Existence and stability of periodic solutions in a neural field equation

TL;DR

This paper investigates the existence and stability of 1-bump periodic solutions in a neural field model with a discontinuous activation function, providing analytical conditions and spectral analysis for stability.

Contribution

It introduces necessary and sufficient conditions for 1-bump periodic solutions and analyzes their stability via spectral properties of the associated operator.

Findings

Conditions for existence of 1-bump solutions derived

Eigenvalues and eigenfunctions explicitly characterized

Stability determined through spectral analysis

Abstract

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.