Fluctuating hydrodynamics approximation of the stochastic Cowan-Wilson model

TL;DR

This paper develops a stochastic hydrodynamics approximation for the Wilson-Cowan neural model, capturing finite population effects and bistability, validated through simulations and analytical progress.

Contribution

It introduces a novel Langevin equation-based approximation for the stochastic Wilson-Cowan model, including analytical insights and validation methods.

Findings

Validates the approximation with Gillespie simulations.

Derives coupled non-linear Langevin equations with multiplicative noise.

Provides analytical progress by simplifying the model to study bistability.

Abstract

We consider a stochastic version of the Wilson-Cowan model which accommodates for discrete populations of excitatory and inhibitory neurons. The model assumes a finite carrying capacity with the two populations being constant in size. The master equation that governs the dynamics of the stochastic model is analyzed by an expansion in powers of the inverse population size, yielding a coupled pair of non-linear Langevin equations with multiplicative noise. Gillespie simulations show the validity of the obtained approximation, for the parameter region where the system exhibits dynamical bistability. We report analytical progress by silencing the retroaction of the activators on the inhibitors, while still assigning the parameters so to fall in the region of deterministic bistability for the excitatory species. The proposed approach forms the basis of a perturbative generalization which applies to the case where a modest degree of coupling is restored.