A Multiscale Analysis of Traveling Waves in Stochastic Neural Fields

TL;DR

This paper investigates how stochastic noise affects traveling wave solutions in neural fields, revealing phase shifts and fluctuations characterized by a stochastic phase and Ornstein-Uhlenbeck processes.

Contribution

The study introduces a multiscale framework in weighted L^2-space to analyze noise effects on neural field traveling waves, including phase shifts and wave profile fluctuations.

Findings

Phase shifts are approximately diffusive under noise.

Wave fluctuations follow a stationary Ornstein-Uhlenbeck process.

The wave stability persists despite stochastic perturbations.

Abstract

We analyze the effects of noise on the traveling wave dynamics in neural fields. The noise influences the dynamics on two scales: first, it causes fluctuations in the wave profile, and second, it causes a random shift in the phase of the wave. We formulate the problem in a weighted $L^2$-space, allowing us to separate the two spatial scales. By tracking the stochastic solution with a reference wave we obtain an expression for the stochastic phase. We derive an expansion of the stochastic wave, describing the influence of the noise to different orders of the noise strength. To first order of the noise strength, the phase shift is roughly diffusive and the fluctuations are given by a stationary Ornstein-Uhlenbeck process orthogonal to the direction of movement. This also expresses the stability of the wave under noise.