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

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; the former feature SAOs or pseudo-plateau bursting either "below" or "above" in their time series, while in the latter, SAOs or pseudo-plateau bursting occur both "below" and "above." We identify a relatively simple prototypical three-timescale system that realizes our mechanism, featuring a one-dimensional S-shaped 2-critical 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 realization 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 a three-dimensional, three-timescale reduction of the Hodgkin-Huxley equations from mathematical neuroscience.