Bifurcation Analysis Of A Class Of Morris-Lecar Models

This paper considers the bifurcation of Morris-Lecar-like model in spinal lamina Ⅰneurons.As we know,the firing patterns of neurons by biophysics,but it is also important toresearch the neuron models by using the bifurcation theory of dynamical systems.There aremany references on these models,such as the Hodgkin-Huxley model,the FitzHugh-Nagumomodel,the Morris-Lecar model and so on.The majority of them have focused on discussingthe codimension one bifurcation diagrams of the model and the waveform in itscorresponding parameters regions.Little attention has been paid to study the codimension twobifurcation analysis of the neuron model and use the results to carry on dynamics descriptionfor the corresponding biolodical phenomena.The model in this paper is derived from theMorris-Lecar model,by means of qualitative theory and bifurcation theory of differentialequations,we explore the codimension one and codimension two bifurcation phenomena ofthe M-L-k model in spinal lamina Ⅰ neurons,and obtain the parameter regions of emergingthe bistability,periodic solutions and homoclinic orbit.By considering the frequency-currentcurve,we distinguish neuronal excitability,and also discuss the bifurcation phenomena nearthe Bogdanov-Takens.Our results show more complex dynamical behaviors duing tocodimension two bifurcation.This artical is divided into three chapters.The first chapter introduces the significance of neuron model and arrangements for thisarticle.The second chapter describes the basic concepts and methods of this paper,lists the toolsfor drawing the bifurcation diagrams,phase portraits and waveforms and so on.In the chapter3.we study the two-dimensional Morris-Lecar-like model.We mainlydiscuss the codimension one and codimension two bifurcation of the model,by observing thephase portraits and waveforms of the codimension one bifurcation diagram,we obtain theregions of bistability and types of excitability of neuron.Finally,the Bogdanov-Takensbifurcation near an equilibrium point is analyzed by using the bifurcation theory,we obtain thecorresponding saddle-node bifurcation curve,Hopf bifurcation curve and homoclinicbifurcation curve,and also give the bifurcation diagram.