Existence and Uniqueness of Traveling Fronts in Lateral Inhibition Neural Fields with Sigmoidal Firing Rates

TL;DR

This paper rigorously proves the existence and uniqueness of traveling fronts in neural field models with lateral inhibition and sigmoidal firing rates, extending previous work from Heaviside functions using a homotopy approach.

Contribution

It introduces a non-monotone homotopy method to establish existence and uniqueness of traveling fronts in neural fields with smooth firing rates, generalizing prior results.

Findings

Existence of traveling fronts for a range of firing rates

Uniqueness of fronts modulo translation in perturbative cases

Comparison techniques between smooth and Heaviside firing rates

Abstract

We rigorously prove the existence of traveling fronts in neural field models with lateral inhibition coupling types and smooth sigmoidal firing rates. With Heaviside firing rates as our base point (where unique traveling fronts exist), we repeatedly apply the implicit function theorem in Banach spaces to provide a non-monotone version of the homotopy approach originally proposed by Ermentrout and McLeod (1993) in their seminal study of monotone fronts in purely excitatory models. By comparing smooth and Heaviside firing rates, we develop global wave speed and profile comparisons that guide our analysis, leading to uniqueness (modulo translation) in the perturbative case. Moreover, we establish a meaningful a priori existence result; we prove existence holds for a range of firing rates, independent of continuation path.