}} \frac{\mathrm{dL}\_{B}\left(2q'\right)}{\mathrm{d}\left(2\mathrm{q'}\right)} \frac{\mathrm{dq'}}{\mathrm{d}\left(2\mathrm{q'}\right)} \mathrm{d}\left(2\mathrm{q'}\right) \right|$$

$$\mathrm{Sync}\_{AB}(q) = \left| 0.5\left(L\_{A}(q) - L\_{A}(0)\right) - 0.5\left(L\_{B}(q) - L\_{B}(0)\right) \right| \quad \text{(A1)}$$

Note that these calculations refer to even values of *q*. Similar equations can be obtained for odd values of *q* (not shown). In addition, fluctuations in the latencies may also enhance the deviation from synchronization.

# Transient dynamics and rhythm coordination of inferior olive spatio-temporal patterns

#### *Roberto Latorre1 \*, Carlos Aguirre1, Mikhail I. Rabinovich2 and Pablo Varona1*

*<sup>1</sup> Grupo de Neurocomputación Biológica, Dpto. de Ingeniería Informática, Escuela Politécnica Superior, Universidad Autónoma de Madrid, Madrid, Spain <sup>2</sup> BioCircuits Institute, University of California San Diego, La Jolla, CA, USA*

#### *Edited by:*

*Guillermo A. Cecchi, IBM Watson Research Center, USA*

#### *Reviewed by:*

*Gabriel B. Mindlin, Universidad de Buenos Aires, Argentina Leandro M. Alonso, The Rockefeller University, USA*

#### *\*Correspondence:*

*Roberto Latorre, Grupo de Neurocomputación Biológica, Dpto. de Ingeniería Informática, Escuela Politécnica Superior, Universidad Autónoma de Madrid, c/ Francisco Tomás y Valiente, 11, 28049 Madrid, Spain e-mail: roberto.latorre@uam.es*

The inferior olive (IO) is a neural network belonging to the olivo-cerebellar system whose neurons are coupled with electrical synapses and display subthreshold oscillations and spiking activity. The IO is frequently proposed as the generator of timing signals to the cerebellum. Electrophysiological and imaging recordings show that the IO network generates complex spatio-temporal patterns. The generation and modulation of coherent spiking activity in the IO is one key issue in cerebellar research. In this work, we build a large scale IO network model of electrically coupled conductance-based neurons to study the emerging spatio-temporal patterns of its transient neuronal activity. Our modeling reproduces and helps to understand important phenomena observed in IO *in vitro* and *in vivo* experiments, and draws new predictions regarding the computational properties of this network and the associated cerebellar circuits. The main factors studied governing the collective dynamics of the IO network were: the degree of electrical coupling, the extent of the electrotonic connections, the presence of stimuli or regions with different excitability levels and the modulatory effect of an inhibitory loop (IL). The spatio-temporal patterns were analyzed using a discrete wavelet transform to provide a quantitative characterization. Our results show that the electrotonic coupling produces quasi-synchronized subthreshold oscillations over a wide dynamical range. The synchronized oscillatory activity plays the role of a timer for a coordinated representation of spiking rhythms with different frequencies. The encoding and coexistence of several coordinated rhythms is related to the different clusterization and coherence of transient spatio-temporal patterns in the network, where the spiking activity is commensurate with the quasi-synchronized subthreshold oscillations. In the presence of stimuli, different rhythms are encoded in the spiking activity of the IO neurons that nevertheless remains constrained to a commensurate value of the subthreshold frequency. The stimuli induced spatio-temporal patterns can reverberate for long periods, which contributes to the computational properties of the IO. We also show that the presence of regions with different excitability levels creates sinks and sources of coordinated activity which shape the propagation of spike wave fronts. These results can be generalized beyond IO studies, as the control of wave pattern propagation is a highly relevant problem in the context of normal and pathological states in neural systems (e.g., related to tremor, migraine, epilepsy) where the study of the modulation of activity sinks and sources can have a potential large impact.

**Keywords: spike wave fronts, subthreshold oscillations, electrical coupling, multifunctional neural networks, cerebellar circuits, source-sink phenomena, rhythm coordination and encoding, activity reverberation**

## **1. INTRODUCTION**

The architecture of the inferior olive (IO) network and the associated circuits of the cerebellar cortex of mammals have been investigated anatomically and physiologically in great detail (De Zeeuw et al., 1998; D'Angelo et al., 2011; De Zeeuw et al., 2011). Experimental recordings, both *in vitro* and *in vivo*, show that IO cells are electrically coupled and display a characteristic behavior with subthreshold oscillations (Llinás and Yarom, 1981; Benardo and Foster, 1986; Lampl and Yarom, 1993; Bal and McCormick, 1997; Long et al., 2002; Chorev et al., 2007; Choi et al., 2010) and spiking activity (Llinás et al., 1974; Sotelo et al., 1974). Their axons transmit synchronous and rhythmic excitatory synaptic input to both the deep cerebellar nuclear cells (CNs) and to the *Purkinje* cells (PCs) of the cerebellar cortex (Uusisaari and De Schutter, 2011). The phasic response of the PCs is transmitted as inhibitory inputs to the CNs. Thus, the nuclear cells are excited by the IO neurons and later inhibited by the PCs. This inhibition leads to rebound excitation. The nuclear cells also send an inhibitory feedback to the IO closing this inhibitory loop (IL) (De Zeeuw et al., 1997; Uusisaari and De Schutter, 2011) (see **Figure 1**).

While much is now known from the anatomical and physiological perspective, the functional role of the IO is still under discussion (Welsh et al., 1995; Llinás et al., 1997; De Zeeuw et al., 1998; Kobayashi et al., 1998; Kistler and De Zeeuw, 2002; Long et al., 2002; Schweighofer et al., 2004; Dean et al., 2010; Bazzigaluppi et al., 2012). It has been proposed as a system that controls and coordinates different rhythms through the intrinsic oscillatory properties of the individual IO neurons and their electrical interconnections (Llinás and Yarom, 1986; Llinás and Welsh, 1993; Manor et al., 1997; De Zeeuw et al., 1998; Hutcheon and Yarom, 2000; Leznik and Llinás, 2005) by means of clusters of functionally interconnected cells. The IO has also been suggested to be implicated in learning (Ito, 1982; Raymond et al., 1996; Ito, 2005; Van Der Giessen et al., 2008; Schweighofer et al., 2013), in comparing tasks of intended and achieved movements as a generator of error signals (Oscarsson, 1980; Llinás, 2009; Schlerf et al., 2012; Ito, 2013), and as a dynamical working memory in the context of the olivo-cerebellar closed-loop (Kistler and De Zeeuw, 2002).

Electrical gap junctions allow the synchronization of the subthreshold oscillations among groups of neurons (Bennett and Zukin, 2004; Connors and Long, 2004), and through these, the synchronization of the spiking activity. Subthreshold oscillations must clearly have a relevant role for information processing in the context of a system with extensive electrical coupling. In such systems, not only the spiking activity can be propagated through the network, but also small voltage differences of hyperpolarized membrane potentials among neighbor cells. Subthreshold oscillations are present in many other neural systems (Leung and Yim, 1991; Gutfreund et al., 1995; Pape et al., 1998; Amir et al., 2002; Reboreda et al., 2003; D'Angelo et al., 2009), as well as gapjunctions (Bennett and Zukin, 2004; Long et al., 2004). However, the IO seems to be a system where the joint presence of these two features can have a special significance for its function. Several theoretical models of the IO network have been proposed to study its properties and behavior (e.g., see Manor et al., 1997; Varona et al., 2002; Velarde et al., 2004; Jacobson et al., 2008; Katori et al., 2010; De Gruijl et al., 2012; Torben-Nielsen et al., 2012). These studies largely contribute to the understanding of the IO function from a network dynamics perspective.

