Cluster Synchronization And Synchronization Suppression Of Module Neuron Networks With Coupling Delay

Recent experimental evidence that the analysis of connectivity between regions of cerebral cortex in macaque monkeys and cats has showed the topology structures of the neuronal networks are modular. Bursting synchronization is a typical form of firing activity of the neuronal network, which is an important mechanism of neuronal information processing. However, recent medical researches have suggested that the abnormal synchronization of neurons plays a key role in the emergence of some neurophysiological diseases, such as Parkinson’s disease, epilepsy et al.. Because of the existence of synaptic cleft and the finite speed of action potential propagating across neuron axons, time delay in the coupling is inevitable in the neuronal network. Therefore, studying syschronized firing behavior of delay-coupled modular neuronal network with modular structure is of important theoretical significance and potential application. In this dissertation, firstly we construct modular neuronal network close to the real nervous system by the map-based neuronal model. Then we investigate the phenomenon of bursting synchronization and synchronization suppression by the technique of nonlinear dynamics and numerical simulation. Furthermore, we explore the nontrivial effects of time-delayed coupling, coupling strength and modular topology structure on the synchronized dynamics of the consiedered modular network. The main contents and conclusions are as follows:1. By construting a delay-coupled modular neuronal network with hybrid chemical and electrical synapses, we study bursting synchronization of the modular neuronal network with hybrid synapses and time-delayed coupling. The results show that the intra- and inter-coupling between different neurons can induce bursting synchronization of the modular neuronal network, which is robust against the number of subnetworks and the number of neurons within each subnetwork. However, time delays in the coupling have found to be negative on the coupling-induced bursting synchronization. Simultaneously, we investigate the effect of the topology structure of the subnetwork on the process of bursting synchronization of the modular neuronal network. The results declare that bursting synchronization of the subnetwork is prior to that of the whole modular network whether the topology structures of the subnetworks are same or not. Moreover, it indicates that a larger inter-probability between subnetworks can induce bursting synchronization of the whole modular network when its subnetworks are of the same topology structure. However, bursting synchronization of the whole modular network can not be improved by a larger interconnection probability when its subnetworks are of different topology structure.2. We study suppression of bursting synchronization on a modular neuronal networld composed by small-world subnetworks with time-delayed coupling. Firstly, it is shown that, irrespective of the numbers of subnetworks, a certain coupling strength can induce bursting synchronization of the modular neuronal network. Then, a linear feedback control (including the direct feedback control and the differential feedback control) is introduced into the modular neuronal network. We detect the control domain of the parameters of feedback strength and feedback delay, by which the linear feedback control can suppress bursting synchronization effectively. The results show control domain of the direct feedback control focused on negative feedback strength and small value of feedback delay. However, control domain of the differential feedback control is distributed in middle of the considered parameters. Moreover, it indicates that the larger are the numbers of subnetworks, the smaller are the control domains of the differential feedback control. Finally, we discuss how the time-delayed coupling affects the linear feedback control to inhibit bursting synchronization. The results declare that small delay in the coupling can enhance the linear feedback control of suppressing bursting synchronization, and there exists an optimal time delay such that this enhancement obtains its maximum.The results obtained in this dissertation are helpful to understand bursting synchronization of the real neuronal system, and also are theoretical significant for curing some neuronal diseases caused by bursting synchronization.