Stochastic collocation schemes for Neural Field Equations with random data

TL;DR

This paper introduces numerical schemes combining spatial projection and stochastic collocation to quantify uncertainty in neural field equations with random data, providing error estimates and convergence analysis.

Contribution

It presents a novel combined approach for spatial discretisation and stochastic collocation in neural field equations with random parameters, including error analysis and convergence guarantees.

Findings

Schemes achieve predicted convergence rates in numerical experiments.

Error estimates depend on the spatial projector and data analyticity.

Applicable to both linear and nonlinear neural field models.

Abstract

We develop and analyse numerical schemes for uncertainty quantification in neural field equations subject to random parametric data in the synaptic kernel, firing rate, external stimulus, and initial conditions. The schemes combine a generic projection method for spatial discretisation to a stochastic collocation scheme for the random variables. We study the problem in operator form, and derive estimates for the total error of the schemes, in terms of the spatial projector. We give conditions on the projected random data which guarantee analyticity of the semi-discrete solution as a Banach-valued function. We illustrate how to verify hypotheses starting from analytic random data and a choice of spatial projection. We provide evidence that the predicted convergence rates are found in various numerical experiments for linear and nonlinear neural field problems.