In this paper we show that large scale networks of electrically coupled IO neurons generate localized spatio-temporal patterns which can easily encode several coexisting rhythms. In our simulations, we have used conductance-based neurons to generate subthreshold oscillations as well as spiking activity in the amplitude and frequency ranges reported for these neurons. We argue that neither the knowledge of the anatomic organization of these neural circuits, nor the study of the individual activity of the cells alone is enough for the identification of their function. However, understanding the collective dynamics gives us important clues about the underlying computational properties and the possible multifunctional nature of the IO. We want to emphasize here the combination of the specific organization of the connections and of the specific dynamics of the neurons as an essential step in understanding the role of the olivo-cerebellar circuits. The use of large population networks of realistic neurons is required for the study of the IO network dynamics and, in particular, the encoding and control of rhythms in the IO transient spatio-temporal activity.

Our results show that the two principal characteristics of the IO, i.e., the subthreshold oscillations of the individual neurons and the electrical gap junctions, make this system a powerful encoder and generator of spatio-temporal patterns with different but coordinated oscillatory rhythms. In our study, we first analyze the factors that shape the patterns in autonomous networks. We show that networks of spiking neurons that do not generate subthreshold oscillations have a more restricted ability to develop dynamical patterns. Then, we study the ability of IO networks to generate and encode reverberating rhythms depending on external stimuli. We also show that the presence of low and high excitability regions in this system creates activity sinks and sources that shape the propagation of coordinated spike wave fronts. Finally, we discuss the modulatory effect that the IL can have on the IO spatio-temporal activity.

## **2. MODELS AND METHODS**

## **2.1. NEURON MODEL**

To model the individual behavior of each IO cell, we have built a conductance-based neuron using a Hodgkin–Huxley type formalism (Hodgkin and Huxley, 1952) that generates the characteristic subthreshold oscillations and spiking activity in the amplitude and frequency ranges observed in the living IO cells. **Figure 2** shows some examples of the different behaviors displayed by the neuron model as a function of its parameters: from subthreshold oscillations to tonic spiking. The results discussed in this paper do not depend on whether individual neurons are intrinsic or network oscillators (Manor et al., 1997; Schweighofer et al., 1999; Manor et al., 2000; Kistler and De Zeeuw, 2002).

Our model is based on Wang's work on subthreshold membrane potential oscillations in cortical pyramidal cells (Wang, 1993). The model uses a single compartment to describe the neuron dynamics. Since we want to build large scale networks of thousands of units, the computational cost is an important issue. Five voltage-dependent ionic currents (*I*Na, *I*Nap, *I*Kd, *I*Ks and *Ih*), a leakage current (*Il*) and a stimulus injected current (*I*inj) define

Equation 1).

**FIGURE 2 | In our simulations, we model the individual behavior of the single IO cells with a Hodgkin–Huxley type model.** The model is able to generate subthreshold oscillations and spiking activity at different frequencies as a function of the parameters (see the text for details). The

the specific behavior of the neuron (see below). Formally, the membrane voltage is described by the following equation:

$$C\_m \frac{dV}{dt} = -(I\_{\rm Na} + I\_{\rm Nap} + I\_{\rm Kd} + I\_{\rm Is} + I\_h + I\_l + I\_{\rm inj} + I\_{\rm elec} + I\_{\rm syn}) \tag{1}$$

(1) where *Cm* = 1 μF/cm2; *Il* = *gl*(*V* − *Vl*) with *gl* = 0.1 ms/cm2 and *Vl* = −60 mV; and *I*inj is a constant depolarizing current. Currents *I*elec and *I*syn are, respectively, the total current from the electrical gap junctions connecting the neurons to build the IO network and the total synaptic current from the IL (see section 2.2 for a detailed description of these two currents).

The general description of the five active ionic currents considered in the model follows the Hodgkin–Huxley formalism:

$$I\_i = \bar{\mathbf{g}}\_i \cdot \mathbf{x}^\rho \cdot \mathbf{y}^\theta \cdot (V - V\_i) \tag{2}$$

where *g*¯*<sup>i</sup>* is the maximal conductance of the current, *V* is the membrane potential, *Vi* is the reversal potential of the current and *x* and *y* are the activation and inactivation variables. **Table 1** provides the specific values of these parameters for each particular current. The activation and inactivation variables, when exist, satisfy the following equations:

$$\frac{d\mathbf{x}}{dt} = \frac{\mathbf{x}\_{\infty} - \mathbf{x}}{\mathbf{r}\_{\mathbf{x}}}, \qquad \qquad \frac{d\mathbf{y}}{dt} = \frac{\mathbf{y}\_{\infty} - \mathbf{y}}{\mathbf{r}\_{\mathbf{y}}} \tag{3}$$

The steady state and time constants of these variables for each current are:

$$\begin{array}{ll} \bullet \, \mathsf{I\_{Na}} & \\ m\_{\infty} = \frac{\alpha\_{m}}{\alpha\_{m} + \beta\_{m}}; & \mathsf{r\_{m}} = \frac{1}{\alpha\_{m} + \beta\_{m}} \\ \alpha\_{m} = 0.1(V + 30 - \sigma)(1 - \exp(-0.1(V + 30 - \sigma))); \\ \beta\_{m} = 4 \exp((-V - 55 + \sigma)/18); \\ \end{array}$$

**Table 1 | Conductance description, maximal conductances and reverse potential (Equation 2) for the ionic currents of the single neuron model.**

figure illustrates different examples of spiking frequencies over the subthreshold oscillations depending on the values of σ (defining the action potential threshold of the model) and *I*inj (constant depolarizing current in


$$\begin{array}{ll} h\_{\lambda} = \frac{\alpha\_{h}}{1 + \beta\_{h}}; & \mathsf{r}\_{h} = \frac{\alpha\_{h}}{\alpha\_{h} + \beta\_{h}} \\ \alpha\_{h} = 19.99 \,\mathrm{exp}((-V - 44 + \sigma) \text{2}0); \\ \beta\_{h} = \frac{28.57}{1 + \exp(-0.1(V + 14 - \sigma))}; \\ \bullet \ \mathrm{I}\_{\mathrm{Xap}} = \Gamma(V, 51, 5); \\ \alpha\_{\infty} = \Gamma(V, 51, 5); \\ \epsilon\_{\infty} = \frac{\alpha\_{\varepsilon}}{\alpha\_{\varepsilon} + \beta\_{\varepsilon}}; \qquad \mathsf{r}\_{\varepsilon} = \frac{1}{\alpha\_{\varepsilon} + \beta\_{\varepsilon}}; \\ \alpha\_{\varepsilon} = 0.2857(V + 34 - \sigma) \langle (1 - \exp(-0.1(V + 34 - \sigma))); \\ \beta\_{\varepsilon} = 3.57 \,\mathrm{exp}((-V - 44 + \sigma) \&\varnothing); \\ \epsilon\_{\mathbb{R}} = 3.57 \,\mathrm{exp}((-V - 44 + \sigma) \text{\&0}); \\ d\alpha = \Gamma(V, 34, 6.5); \qquad \mathsf{r}\_{d} = 50 \,\mathrm{ms} \\ \epsilon\_{\mathbb{R}} = \Gamma(-V, -65, 6.6); \qquad \mathsf{r}\_{\varepsilon} = 200 + 220 \Gamma(V, 71, 6.685); \\ \epsilon\_{\mathbb{R}} = \Gamma(-V, -65, 6.6); \qquad \mathsf{r}\_{f} = 200 + 320 \Gamma(V, 63.6, 4); \\ \mathsf{\texttt{\texttt{\$$

where ρ and σ are parameters for the fine tuning of the action potential threshold of the model; and (*X*, *Y*, *Z*) = 1 <sup>1</sup> <sup>+</sup> exp(−(*<sup>X</sup>* <sup>+</sup> *<sup>Y</sup>*)/*Z*) .

