Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example

TL;DR

This paper investigates the bifurcations of mixed-mode oscillations in three-timescale systems using geometric singular perturbation theory, revealing a new mechanism for transitions between different oscillation patterns and connecting it to known models like Koper and Hodgkin-Huxley.

Contribution

The paper introduces a novel geometric mechanism explaining transitions in mixed-mode oscillations in three-timescale systems, supported by a prototypical model and applications to existing biological models.

Findings

Identified a new geometric mechanism for MMO bifurcations.

Demonstrated the prototypical system's relevance to the Koper model.

Connected the mechanism to Hodgkin-Huxley dynamics.

Abstract

We study a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. We apply geometric singular perturbation theory to explore the dependence of the geometry of these systems on their parameters, with a focus on mixed-mode oscillations (MMOs) and their bifurcations. In particular, we uncover a novel geometric mechanism that encodes the transition from MMOs with single epochs of small-amplitude oscillations (SAOs) to those with double-epoch SAOs. We identify a relatively simple prototypical three-timescale system that realises our mechanism, featuring a one-dimensional $S$-shaped supercritical manifold that is embedded into a two-dimensional $S$-shaped critical manifold in a symmetric fashion. We show that the Koper model from chemical kinetics is merely a particular realisation of that prototypical system for a specific choice of parameters; in particular, we explain the robust occurrence of mixed-mode dynamics with double epochs of SAOs therein. Finally, we argue that our geometric mechanism can elucidate the mixed-mode dynamics of more complicated systems with a similar underlying geometry, such as of a three-dimensional, three-timescale reduction of the Hodgkin-Huxley equations from mathematical neuroscience.