Coupled fractional spiking neurons

We propose a fractional-order (FO) model of two symmetrically coupled Hodgkin-Huxley equations and study the patterns of the neurons' firing rates, for distinct values of the order of the fractional derivative, α, and the temperature, T. We find that, for positive values of the coupling, the neurons exhibit in-phase periodic solutions (neurons fire at the same time). Moreover, the spike amplitude decreases with α, meaning that the neuron stops firing below some threshold. This is observed for the three values of T studied here. For smaller T, the periodic solutions are sustained for smaller values of α. For negative values of the coupling the neurons show anti-phase synchronization for the integer-order model (neurons fire periodically with a half-period phase shift). In the case of the FO model, these anti-phase symmetric solutions disappear as α decreases from 1, for fixed T. Another bifurcation seems thus to occur being α again a bifurcation parameter. This feature occurs only in the FO system, which seems to behave as an asymmetrically coupled HH system previously studied. Further analyses is required.