The equations were numerically solved with a Runge-Kutta6(5) variable step method with a maximum error of 10<sup>−</sup>13. In all the simulations presented in this paper ρ = 0.6 and σ = 1, unless another value is specified in the simulation description to change the excitability level in different regions of the network. The initial conditions were selected randomly from a set of 10,000 coherent values for all the dynamic variables (10,000 different final conditions in simulations of a single cell) and the stimulus injected current (*I*inj in Equation 1), unless a specific value is indicated to implement external stimuli, was selected randomly between 0.0 and 0.35μA/cm2.

## **2.2. NETWORK MODEL**

To simulate the IO network we have built two-dimensional networks of 50 × 50 IO neurons connected with gap junctions among close neighbors. The term *I*elec in Equation (1) denotes the current received by each neuron through these connections. Therefore, *I*elec = *gc* - *i* (*V* − *Vi*), where index *i* runs over the neighbors of each neuron and *gc* is the electrical coupling conductance. The number of electrically coupled neighbors varied from 4 to 12. We imposed periodic boundary conditions within the network to avoid border effects.

With this network topology we simulate the autonomous behavior of the IO network. However, as **Figure 1** illustrates, the IO forms part of an IL with the deep cerebellar nuclei and the PCs of the cerebellar cortex. To test the effect of this inhibition, we implemented a simplified model of the cerebellar IL that takes into account the timing of these inhibitory inputs omitting the details of the cerebellar neurons and circuits. The inhibitory connections were modeled without the detailed implementation of the cell types involved in the IL (PCs and cerebellar nuclei). The action of the cerebellar IL was built through an inhibitory feedback in the IO networks from a bidimensional network of simple integrate and fire (IF) neurons. Each IF neuron was connected to the IO network bidirectionally as depicted in **Figure 3** in one dimension for simplicity. These connections preserved the topology of the IO network, i.e., neighbor cells in the IO network sent and received connections to neighbor cells in the IF layer. The connection probability was 75% in both directions. Each IF neuron took into account whether a group of neighbor IO neurons (up to 10 neighbor cells) had a synchronous spiking event (in a time window of 5 ms), if so, the IF cells evoked a delayed IPSP (10 ms) back to a cluster of up to 10 IO neurons. Then the IF neuron had a refractory period where it could not fire for a short time (10 ms). Inhibitory synaptic currents from the action of the integrate and fire neurons were implemented using the model and parameters described in Destexhe et al. (1994) (*g*¯syn = 0.1 nS), and were added into the term *I*syn of Equation (1).

## **2.3. GRAPHICAL REPRESENTATION OF THE SPATIO-TEMPORAL PATTERNS**

The spatio-temporal patterns generated by the IO networks consist of propagating wave fronts of spiking activity that can remain bounded in a region of the network. To illustrate the propagating waves of activity in the simulations, we have generated movies of square-shaped networks to represent their evolving dynamics. Each point in the 50 × 50 square represents the evolution in time of the activity of a given neuron within the network. The neural activity is represented with a color scale, where warm colors (red) correspond to neurons with a membrane potential over the spiking threshold (around −47 mV in our model) and cool colors (blue) correspond to hyperpolarized neurons. Intermediate colors represent subthreshold activity. Regions with the same color in the movies have synchronous behavior. Although in the paper we provide snapshots of the evolution of the network activity, the described phenomena are better appreciated in the movies included as supplementary material. To better appreciate the dynamics in slow motion, the temporal scale in the movies does not correspond to the neuron time in ms. The videos are generated at a 25 Hz frame rate and each frame corresponds to 0.5 ms in neuron time.

## **2.4. WAVELET ANALYSIS OF THE PATTERNS**

In one dimensional signals, spectral methods are suitable for the detection of rhythms present in the signal. However, in higher dimensions, the coefficients produced by the multidimensional Fourier transform are hard to interpret as they present a number of artifacts not directly related with the behavior of the signal, but to sampling (aliasing effects) or border conditions. Thus, to characterize quantitatively the localized spatio-temporal patterns of the IO network models we did not use spectral methods. Instead, we propose the study of the IO spatio-temporal patterns as a sequence of images evolving in time by means of a wavelet based

the IO network preserving the topology, i.e., neighbor cells in the IO network sent and received connections to/from neighbor cells in the IL layer.

compression scheme. Wavelet based techniques have proven to be a useful tool for signal analysis (Mallat, 1999) and, in particular, for the study of images or sequences of images (Stollnitz et al., 1996). Unlike the Fourier transform coefficients, where the "frequency" content of the signal cannot be localized in time (or space), the wavelet transform coefficients are determined both by a resolution component and a time (or space) component and, therefore, they represent the resolution content at a given portion of the original signal. The number of coefficients of the wavelet transform that are higher than a given threshold, or alternatively, those that comprise a given percentage of the total energy of the signal, characterizes the whole complexity of the signal. Wavelet based compression schemes are based on this complexity. On one hand, when the number of wavelet coefficients larger than a fixed threshold is small, the corresponding signal can be represented only with a few low resolution components and high compression can be achieved without highly distorting the original data. On the other hand, if the number of coefficients larger than the fixed threshold is high, we have a complex signal, so we will need both high resolution (details) and low resolution components to represent it and, therefore, low compression can be performed. Wavelet based compression schemes have been used, for example, for one dimensional signal segmentation through "wavelet probing" techniques (Deng et al., 1993).

The Wavelet Transform (WT) is related with multiresolution analysis and presents a hierarchical structure that is particularly suited for fast numerical algorithms (Daubechies, 1992). In particular, the multiresolution process allows the computation of the coefficients of the WT by means of the Discrete Wavelet Transform (DWT) with a low computational cost. The two dimensional wavelet transform has been used for image compression, as it presents high compression levels and a low computational cost (*O*(*w* × *h* × *t*) vs. *O*(*w* × *h* × max{log(*w*) × log(*h*)} × *t*) for similar compression schemes based on spectral techniques; where *w* and *h* are the image width and height in pixels, respectively, and *t* is the number of images). Briefly, the idea behind the image compression techniques based on the WT is that wavelet coefficients that correspond to parts of the image that are smooth have a small value (low spatial complexity), in contrast with complex images that present a low number of small parameters of the two dimensional WT and their compression ratios are lower.

