Hidden dynamics in a fractional-order memristive Hindmarsh–Rose model

By showing that there is no any memristor in the integer-order Hindmarsh–Rose (H–R) model, the slow ion channel is remodeled by a fractional-order memristor, where the fractional-order derivative is related to the memory effect and the decay rate of the ion current. Then a fractional-order memristive (FOM) H–R model without equilibrium is proposed. Due to a small parameter, the FOM H–R model is a fast–slow system which can be taken as the small perturbation system of the fast subsystem. By Lyapunov direct method, the boundedness of the fast subsystem is proved. With the help of the boundedness and stability of the fast system, the hidden dynamics of the FOM H–R model is discussed. As the small parameter is fixed, it is shown that the membrane potential of the FOM H–R model limits to a quiescent state as the fractional order is less than 0.5 and limits to a periodic spiking as the fractional order is greater than 0.5 less than 1. Without the constraint of the small parameter, the periodic and chaotic bursting appears in the FOM H–R model, which implies the small parameter makes the membrane potential activity simpler. It can be drawn a conclusion that the ion current with the slow decay rate can inhibit the neurons.