Research On Dynamic Analysis And Collective Behavior Of Map-based Neuron Systems

In this thesis,the dynamic behavior and collective behavior of of the map-based neuron system are taken into account.It mainly includes the following three aspect-s:Firstly,the qualitative theory analysis and numerical analysis are used to study the dynamic behavior of the chaotic Rulkov neuron model;Secondly,the complete syn-chronization,the anticipated synchronization and the lag synchronization of the simple network composed of two identical or non-identical Rulkov neurons are investigated;Finally,complete synchronization,synchronization transitions and spatio-temporal pat-tern evolution of N coupled Rulkov neuron network are studied carefully.The details are as follows:In Chapter 1,the research background and significance,the research status at home and abroad,and the research contents and methods of this thesis are introduced.In Chapter 2,the basic knowledge used in this paper is introduced,including the brief introduction of neuron system and dynamic system.In Chapter 3,firstly,the stability of the fixed point and the analytical expressions of possible bifurcation curves are studied,and the first kind of parameter plane diagram is depicted.Also,the existence of period-doubling bifurcation of Rulkov model is stud-ied by using the center manifold theorem.Secondly,the other two kinds of parameter plane graphs are obtained by numerical calculation,and the one is called iso-periodic diagram,which presents visually the stable domains of periodic points,quasi-periodic points and chaotic orbits.Three types of the classical routes to chaos,i.e.period-doubling bifurcation,intermittency,and quasi-periodicity,are all found in the chaotic Rulkov neuron model.Attractor coexistence phenomenon is also found.The other displays directly the values of the largest Lyapunov exponent,in which an interesting phenomenon that a comb-shaped chaotic region immersed in a periodic region is dis-covered.Moreover,the phenomena that the periodic region separated by the chaotic comb-shaped region organizes itself in period-adding bifurcations and period-doubling bifurcations occur in each periodic window is discovered.In Chapter 4,firstly,based on the master stability function(MSF)analysis,the complete synchronization of two electrical coupled chaotic Rulkov neurons is investi-gated in detail.The parameter plane plot that displays directly the values of the MSF in different colors is numerically obtained.The numerical values of the MSF show that the two electrical coupled Rulkov neurons may achieve the complete synchronization when each single neuron is in a silent state or a period-1 bursting state,while can not reach the complete synchronous state when each single neuron is in a chaotic bursting state or a spiking state.Secondly,Pearson’s correlation coefficient is employed to mea-sure the synchronization degree,which demonstrates the nonexistence of the complete synchronization for non-identical electrical coupled Rulkov neurons.Importantly,the complete synchronization can not be reached with the increase of the electrical cou-pling strength,which is different from the continuous-time neuronal models.Finally,based on the active control method,a synchronization scheme is presented to study the complete synchronization for two Rulkov neurons no matter whether they are identical or not.The scheme is also applied to investigate the anticipated synchronization and the lag synchronization.Numerical simulations verify the correctness of our analytical results and the effectiveness of our methods.Chapter 5 extends the contents of Chapter 4.The complete synchronization of N coupled chaotic Rulkov neuron networks is investigated in detail.The parameter plane plots that display directly the values of the MSF in different colors are numerically ob-tained.For the electrical coupled Rulkov neuron network,all positive values of MSF show that complete synchronization of the electrical coupled Rulkov neuron network can not attain when the single Rulkov neuron is in chaotic bursting state or spiking state.Importantly,a specific inner linking function is found to make the values of the MSF negative,which means that the necessary condition of complete synchronization is sat-isfied.Through numerical simulations,the existence of complete synchronization and the interesting "emergent phenomenon" is verified for Rulkov neuron network with the very specific inner linking function that has been employed.In addition,the maximum number of nodes included in the network and the explicit thresholds for the coupling strength that satisfy the necessary condition of complete synchronization are derived in three typical regular networks.More interestingly,the same route of spatiotemporal patterns transition is found for Rulkov neurons in three typical regular networks.