A showcase of torus canards in neuronal bursters

TL;DR

This paper demonstrates that torus canards are a widespread phenomenon in neuronal models, explaining the transition from spiking to bursting across various types of neuronal dynamics.

Contribution

It extends the understanding of torus canards by showing their occurrence in multiple well-known neuronal models with different bursting behaviors.

Findings

Torus canards arise in diverse neuronal models with multiple time scales.

The transition from spiking to bursting involves an explosion of torus canards.

Torus canards are proposed as a common mechanism separating spiking and bursting regimes.

Abstract

Rapid action potential generation --- spiking --- and alternating intervals of spiking and quiescence --- bursting --- are two dynamic patterns observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which was originally observed in a simple biophysical model of a Purkinje cell. In this article, we expand on that original result by showing that torus canards arise in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.