Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System

This paper investigates travelling wave solutions of the FitzHugh-Nagumo equation from the viewpoint of fast-slow dynamical systems. These solutions are homoclinic orbits of a three dimensional vector field depending upon system parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al. [SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 663-693] observed sharp turns in the curves of homoclinic bifurcations in a two dimensional parameter space. This paper demonstrates numerically that these turns are located close to the intersection of two curves in the parameter space that locate nontransversal intersections of invariant manifolds of the three dimensional vector field. The relevant invariant manifolds in phase space are visualized. A geometrical model inspired by the numerical studies displays the sharp turns of the homoclinic bifurcations curves and yields quantitative predictions about multipulse and homoclinic orbits and periodic orbits that have not been resolved in the FitzHugh-Nagumo model. Further observations address the existence of canard explosions and mixed-mode oscillations.

[1]  Karline Soetaert,et al.  Solving Ordinary Differential Equations in R , 2012 .

[2]  Christopher K. R. T. Jones,et al.  Tracking invariant manifolds with di erential forms in singularly per-turbed systems , 1994 .

[3]  John Guckenheimer,et al.  Computing Slow Manifolds of Saddle Type , 2012, SIAM J. Appl. Dyn. Syst..

[4]  L. Peletier,et al.  Nonlinear diffusion in population genetics , 1977 .

[5]  Christopher Jones,et al.  Stability of the travelling wave solution of the FitzHugh-Nagumo system , 1984 .

[6]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[7]  N. Kopell,et al.  Construction of the Fitzhugh-Nagumo Pulse Using Differential Forms , 1991 .

[8]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[9]  J. Guckenheimer,et al.  HOMOCLINIC ORBITS OF THE FITZHUGH-NAGUMO EQUATION: THE SINGULAR-LIMIT , 2009, 1201.5901.

[10]  John A. Feroe,et al.  Double Impulse Solutions in Nerve Axon Equations , 1982 .

[11]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[12]  Jozsi Z. Jalics,et al.  A novel canard-based mechanism for mixed-mode oscillations in a neuronal model , 2008, 0804.0829.

[13]  Edgar Knobloch,et al.  Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle , 2009, SIAM J. Appl. Dyn. Syst..

[14]  Bernd Krauskopf,et al.  Numerical continuation of canard orbits in slow–fast dynamical systems , 2010 .

[15]  John Guckenheimer,et al.  Singular Hopf Bifurcation in Systems with Two Slow Variables , 2008, SIAM J. Appl. Dyn. Syst..

[16]  Pierre Gaspard,et al.  Complexity in the bifurcation structure of homoclinic loops to a saddle-focus , 1997 .

[17]  Christopher K. R. T. Jones,et al.  Invariant manifolds and singularly perturbed boundary value problems , 1994 .

[18]  B. Sandstede,et al.  Fast and Slow Waves in the FitzHugh–Nagumo Equation , 1997 .

[19]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[20]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[21]  Peter Szmolyan,et al.  Multiple Time Scales and Canards in a Chemical Oscillator , 2001 .

[22]  G. Carpenter.  A geometric approach to singular perturbation problems with applications to nerve impulse equations , 1977 .

[23]  John Guckenheimer,et al.  A Survey of Methods for Computing (un)Stable Manifolds of Vector Fields , 2005, Int. J. Bifurc. Chaos.

[24]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[25]  M. Koper.  Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram , 1995 .

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

[27]  N. Kopell,et al.  Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron. , 2008, Chaos.

[28]  S. Hastings.  ON THE EXISTENCE OF HOMOCLINIC AND PERIODIC ORBITS FOR THE FITZHUGH-NAGUMO EQUATIONS , 1976 .

[29]  Edgar Knobloch,et al.  When Shil'nikov Meets Hopf in Excitable Systems , 2007, SIAM J. Appl. Dyn. Syst..

[30]  Bernd Krauskopf,et al.  Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system. , 2008, Chaos.

[31]  J. Rubin,et al.  The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. , 2008, Chaos.

[32]  Horacio G. Rotstein,et al.  Canard Induced Mixed-Mode Oscillations in a Medial Entorhinal Cortex Layer II Stellate Cell Model , 2008, SIAM J. Appl. Dyn. Syst..