Pattern formation, i.e., the generation of an inhomogeneous spatial activity distribution in a dynamical system with translation invariant structure, is a well-studied phenomenon in neuronal network dynamics, specifically in neural field models. These are population models to describe the spatio-temporal dynamics of large groups of neurons in terms of macroscopic variables such as population firing rates. Though neural field models are often deduced from and equipped with biophysically meaningful properties, a direct mapping to simulations of individual spiking neuron populations is rarely considered. Neurons have a distinct identity defined by their action on their postsynaptic targets. In its simplest form they act either excitatorily or inhibitorily. When the distribution of neuron identities is assumed to be periodic, pattern formation can be observed, given the coupling strength is supracritical, i.e., larger than a critical weight. We find that this critical weight is strongly dependent on the characteristics of the neuronal input, i.e., depends on whether neurons are mean- or fluctuation driven, and different limits in linearizing the full non-linear system apply in order to assess stability. In particular, if neurons are mean-driven, the linearization has a very simple form and becomes independent of both the fixed point firing rate and the variance of the input current, while in the very strongly fluctuation-driven regime the fixed point rate, as well as the input mean and variance are important parameters in the determination of the critical weight. We demonstrate that interestingly even in “intermediate” regimes, when the system is technically fluctuation-driven, the simple linearization neglecting the variance of the input can yield the better prediction of the critical coupling strength. We moreover analyze the effects of structural randomness by rewiring individual synapses or redistributing weights, as well as coarse-graining on the formation of inhomogeneous activity patterns.
Keywords: pattern formation, spiking neurons, linear model, mean-driven, fluctuation driven, ring networks, small-world networks
Citation: Kriener B, Helias M, Rotter S, Diesmann M and Einevoll GT (2014) How pattern formation in ring networks of excitatory and inhibitory spiking neurons depends on the input current regime. Front. Comput. Neurosci. 7:187. doi: 10.3389/fncom.2013.00187
Received: 12 April 2013; Accepted: 09 December 2013;
Published online: 07 January 2014.
Edited by:Klaus R. Pawelzik, University Bremen, Germany
Reviewed by:Klaus R. Pawelzik, University Bremen, Germany
Copyright © 2014 Kriener, Helias, Rotter, Diesmann and Einevoll. 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.
*Correspondence: Birgit Kriener, Computational Neuroscience Group, Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, PO Box 5003, 1432 Ås, Norway e-mail: firstname.lastname@example.org
†These authors have contributed equally to this work.