Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations

TL;DR

This paper investigates mixed-mode oscillations in a hybrid neuronal model, revealing how the interplay of continuous dynamics and resets generates complex oscillatory patterns and classifying these behaviors geometrically.

Contribution

It introduces a novel geometric and dynamical systems framework for understanding MMOs in hybrid neuron models, emphasizing the role of invariant manifolds and a singular adaptation map.

Findings

MMOs arise from the hybrid structure of the model.

A classification of reset points associated with small oscillations.

Complex behaviors are analyzed using rotation theory for discontinuous maps.

Abstract

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.