The Phase Synchronization Of Coupled Neurons

The collective behavior of the neuronal network is the key to understand the mechanism of the information processing of the brain,and the neurons in the network would fire synchronously.In real life,the cooperativity of the neuronal network behaves as the form of phase synchronization.However,the results about the phase synchronization were mostly derived by numerical simulation.So it is necessary to analyze the phase synchronization theoretically.In this thesis,we mainly study the phase synchronization of two coupled neurons with different coupling modes and various firing patterns by the interaction functions.In chapter one,some results about the synchronization of coupled neurons in recent years are introduced.In chapter two,a series of basic knowledge is presented,including the action potential of a single neuron,the phase of a neuron,and the definition of the phase response curve.Then the concepts of synchronization and phase locking are described.In chapter three,the phase response curve(PRC)and the burst phase response curve(BPRC)of a single Hindmarsh-Rose(HR)neuron are investigated.It is concluded that the BPRC of the bursting pattern is quite different to the PRC of the spiking pattern.And the BPRC becomes regular with the increase of the number of spiking within a burst.In chapter four,the interaction functions and phase synchronization of coupled HR neurons for different firing patterns are investigated.By applying the phase reduction technique,the system of a pair of weakly coupled neurons is reduced to a simple phase equation.The equation of the difference of the phases is then derived and its solution is determined by the interaction function.The interaction function of two coupled neurons can be calculated numerically according to the PRC(or BPRC)and the voltage time course of the neurons.Then the phase synchronization state is studied by the equation of the difference of the phases of the coupled neurons.We derived the following results:firstly,the bursting case is more complicated than the spiking one,and some new equilibrium will generate with the increase of the number of spiking within a burst.The stability of the original equilibrium could change and the system becomes more and more complicated and uncertain.Secondly,the chemically coupled case is more complicated than the electrically coupled case.The number and the stability of the equilibrium could be changed easily in the chemically coupled situation.The network becomes simpler and more stable in the bi-directionally inhibitorily coupled case.Thirdly,two certain things are unchanged in the electrically coupled case:the equilibrium which corresponds to the in-phase synchronization state is always stable,while the equilibrium corresponding to the anti-phase synchronization state is unstable.