Study On Synchronization In Networks Of Hindmarsh-Rose Neurons

It is well known that the neural network of the brain is a complex network with hierarchical and modular structure, and the connection between neurons in the neuronal network has been proved to have characteristics of small-world networks. With the establishment of various neuronal models, neuroscientists have made an extensive and in-depth research on synchronization dynamics of coupled neuronal networks by using various neuromime, trying to uncover the secrets hidden in the brain. The study on synchronization dynamics of coupled neuronal network not only helps us to understand the plasticity of synapses in brain, but also is expected to decipher the neural coding rules. The investigation has important medical value for prevention and treatment of the nervous system disease, such as epilepsy and Parkinson. At present, there are a lot of domestic research works on synchronization of neuronal networks. The research works on synchronization of coupled neural system represented by Wang Qingyun are remarkable, and not only elaborate on the synchronization theory of coupled neuronal system and its develpment, but also show a plenty of research achievements. Nevertheless, there are still many unresolved problems in this field, such as the topological structures and characteristics of optimal synchronization networks and synchronization mechanism in clustered networks. Those issues need to study further. Therefore, in this paper, the Hindmarsh-Rose (HR) neuron model is used to study the above issues. The contents are arranged as following:The first chapter mainly introduces some conceptual frameworks, including basic conceptions of network and synchronization, basic knowledge of neurons, relevant research background and achievements.In chapter2, we study the synchronization in NW networks. The topological structures of some optimal synchronization networks are given. Numerical simulation results show that the synchronizability of the NW network is mainly determined by the distribution of coupling points in coupling space. The network with even distribution of coupling points generally has the strongest synchronizability. For a given number of edges there may are several optimal synchronization networks with different topological structures. Optimal synchronization network has the strongest synchronizability, homogeneous degree distribution and better symmetry, but its symmetry is not necessarily the best. In general, optimal synchronization network is the irregular network. However, in a few cases, the regular network may be an optimal synchronization network. A new type of network, the ergodic network, is introduced in this paper, and it has strong synchronizability as well as the characteristics of the optimal synchronization network.In chapter3, we study the synchronization of clustered network composed by three NW subnets with uniform degree. The investigation found that these three different subnets have the similar dynamical behaviors, and they could have the same or different synchronizability when the internal coupling strengths of subnets are appropriately chosen. Numerical simulation results show that the synchronizability of the sparse clustered networks is not greater than its subnets. There exists matching combination of subnets, which could make the sparse clustered network achieve bursting synchronization and reach the synchronizability of subnets without external coupling, and make the dense clustered network get global synchronization under the least density of intercluster links. If the combination of subnets does not match, it will reduce synchronizability of the clustered network, or even cause asynchronization of the clustered network. The condition that matching combination of subnets meets is that the three subnets have the same and proper synchronizability. For a given the density of intercluster links and external coupling strength, there exists an optimal internal coupling strength, at which the synchronizability of sparse cluster network and matching subnets reach the maximum at the same time. In sparse clustered network, the reversal phenomenon of correlation coefficient is observed. When the internal coupling strength is small and external coupling strength is peoperly chosen, the dense clustered netwok exhibits strong and weak synchronizability in an alternating manner as the density of intraciuster links is increased. In particular, when the number of intercluster links is equal to the number of intracluster links, the phenomenon that synchronizability of dense clusterd network is enhanced or suppressed is observed. The physical mechanisms underlying above phenomena are analyzed.