Neural Field Equations with random data

TL;DR

This paper investigates neural field equations with random data, establishing conditions for solutions' existence, uniqueness, and regularity, and analyzing both continuous and discretized models to support uncertainty quantification.

Contribution

It provides a rigorous mathematical framework for neural fields with randomness, including existence, uniqueness, and regularity results, and sets the stage for uncertainty quantification methods.

Findings

Conditions for existence and uniqueness of solutions

Regularity results relating inputs and solutions

Framework for analyzing discretized neural fields

Abstract

We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view this spatially-extended, nonlocal evolution equation as a Cauchy problem on abstract Banach spaces, with randomness in the synaptic kernel, firing rate function, external stimuli, and initial conditions. We determine conditions on the random data that guarantee existence, uniqueness, and measurability of the solution in an appropriate Banach space, and examine the regularity of the solution in relation to the regularity of the inputs. We present results for linear and nonlinear neural fields, and for the two most common functional setups in the numerical analysis of this problem. In addition to the continuous problem, we analyse in abstract form neural fields that have been spatially discretised, setting the foundations for analysing uncertainty quantification (UQ) schemes.