Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model.

By employing a semianalytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an N -unit ensemble described by Langevin equations with delays. In an AMM, original N -dimensional stochastic delay differential equations (SDDEs) are transformed to infinite-dimensional deterministic DEs for means and correlations of local as well as global variables. Infinite-order DEs arising from the non-Markovian property of SDDE, are terminated at the finite level m in the level-m AMM (AMMm), which yields (3+m)-dimensional deterministic DEs. Model calculations have been made for linear and nonlinear Langevin models. The stationary solution of AMM for the linear Langevin model with N=1 is nicely compared to the exact result. In the nonlinear Langevin ensemble, the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states. Results calculated by AMM6 are in good agreement with those obtained by direct simulations.

[1]  M.-P. Zorzano,et al.  Emergence of synchronous oscillations in neural networks excited by noise , 2003 .

[2]  K Pakdaman,et al.  Response of an ensemble of noisy neuron models to a single input. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Seon Hee Park,et al.  MULTISTABILITY IN COUPLED OSCILLATOR SYSTEMS WITH TIME DELAY , 1997 .

[4]  Ohira,et al.  Delayed stochastic systems , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  J H Cartwright,et al.  Emergent global oscillations in heterogeneous excitable media: the example of pancreatic beta cells. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[7]  Roman Borisyuk,et al.  Oscillatory activity in the neural networks of spiking elements. , 2002, Bio Systems.

[8]  Longtin,et al.  Rate processes in a delayed, stochastically driven, and overdamped system , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  E. Louis,et al.  Triggering synchronized oscillations through arbitrarily weak diversity in close-to-threshold excitable media. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M. Mackey,et al.  Solution moment stability in stochastic differential delay equations. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Hideo Hasegawa,et al.  Dynamical mean-field theory of spiking neuron ensembles: response to a single spike with independent noises. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  K Pakdaman,et al.  Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Khashayar Pakdaman,et al.  Noise-induced transition in excitable neuron models , 2001, Biological Cybernetics.

[14]  Foss,et al.  Multistability and delayed recurrent loops. , 1996, Physical review letters.

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

[16]  P J Beek,et al.  Fokker-Planck perspective on stochastic delay systems: exact solutions and data analysis of biological systems. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  H. Kalmus.  Biological Cybernetics , 1972, Nature.

[18]  T Fujii,et al.  Time delay effect in a living coupled oscillator system with the plasmodium of Physarum polycephalum. , 2000, Physical review letters.

[19]  P J Beek,et al.  Stationary solutions of linear stochastic delay differential equations: applications to biological systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  H. Tuckwell,et al.  Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  C. Masoller.  Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback. , 2001, Physical review letters.

[22]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[23]  K. Ikeda,et al.  Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity , 1980 .

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

[25]  L. Tsimring,et al.  Noise-induced dynamics in bistable systems with delay. , 2001, Physical review letters.

[26]  Hideo Hasegawa.  Dynamical mean-field theory of noisy spiking neuron ensembles: application to the Hodgkin-Huxley model. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  A. Longtin,et al.  Small delay approximation of stochastic delay differential equations , 1999 .

[28]  T. Tsumoto,et al.  Change of conduction velocity by regional myelination yields constant latency irrespective of distance between thalamus and cortex , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[29]  NOISE AMPLIFICATION IN A STOCHASTIC IKEDA MODEL , 1996 .

[30]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[31]  Henry C. Tuckwell,et al.  Analytical and Simulation Results for Stochastic Fitzhugh-Nagumo Neurons and Neural Networks , 1998, Journal of Computational Neuroscience.

[32]  Hideo Hasegawa,et al.  Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[34]  L S Tsimring,et al.  Dynamics of an ensemble of noisy bistable elements with global time delayed coupling. , 2003, Physical review letters.

[35]  Longtin,et al.  Noise and critical behavior of the pupil light reflex at oscillation onset. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[36]  K. Ikeda,et al.  High-dimensional chaotic behavior in systems with time-delayed feedback , 1987 .

[37]  Uwe Küchler,et al.  Langevins stochastic differential equation extended by a time-delayed term , 1992 .

[38]  Yanqing Chen,et al.  Long Memory Processes ( 1 / f α Type) in Human Coordination , 1997 .