Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris-Lecar neuronal model

TL;DR

This paper establishes stability criteria for a two-dimensional system with a Caputo fractional derivative and applies these results to analyze a fractional-order Morris-Lecar neuronal model, including bifurcation and numerical simulations.

Contribution

It generalizes Routh-Hurwitz stability conditions to fractional systems and applies them to a biologically relevant neuronal model.

Findings

Derived stability and instability conditions for fractional systems.

Identified conditions for Hopf bifurcations in the fractional model.

Numerical simulations show rich spiking behavior and compare fractional and integer-order models.

Abstract

Necessary and sufficient conditions are given for the asymptotic stability and instability of a two-dimensional incommensurate order autonomous linear system, which consists of a differential equation with a Caputo-type fractional order derivative and a classical first order differential equation. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional order of the Caputo derivative. In this setting, we obtain a generalization of the well known Routh-Hurwitz conditions. These theoretical results are then applied to the analysis of a two-dimensional fractional-order Morris-Lecar neuronal model, focusing on stability and instability properties. This fractional order model is built up taking into account the dimensional consistency of the resulting system of differential equations. The occurrence of Hopf bifurcations is also discussed. Numerical simulations exemplify the theoretical results, revealing rich spiking behavior. The obtained results are also compared to similar ones obtained for the classical integer-order Morris-Lecar neuronal model.