Theoretical / numerical study of modulated traveling waves in inhibition stabilized networks

TL;DR

This paper provides a comprehensive theoretical and numerical analysis of modulated traveling waves in inhibition-stabilized neural networks, including stability, bifurcations, and numerical computation methods.

Contribution

It introduces a stability principle, constructs a finite-dimensional center manifold, and investigates bifurcations and spectral properties of modulated traveling waves in neural field models.

Findings

Existence of bifurcations such as Fold, Hopf, and Bogdanov-Takens in neural traveling pulses.

Numerical evidence for snaking behavior of modulated traveling pulses.

Development of numerical schemes for computing modulated traveling waves.

Abstract

We prove a principle of linearized stability for traveling wave solutions to neural field equations posed on the real line. Additionally, we provide the existence of a finite dimensional invariant center manifold close to a traveling wave, this allows to study bifurcations of traveling waves. Finally, the spectral properties of the modulated traveling waves are investigated. Numerical schemes for the computation of modulated traveling waves are provided. We then apply these results and methods to study a neural field model in a inhibitory stabilized regime. We showcase Fold, Hopf and Bodgdanov-Takens bifurcations of traveling pulses. Additionally, we continue the modulated traveling pulses as function of the time scale ratio of the two neural populations and show numerical evidences for snaking of modulated traveling pulses.