Our metric to characterize the IO spatio-temporal patterns is based on the previous compression technique. The method consists in considering the spatio-temporal patterns generated by our IO models as sequences of images and estimating the compression rate of each of them by calculating the number of DWT coefficients higher than a given threshold. In this way, we translate the spatio-temporal pattern to a new one dimensional signal, *C*(*t*), which represents the evolution in time of the spatio-temporal pattern *complexity*. As a first step in the characterization, a twodimensional basis was generated by direct Cartesian product of the one-dimensional Haar basis (Stollnitz et al., 1996). Then, we calculated the two dimensional non-standard DWT for each frame of network activity and counted the number of coefficients, *C*(*t*), that were larger in absolute value than a given threshold (*th* = 1 in the simulations shown here). The new one dimensional signal provides a useful characterization of the spatio-temporal patterns in which both the frequencies and the spatial complexity can be discussed. At a given time *t*, a high value for *C*(*t*) means that the network has a complex spatial structure, while a low value indicates a uniform space (synchronized activity). The time evolution of *C*(*t*) provides information about the frequency of the spatio-temporal patterns.

## **2.5. WAVE FRONT PROPAGATION CHARACTERIZATION**

The IO wave fronts in our simulations have the shape of circles or arcs centered at a given region and propagating through the network. To characterize the evolution of these wave fronts, we have developed an algorithm for the detection of arcs of propagating spiking activity through the analysis of each video frame. Several methods have been developed for the detection of circles in images, most of them based on the Hough Transform (Duda and Hart, 1972). The Hough transform provides accurate results as long as the circle radius is known, and the image has low noise and low density of edge pixels (pixels with value 1 in a binary representation of the image). Furthermore, the Hough transform requires a high amount of memory and computational resources as a non-linear optimization process is involved in the method.

Our method for detection of arcs is based on the idea that a circle is completely determined by the position of three edge pixels in the image representing a given frame of IO network activity. To apply this algorithm, we convert the membrane potential time series to binary time series where 0 means that the neuron is under the firing threshold, and 1 that it is over the threshold. The algorithm iteratively searches for a set of three edge pixels in the frame *F*. Once a set of three edge pixels is found [*p*<sup>1</sup> = (*x*1, *y*1), *p*<sup>2</sup> = (*x*2, *y*2), *p*<sup>3</sup> = (*x*3, *y*3)], the center of the unique circle that contains them is calculated in the following way:

• The general equation of the circle can be written as *x*<sup>2</sup> + *y*<sup>2</sup> + *Ax* + *By* + *C* = 0 with center at the point *p* = −<sup>1</sup> <sup>2</sup> (*A*, *B*) and radius *<sup>r</sup>* <sup>=</sup> <sup>√</sup> *C* − *A*<sup>2</sup> − *B*2. Then the three edge pixels coordinates are used in the previous equation, obtaining the following linear system:

$$\begin{array}{l} \left(\varkappa\_1^2 + \mathcal{y}\_1\right)^2 + A\varkappa\_1 + B\mathcal{y}\_1 + C = 0\\ \left(\varkappa\_2^2 + \mathcal{y}\_2\right)^2 + A\varkappa\_2 + B\mathcal{y}\_2 + C = 0\\ \varkappa\_3^2 + \mathcal{y}\_3^2 + A\varkappa\_1 + B\mathcal{y}\_3 + C = 0 \end{array}$$

The values *A*, *B*, *C* can now be obtained by methods such as Gauss elimination or Cramer's rule. If these methods cannot solve the system, then the three points lie on a straight line.


This procedure is repeated until all different circles are found in frame *F*. By analyzing the evolution of centers at different time frames, and the change in the value of the radii of the arcs, we can decide whether each center is a source or a sink.

## **3. RESULTS**

## **3.1. ORIGIN AND PROPAGATION OF THE SPATIO-TEMPORAL PATTERNS**

## *3.1.1. Gap-junction mediated quasi-synchronized IO activity*

*In vitro* experiments demonstrate that IO cells generate spatiotemporal patterns of network activity (Devor and Yarom, 2002b; Leznik et al., 2002; Leznik and Llinás, 2005). In our analysis of these patterns, we first discuss the spontaneous activity of autonomous IO network models without any external stimuli. We have built two-dimensional networks of 50 × 50 IO model neurons connected with gap junctions among close neighbors (see section 2.2 for details). The parameters of the IO cells were set so that they could generate subthreshold oscillations and spiking activity (section 2.1). Different simulations were performed varying the magnitude of the electrical coupling conductance among neurons and the number of electrotonically coupled neighbors.

In a neural media with the features of the IO, currents arising from incoming inputs are invested to increase/decrease the excitability level of each unit and to be shared among neighbors through the diffusive coupling. To address the effect of the strength of the electrical coupling in the spatio-temporal activity of the IO network, we discuss four representative cases illustrated in **Movies S1**–**S4**: weak coupling, strong coupling, moderate coupling and weak coupling extended to further neighbors. Sequences of network activity in the form of snapshots of these movies are shown in **Figures 4A–D**. Sequences develop in time from left to right with a time interval between frames of 3 ms. Both in the videos and in the snapshots, the level of activity

Color bar maps the membrane potential. When the coupling is moderate (panels **C** and **D**), there exist well-defined spatio-temporal patterns of spiking activity traveling over the network. However, when the coupling is either too weak or too strong these spatial structures do not appear. **(E–H)** Membrane potential time series of four randomly chosen neurons from the IO networks whose activity is represented in panels **(A–D)**. Units are mV. The inset in

oscillations in these cases. These transient small phase shifts create the spatio-temporal patterns observed in panels **(C,D)** and the corresponding activity movies in the supplementary material. As the strength of the coupling grows **(F)**, the global spiking frequency decreases and, if the electrical coupling is high enough, the individual activity of the cells is completely synchronized (cf. panel **B**).

of each neuron is represented with the color scale described in section 2.3: warm colors correspond to neurons with a membrane potential over the spiking threshold and cool colors correspond to hyperpolarized neurons.

In these four representative cases, we observe that the activity of the autonomous IO network model strongly depends on the magnitude of the electrical coupling. As expected, when the coupling between close neighbors is too weak, i.e., *gc* < 0.01 mS/cm2, the IO neuron activity is nearly independent and thus no coherent patterns are formed (see **Movie S1** and the corresponding snapshots in **Figure 4A**). Strong coupling, *gc* > 0.7 mS/cm2, induces almost total synchronization and also avoids the formation of spatial structure in the patterns, as seen in **Movie S2** and **Figure 4B**. However, networks with moderate values of the coupling always show evolving spatio-temporal patterns (see **Movie S3** and **Figure 4C**). The spatio-temporal patterns consist of transient spiking activity wave fronts that propagate throughout different regions of the IO network. Physiological experiments have revealed that each IO cell can be coupled to a large number of neighbors (Devor and Yarom, 2002b; Hoge et al., 2011). This situation corresponds to our fourth representative case. The simulations show that increasing the extent of the connections has a similar effect in the network dynamics as an increase in the coupling strength. As an example, **Figure 4D** shows the spatio-temporal patterns corresponding to an IO network with electrical coupling among 12 nearest neighbors with *gc* = 0.01 mS/cm2 (see also **Movie S4**). Note that a network with the same coupling strength but electrical connectivity just among four close neighbors displays nearly independent activity (not shown here). Thus, increasing the number of connections among cells in the IO network model leads to equivalent dynamics as those generated with less connections but larger coupling strength (cf. **Figures 4C,D**).

