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 stabilityboundary of the synchronized states is analyzed using the Master Stability Functionapproach for the case of identical oscillators (complete synchronization) and the Kuramotoorder parameter for the disordered case (phase synchronization). We find that, through alinear coupling modeling electrical synapses, complete synchronization occurs in a systemof many nearest-neighbor or globally coupled identical oscillators, and in the case of nonidenticalneurons it is stable even in the presence of a spread of the parameters. We findthat the Hindmarsh–Rose neuronal models can synchronize when coupled through theaction of potential variable or through the interaction by rapid flows of ions through themembrane. The degree of connectivity of the network favors synchronization: in the globalcoupling case, the threshold for the in-phase state stabilizes when the number of dynamicalunits increases. The transition from disordered to the ordered state is a second orderdynamical phase transition, although very sharp.