The diversity of neuron models used in contemporary theoretical neuroscience to investigate specific properties of covariances in the spiking activity raises the question how these models relate to each other. In particular it is hard to distinguish between generic properties of covariances and peculiarities due to the abstracted model. Here we present a unified view on pairwise covariances in recurrent networks in the irregular regime. We consider the binary neuron model, the leaky integrate-and-fire (LIF) model, and the Hawkes process. We show that linear approximation maps each of these models to either of two classes of linear rate models (LRM), including the Ornstein–Uhlenbeck process (OUP) as a special case. The distinction between both classes is the location of additive noise in the rate dynamics, which is located on the output side for spiking models and on the input side for the binary model. Both classes allow closed form solutions for the covariance. For output noise it separates into an echo term and a term due to correlated input. The unified framework enables us to transfer results between models. For example, we generalize the binary model and the Hawkes process to the situation with synaptic conduction delays and simplify derivations for established results. Our approach is applicable to general network structures and suitable for the calculation of population averages. The derived averages are exact for fixed out-degree network architectures and approximate for fixed in-degree. We demonstrate how taking into account fluctuations in the linearization procedure increases the accuracy of the effective theory and we explain the class dependent differences between covariances in the time and the frequency domain. Finally we show that the oscillatory instability emerging in networks of LIF models with delayed inhibitory feedback is a model-invariant feature: the same structure of poles in the complex frequency plane determines the population power spectra.
Keywords: correlations, linear response, Hawkes process, leaky integrate-and-fire model, binary neuron, linear rate model, Ornstein–Uhlenbeck process
Citation: Grytskyy D, Tetzlaff T, Diesmann M and Helias M (2013) A unified view on weakly correlated recurrent networks. Front. Comput. Neurosci. 7:131. doi: 10.3389/fncom.2013.00131
Received: 22 April 2013; Accepted: 10 September 2013;
Published online: 18 October 2013.
Edited by:Ruben Moreno-Bote, Foundation Sant Joan de Deu, Spain
Copyright © 2013 Grytskyy, Tetzlaff, Diesmann and Helias. 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: Dmytro Grytskyy, Jülich Research Centre and JARA, Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6), Building 15.22, 52425 Jülich, Germany e-mail: firstname.lastname@example.org