Bursting Oscillations in Two Coupled FitzHugh-Nagumo Systems

The purpose of this short article is to show that bursting oscillations occur naturally in two coupled FitzHugh-Nagumo systems. As far as we know, this has not been observed previously. We gradually came to this conclusion after a discussion (reproduced herein) of known examples where these oscillations have been described in biological systems. Bursting activity typically is the result of the interaction of a fast excitatory subsystem and a slow subsystem. We consider here three different time scales. The fast excitatory subsystem is itself a ‘fast-slow dynamics. The special pattern of the bursting oscillations obtained (appearance of a transitory bump between the spiking phase and the silent phases) is discussed in the framework of qualitative plane analysis and geometric singular perturbation theory.

[1]  A. Arvanitaki,et al.  Recherches sur la Réponse Oscillatoire Locale de L'axone Géant Isolé de « Sepia » , 1939 .

[2]  Thomas Erneux,et al.  Slow Passage Through a Hopf Bifurcation: From Oscillatory to Steady State Solutions , 1993, SIAM J. Appl. Math..

[3]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[4]  A. Goldbeter,et al.  Biochemical Oscillations And Cellular Rhythms: Contents , 1996 .

[5]  J. Keizer,et al.  Minimal model for membrane oscillations in the pancreatic beta-cell. , 1983, Biophysical journal.

[6]  J. Rinzel,et al.  Oscillatory and bursting properties of neurons , 1998 .

[7]  B. Braaksma,et al.  Singular Hopf Bifurcation in Systems with Fast and Slow Variables , 1998 .

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

[9]  G. de Vries,et al.  Multiple Bifurcations in a Polynomial Model of Bursting Oscillations , 1998 .

[10]  A. Arvanitaki,et al.  EFFECTS EVOKED IN AN AXON BY THE ACTIVITY OF A CONTIGUOUS ONE , 1942 .

[11]  R. Bertram,et al.  Topological and phenomenological classification of bursting oscillations. , 1995, Bulletin of mathematical biology.

[12]  D. Chialvo,et al.  Non-linear dynamics of cardiac excitation and impulse propagation , 1987, Nature.

[13]  R. Guttman,et al.  Frequency entrainment of squid axon membrane , 1980, The Journal of Membrane Biology.

[14]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

[15]  J. Rinzel,et al.  Dissection of a model for neuronal parabolic bursting , 1987, Journal of mathematical biology.

[16]  Mark Pernarowski,et al.  Fast Subsystem Bifurcations in a Slowly Varying Liénard System Exhibiting Bursting , 1994, SIAM J. Appl. Math..

[17]  John Guckenheimer,et al.  Asymptotic analysis of subcritical Hopf-homoclinic bifurcation , 2000 .

[18]  M. Deschenes,et al.  Morphology and electrophysiological properties of reticularis thalami neurons in cat: in vivo study of a thalamic pacemaker , 1986, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[19]  Nancy Kopell,et al.  Subcellular oscillations and bursting , 1986 .

[20]  A. V. Holden,et al.  The response of excitable membrane models to a cyclic input , 2004, Biological Cybernetics.

[21]  Hans G. Othmer,et al.  On the resonance structure in a forced excitable system , 1990 .

[22]  E. Rojas,et al.  The nature of the oscillatory behaviour in electrical activity from pancreatic beta-cell. , 1980, Hormone and metabolic research. Supplement series.

[23]  John Rinzel,et al.  A Formal Classification of Bursting Mechanisms in Excitable Systems , 1987 .

[24]  V. Arnold.  LECTURES ON BIFURCATIONS IN VERSAL FAMILIES , 1972 .

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