Mean Field Dynamics of a Network of Wilson-Cowan Neurons with Electrical Synapses

TL;DR

This paper proves that in large networks of Wilson-Cowan neurons with electrical synapses, neurons behave independently, using Gaussian mean-field equations and probabilistic inequalities to handle nonlinear synaptic interactions.

Contribution

It establishes propagation of chaos for a Wilson-Cowan neuron network with nonlinear electrical synapses, extending classical results to more complex synaptic models.

Findings

Neurons become independent as network size grows.

Mean-field equations are Gaussian, enabling tail decay analysis.

Propagation of chaos holds despite nonlinear synaptic connections.

Abstract

In this paper we prove the propagation of chaos property for an ensemble of interacting neurons subject to independent Brownian noise. The propagation of chaos property means that in the large network size limit, the neurons behave as if they are probabilistically independent. The model for the internal dynamics of the neurons is taken to be that of Wilson and Cowan, and we consider there to be multiple different populations. The synaptic connections are modelled with a nonlinear `electrical' model. The nonlinearity of the synaptic connections means that our model lies outside the scope of classical propagation of chaos results. We obtain the propagation of chaos result by taking advantage of the fact that the mean-field equations are Gaussian, which allows us to use Borell's Inequality to prove that its tails decay exponentially.