Stochastic Wilson-Cowan models of neuronal network dynamics with memory and delay

TL;DR

This paper introduces a stochastic Wilson-Cowan model with memory and delay to study neuronal network dynamics, revealing robust avalanche size power laws and complex non-Markovian effects through numerical simulations.

Contribution

It develops a Markovian framework for non-Markovian neuronal dynamics with refractory delay, analyzing avalanche phenomena in balanced networks.

Findings

Power law in avalanche size distribution with exponent around -1.16

Robustness of power law across refractory timescales

Avalanche durations follow a biexponential distribution

Abstract

We consider a simple Markovian class of the stochastic Wilson-Cowan type models of neuronal network dynamics, which incorporates stochastic delay caused by the existence of a refractory period of neurons. From the point of view of the dynamics of the individual elements, we are dealing with a network of non-Markovian stochastic two-state oscillators with memory which are coupled globally in a mean-field fashion. This interrelation of a higher-dimensional Markovian and lower-dimensional non-Markovian dynamics is discussed in its relevance to the general problem of the network dynamics of complex elements possessing memory. The simplest model of this class is provided by a three-state Markovian neuron with one refractory state, which causes firing delay with an exponentially decaying memory within the two-state reduced model. This basic model is used to study critical avalanche dynamics (the noise sustained criticality) in a balanced feedforward network consisting of the excitatory and inhibitory neurons. Such avalanches emerge due to the network size dependent noise (mesoscopic noise). Numerical simulations reveal an intermediate power law in the distribution of avalanche sizes with the critical exponent around -1.16. We show that this power law is robust upon a variation of the refractory time over several orders of magnitude. However, the avalanche time distribution is biexponential. It does not reflect any genuine power law dependence.