Theorical Studies Of Spatiotemporal Dynamics On Complex Networks

In Recent decades,as an important interdisciplinary science,the concept of complex networks has been applied in many subjects.Ranging from the whole human society to protein in cells,the concept of complex networks provides us a new point of view in the research of complex systems.Basically,the complexity of complex systems can be treated as two parts:One is the complexity of the components.The other one is the complexity of interactions. The complex system will emerge a lot of collective behavior even it is composed of the simplest components.Therefore,the relationship between the interaction of the components and the complex behavior of the whole system is one of the main topics in the research on complex networks.A coupled neuron network is a typical complex system.First,a neuron is a excitable nonlinear dynamical system of multi-time scale.It shows different dynamical behaviors while the outer stimulations are different,and it also have abundant types of bifurcations during the changing of outer stimulations.Second,the connections between neurons are also complicated.On one hand,the dynamics of the neurons' synapses are complex.E.g.When a neuron transmits a signal to another neuron via chemical synapse.The axon of the first neuron releases a neurotransmitter, adjusted by the astrocytes,and then the neurotransmitter reaches the dendrite of the second neuron.During this process,the energy is provided by ATP moleculars.This process has so many biological and chemical reactions that the process itslef is complicated enough.On the other hand,A lot of experiment results indicate that either the function or the structure of a neuron system is not a regular or random network. Actually,the structure or the function network of the brain has both small world and scale free characteristics.Thus,the neurons complexity and the connections complexity make the coupled neuron networks have abundant of spatio-temporal dynamical phenomena.One of these nontrivial dynamical phenomena is burst synchronization(BS),which is one of this essay's topics.As well as the differential equation models,the coupled map lattice model is also a useful tool to study the spatio-temporal behavior of complex systems.The coupled map lattice model is more suitable for numeral study because it has less calculated amount than differential equation models.The other topic of this essay is focused on the behavior of spatio-temporal patterns of coupled logistic maps.Overall,The spatio-temporal dynamical behaviors of coupled neurons networks and coupled map lattice are studied in this essay,which includes the following parts:The bursting synchronization process on complex networksThe full synchronization state of homogeneous oscillators is a synchronous manifold,but the BS state is not.Therefore the Master Stability Function(MSF) can not be used to analyse the BS behavior.Here,we studied the electrical coupled Hindmash-Rose neurons on small world networks proposed by Newman arts Watts. With the increasing coupling strength,the spatio-temporal chaos of the system can be tamed into one type of BS states with fold-homoclinic bursting,and then undergoes spiking-adding and transits into another type of BS states with fold-Hopf bursting. During the latter transition,The neuron's degree plays a key role.All the neurons with degree k＞k_c change to show fold-Hopf bursting,withεk_c nearly a constant.This is a dynamic cluster separation process,which can be explained by using local mean field aproximation.The case when the neurons are coupled via chemical synapses is also briefly discussed.Based on this result,we studied the transiion process of coupled HR neurons on a series of complex networks by adding random shortcuts to a regular ring.Using numeral simulation,we got the phase diagram in the parameter space of coupling strength and random links probability.The numeral result is capable with theorical result about complete synchronization and BS state.More interestingly,when the network is of small world type,the system is most likely to be at BS state,which is spatio-temporal ordered.Revealing Network's topology property by investigating the spike and burst behaviorsFor many real systems,especially neuron systems,sometimes it is not easy to figure out the networks structure directly.Therefore,it is interesting to establish a 'reverse engineering' method to probe the network topology properties by studing the various dynamic behaviors of the system.Here we present two methods to reveal topology property based on the information of spike amplitude and frequency, respectively.First we studied a Morris-Lecar neuron network coupled via electrical synapse. Under the weak coupling condition,where the system is at BS state,the spike amplitude of a given neuron have a excellent linear relationship with its degree.The keypoint of this relationship is the incoherence of the spikes inside the burst among the neurons.Based on this relationship,we proposed a possible method to estimate the degree distribution of the network by simple statistics of the spike amplitudes.We demonstrate the validity of this scheme on scale free as well as small world networks.Second we studied the electrical coupled HR neuron network.During the transition to BS of coupled HR neurons,We found out the spike number per burst(SPB) is quite different between fold-homoclinic neurons and fold-Hopf neurons. and the critical coupling strength for a neuron to become fold-Hopf type has a reciprocal relationship with the neuron's degree.Thus,combining the information of the SPB and critical coupling strength,we can get the degree property of the network.Synchronization and pattern dynamics of coupled map lattice on complex networks.Coupled map lattice is another important spatio-temporal model.Here we studied the coupled logistic maps on complex networks of N-W type.First,combining numerical simulations and master function analysis,we figured out the synchronization zone in the parameter space of coupling strength and the fraction of random shortcuts.Second,we studied the pattern formation and selection process. Under strong coupling condition,the system will change from spatio-temporal order pattern to two different patterns and finally changed to the full synchonized chaotic pattern.Under weak coupling condition,the system undergoes the patterns of three types and finelly stays at the spatio-temporal ordered pattern,which is one of the former patterns.During these processes,there are two new pattems,spatio-temporal ordered and synchronized chaotic patterns,which are not existed in regular networks. Hence,is seems that the topological disorder has induced somewhat pattern formation and has an effect of'random shortcut can tame spatiotemporal chaos'.