Noise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fields

TL;DR

This paper investigates how noise influences bifurcations in nonlinear Fokker-Planck equations modeling stochastic neural fields, revealing multiple bifurcations and pattern formations as noise levels vary, supported by numerical analysis.

Contribution

It extends bifurcation analysis to stochastic neural field models, characterizing noise-driven bifurcations and pattern formations in a nonlinear, non-local PDE framework.

Findings

Multiple bifurcations occur as noise decreases.

Bifurcation branches and their shapes are characterized.

Hexagonal patterns are identified as prevalent stable states.

Abstract

The existence and characterisation of noise-driven bifurcations from the spatially homogeneous stationary states of a nonlinear, non-local Fokker--Planck type partial differential equation describing stochastic neural fields is established. The resulting theory is extended to a system of partial differential equations modelling noisy grid cells. It is shown that as the noise level decreases, multiple bifurcations from the homogeneous steady state occur. Furthermore, the shape of the branches at a bifurcation point is characterised locally. The theory is supported by a set of numerical illustrations of the condition leading to bifurcations, the patterns along the corresponding local bifurcation branches, and the stability of the homogeneous state and the most prevalent pattern: the hexagonal one.