Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

TL;DR

This paper establishes well-posedness and stability results for a complex stochastic neural field model expressed as a nonlinear PDE, transforming it into a Stefan-like problem to analyze solution behavior over time.

Contribution

It introduces a novel transformation of the stochastic neural field PDE into a Stefan-like free boundary problem, enabling the proof of local and global well-posedness and stability.

Findings

Constructed local-in-time smooth solutions under mild hypotheses.

Proved global-in-time solutions for certain initial conditions.

Enhanced stability results to include nonlinear asymptotic stability.

Abstract

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.