The network dynamics is better understood by looking at the membrane potential of single neurons together with the collective activity shown in the movies. **Figures 4E–H** plot the membrane potential time series of four representative neurons within the IO networks whose spatio-temporal activity is shown in panels **(A–D)**, correspondingly. The analysis of panels in **Figure 4** and the activity movies shows that the patterns arise from small transient phase shifts in the quasi-synchronized subthreshold oscillations for moderate values of the electrical coupling [see the time series of panels **(G)** and **(H)** in **Figure 4**, and compare them with panels **(E)** and **(F)**]. The occurrence of a spike induces new phase shifts and fast propagating waves that shape the patterns within the quasi-synchronized (period locked) subthreshold activity. However, when the coupling is too weak, the spatio-temporal patterns are absent (see **Movie S1** and **Figure 4A**) since, as can be observed in **Figure 4E**, the activity of each IO cell is nearly independent because the low electrical current cannot provide coherence to the subthreshold oscillations. Strong coupling also avoids the formation of a spatial structure in the patterns since all the neurons are almost synchronized and follow each other (see **Movie S2** and **Figure 4F**). Higher values of the coupling strength increase the synchronization level but diminish the frequency of the global spiking behavior (cf. **Figures 4A–C**). **Figure 5** quantifies this decrease by showing the average firing rate of the IO network model as a function of the coupling conductance. Stronger electrical coupling has a shunting effect that reduces the excitability of the neurons. Similarly, a larger extent of the electrotonic connections also decreases the firing rates for a strong enough coupling (see **Figure 5**).

Thus, the autonomous IO network model with the topology discussed above and moderate values of the electrical coupling is able to generate well-defined spatio-temporal patterns based on quasi-synchronized subthreshold activity. The patterns consist of transient propagating wave fronts of spiking activity that can remain bounded in a region of the network. In all the cases discussed so far, the degree of synchrony among cells changes as a function of the coupling conductance although the frequency of the subthreshold oscillations remains nearly constant.

The characterization of the spatio-temporal patterns generated by the autonomous IO network models with a discrete wavelet transform analysis (DWT, see section 2.4) corroborates these results. **Figure 6** shows the evolution of the number of DWT coefficients for different simulations of autonomous IO networks (spontaneous activity). The red traces correspond to a network with very small coupling among the IO cells. The number of DWT coefficients remains high during the simulation revealing a complex spatial structure in the patterns (i.e., independent single neuron activity). Nevertheless, the homogeneous frequency of the subthreshold oscillations is captured by the DWT analysis. The magenta trace shows the opposite case, a network with a high electrical coupling showing a high degree of synchronization (no complexity in the spatial structure and thus low number of DWT coefficients), only broken briefly at each spike event [see **Movie S2** and **Figure 4B**]. The other two traces (green and blue) correspond to a moderate value of the coupling where the evolving spatio-temporal patterns can be observed with the dominant frequency of the quasi-synchronized subthreshold oscillations. Note the sustained broad range (∼200–1500) in the number of DWT coefficients characterizing these patterns.

## *3.1.2. Activity phase locks in the absence of subthreshold oscillations*

To investigate to what extent the presence of the subthreshold oscillations influences the structure of the patterns, we also simulated autonomous networks of tonically spiking neurons without subthreshold oscillations. This behavior can be easily induced in the model by adjusting the spiking threshold through the kinetics of the ionic channels of the model (parameter σ) or by applying a constant current injection (*I*inj) in the whole population (see section 2.1). When the subthreshold oscillations were not present, the spatial topology of the patterns remained nearly constant (see **Movie S5** and **Figure 7** with snapshots of the evolution of the network activity in this situation). The fixed spatial organization of the patterns is due to a high degree of sustained phase-locking among neighbor units in the absence of subthreshold oscillations. **Movie S5** shows that the propagation of the wave fronts is faster over this nearly fixed spatial shape. In all our simulations, the characteristic spatial structure of the patterns changed significantly in time only when subthreshold oscillations were present in the model.

## **3.2. RHYTHM ENCODING AND COORDINATION**

The simulations described so far implemented neurons with spontaneous spiking activity over the subthreshold oscillations. A major point of interest in this study was the analysis of the response of the IO network model to stimuli that could induce different coherent spiking frequencies in the IO neurons.

The single neuron model can generate different spiking frequencies depending on the current injection (*I*inj). The spiking frequencies are commensurate with the subthreshold oscillation frequencies up to the tonic firing (see section 2.1 for details).

## *3.2.1. Coexistence of stimulus induced rhythms in the IO media*

To study the spatio-temporal patterns induced by stimuli to the IO network model, we performed simulations where different external currents were injected in different clusters of neighbor neurons. The stimulus clusters are surrounded by neurons that have no stimuli. The stimulus induces a higher spiking frequency in the neurons of these clusters while sustaining a similar subthreshold oscillation, as compared to the neurons without stimuli. The larger the input current, the higher the spiking frequency. Rhythms induced by the stimuli could then be observed in the network of IO neurons. As an example, **Movie S6** shows the spatio-temporal patterns produced in an IO network model when two external stimuli evoking two different spiking frequencies were applied to two clusters of 6 × 6 cells within the network. **Figure 8A** displays snapshots of this movie showing the coexistence of wave fronts with different spiking frequencies. The right panel in this figure indicates the approximate location of each cluster in the network. The coherent wave fronts originate in the regions with stimuli and generate the spatio-temporal patterns. The spatial scale of the patterns evoked by stimuli in the IO network depends on the frequency of response of the clusters (normal dispersion) and the strength of the coupling. Multiple spatiotemporal structures with different spiking frequencies may coexist simultaneously in the IO networks. For example, **Movie S7** (and the corresponding snapshots in **Figure 8B**) corresponds to a simulation with 25 clusters of 6 × 6 neurons each with different stimuli.

**Figure 8C** shows the wavelet analysis of three representative examples of networks with several coexisting frequencies of oscillations induced by stimuli. In all the traces, the frequency of the subthreshold oscillations can be observed. However, the spiky waveform indicates the presence of multiple coexisting frequencies. This can be better noted in the blue trace corresponding to the evolution of the DWT coefficients for the network with

25 input clusters. It is important to emphasize that any input to the IO clusters is encoded into a spiking frequency that is commensurate with the subthreshold oscillations, which results in a coordinated network activity.

## *3.2.2. Sort term memory through stimulus reverberation*

In the IO network simulations we observe that the stimulus induced spatio-temporal patterns can survive for several seconds after the excitation is over. This *stimulus reverberation effect* depends on the coupling strength in the network and is illustrated in **Movie S8**. In the simulation shown in this video, initially the network dynamics evolves freely as in the previously studied autonomous IO networks. Then, the external stimulation starts (at instant 0:40 in the movie) and lasts for 2 s (until the instant 2:00 in the movie). During this interval, we apply two external stimuli to two different clusters of 6 × 6 cells. Note that from the beginning of the stimulation, stepwise, the stimulated clusters become the two principal sources of the spatio-temporal patterns. When the external stimulus is over, this behavior continues for a long period and the IO network generates the same wave fronts that were induced by the stimuli.

