Synchronization of coupled chaotic FitzHugh-Nagumo neurons via Lyapunov functions

After investigating the effect of the frequency of an external electrical stimulation on the chaotic dynamics of a single FitzHugh-Nagumo (FHN) neuron, this paper derives both a sufficient and a necessary condition of the coupling coefficient for self-synchronization of two interacting FHN neurons by using the Lyapunov function method and the largest transverse Lyapunov exponent, respectively. Also, for the cases that self-synchronization is not achieved through the coupling coefficient, a feedback control law for synchronization using the Lyapunov method is investigated. The performance of the proposed control law is compared with that of an existing one in the literature. Simulation results are provided.

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

[2]  Kazuyuki Aihara,et al.  An alternating periodic-chaotic sequence observed in neural oscillators , 1985 .

[3]  C. Morris,et al.  Voltage oscillations in the barnacle giant muscle fiber. , 1981, Biophysical journal.

[4]  Carroll,et al.  Desynchronization by periodic orbits. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  S. Coombes,et al.  Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Erik Mosekilde,et al.  Effects of a parameter mismatch on the Synchronization of Two Coupled Chaotic oscillators , 2000, Int. J. Bifurc. Chaos.

[7]  L. Glass,et al.  Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. , 1981, Science.

[8]  Donghai Li,et al.  Nonlinear-estimator-based robust synchronization of Hodgkin-Huxley neurons , 2008, Neurocomputing.

[9]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[10]  L. Chua,et al.  A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS , 1994 .

[11]  G. Matsumoto,et al.  Periodic and Nonperiodic Responses of Membrane Potentials in Squid Giant Axons during Sinusoidal Current Stimulation , 1984 .

[12]  V. N. Belykh,et al.  Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models , 2000 .

[13]  Miguel A F Sanjuán,et al.  Analysis of the noise-induced bursting-spiking transition in a pancreatic beta-cell model. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Emad E. Mahmoud,et al.  Synchronization and control of hyperchaotic complex Lorenz system , 2010, Math. Comput. Simul..

[15]  José M. Casado,et al.  Synchronization of two Hodgkin–Huxley neurons due to internal noise , 2003 .

[16]  Alexander B Neiman,et al.  Synchronization of noise-induced bursts in noncoupled sensory neurons. , 2002, Physical review letters.

[17]  Huiyan Li,et al.  Synchronization of FitzHugh-Nagumo systems in EES via H∞ variable universe adaptive fuzzy control , 2008 .

[18]  Tambe,et al.  Driving systems with chaotic signals. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[19]  Maria E. Schonbek,et al.  Fitzhugh-nagumo Revisited: Types of bifurcations, Periodical Forcing and Stability Regions by a Lyapunov Functional , 2004, Int. J. Bifurc. Chaos.

[20]  Keum-Shik Hong,et al.  LMI-based robust adaptive synchronization of FitzHugh–Nagumo neurons with unknown parameters under uncertain external electrical stimulation , 2011 .

[21]  Teresa Ree Chay,et al.  Chaos in a three-variable model of an excitable cell , 1985 .

[22]  M. Chou,et al.  Exotic dynamic behavior of the forced FitzHugh-Nagumo equations , 1996 .

[23]  Jitao Sun,et al.  Adaptive-impulsive synchronization of chaotic systems , 2011, Math. Comput. Simul..

[24]  John R. Terry,et al.  Blowout bifurcation in a system of coupled chaotic lasers , 1998 .

[25]  Reinhard Eckhorn,et al.  Neural mechanisms of scene segmentation: recordings from the visual cortex suggest basic circuits for linking field models , 1999, IEEE Trans. Neural Networks.

[26]  K. Schäfer,et al.  Oscillation and noise determine signal transduction in shark multimodal sensory cells , 1994, Nature.

[27]  Teresa Ree Chay,et al.  Generation of periodic and chaotic bursting in an excitable cell model , 1994, Biological Cybernetics.

[28]  Keum Shik Hong,et al.  Synchronization of multiple chaotic FitzHugh-Nagumo neurons with gap junctions under external electrical stimulation , 2011, Neurocomputing.

[29]  Hassan Salarieh,et al.  Adaptive chaos synchronization in Chua's systems with noisy parameters , 2008, Math. Comput. Simul..

[30]  E Mosekilde,et al.  Bifurcation structure of a model of bursting pancreatic cells. , 2001, Bio Systems.

[31]  K. Hong,et al.  Synchronization of coupled chaotic FitzHugh-Nagumo systems , 2012 .

[32]  Dmitry E. Postnov,et al.  Noise Controlled Synchronization in potassium Coupled Neural Models , 2007, Int. J. Neural Syst..

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

[34]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[35]  E. Ott,et al.  Blowout bifurcations: the occurrence of riddled basins and on-off intermittency , 1994 .

[36]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[37]  Wang Jiang,et al.  Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control , 2006 .

[38]  R. FitzHugh.  Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations , 1960, The Journal of general physiology.

[39]  N. Rulkov,et al.  Robustness of Synchronized Chaotic Oscillations , 1997 .

[40]  Dmitry E. Postnov,et al.  Neural Synchronization via Potassium Signaling , 2006, Int. J. Neural Syst..

[41]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[42]  R. Plant,et al.  Bifurcation and resonance in a model for bursting nerve cells , 1981, Journal of mathematical biology.

[43]  F. M. Moukam Kakmeni,et al.  Chaos synchronization and duration time of a class of uncertain chaotic systems , 2006, Math. Comput. Simul..

[44]  L. Tsimring,et al.  Generalized synchronization of chaos in directionally coupled chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[45]  Ricardo Femat,et al.  Unidirectional synchronization of Hodgkin–Huxley neurons , 2005 .

[46]  Colin J. Thompson,et al.  Nonlinear Cable Models for Cells Exposed to Electric Fields I. General Theory and Space-Clamped Solutions , 1999 .