Firing Activities in Fractional-Order Hindmarsh-Rose Neuron with Multistable Memristor as Autapse

Abstract

Compared with integer order neurons, fractional-order neuron model can more accurately describe the firing behavior of biological neurons. Considering the fact that memristors have the characteristics similar to biological synapses, a fractional-order multistable memristor is firstly proposed in this study. It is verified that the fractional-order memristor has multiple local active regions and multiple stable hysteresis loops, and the influence of fractional order on its nonvolatility is also revealed. Then by considering the fractional-order memristor as an autapse of HR neuron model, a fractional-order memristive neuron model is developed. The effects of the initial value, external excitation current, coupling strength and fractional order on the firing behavior are discussed by time series, phase diagrams, Lyapunov exponents and inter spike interval (ISI) bifurcation diagrams. Three coexisting firing patterns, including irregulate A-periodic bursting, A-periodic bursting and chaotic bursting, dependent on the memristor initial values are observed. It is also revealed that the fractional order can not only induce the transition of firing patterns, but also change the firing frequency of the neuron. Finally, a neuron circuit with variable fractional order is designed to verify the numerical simulations.