Pair of excitable FitzHugh-Nagumo elements: synchronization, multistability, and chaos.

We analyze a pair of excitable FitzHugh-Nagumo elements, each of which is coupled repulsively. While the rest state for each element is globally stable for a phase-attractive coupling, various firing patterns, including cyclic and chaotic firing patterns, exist in an phase-repulsive coupling region. Although the rest state becomes linearly unstable via a Hopf bifurcation, periodic solutions associated to the firing patterns is not connected to the Hopf bifurcation. This means that the solution branch emanating from the Hopf bifurcation is subcritical and unstable for any coupling strength. Various types of cyclic firing patterns emerge suddenly through saddle-node bifurcations. The parameter region in which different periodic solutions coexist is also found.

[1]  F Moss,et al.  Synchronization of hyperexcitable systems with phase-repulsive coupling. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[3]  DESYNCHRONIZATION AND SYNCHRONIZATION PROCESSES IN A RANDOMLY COUPLED ENSEMBLE OF NEURONS , 1998 .

[4]  A. Winfree.  Spiral Waves of Chemical Activity , 1972, Science.

[5]  R. Pérez,et al.  Fine Structure of Phase Locking , 1982 .

[6]  K. Aihara,et al.  Chaos and phase locking in normal squid axons , 1987 .

[7]  Leon O. Chua,et al.  Edge of Chaos and Local Activity Domain of FitzHugh-Nagumo Equation , 1998 .

[8]  Hatsuo Hayashi,et al.  Phase Dependent Transition between multistable States in a Neural Network with Reciprocal Inhibition , 2004, Int. J. Bifurc. Chaos.

[9]  John R. Clay,et al.  A Novel Mechanism for Irregular Firing of a Neuron in Response to Periodic Stimulation: Irregularity in the Absence of Noise , 2003, Journal of Computational Neuroscience.

[10]  S Kunec,et al.  Role of synaptic delay in organizing the behavior of networks of self-inhibiting neurons. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Kaplan,et al.  Subthreshold dynamics in periodically stimulated squid giant axons. , 1996, Physical review letters.

[12]  S. Kondo,et al.  A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus , 1995, Nature.

[13]  Y. Nishiura,et al.  Signal propagation and failure in one-dimensional FitzHugh-Nagumo equations with periodic stimuli. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  N. Kopell,et al.  Dynamics of two mutually coupled slow inhibitory neurons , 1998 .

[15]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[16]  Yasuomi D. Sato,et al.  Spiking neuron models with excitatory or inhibitory synaptic couplings and synchronization phenomena. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  A. Zhabotinsky,et al.  Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System , 1970, Nature.

[18]  Ichiro Tsuda,et al.  Itinerant Dynamics of Class I* Neurons Coupled by Gap Junctions , 2003, Summer School on Neural Networks.