Firing properties and synchronization rate in fractional-order Hindmarsh-Rose model neurons

We find that the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of the fractional order in this study. There exists a clear difference in the bifurcation diagram between the fractional-order Hindmarsh-Rose model and the corresponding integer-order model even though the neuron undergoes a Hopf bifurcation to oscillation and then starts a period-doubling cascade to chaos with the decrease of the externally applied current. Interestingly, the discharge frequency of the fractional-order Hindmarsh-Rose model neuron is greater than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic or chaotic firing. Then we demonstrate that the firing behavior of the fractional-order Hindmarsh-Rose model neuron has a higher complexity than that of the integer-order counterpart. Also, the synchronization phenomenon is investigated in the network of two electrically coupled fractional-order model neurons. We show that the synchronization rate increases as the fractional order decreases.