Synchronized nonlinear patterns in electrically coupled Hindmarsh–Rose neural networks with long-range diffusive interactions

Highlights: • Modulated synchronized states are investigated in two electrically coupled Hindmarsh–Rose networks with long-range intra-neuron coupling. • The condition for direct and indirect synchronization is found via the activation of modulational instability. • Indirect synchronization between networks appears to depend on the local synchronization among neurons of the same network. - Abstract: Two electrically coupled Hindmarsh–Rose neural networks are considered, each including power-law long-range dispersive interactions. The whole dynamics of the system is reduced to a set of two coupled complex Ginzburg–Landau equations. The linear stability analysis of the plane wave solutions brings about the existence of two dynamical regimes that predict direct and indirect synchronization of the two networks, under the activation of modulational instability. The conditions for the latter to develop are discussed and used to observe numerically the synchronized longtime dynamics of action potentials, under the effect of both long-range intra-coupling and electrical inter-coupling parameters. Mainly, the synchronization criterion depends on the plane wave amplitudes and for some of their values, perfect and partial inter-network synchronization phenomena are observed. It is also found that indirect synchronization between adjacent networks requires local synchronization among neurons of the same fiber. This is discussed based on some further formulation of the synchronization error, additionally to the time series of action potentials. Some spatiotemporal behaviors of the corresponding bursts of spikes are also discussed using coupling parameters.