Canonical models for torus canards in elliptic bursters

TL;DR

This paper analyzes elliptic bursting dynamics through torus canards, introduces a new canonical model called Leidenator that supports various bursting transitions, and uses singular flow analysis to predict complex neuronal behaviors.

Contribution

The authors identify limitations of existing canonical models and develop the Leidenator, a modified model that accurately captures transitions involving torus canards in elliptic bursters.

Findings

Classical and mixed-type torus canards occur at bursting transitions.

The Leidenator model supports both types of torus canards and transitions.

Singular flow analysis effectively predicts bursting dynamics.

Abstract

We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards can appear, the difference between the two being the fast subsystem bifurcation that they approach, saddle-node of cycles for the former and subcritical Hopf for the latter. We first showcase such dynamics in a Wilson-Cowan type elliptic bursting model, then we consider minimal models for elliptic bursters in view of finding transitions to and from bursting solutions via both kinds of torus canards. We first consider the canonical model proposed by Izhikevich (ref. [22] in the manuscript) and adapted to elliptic bursting by Ju, Neiman, Shilnikov (ref. [24] in the manuscript), and we show that it does not produce mixed-type torus canards due to a nongeneric transition at one end of the bursting regime. We therefore introduce a perturbative term in the slow equation, which extends this canonical form to a new one that we call Leidenator and which supports the right transitions to and from elliptic bursting via classical and mixed-type torus canards, respectively. Throughout the study, we use singular flows ($\varepsilon=0$) to predict the full system's dynamics ($\varepsilon>0$ small enough). We consider three singular flows: slow, fast and average slow, so as to appropriately construct singular orbits corresponding to all relevant dynamics pertaining to elliptic bursting and torus canards.