Diffusion approximation of the stochastic Wilson–Cowan model

• Published in: Chaos, solitons and fractals solitons and fractals, 2017-10, Vol.103, p.504-512
• Main Authors: Zankoc, Clément, Biancalani, Tommaso, Fanelli, Duccio, Livi, Roberto
• Format: Article
• Language: English
• Subjects: Bistability; Kramers Moyal; Neuronal dynamics; Stochastic models
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2017.07.010

•Stochastic Cowan–Wilson model with finite volume constraint.•Kramers Moyal expansion to derive the fuctuating hydrodynamics approximation for the stochastic Wilson Cowan model.•Target the study to the region of bistability.•Simple approximation of the global dynamics in terms of one Langevin equation for the excitatory species: the population of inhibitors acts as a source of intrinsic noise, shaking the discrete ensemble of excitators from the inside.•This is a minimal framework for modelling the dynamics of distinct species of neurons in mutual interaction and subject to the endogenous noise. 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 excitatory neurons on inhibitory neurons, 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.