Generic Torus Canards

TL;DR

This paper develops an analytical framework combining Floquet, averaging, and geometric singular perturbation theories to identify and classify generic torus canards in higher-dimensional slow/fast systems, with applications to neural and calcium dynamics.

Contribution

It introduces a rigorous method for analyzing and classifying torus canards in four-dimensional systems, extending prior numerical and heuristic approaches.

Findings

Established that the average of a torus canard is a folded singularity canard.

Provided an analytic scheme for topological classification of torus canards.

Applied the theory to calcium dynamics, explaining amplitude-modulated bursting.

Abstract

Torus canards are solutions of slow/fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. A relatively new dynamic phenomenon, torus canards have been found in neural applications to mediate the transition from spiking to bursting via amplitude-modulated spiking. In $\mathbb{R}^3$, torus canards are degenerate: they require one-parameter families of 2-fast/1-slow systems in order to be observed and even then, they only occur on exponentially thin parameter intervals. The addition of a second slow variable unfolds the torus canard phenomenon, making them generic and robust. That is, torus canards in slow/fast systems with (at least) two slow variables occur on open parameter sets. So far, generic torus canards have only been studied numerically, and their behaviour has been inferred based on averaging and canard theory. This approach, however, has not been rigorously justified since the averaging method breaks down near a fold of periodics, which is exactly where torus canards originate. In this work, we combine techniques from Floquet theory, averaging theory, and geometric singular perturbation theory to show that the average of a torus canard is a folded singularity canard. In so doing, we devise an analytic scheme for the identification and topological classification of torus canards in $\mathbb{R}^4$. We demonstrate the predictive power of our results in a model for intracellular calcium dynamics, where we explain the mechanisms underlying a novel class of elliptic bursting rhythms, called amplitude-modulated bursting, by constructing the torus canard analogues of mixed-mode oscillations. We also make explicit the connection between our results here with prior studies of torus canards and torus canard explosion in $\mathbb{R}^3$, and discuss how our methods can be extended to slow/fast systems of arbitrary (finite) dimension.