Kinetic Analysis Of A Neuron Model

It is focused on the analysis of the dynamics of the neuron network proposed by Rulkov.And it is included that the analysis and proof of the properties of the fixed points and the existing conditions of the bifurcations with the relationship of the parameters.And the numerical simulation is the way I use in the analysis of the synchronization of the neuron networks with the various coupling ways.The preface is the ChapterⅠ,in which,the development of the researches of neuron networks are mentioned,from the linear methods to the nonlinear methods. Since there exist many phenomena of chaos in real neuron networks and linear models are not good ways to simulate the phenomena of the bursting and chaos,nonlinear models are considered.That is the background of the model proposed by Rulkov.In the Chapter II,it is based on the assumption that slow variable y of the Rulkov model is constant and there are no interacting between any neurons in the network.The properties of the fixed points of a single-neuron model under various conditions are studied.And the influence of the parameters on the fixed points and the kinds of the bifurcations are analyzed.Finally,the existence conditions of the chaos are found.In the first paragraph of the ChapterⅡ,the main parameter,a,is discussed. The smallest alpha that leads to chaos,α₀,is found.As per the proof,ifαis bigger thanα₀,the fixed points of the system may be stable or unstable.There may exist one,two or three fixed points and bifurcations.Ifαis equal toα₀,except two fixed points with the slope of 1 or-1,there only exist one stable fixed point.Otherwise,αis smaller thanα₀ and there is only one stable fixed point in the system.In the second paragraph of the ChapterⅡ,βis discussed to show the properties of the fixed points in the systems.And the existence conditions of the Period Doubling Bifurcation and Saddle-Node Bifurcation are studied.With the change of the value ofβunder a fixed a,the system will show different characteristics of fixed points.In the last paragraph of the ChapterⅡ,on the basis of the above analysis,the relationship ofαandβis shown completely and the conclusion is provided,while (?)Rulkov[]and Vries[]are only showed part of it.In the last chapter,numerical simulation by Matlab is used to study the synchronization of the coupling neuron networks on the basis of the theory of Scale-free Network.Several ways of coupling are simulated with or without slow variable y.As per the solutions of the simulations,without the slow variable,the neuron network will get relatively synchronized,skipping between two stable status.With the slow variable,the values of every neurons will change out of order in a small extent via a long-time evolvement.