On Local Bifurcations in Neural Field Models with Transmission Delays

TL;DR

This paper analyzes stability and bifurcations in neural field models with transmission delays using dual semigroup theory, deriving conditions for Hopf bifurcations and double Hopf points.

Contribution

It introduces a functional analytic framework for neural field models with delays and derives explicit formulas for bifurcation analysis.

Findings

Identification of conditions leading to Hopf bifurcations.

Derivation of formulas for double Hopf points.

Numerical evaluation of bifurcation coefficients.

Abstract

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.