Dynamics of delayed neural field models in two-dimensional spatial domains

TL;DR

This paper analyzes the spectral properties of delayed neural field models in two-dimensional domains, transforming the problem into a PDE boundary value problem and exploring bifurcation phenomena.

Contribution

It introduces a method to characterize the spectrum of delayed neural field models by converting the DDE into a PDE boundary value problem and analyzes bifurcations.

Findings

Spectral properties are characterized for specific connectivity and delay functions.

Eigenvalues are obtained via solutions to a boundary value problem.

An example of Hopf bifurcation and Lyapunov coefficient calculation is provided.

Abstract

Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the spectral properties of the linearized equation. We transform the characteristic integral equation for the delay differential equation (DDE) into a linear partial differential equation (PDE) with boundary conditions. We demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent with obtaining nontrivial solutions of this boundary value problem (BVP). When the connectivity kernel consists of a single exponential, we construct a basis of the solutions of this BVP that forms a complete set in $L^2$. This gives a complete characterization of the spectrum and is used to construct a solution to the resolvent problem. As an application we give an example of a Hopf bifurcation and compute the first Lyapunov coefficient.