Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

TL;DR

This paper establishes a rigorous mathematical framework for analyzing stochastic neural field equations with spatial noise, providing stability results and new representations for stochastic convolutions.

Contribution

It introduces a comprehensive framework for stochastic neural fields, including a novel representation formula and a dynamic phase-adaption process.

Findings

Rigorous stability results for stochastic neural fronts.

New representation formula for stochastic convolutions.

Effective multiscale analysis of stochastic neural dynamics.

Abstract

We develop a complete and rigorous mathematical framework for the analysis of stochastic neural field equations under the influence of spatially extended additive noise. By comparing a solution to a fixed deterministic front profile it is possible to realise the difference as strong solution to an $L^2(\mathbb{R})$-valued SDE. A multiscale analysis of this process then allows us to obtain rigorous stability results. Here a new representation formula for stochastic convolutions in the semigroup approach to linear function-valued SDE with adapted random drift is applied. Additionally, we introduce a dynamic phase-adaption process of gradient type.