A gradient flow formulation for the stochastic Amari neural field model

TL;DR

This paper establishes a gradient flow formulation for stochastic Amari neural field models, enabling new analytical methods and providing results on well-posedness, regularity, and ergodic properties of solutions.

Contribution

It introduces a rigorous gradient flow framework for stochastic neural fields, which was previously unknown, and analyzes their well-posedness and long-term behavior.

Findings

Neural field equations can be viewed as gradient flows in a nonlocal Hilbert space.

Solutions remain in the Hilbert space for all times, ensuring well-posedness.

The paper discusses ergodic properties and invariant measures for the models.

Abstract

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.