The Bifurcation Analysis Of Fitzhugh-Nagumo-Rinzel Model

With the continuous development and improvement of neural dynamics, many types ofneuron model have been studied widely, for example, the Hodgkin-Huxley model, theMorris-Lecar model, the Hindmarsh-Rose model and the FitzHugh-Nagumo model. ThenRinzel extended the FHN model into a three-dimensional dynamical system, calledFitzHugh-Nagumo-Rinzel model, this model also reflects different discharge modes anddynamic behaviors when the neurons received stimulation. It showing important theoreticalsignificance and practicality.In this paper, we uses the qualitative analysis and the bifurcation theory of differentialequations to study the FitzHugh-Nagumo-Rinzel model. We study the equilibriums and theconditions for local asymptotic stability of the model. Also we discussed about the three kindof bifurcations near the equilibria：the Hopf bifurcation, the Bogdanov-Takens bifurcation, theFold-Hopf bifurcation.The first chapter introduces the significance and situations of studying theFitzHugh-Nagumo-Rinzel model, then the main works of the thesis are introduced. Thesecond chapter describes some basic knowledge in the qualitative and bifurcation theories ofdifferential equations. In the third chapter, we study the FitzHugh--Nagumo-Rinzel model.We analysis the existence and the number of the equilibriums and discuss the local stability ofthe equilibriums. In chapter four, firstly, we study the Hopf bifurcation of codimension1andcalculate the stability coefficient of the periodic orbit near the Hopf bifurcation point;secondly, we study the Bogdanov-Takens bifurcation of codimension2, obtaining thesaddle-node bifurcation curve, the Hopf bifurcation curve and the homoclinic bifurcationcurve; finally, we study the Fold-Hopf bifurcation, and we get the cylindrical polar systemwhich is topologically equivalent to the original model, with this system, we can obtain richerdynamical properties.