Eventual dissipativeness and synchronization of nonlinearly coupled dynamical network of Hindmarsh-Rose neurons

In this paper, we study the asymptotic collective behavior of nonlinearly coupled dynamical network of Hindmarsh-Rose neurons, where the neurons are asymmetrically interconnected through a sigmoidal coupling function. We first show that the nonlinearly coupled dynamical network with a certain asymmetric connection topology is eventually dissipative and hence all solutions are eventually bounded. Furthermore, under some mild conditions on the system parameters, we derive an eigenvalue-related criterion that ensures the nonlinearly coupled dynamical network to be globally exponentially synchronized. Numerical experiments for the modular network of Hindmarsh-Rose neurons with or without the small-world property are given to demonstrate the theoretical results.