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

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; 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 onedimensional 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.

Geometric singular perturbation analysis of the multiple-timescale Hodgkin-Huxley equations

“…We then consider the scenario where Equation ( 18) exhibits dynamics on three timescales, in which case the full system in (5) -or, equivalently, in (15) -exhibits dynamics on four timescales. In Section 3 and Section 4, the variables h and n are taken to be slow, respectively; we demonstrate that the geometric mechanisms proposed in [10] can explain bifurcations of MMOs that have been previously documented, but not emphasised in the context of GSPT, in the literature [4]. We conclude the article in Section 5 with a summary, and an outlook to future research.…”

“…We show that these two regimes are not fundamentally different in terms of their singular geometry and of the resulting mixed-mode dynamics. In particular, we demonstrate that, when h is taken to be the slowest variable, the geometric mechanisms introduced in [10] can reproduce the various firing patterns observed in (5), and the bifurcations between those, with the rescaled applied current Ī the relevant bifurcation parameter. We explain the transition from MMOs with double epochs of SAOs to MMOs with single epochs of SAOs and then to relaxation oscillation, cf.…”

“…Moreover, the O(ε)-terms in (25a) and (27a) contribute to local qualitative phenomena, which in turn affect the resulting global mixed-mode dynamics. These terms can therefore not be neglected in our analysis; see Appendix A and [10] for details.…”

“…recall (10) for the definition of V (v, m, h, n). We emphasise that Equation (27) and the centre manifold reduction in (17) that was introduced in [15] admit the same critical manifold M 2 as is defined by (29) -or, equivalently, by (30).…”

“…1 exhibit double epochs of SAOs, as their time series feature SAOs both "above" and "below". In [10], geometric mechanisms were described that distinguish MMOs with double epochs of SAOs from those with single SAO epochs -as shown in panels (d) and (e) -as well as from relaxation oscillation in a general family of three-timescale systems; see panel (f). A further distinction can be made between the MMO trajectories with double epochs in panels (a) and (b) and the transitive mixed-mode dynamics observed in panel (c) in which SAO epochs above and below are separated by LAOs; see Section 3 for details.…”