Abstract

Phase oscillator ensembles exhibiting a preferred direction of coupling are a much studied topic of mathematical neuroscience, where they provide a framework for understanding the generation and propagation of electrical impulses across brain tissues. We report here on some recent successes in establishing stability of traveling wave solutions (TW) in feedforward chains of idealized neural oscillators featuring a pulse emission / Type-I phase response (PRC) interaction. In prior work, the smooth version of this model was studied through a combined numerical / analytical methodology: an iterative fixed point scheme was used to verify existence of TW; these findings then supported a relevant abstract hypothesis that proved sufficient to establish global stability of the solution. We have since completed a full mathematically-rigorous study of a piecewise affine version of this model, establishing all of the observed phenomenology (both existence and stability of TW) analytically. In addition, we now have an understanding of how a robust TW solution in this setting may be generated with a variety of external forcing stimuli, including such that are structurally distinct from the consequent wave. This is an apparent feature of this class of models.

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