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

L. Spek, K. Dijkstra, S.A. van Gils, M. Polner

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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.

Original languageEnglish
Pages (from-to)439-473
Number of pages35
JournalJournal of differential equations
Volume317
Early online date15 Feb 2022
DOIs
Publication statusPublished - 25 Apr 2022

Keywords

  • Neural field equation
  • Delay differential equation
  • Sturm-Liouville problem
  • Completeness of eigenfunctions
  • Hopf bifurcation
  • UT-Gold-D

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