Diverse Dynamic Behaviors and Firing Activities of the Modified Fractional-Order Hindmarsh–Rose Neuronal Model Induced by Fractional-Order

It is important to investigate the firing activities of neurons, and previous experimental works have shown that fractional-order neuronal models depict the firing rate of neurons more verifiably. In this study, a modified fractional-order Hindmarsh–Rose neuronal model is proposed, and the dynamics and firing activities are investigated. Some novel phenomenon can be found. First, by analyzing numerically and theoretically, the Hopf bifurcation is found to occur when the external direct current stimulus is chosen appropriately. The effects of fractional-order on the bifurcation are also studied. Second, when injecting a direct current stimulus, compared with the integer-order model, the system has more varying dynamic behaviors and firing pattern transitions. Under different external current stimulus, periodic firing patterns and chaotic firing patterns occur when fractional-order changes, but the regions of chaotic firing patterns are different. In other words, the transition mode of periodic firing and chaotic firing induced by fractional-order is different under different external current stimulus. The two-dimensional colored diagram of firing patterns is also investigated. Finally, when injecting periodic current stimulus, regular/irregular bursting, multiple spiking, regular\irregular square wave bursting, and mixed firing mods are found by setting the appropriate fractional-order, amplitude, and frequency of the external current stimulus. Some firing patterns cannot be found in integer-order models. When the amplitude is chosen at appropriate values, the region of frequency when the system displays the mixed firing modes decreases with increasing fractional-order.