Slow passage through a Hopf-like bifurcation in piecewise linear systems: application to elliptic bursting

TL;DR

This paper investigates the slow passage through a Hopf-like bifurcation in piecewise linear systems, providing new conditions for this phenomenon and applying it to neuronal models exhibiting elliptic bursting.

Contribution

It introduces the first analysis of slow passage through a Hopf bifurcation in PWL systems, with conditions linked to system structure and manifold connections.

Findings

Conditions for Hopf-like bifurcation in PWL systems derived

Analysis of way-in/way-out function in PWL context completed

Application to neuronal bursting model demonstrating practical relevance

Abstract

The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly-varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of a Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems. In the present work, we study for the first time this phenomenon in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with the number of linearity zones considered in the system and possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.