Bottom-up approach to torus bifurcation in neuron models

TL;DR

This paper investigates the transition phenomena between different neuronal activity states in Hodgkin-Huxley models, focusing on torus bifurcations and their role in quasi-periodic dynamics.

Contribution

It introduces a geometric slow-fast dissection and parameter continuation method to analyze torus bifurcations in neuron models, revealing new insights into their bifurcation structure.

Findings

Identified torus bifurcations as key in activity transitions

Described various types of torus bifurcations including stable and saddle torus-canards

Linked torus breakdown to complex and bistable neuronal dynamics

Abstract

We study the quasi-periodicity phenomena occurring at the transition between tonic spiking and bursting activities in exemplary biologically plausible Hodgkin-Huxley type models of individual cells and reduced phenomenological models with slow and fast dynamics. Using the geometric slow-fast dissection and the parameter continuation approach we show that the transition is due to either the torus bifurcation or the period-doubling bifurcation of a stable periodic orbit on the 2D slow-motion manifold near a characteristic fold. We examine various torus bifurcations including stable and saddle torus-canards, resonant tori, the co-existence of nested tori and the torus breakdown leading to the onset of complex and bistable dynamics in such systems.