Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactions

TL;DR

This paper studies how different discretisation schemes affect the behavior of reaction-diffusion equations, including neural field models, by constructing travelling wave solutions in the fully discrete setting using spectral convergence techniques.

Contribution

It extends previous methods to construct travelling wave solutions for fully discrete neural field models with infinite-range interactions, refining spectral convergence techniques.

Findings

Constructed travelling wave solutions for fully discrete systems.

Analyzed impact of spatial-temporal discretisation on wave dynamics.

Extended spectral convergence approach to neural field models.

Abstract

We investigate the impact of spatial-temporal discretisation schemes on the dynamics of a class of reaction-diffusion equations that includes the FitzHugh-Nagumo system. For the temporal discretisation we consider the family of six backward differential formula (BDF) methods, which includes the well-known backward-Euler scheme. The spatial discretisations can feature infinite-range interactions, allowing us to consider neural field models. We construct travelling wave solutions to these fully discrete systems in the small time-step limit by viewing them as singular perturbations of the corresponding spatially discrete system. In particular, we refine the previous approach by Hupkes and Van Vleck for scalar fully discretised systems, which is based on a spectral convergence technique that was developed by Bates, Chen and Chmaj.