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

Abstract

We set up a system of Caputo-type fractional differential equations for a reduced neuron model known as the denatured Morris–Lecar (dML) system. 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. Fractional order systems induce memory effects to excitable cells, thus providing an efficient and biophysically more realistic scenario. All theoretical analyses in this study are supported by rigorous numerical simulations.