Synchronized Firings in the Networks of Class 1 Excitable Neurons with Excitatory and Inhibitory Connections and Their Dependences on the Forms of Interactions

Synchronized firings in the networks of class 1 excitable neurons with excitatory and inhibitory connections are investigated, and their dependences on the forms of interactions are analyzed. As the forms of interactions, we treat the double exponential coupling and the interactions derived from it: pulse coupling, exponential coupling, and alpha coupling. It is found that the bifurcation structure of the networks depends mainly on the decay time of the synaptic interaction and the effect of the rise time is smaller than that of the decay time.

[1]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[2]  Bard Ermentrout,et al.  When inhibition not excitation synchronizes neural firing , 1994, Journal of Computational Neuroscience.

[3]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[4]  Eugene M. Izhikevich,et al.  Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models , 1999, IEEE Trans. Neural Networks.

[5]  William H. Press,et al.  Numerical Recipes in FORTRAN - The Art of Scientific Computing, 2nd Edition , 1987 .

[6]  G. Buzsáki,et al.  High-frequency network oscillation in the hippocampus. , 1992, Science.

[7]  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.

[8]  Abbott,et al.  Asynchronous states in networks of pulse-coupled oscillators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Nancy Kopell,et al.  Synchronization in Networks of Excitatory and Inhibitory Neurons with Sparse, Random Connectivity , 2003, Neural Computation.

[10]  Masatoshi Sekine,et al.  Analysis of globally connected active rotators with excitatory and inhibitory connections using the Fokker-Planck equation. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[12]  Y. Kuramoto,et al.  Phase transitions in active rotator systems , 1986 .

[13]  Nicolas Brunel,et al.  Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons , 2000, Journal of Computational Neuroscience.

[14]  Wulfram Gerstner,et al.  Spiking Neuron Models: Formal spiking neuron models , 2002 .

[15]  T. Kanamaru.  Analysis of Synchronization Between Two Modules of Pulse Neural Networks with Excitatory and Inhibitory Connections , 2006 .

[16]  H. Sompolinsky,et al.  Chaos in Neuronal Networks with Balanced Excitatory and Inhibitory Activity , 1996, Science.

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

[18]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[19]  Charles M. Gray,et al.  Synchronous oscillations in neuronal systems: Mechanisms and functions , 1994, Journal of Computational Neuroscience.

[20]  K. Schulten,et al.  Noise-induced synchronous neuronal oscillations. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  G B Ermentrout,et al.  Bistability in pulse propagation in networks of excitatory and inhibitory populations. , 2001, Physical review letters.

[22]  William H. Press,et al.  Numerical recipes in C , 2002 .

[23]  Vreeswijk,et al.  Partial synchronization in populations of pulse-coupled oscillators. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  Masatoshi Sekine,et al.  An analysis of globally connected active rotators with excitatory and inhibitory connections having different time constants using the nonlinear Fokker-Planck equations , 2004, IEEE Transactions on Neural Networks.

[25]  K Pakdaman,et al.  Response of coupled noisy excitable systems to weak stimulation. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  Cholinergic induction of networkoscillations at 40Hz in the hippocampus in vitro , 1998 .

[27]  Germán Mato,et al.  Synchrony in Excitatory Neural Networks , 1995, Neural Computation.

[28]  Germán Mato,et al.  Asynchronous States and the Emergence of Synchrony in Large Networks of Interacting Excitatory and Inhibitory Neurons , 2003, Neural Computation.

[29]  Shigeru Shinomoto,et al.  Phase Transitions and Their Bifurcation Analysis in a Large Population of Active Rotators with Mean-Field Coupling , 1988 .

[30]  Y. Kuramoto.  Collective synchronization of pulse-coupled oscillators and excitable units , 1991 .

[31]  Wulfram Gerstner,et al.  Spiking Neuron Models: An Introduction , 2002 .

[32]  Bard Ermentrout,et al.  Type I Membranes, Phase Resetting Curves, and Synchrony , 1996, Neural Computation.

[33]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[34]  W. Singer,et al.  Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. , 1989, Proceedings of the National Academy of Sciences of the United States of America.

[35]  Sompolinsky,et al.  Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions. , 1993, Physical review letters.

[36]  S. A. Robertson,et al.  NONLINEAR OSCILLATIONS, DYNAMICAL SYSTEMS, AND BIFURCATIONS OF VECTOR FIELDS (Applied Mathematical Sciences, 42) , 1984 .