Wave generation in unidirectional chains of idealized neural oscillators

TL;DR

This paper provides a rigorous mathematical proof of wave generation in unidirectional chains of idealized neural oscillators, demonstrating existence, stability, and robustness of traveling waves under uniform forcing.

Contribution

It offers the first explicit proof of wave generation in such neural oscillator chains, including stability and robustness results in a simplified setting.

Findings

Existence of traveling waves for all parameters in the model.

Global stability of the generated waves.

Robustness of wave solutions to perturbations in forcing.

Abstract

We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the far-downstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems. Here, we give a full mathematical proof of generation under uniform forcing in a simple piecewise affine setting for which the dynamics can be solved explicitly. In particular, our analysis proves existence, global stability, and robustness with respect to perturbations of the forcing, of families of waves with arbitrary period/wave number in some range, for every value of the parameters in the system.