Chaos, spike-adding, and mixed-mode oscillations in a nonlinear neuronal model with resets

Abstract: Neuronal models with resets, or hybrid models, can capture complicated dynamics in a relatively low-dimensional setting. I will discuss two activity patterns, chaos and mixed-mode oscillations (MMOs), in a general class of these hybrid models. I will explain the mechanisms underlying chaos and MMOs as well as the pathways through which various model parameters can modulate the resulting model dynamics. The analysis makes use of a one-dimensional adaptation map. I will discuss how under parameter variations, chaos emerges in association with period-incrementing transitions, and how, in another regime, rotation theory for circle maps can be used to elucidate the MMO dynamics.