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Front. Comput. Neurosci. | doi: 10.3389/fncom.2018.00009

Self-consistent scheme for spike-train power spectra in heterogeneous sparse networks

  • 1Física, Universidade de São Paulo Ribeirão Preto, Brazil
  • 2Bernstein Center for Computational Neuroscience, Germany
  • 3Physics, Humboldt-Universität zu Berlin, Germany

Recurrent networks of spiking neurons can be in an asynchronous state characterized by low or absent cross-correlations and spike statistics which resemble those of cortical neurons. Although spatial correlations are negligible in this state, neurons can show pronounced temporal correlations in their spike trains that can be quantified by the autocorrelation function or the spike-train power spectrum. Depending on cellular and network parameters, correlations display diverse patterns (ranging from simple refractory-period effects and stochastic oscillations to slow fluctuations) and it is generally not well understood how these dependencies come about. Previous work has explored how the single-cell correlations in a homogeneous network (excitatory and inhibitory integrate-and-fire neurons with nearly balanced mean recurrent input) can be determined numerically from an iterative single-neuron simulation. Such a scheme is based on the fact that every neuron is driven by the network noise (i.e. the input currents from all its presynaptic partners) but also contributes to the network noise, leading to a self-consistency condition for the input and output spectra. Here we first extend this scheme to homogeneous networks with strong recurrent inhibition and a synaptic filter, in which instabilities of the previous scheme are avoided by an averaging procedure. We then extend the scheme to heterogeneous networks in which i) different neural subpopulations (e.g. excitatory and inhibitory neurons) have different cellular or connectivity parameters; ii) the number and strength of the input connections are random (Erdős–Rényi topology) and thus different among neurons. In all heterogeneous cases, neurons are lumped in different classes each of which is represented by a single neuron in the iterative scheme; in addition, we make a Gaussian approximation of the input current to the neuron. These approximations seem to be justified over a broad range of parameters as indicated by comparison with simulation results of large recurrent networks. Our method can help to elucidate how network heterogeneity shapes the asynchronous state in recurrent neural networks.

Keywords: neural noise, recurrent neural networks, neural dynamics, spike-train statistics, spike-train power spectrum, non-Poissonian spiking

Received: 29 Sep 2017; Accepted: 07 Feb 2018.

Edited by:

Markus Diesmann, Forschungszentrum Jülich, Germany

Reviewed by:

Maurizio Mattia, Istituto Superiore di Sanità, Italy
Alex Roxin, Centre de Recerca Matemàtica, Spain  

Copyright: © 2018 De Oliveira Pena, Vellmer, Bernardi, Roque and Lindner. 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) and the copyright owner 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: Mr. Rodrigo Felipe De Oliveira Pena, Universidade de São Paulo Ribeirão Preto, Física, Av Bandeirantes 3900, Monte Alegre, Ribeirão Preto, 14040901, SP, Brazil, rfdop@uol.com.br