Dwell time for local stability of switched systems with application to non-spiking neuron models

TL;DR

This paper derives explicit formulas for dwell time in switched systems to ensure local stability, with applications to non-spiking neuron models, including both linear and nonlinear cases.

Contribution

It introduces the first explicit dwell-time conditions for non-spiking in linear neuron models with periodic switching, extending to nonlinear systems.

Findings

Closed-form formulas for dwell time ensure local stability.

Explicit dwell-time condition prevents spiking in linear neuron models.

Extension of stability analysis to nonlinear switched systems.

Abstract

For switched systems that switch between distinct globally stable equilibria, we offer closed-form formulas that lock oscillations in the required neighborhood of the equilibria. Motivated by non-spiking neuron models, the main focus of the paper is on the case of planar switched affine systems, where we use properties of nested cylinders coming from quadratic Lyapunov functions. In particular, for the first time ever, we use the dwell-time concept in order to give an explicit condition for non-spiking of linear neuron models with periodically switching current. An extension to the general nonlinear case is also given.