Adaptive synchronization in the complex heterogeneous networks of Hindmarsh–Rose neurons

Abstract

• The limit set for trajectories of the heterogeneous Hindmarsh–Rose network is defined. • Sufficient conditions of synchronization in the heterogeneous Hindmarsh–Rose networks with linear couplings are obtained. • The relationship between connectivity of the topological graph and synchronization is established. • The simulation coincides with the obtained theoretical results. This paper is devoted to the adaptive synchronization problem in the heterogeneous Hindmarsh–Rose neuronal networks. Heterogeneity is a natural property of biological neuronal networks, as each neuron has its own physiological characteristics, which may differ from other neurons within the population. Therefore, the study of the effect of heterogeneity on the synchronization in the biological neuronal network is an important problem. In order to solve this problem, the ultimate boundedness of the network trajectories is established, and also the limit set is defined for these trajectories. Based on the boundedness analysis and the Speed Gradient method, the adaptive algorithm for adjusting the coupling strength is developed. It is proved mathematically that the developed algorithm provides synchronization in the network under study. The obtained theoretical results are confirmed by the simulations.