Two-parameter bifurcation in a two-dimensional simplified Hodgkin-Huxley model

Abstract The dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields are investigated based on the qualitative analysis and numerical simulation. A necessary and sufficient condition is given for the existence of the Hopf bifurcation. The stability of equilibrium points and limit cycles is also studied. Moreover, the canards and bifurcation are discussed in the simplified model and original model. The dynamical behaviors of the simplified model are consistent with the original model. It would be a great help to further investigations of the original model.

[1]  Jeff Moehlis,et al.  Canards in a Surface Oxidation Reaction , 2002, J. Nonlinear Sci..

[2]  Li Li,et al.  Multi-parameter Hopf-bifurcation in Hodgkin–Huxley model exposed to ELF external electric field , 2005 .

[3]  S. Baer,et al.  Sungular hopf bifurcation to relaxation oscillations , 1986 .

[4]  Yanqiu Che,et al.  Analysis of Hopf bifurcation caused by leakage conductance gl in the Hodgkin–Huxley model in muscles , 2006 .

[5]  A. Hodgkin,et al.  The components of membrane conductance in the giant axon of Loligo , 1952, The Journal of physiology.

[6]  Terrence J. Sejnowski,et al.  The Computational Brain , 1996, Artif. Intell..

[7]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[8]  K. Bar-Eli,et al.  Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction , 1991 .

[9]  J. Rinzel.  Excitation dynamics: insights from simplified membrane models. , 1985, Federation proceedings.

[10]  A. Hodgkin,et al.  Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo , 1952, The Journal of physiology.

[11]  John Guckenheimer,et al.  Numerical Computation of Canards , 2000, Int. J. Bifurc. Chaos.

[12]  James P. Keener,et al.  Mathematical physiology , 1998 .

[13]  J. Callot,et al.  Chasse au canard , 1977 .

[14]  Jianzhong Su,et al.  Analysis of a Canard Mechanism by Which Excitatory Synaptic Coupling Can Synchronize Neurons at Low Firing Frequencies , 2004, SIAM J. Appl. Math..

[15]  Jiang Wang,et al.  Two-parameters Hopf bifurcation in the Hodgkin–Huxley model , 2005 .

[16]  Jiang Wang,et al.  Bifurcation control of the Hodgkin-Huxley equations , 2007 .

[17]  Yanqiu Che,et al.  Phase-locking and chaos in a silent Hodgkin–Huxley neuron exposed to sinusoidal electric field , 2009 .

[18]  P. Szmolyan,et al.  Canards in R3 , 2001 .

[19]  Jeff Moehlis,et al.  Canards for a reduction of the Hodgkin-Huxley equations , 2006, Journal of mathematical biology.

[20]  M. Krupa,et al.  Relaxation Oscillation and Canard Explosion , 2001 .

[21]  Michael A. Arbib,et al.  The handbook of brain theory and neural networks , 1995, A Bradford book.

[22]  Jane Cronin,et al.  Mathematical aspects of Hodgkin–Huxley neural theory: Mathematical theory , 1987 .

[23]  Ran Zhao,et al.  The dynamical analysis of a two-dimensional simplified Hodgkin-Huxley model , 2010, 2010 Sixth International Conference on Natural Computation.

[24]  Wiktor Eckhaus,et al.  Relaxation oscillations including a standard chase on French ducks , 1983 .

[25]  B. John Oommen,et al.  Spikes annihilation in the Hodgkin-Huxley neuron , 2008, Biological Cybernetics.

[26]  A. Hodgkin,et al.  The dual effect of membrane potential on sodium conductance in the giant axon of Loligo , 1952, The Journal of physiology.

[27]  Jiang Wang,et al.  Analysis and control of the bifurcation of Hodgkin-Huxley model , 2007 .