**Figure 9** analyzes in detail this phenomenon by comparing an IO network of nearly isolated cells (panels **A.1, A.2**) and a network with moderate coupling (panels **B.1–B.4**). To highlight the reverberant effect, for this analysis the parameters of each individual IO cell were set to generate just subthreshold oscillations without spiking activity in isolation. Therefore, the spiking activity without an external stimulus is a network effect (cf. the neuron activity before the stimuli in **Figures 9A.2,B.3**). **Figures 9B.1,B.2** show, respectively, the evolution of the DWT coefficients of the whole network and the stimulated clusters in the simulation with moderate coupling. This DWT analysis shows that the global network dynamics changes when the external stimuli are applied. Note that the change due to the excitation lasts for several seconds after the stimulation is over, and then the network goes back to the autonomous activity. Conversely, in the network with very weak

coupling (**Figures 9A.1,A.2**), the change in the dynamics induced by the external stimulus is not sustained when the stimulus is over.

## *3.2.3. Activity source-sink phenomena*

Another remarkable feature arising in the IO model networks is the activity *source-sink* phenomena when at least two specific clusters of neurons are present: one cluster with a higher rate of spiking activity than the average population and the other with no intrinsic spiking activity (subthreshold oscillating neurons) or low excitability. In this situation, the wave fronts generated in the cluster with high excitability (source) travel to the cluster with low excitability (sink). To identify the sources and sinks we use the algorithm described in section 2.5 which can characterize the wave front propagation. To apply this algorithm, first we convert the membrane potential time series to binary time series where 0 means that the neuron is under the firing threshold (−47 mV), and 1 that it is over the threshold. Then, to identify the wave front sources and sinks, we search for arcs centered in a given region and analyze the evolution of their mean radius. In the activity sources, the arc radius grows; while in the sinks the radius decreases. **Table 2** shows the result of this analysis in two simulations where the source and the sink are located in different regions. In both cases, each IO neuron is connected to 12 neighbors with electrical coupling *gc* = 0.01 mS/cm2. **Figure 10** displays snapshots of these simulations. The approximate location of each cluster is shown in the right panels. In these simulations, more than 50% of the wave-front arcs whose radii increase are centered in the cluster with a higher spiking rate. Near 60% of the arcs whose radii decrease travel to the cluster of subthreshold oscillating cells.

The wave fronts originated in the source generate secondary wave fronts and travel through the IO network following different trajectories depending on the sink location where they finally die. **Figure 11** illustrates how the source-sink phenomena allows the IO network models to generate spiking wave fronts in different regions and attract them to specific locations by modulating the excitability of the IO cells. The figure compares the pathways followed by the spatio-temporal patterns produced in a network of strongly coupled cells (*gc* = 0.8 mS/cm<sup>2</sup> among eight neighbors) as a function of the source and sink location. Insets in each panel indicate the approximate position of both regions in each situation. The strong level of coupling facilitates the analysis of the traveling spiking activity since in this case only one wave front is active at a give time. The pathways in **Figure 11** correspond to a simulation where both the sink and source location change in time. Note that the trajectories are similar when the source and the sink are in the same position



*The table shows the evolution of the mean radius of the IO wave fronts in two representative cases to illustrate the activity source-sink phenomena. In a simulation where a cluster of neurons has a higher rate of spiking activity than the average population (CA) and another cluster has only subthreshold oscillating neurons (CB), the evolution of the mean radius of the arcs with center in these clusters shows that the wave fronts generated in CA travel to CB. The table characterizes the wave front propagation corresponding to the two simulations (SA and SB) illustrated in Figure 10. The number of connections among the nearest neighbors is 12 with gc* = *0.01 mS*/*cm2. Both clusters consist of 6* × *6 neighbor neurons. In the cluster with highly excitable neurons (source) Iinj* = *0.5* μ*A*/*cm2, while in the cluster with low excitable neurons (sink)* σ = *2 and Iinj* = *0*μ*A*/*cm2. To calculate the mean radius, the interval from the wave front birth to the wave front death was divided in five subintervals (ti ) with the same duration. Dashes indicate that no arcs were detected for that interval. Units are dimensionless as the radii were calculated in terms of the number of adjacent neurons covered by a spatio-temporal pattern from a given center detected with the wave front characterization algorithm described in section 2.5.*

(cf. left and right columns). In particular, they have the same origin and die in the same destination. **Movie S9** is a movie of this simulation. Note that, to better show the propagation of spike wave fronts, in this activity movie the color scale changes. Each time the source/sink location changes in the simulation, the new position is pointed out in the video. The spatio-temporal patterns mostly travel from the source to the sink (**Figure 11**). Nevertheless, the video shows the competition between the global intrinsic IO network dynamics and the source dynamics. This competition allows the generation of wave fronts in a location different from the source traveling to the sink (e.g., the wave front generated in the left-upper corner at instant 0:43 in the movie). Finally, the video also shows that, due to the stimulus reverberation effect, after each change in the excitability of a group of cells, it may exist a short interval where the spatio-temporal patterns do not travel to the sink region (e.g., the first wave front generated after the sink/source location change at instant 0:32). After this adaptation period, the wave fronts are attracted to the new sink.

Different stimulus can shape the presence of sources and sinks in the spatio-temporal patterns of the IO. The ability to attract the wave fronts from one region to another by modulating the excitability can be an important feature for a system with topology preserving connections as those found in the cerebellar circuits.

## **3.3. EFFECT OF THE INHIBITORY LOOP**

As **Figure 1** illustrates, the IO is part of an IL with the deep cerebellar nuclei and the PCs of the cerebellar cortex. The terminals of the inhibitory synapses from the cerebellar nuclei are located close to the gap junctions of the IO (Sotelo et al., 1986) and this can produce a transient decoupling of neighbor neurons. The effect of inhibitory synapses into the IO network could in principle destroy the quasi-synchronization of the subthreshold oscillations observed in the previous simulations, and thus destroy or largely affect the dynamics of the spatio-temporal patterns. To test the effect of this inhibition we have performed simulations using a simplified model of cerebellar IL (see section 2.2).

The presence of inhibitory chemical synapses coming from the IL changed both the spiking frequency and the frequency of the subthreshold oscillations in the IO network simulations. Each synapse induced a transient desynchronization of the subthreshold oscillations among neighbor cells (**Figure 12A**). The synchronization was recovered later for close enough cells and the spatio-temporal patterns were not destroyed, but received an additional modulation (see wavelet analysis below). **Movies S10**, **S11** illustrate the dynamics of two IO networks where the IL is present: one without stimuli and the other with several stimuli (**Figures 12B,C** show snapshots of these movies). As in the autonomous networks, several frequencies for the oscillations could also be distributed in different clusters with different stimuli in the presence of the IL (see **Movie S11** and the corresponding snapshots in **Figure 12C**). The inhibitory connections affect the extent of the propagation of the patterns in the network and a larger coupling conductance or number of connections is needed to reproduce the extent of the patterns without the inhibition.

**FIGURE 10 | The source-sink phenomena appears in the IO network model when a cluster of neurons is set to have a high rate of spiking activity while another is set in a subthreshold oscillatory regime.** In the networks illustrated in the figure, the number of connections among the nearest neighbors is 12 with *gc* = 0.01 mS/cm2. Sequences develop in time from left to right with a time interval between frames of 10 ms. Right panel shows the approximate location of the source (cluster with highly excitable neurons, *I*inj = 0.5μA/cm2) and the sink, (cluster with low excitable neurons, σ = 2 and *I*inj = 0μA/cm2). The difference between the top and bottom panel is the location of the sink cluster (the source is the same in both cases).

