Synchronizing Hindmarsh-Rose neurons over Newman-Watts networks.

In this paper, the synchronization behavior of the Hindmarsh-Rose neuron model over Newman-Watts networks is investigated. The uniform synchronizing coupling strength is determined through both numerically solving the network's differential equations and the master-stability-function method. As the average degree is increased, the gap between the global synchronizing coupling strength, i.e., the one obtained through the numerical analysis, and the strength necessary for the local stability of the synchronization manifold, i.e., the one obtained through the master-stability-function approach, increases. We also find that this gap is independent of network size, at least in a class of networks considered in this work. Limiting the analysis to the master-stability-function formalism for large networks, we find that in those networks with size much larger than the average degree, the synchronizing coupling strength has a power-law relation with the shortcut probability of the Newman-Watts network. The synchronization behavior of the network of nonidentical Hindmarsh-Rose neurons is investigated by numerically solving the equations and tracking the average synchronization error. The synchronization of identical Hindmarsh-Rose neurons coupled over clustered Newman-Watts networks, networks with dense intercluster connections but sparsely in intracluster linkage, is also addressed. It is found that the synchronizing coupling strength is influenced mainly by the probability of intercluster connections with a power-law relation. We also investigate the complementary role of chemical coupling in providing complete synchronization through electrical connections.