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

• Published in: Applied mathematical modelling Vol. 39; no. 21; pp. 6631 - 6644
• Main Authors: Li, Chun-Hsien, Yang, Suh-Yuh
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2015
• Subjects: Coupled dynamical network; Eventual dissipativeness; Global synchronization; Hindmarsh–Rose neuron; Lyapunov function; Modular network; Modular network; Eventual dissipativeness; Global synchronization; Hindmarsh–Rose neuron; Coupled dynamical network; Lyapunov function
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2015.02.017

•We study the nonlinearly coupled Hindmarsh–Rose neurons with a sigmoid coupling function.•We show that the nonlinearly coupled system is eventually dissipative.•We derive eigenvalue-related criteria to ensure the global exponential synchronization.•Numerical experiments for the modular network with or without the small-world property are presented. 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.