
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 , 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…
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