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

Effects of network structure on the synchronizability of nonlinearly coupled Hindmarsh–Rose neurons

“…In our recent work [22], we have developed a sufficient condition to ensure the global exponential synchronization of the nonlinearly coupled dynamical network (3) of Hindmarsh-Rose neurons. Here, the coupled dynamical network (3) is said to be globally exponentially synchronized if there exist constants ε > 0 and…”

“…: +886 3 4227151x65130; fax: +886 3 4257379. dynamical networks to be locally or globally synchronized or serve as necessary conditions for synchronization. To this aim, a variety of methods have been proposed such as the master stability function method [11], the connection graph stability method [12][13][14], the linear matrix inequality [15][16][17][18], the matrix measure approach [19][20][21] and the eigenvalue-based approach [22][23][24]. Most of these studies are predominately concentrated on the oscillators which are interconnected using linear coupling functions.…”

“…However, it has been shown that neurons can also be interconnected via chemical synapses and the chemical interaction is described by a nonlinear coupling function [25]. Thus, the study of synchronization in nonlinearly coupled dynamical networks is always an important and ongoing issue [15,16,20,[22][23][24][26][27][28][29][30].…”

“…In Section 2, we first introduce the nonlinearly coupled dynamical network of Hindmarsh-Rose neurons with a sigmoidal coupling function, and then briefly review the synchronization criterion developed in [22]. In Section 3, we compare the synchronizability of three specific network ensembles that have the same global network properties.…”

“…We will mainly focus on the networks that exhibit the small-world character or scale-free property. Based on the synchronization criterion developed in [22], which is closely related to the first nonzero eigenvalue of the outer-coupling matrix, we investigate the synchronizability of several network ensembles in which networks have the same global properties. We examine various networks such as the modular networks [44], the small-world networks [31] and the scale-free networks [32].…”