Noise Sharing and Mexican Hat Coupling in a Stochastic Neural Field

TL;DR

This paper investigates how Mexican Hat coupling and spatially-smoothed noise influence pattern formation in stochastic neural fields, revealing conditions under which noise sustains or amplifies spatial patterns.

Contribution

It provides a quantitative analysis of the effects of coupling parameters and noise smoothing on pattern formation in stochastic neural fields, highlighting noise's role in pattern sustenance.

Findings

Spatially-smoothed noise can induce pattern formation without coupling.

Certain parameter combinations optimize pattern emergence.

Stochasticity can sustain and amplify patterns damped in deterministic systems.

Abstract

A diffusion-type coupling operator biologically significant in neuroscience is a difference of Gaussian functions (Mexican Hat operator) used as a spatial-convolution kernel. We are interested in pattern formation by \emph{stochastic} neural field equations, a class of space-time stochastic differential-integral equations using the Mexican Hat kernel. We explore, quantitatively, how the parameters that control the shape of the coupling kernel, coupling strength, and aspects of spatially-smoothed space-time noise, influence the pattern in the resulting evolving random field. We confirm that a spatial pattern that is damped in time in a deterministic system may be sustained and amplified by stochasticity. We find that spatially-smoothed noise alone causes pattern formation even without direct spatial coupling. Our analysis of the interaction between coupling and noise sharing allows us to determine parameter combinations that are optimal for the formation of spatial pattern.