A general framework for stochastic traveling waves and patterns, with application to neural field equations

TL;DR

This paper introduces a comprehensive framework for analyzing how spatio-temporal noise influences traveling waves and patterns in neural field equations, including stability and long-term behavior.

Contribution

It develops a rigorous stochastic framework that models wave position jumps and stability, extending deterministic results to noisy neural field equations.

Findings

Framework captures wave position jumps due to noise

Stability analysis recovers deterministic results in small-noise limit

Long-term behavior of stochastic waves characterized

Abstract

In this paper we present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses or stationary patterns. To do this, we first formulate a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump. We then study the local stability of this stochastic front, obtaining a result that recovers a well-known deterministic result in the small-noise limit. We finish with a study of the long-time behavior of the stochastic wave.