The Double Parameter Bifurcation Characteristic And Coupling Synchronization Of Hindmarsh-Rose Neural Model

The human nervous system is a very complex system composed of billions of neurons,and it is also an important regulatory mechanism for the physiological function of the human body,which enables the body to become a complete unity and maintain the balance of the environment inside and outside the body to maintain the stability and coordination of a variety of functional activities.Neurons are the most basic structural and functional units in the nervous system.They are the basic unit of work of the brain.They are responsible for the transmission of information in the nervous system and have the function of sensing stimulation and promoting excitement.There is a very complex nonlinear dynamic behavior of neurons in information coding and discharge activities.It is impossible to explain the experimental phenomena by using the statistical method and the traditional linear view,and the description of the experimental results can not meet the requirements of neuroscience.The neurohynamics is a combination of neuroscience and nonlinear dynamics.The study of neuronal discharges is a frontier issue in the field of neuroscience and non-linear interdisciplinarity.With the development of neuroscience,neuro-engineering morphology and nonlinear dynamics,the use of nonlinear dynamics theory and methods to understand the neural system information generation mechanism and conduction process to calculate the parameters of the neural model changes and the impact of external stimuli on the neurohynamic behavior and study the coupled neurons chaos synchronization problem.It can not only guide the experiment to study the enhancement or elimination of synchronization methods,but also help to reveal the brain's storage and coding mechanism,which has a very important significance for the chaotic confidential information transmission.In the dissertation,Hindmarsh-Rose neurons were used as the object of study.Based on the improved mathematical model of Hindmarsh-Rose neurons and numerical computation,a single parameter bifurcation diagram,a two-parameter bifurcation diagram,a phase plan and a Lyapunov exponent are used to analyze in detail the dynamic characteristics of the Hindmarsh-Rose neuron model with different parameters.The periodic motion and chaotic motion of the Hindmarsh-Rose neuron model are analyzed.The discharge state of the Hindmarsh-Rose neuron model under different parameters is also analyzed and the appropriate DC current is added to analyze the effect of DC current on the discharge activity of the Hindmarsh-Rose neuron model.In this paper,two mathematical models of coupled Hindmarsh-Rose neurons are established.The discharge patterns and synchronous behaviors of electrical synaptic coupling and chemical synaptic coupled neurons are studied respectively.When there is either time delay or noise or both exist simultaneously,the influence on coupling synchronization is analyzed,and a nonlinear adaptive controller is designed.The feasibility and validity of the designed controller are proved by theoretical analysis and simulation results,so the effective confidential transmission of information is achieved.The main work of this paper is as follows:First,based on the Hindmarsh-Rose model,the neuron model is calculated and simulated with C language programming and the graph,and then the effect of different parameters on the dynamic behavior of the Hindmarsh-Rose neuron system are examined through analyzing the single parameter as well as two parameter bifurcation diagram.Generally,the following three aspects can be observed from bifurcation diagram of one or two variables as control parameters:(1)When two parameters are taken as control parameters,the bifurcation diagram is an integrated diagram of single parameter bifurcation diagram,that is,the cross or vertical section of the diagram is the bifurcation diagram with one parameter being unchanged and the other as control parameter in the system;(2)When two parameters are taken as control parameters,period adding bifurcation(with or without chaos),doubling period bifurcation and intermittent chaos(periodic and intermittent chaotic)can be observed intuitively in the corresponding bifurcation diagram;(3)In the bifurcation diagram,where one or two variables are control parameters,it is easy to determine the period number and the occurrence time point of the burst firing in the system,as well as the corresponding interval within which the parameter values vary.Secondly,on the basis of Hindmarsh-Rose neuron model,this dissertation joins a depolarization direct current to investigate the dynamics of Hindmarsh-Rose model under action of direct current with numerical computing method by adopting bifurcation diagram of inter-spike interval(ISI),the time responses diagram,the phase diagram and the two-parameter bifurcation diagram.These results suggest period adding bifurcation,period-doubling bifurcation as well as paroxysmal and intermittent chaos phenomenon in spiking neuron models can be observed clearly and intuitively,and not bifurcation structure of neuron model but ranges of parameters of various dynamic characteristics after adding direct current can be changed from the two-parameter bifurcation diagram.Thirdly,the dissertation establishes electrical synaptic coupled and chemical synaptic coupled Hindmarsh-Rose neuron model,studies the basic phenomenon of coupled neurons,observes the firing activity and changes of system synchronization,and analyzes the influence of coupled neurons system in terms of the existence of time delay,noise or both.Research has found that appropriate time delay and noise can promote the synchronization behavior for the non-synchronous electrical synapse coupling Hindmarsh-Rose neuron system.It is also found that appropriate Gauss white noise can induce the synchronization behavior for the chemical synaptic coupling,and appropriate time-delay can eliminate the synchronization for chemical coupling,which has certain practical significance and provides theory basis for medical study of diseases caused by eliminating the synchronization.Finally,using the Lyapunov stability theorem,a model system with adaptive control synchronization is designed.The system uses the pseudo-randomness of the chaotic signal to hide the signal that needs to be transmitted in the chaotic chaotic signal,the small input signal is superimposed on the chaotic signal in the input end,and the receiving end demodulates the useful information with a synchronous chaotic signal.The system can dynamically adjust the value of the controller according to the different initial values of the neuron model so that the two coupled HR neurons can be better in a synchronous state,with good stability and adaptability.In this dissertation,the controller is applied to the transmission of confidential information.The simulation results verify the feasibility and effectiveness of the designed controller,and can well realize the transmission of confidential information.