Stability of the synchronized network of Hindmarsh–Rose neuronal models with nearest and global couplings

We analytically and numerically investigate a set of N identical and non-identical Hindmarsh–Rose neuronal models with nearest-neighbor and global couplings. The stability boundary of the synchronized states is analyzed using the Master Stability Function approach for the case of identical oscillators (complete synchronization) and the Kuramoto order parameter for the disordered case (phase synchronization). We find that, through a linear coupling modeling electrical synapses, complete synchronization occurs in a system of many nearest-neighbor or globally coupled identical oscillators, and in the case of non-identical neurons it is stable even in the presence of a spread of the parameters. We find that the Hindmarsh–Rose neuronal models can synchronize when coupled through the action of potential variable or through the interaction by rapid flows of ions through the membrane. The degree of connectivity of the network favors synchronization: in the global coupling case, the threshold for the in-phase state stabilizes when the number of dynamical units increases. The transition from disordered to the ordered state is a second order dynamical phase transition, although very sharp.