**FIGURE 11 | (A–C)** The source-sink phenomena allows the IO network models to generate spiking wave fronts from different regions and attract them to specific regions. To represent the pathways followed by the spatio-temporal patterns generated in a network model, we plot the IO cells that are over the firing threshold in each moment, from the wave front birth to its death. *y* and *z* axes represent the neuron coordinates in the IO network (50 × 50 square shaped), while *x* axis represents time evolution. Note that time is counted in terms of frames. The color code is used to illustrate the evolution of the wave fronts, blue corresponds to moments near their birth and red to moments

near their death. We have selected three different pairs of locations for a single source and a single sink in the same IO network (also shown in **Movie S9**). A total of six pathways are shown in the figure, two examples for each pathway. The insets indicate the approximate location of the source and the sink in each case. The figure shows that the wave fronts are generated in the regions with a higher rate of spiking activity than the average population (sources), then travel through the IO network in different trajectories depending on the sink location and finally die in this region. Note the effect of the periodic boundary conditions of the network in this representation.

The inhibitory modulation is hardly appreciated by an eye inspection (e.g., see **Movies S10**, **S11**), but can be seen in the wavelet analysis. **Figure 12D** corresponds to the wavelet analysis of networks with the IL. The evolution of DWT coefficients in these networks clearly shows the slow modulatory effect induced by the IL in two networks with *gc* = 0.05 mS/cm2 in the absent or present of stimuli. Note that the more spiky trace (green trace) corresponds to a network with an external stimuli. Thus,

the simulations indicate the IL can introduce an additional modulation in the IO network activity without destroying the patterns.

the IL (snapshots of **Movie S10**). **(C)** Activity of an IO network with two clusters of neurons with an external stimuli (snapshots of **Movie S11**). In

## **4. DISCUSSION**

While the anatomy and physiology of the cerebellar circuits has been studied for more than a century now, the possible roles of the IO are still under discussion. The activity of this neural system has been analyzed mainly at the level of single-cell recordings, from which network properties were then inferred. In particular, electrophysiological and imaging techniques have allowed injection of 0.25μA/cm<sup>2</sup> and 0.5μA/cm2. Note the IL induces an additional spatial modulation in the IO network activity. Nevertheless, the spatio-temporal patterns generated by the IO network do not disappear. the direct study of the IO network activity in *in vitro* (Devor

and Yarom, 2002b; Leznik et al., 2002; Leznik and Llinás, 2005; Chorev et al., 2007; Hoge et al., 2011) and *in vivo* (Chorev et al., 2007) experiments. However, the dynamical properties of IO networks *in vivo* have not been explored in detail. Two major hypothesis have been proposed about the IO: (1) the learning hypothesis, in which IO activity modifies through long-term depression the cerebellar input and output (Ito, 1982; Kobayashi et al., 1998; Ito, 2005; Swain et al., 2011); and (2) the IO activity contributes to motor control in real time through its intrinsically rhythmic synchronous activity (Welsh et al., 1995; Jacobson et al., 2008). Another proposal brings together these two views and postulates that the major role of the IO is to reduce the firing rate carrying the error signal for cerebellar learning while maintaining its information content (Schweighofer et al., 2004, 2013). All these hypotheses are plausible for this neural system with very rich dynamical properties and likely to be multifunctional.

In this paper we have studied for the first time the IO dynamics using a large scale network with conductance-based models. This type of model is necessary to address the dynamics that arises from the interaction between the spiking activity and the subthreshold oscillations in the context of a diffusive neural media built on gap-junctions. Electrical gap junctions have been suggested as a key factor for the characteristic rhythmic dynamics in the IO network (Blenkinsop and Lang, 2006; Marshall et al., 2007). In our simulations, both the subthreshold oscillations and the spiking activity, propagated through the gap junctions, strongly contribute to the generation of coherent and coordinated spatio-temporal patterns for a large range of coupling strengths. The coordination arises from the subthreshold oscillations that keep a high degree of synchronization due to the extensive electrical connectivity while allowing different spiking frequencies in distinct regions of the IO network.

In the presence of stimuli, different rhythms can be encoded in the spiking activity of the model IO neurons that nevertheless remains constrained to a commensurate value of the subthreshold frequency. Experimental recordings show that subthreshold oscillations in the living IO cells are very precise (Devor and Yarom, 2002a), although their frequency can change at different moments or between different groups of cells (Devor and Yarom, 2002b; Chorev et al., 2007). In this context, the climbing fibers to PCs in the cerebral cortex could carry motor signals beating on the rhythm of the subthreshold oscillations being locally propagated through the precisely timed wave fronts of the IO spiking activity. It is also possible in this system the organization of a context dependent coordination of the spatio-temporal patterns that are coming from different sources. Both these functions could provide, from the commensurability of the different incoming frequencies, a convenient representation of motor rhythms for the next processing levels.

Several transient dynamical phenomena were identified in the simulations of IO networks that can be useful for a precise encoding and coordination of rhythms. The specific properties of the dynamic organization of the IO patterns observed in our simulations can be summarized in the following points:


enough coupling. In this case the degree of synchrony among cells was higher although the frequency of the subthreshold oscillations remained nearly constant under all these changes.


The spatio-temporal patterns in our simulations were similar to those observed in imaging recording of IO slices reported in Manor et al. (2000), Leznik et al. (2002), and Leznik and Llinás (2005). The large scale modeling of IO networks is a powerful tool to interpret the imaging recordings and to overcome the restricted amount of experiments that can be done in these setups (Varona et al., 2002; Torben-Nielsen et al., 2012). In particular, the models can tackle the study of the effect of the IL arriving from the cerebellar nuclei, which is difficult to assess through *in vitro* experimental recordings. In short, IO network simulations can help us to test hypotheses related to the role of cellular and network processes in the genesis of neuronal spatio-temporal patterns, as well as to understand how the IO oscillations encode and control several simultaneous rhythms.

In the context of the study of spatio-temporal dynamics in brain circuits, an important question is how detailed the single neuron model has to be (Rabinovich et al., 2006c). The answer depends on what we are planning to model, the functions of some brain network or a specific system. In our case, our IO network model had to display subthreshold oscillations, spiking activity and input-specific excitability modulations. As we addressed the effect of the interaction between the subthreshold and spiking transient activity in the IO networks, our study required a model that could describe the generation and propagation of currents during the action potentials. Many morphological and physiological details of the IO neurons and the IL were not considered in the model discussed in this paper, but the fundamental dynamical phenomena observed here does not likely depend on these details.

