On a stable torus in a 3D system with a saddle-focus

TL;DR

This paper introduces a model explaining how stable tori can form near saddle-focus points in 3D systems, relevant for neuronal bursting, and discusses related bifurcations.

Contribution

It presents a conceptual model for stable torus onset near saddle-focus equilibria in slow-fast systems, applicable to neuronal dynamics.

Findings

Stable tori can emerge near saddle-focus equilibria.

The model captures various dynamical regimes in neuronal models.

Homoclinic bifurcations are also analyzed.

Abstract

This paper proposes a conceptual model for the onset of a stable torus near a saddle-focus equilibrium. This bifurcation scenario is typical of slow-fast systems that generate elliptic bursting in a variety of neuronal models in mathematical neuroscience. Variants of the model also capture other dynamical regimes recurring in a neighborhood of the saddle-focus. We also discuss homoclinic bifurcations for which the model assumptions are feasible.