Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

TL;DR

This paper investigates bifurcations in a class of 2D hybrid systems under periodic forcing, revealing complex dynamics including cascades of gluing bifurcations and multistability through analytical and numerical methods.

Contribution

It introduces a semi-rigorous analysis of periodic orbits and bifurcations in a 2D integrate-and-fire model with dynamic threshold under periodic forcing.

Findings

Existence of periodic orbits with grazing bifurcations.

Identification of cascades of gluing bifurcations.

Presence of multistability between different periodic orbits.

Abstract

In this work we consider a general class of $2$-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a $2$-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.