Fractional order induced bifurcations in Caputo-type denatured Morris-Lecar neurons

TL;DR

This paper investigates how fractional calculus influences bifurcations and oscillatory behaviors in a reduced-order neuron model, revealing new dynamical phenomena and stability conditions through analysis and simulations.

Contribution

It introduces a fractional-order Caputo model for denatured Morris-Lecar neurons and analyzes bifurcations and oscillations with respect to fractional order and coupling strength.

Findings

Identification of bifurcation types influenced by fractional order

Existence of various oscillatory phenomena like tonic spiking and mixed-mode oscillations

Derivation of stability conditions for equilibrium points

Abstract

We set up a system of Caputo-type fractional differential equations for a reduced-order model known as the {\em denatured} Morris-Lecar (dML) neurons. This neuron model has a structural similarity to a FitzHugh-Nagumo type system. We explore both a single-cell isolated neuron and a two-coupled dimer that can have two different coupling strategies. The main purpose of this study is to report various oscillatory phenomena (tonic spiking, mixed-mode oscillation) and bifurcations (saddle-node and Hopf) that arise with variation of the order of the fractional operator and the magnitude of the coupling strength for the coupled system. Various closed-form solutions as functions of the system parameters are established that act as the necessary and sufficient conditions for the stability of the equilibrium point. The theoretical analysis are supported by rigorous numerical simulations.