Synchronization at different levels is one of the most discussed phenomena in relation to neural coding (Engel et al., 1992; Diesmann et al., 1999) and neural dynamics (Chow and Kopell, 2000; Rabinovich et al., 2000b, 2006a; Engel et al., 2001), particularly in the context of electrically coupled neurons (Bennett and Zukin, 2004; Connors and Long, 2004). Sustained or transient phase locks and phase synchronization have been extensively studied both experimentally and theoretically (e.g., see Chow and Kopell, 2000; Rabinovich et al., 2006b; Rabinovich and Varona, 2011; Latorre et al., 2013). Several features of transient spatio-temporal pattern activity are universal for excitable systems of different nature (biological, chemical, physical). For example, the emergence of large-scale spatio-temporal patterns in the form of synchronized spirals is typical for epileptic brains (Stacey, 2012), termo-convection and many other media (Rabinovich et al., 2000a). The results of this work can be generalized beyond IO studies, as the control of wave pattern propagation is a highly relevant problem in the context of normal and pathological states in neural systems (e.g., related to tremor, migraine, epilepsy, etc.) where the study of the modulation of activity sinks and sources can have a potential large impact.

Our modeling has shown important phenomena observed in IO *in vitro* and *in vivo* experiments and produced new predictions regarding the computational properties of this network. IO spatio-temporal patterns demonstrate specific features because the IO is a two-level neural media consisting of subthreshold oscillations and spiking activity in the context of diffusive electrical coupling. These two kinds of activity mutually interact: spikes influence the phase of the subthreshold oscillation and at the same time these oscillations determine the probability of the spikes to occur and coordinate the coherency of large-scale patterns. Together with the inhibitory feedback, such specificity of the IO system demonstrates a unique long-lasting encoding and highly shapeable spatio-temporal patterns that can participate in functions related to timing control, learning and motor memory.

Excitatory inputs to the IO neurons coming from the deep cerebellar nuclei or mesodiencephalic junction, which is innervated by excitatory projection neurons of the cerebellar nuclei (De Zeeuw and Ruigrok, 1994), have not been discussed in this paper. Some of the neurons of the mesodiencephalic junction project directly to motoneurons and interneurons in the spinal cord responsible for motor activity. Interestingly, the olivocerebellar loops appears to be topographically organized (De Zeeuw et al., 1998), and they surely react to each of the local spiking frequencies in the IO patterns. This means that if different patterns are clustered in the IO encoding different rhythms, they could be coordinated, transported and controlled through the intrinsic dynamical properties discussed above.

## **ACKNOWLEDGMENTS**

*Funding*: Roberto Latorre, Carlos Aguirre, and Pablo Varona were supported by MINECO TIN2012-30883 and Mikhail I. Rabinovich by ONR Grant N00014310205.

## **SUPPLEMENTARY MATERIAL**

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/Neural\_Circuits/ 10.3389/fncir.2013.00138/abstract

**Movie S1 | Nearly independent activity in an autonomous weakly coupled IO network model.** Each neuron is connected to four neighbors with a weak electrical coupling (*gc* < 0.001 mS/cm2). See section 2.3 for a description of the graphical representation. Note that the time scale does not correspond with the simulation time. For low coupling conductances, the activity of the neurons is nearly independent (see **Figure 4E**) and the spatio-temporal patterns are absent.

**Movie S2 | Synchronized activity in an autonomous strongly coupled IO network model.** The network displayed in this video is equivalent to the network in **Movie S1** but with a strong electrical coupling (*gc* < 0.8 mS/cm2). In this situation, there is almost total synchronization among neurons (see **Figure 4F**), only broken briefly when spiking behavior occurs, and no spatio-temporal patterns are generated.

**Movie S3 | Spatio-temporal patterns of coordinated activity in an autonomous moderately coupled IO network model.** The network displayed in this video is analogous to the network in **Movies S1**, **S2** but with a moderate electrical coupling among cells (*gc* < 0.08 mS/cm2). In this case, the individual neurons have quasi-synchronized subthreshold activity (see **Figure 4G**) and the network displays well-defined spatio-temporal patterns consisting of propagating wave fronts of spiking activity from transient phase shifts in the subthreshold oscillations.

**Movie S4 | Spatio-temporal patterns in an autonomous IO network model with weak coupling extended to further neighbors.** Each neuron is connected to 12 neighbors with a weak electrical coupling (*gc* < 0.01 mS/cm2). The effect of increasing the number of connections between neighbors is equivalent to increasing the coupling strength to a moderate magnitude (cf. **Movie S3**).

**Movie S5 | Nearly constant spatial pattern topology in the absence of subthreshold oscillations.** The video shows the activity of an autonomous IO network of tonically spiking neurons without subthreshold oscillations where each unit is connected to four neighbors with a weak electrical coupling (*gc* < 0.05 mS/cm2). The spiking behavior is induced in all neurons of the network by injecting a constant current (see section 2.1 and **Figure 2** for details). In this situation, the IO network generates spatio-temporal patterns, but their spatial topology remains nearly constant (cf. **Movie S3**).

**Movie S6 | Coexistence and coordination of spatio-temporal patterns induced by stimuli.** Activity of an IO network with two external stimuli. The number of connected neighbors is eight with *gc* = 0.05 mS/cm2. External stimuli are introduced as a constant current injected in clusters of closed neurons. In this case, we consider two clusters of 6 × 6 cells with *I*inj1 = 0.75μA/cm2 and *I*inj2 = 0.25μA/cm2. **Figure 8A** shows the approximate position of each cluster. The stimulated neurons have a higher spiking frequency. We observe that each stimulated cluster is the source of a wave front with different frequencies. The IO network is able to simultaneously encode several coexisting spiking rhythms.

**Movie S7 | Encoding of multiple simultaneous rhythms.** The network in this video is equivalent to the network in **Movie S6**, but with 25 stimulated clusters of 6 × 6 cells with different current injections. **Figure 8B** shows the approximate position of each cluster. The effect observed here is the same observed in **Movie S6**, but with many more coexisting spiking rhythms that are nevertheless coordinated through the subthreshold oscillations. Different current injections in the different clusters are encoded in different spiking rhythms.

#### **Movie S8 | Stimulus reverberation allows short memory mechanisms in**

**the IO models.** Activity movie showing the stimulus reverberation effect in an IO network model where each neuron is connected to four nearest neighbors with electrical coupling *gc* = 0.05mS/cm2. Individual neuron parameters in this simulation are σ = 2 and *I*inj = 0.35μA/cm2. Note that parameters are set so that each individual neuron only displays subthreshold oscillations in isolation. The simulation starts with a period without stimulation (from 0:00 to 0:40). After this interval, two clusters of 6 × 6 neurons receive an external stimuli (from 0:40 to 2:00). Here on, the IO network does not receive any external stimulus again. Note that the stimulus induced spatio-temporal patterns survive long after the end of the excitation.

#### **Movie S9 | The source-sink phenomena allows the IO models to attract**

**the wave fronts to specific locations.** Note that in this video the color scale changes with respect to the one used in the rest of IO activity movies. Here, blue color means that the corresponding neuron is under the firing

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**Movie S11 | Modulatory effect of the IL in the presence of stimuli.** Activity movie showing the modulatory effect of the IL in the same network model as in **Movie S10** but in the presence of two external stimuli (cf. **Figures 8C** and **Figures 12D**, which shows the DWT analysis).

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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 11 June 2013; accepted: 09 August 2013; published online: 05 September 2013.*

*Citation: Latorre R, Aguirre C, Rabinovich MI and Varona P (2013) Transient dynamics and rhythm coordination of inferior olive spatiotemporal patterns. Front. Neural Circuits 7:138. doi: 10.3389/fncir. 2013.00138*

*This article was submitted to the journal Frontiers in Neural Circuits.*

*Copyright © 2013 Latorre, Aguirre, Rabinovich and Varona. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

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