Synchronization in networks with random interactions: theory and applications.

Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to systems with time delayed coupling and then nonlinear systems where the synchronized dynamics can be periodic or chaotic. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.

[1]  Jianfeng Feng,et al.  Computational neuroscience , 1986, Behavioral and Brain Sciences.

[2]  Jianfeng Feng,et al.  Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  S. Boccaletti,et al.  Synchronization is enhanced in weighted complex networks. , 2005, Physical review letters.

[4]  J. Kurths,et al.  Enhancing complex-network synchronization , 2004, cond-mat/0406207.

[5]  E. Marder,et al.  Plasticity in single neuron and circuit computations , 2004, Nature.

[6]  Mingzhou Ding,et al.  Will a large complex system with time delays be stable? , 2004, Physical review letters.

[7]  Owen L. Petchey,et al.  Interaction strengths in food webs: issues and opportunities , 2004 .

[8]  E. Ott,et al.  Desynchronization waves and localized instabilities in oscillator arrays. , 2004, Physical review letters.

[9]  Mingzhou Ding,et al.  Enhancement of neural synchrony by time delay. , 2004, Physical review letters.

[10]  John Rinzel,et al.  Short duty cycle destabilizes a half-center oscillator, but gap junctions can restabilize the anti-phase pattern. , 2004, Journal of neurophysiology.

[11]  Jianfeng Feng,et al.  Synchronization in stochastic coupled systems: theoretical results , 2004 .

[12]  Fan Chung Graham,et al.  The Spectra of Random Graphs with Given Expected Degrees , 2004, Internet Math..

[13]  F. Chung,et al.  Spectra of random graphs with given expected degrees , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[14]  F. Atay.  Distributed delays facilitate amplitude death of coupled oscillators. , 2003, Physical review letters.

[15]  G. Rangarajan,et al.  General stability analysis of synchronized dynamics in coupled systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Hildebrando M. Rodrigues,et al.  Synchronization of coupled equations of Hodgkin-Huxley type , 2003 .

[17]  K. Aihara,et al.  Stability of genetic regulatory networks with time delay , 2002 .

[18]  J. Jost,et al.  Evolving networks with distance preferences. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  G. Rangarajan,et al.  Stability of synchronized chaos in coupled dynamical systems , 2002, nlin/0201037.

[20]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[21]  J. Jost,et al.  Spectral properties and synchronization in coupled map lattices. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  T. Yamada.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems _ IV-- Instability of Synchronized Chaos and New Intermittency - , 2002 .

[23]  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.

[24]  J. E. Cohen,et al.  Transient dynamics and food–web complexity in the Lotka–Volterra cascade model , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[25]  V N Belykh,et al.  Cluster synchronization modes in an ensemble of coupled chaotic oscillators. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Belykh,et al.  Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  Johnson,et al.  Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  Chern.,et al.  Synchronization of mutually coupled self-mixing modulated lasers , 2000, Physical review letters.

[29]  J Feng,et al.  Synchronization due to common pulsed input in Stein's model. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  P C Bressloff,et al.  Mean-field theory of globally coupled integrate-and-fire neural oscillators with dynamic synapses. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Jianfeng Feng,et al.  Variability of Firing of Hodgkin-Huxley and FitzHugh-Nagumo Neurons with Stochastic Synaptic Input , 1999 .

[32]  Frank Pasemann,et al.  Synchronized chaos and other coherent states for two coupled neurons , 1999 .

[33]  G. Hu,et al.  Instability and controllability of linearly coupled oscillators: Eigenvalue analysis , 1998 .

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

[35]  Chungyong Lee,et al.  Generalized iterative methods for enhancing contaminated chaotic signals , 1997 .

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

[37]  Nikolai F. Rulkov,et al.  Designing a Coupling That Guarantees Synchronization between Identical Chaotic Systems , 1997 .

[38]  Ott,et al.  Transitions to Bubbling of Chaotic Systems. , 1996, Physical review letters.

[39]  David Hansel.  Synchronized Chaos in Local Cortical Circuits , 1996, Int. J. Neural Syst..

[40]  Gauthier,et al.  Intermittent Loss of Synchronization in Coupled Chaotic Oscillators: Toward a New Criterion for High-Quality Synchronization. , 1996, Physical review letters.

[41]  Gade,et al.  Synchronization of oscillators with random nonlocal connectivity. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  Parlitz,et al.  Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. , 1996, Physical review letters.

[43]  Gupte,et al.  Synchronization in coupled sine circle maps. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[44]  Stanislaw Jankowski,et al.  Synchronization and association in a large network of coupled Chua's circuits† , 1995 .

[45]  I. Stewart,et al.  Bubbling of attractors and synchronisation of chaotic oscillators , 1994 .

[46]  Carroll,et al.  Synchronous chaos in coupled oscillator systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[47]  R. Lima,et al.  Exact Lyapunov exponent for infinite products of random matrices , 1994, chao-dyn/9407013.

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

[49]  Roy,et al.  Experimental synchronization of chaotic lasers. , 1994, Physical review letters.

[50]  Li,et al.  Bifurcation to standing and traveling waves in large arrays of coupled lasers. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[51]  Thomas Erneux,et al.  Stability conditions for coupled lasers: series coupling versus parallel coupling , 1993 .

[52]  Harold M. Hastings,et al.  Stability of structured random matrices , 1992, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[53]  Paladin,et al.  Analytic solution of the random Ising model in one dimension. , 1992, Physical review letters.

[54]  Hansel,et al.  Synchronization and computation in a chaotic neural network. , 1992, Physical review letters.

[55]  Charles M. Newman,et al.  The triangle law for Lyapunov exponents of large random matrices , 1992 .

[56]  Y. Peres.  Domains of analytic continuation for the top Lyapunov exponent , 1992 .

[57]  Charles M. Newman,et al.  Stochastic structure and nonlinear dynamics of food webs: qualitative stability in a Lotka-Volterra cascade model , 1990, Proceedings of the Royal Society of London. B. Biological Sciences.

[58]  Vi︠a︡cheslav Leonidovich Girko,et al.  Theory of random determinants , 1990 .

[59]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990, Bulletin of mathematical biology.

[60]  B. Huberman,et al.  The stability of ecosystems , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

[61]  C. Newman,et al.  Host-Parasite Relations and Random Zero-Sum Games: The Stabilizing Effect of Strategy Diversification , 1989, The American Naturalist.

[62]  J. Eckmann,et al.  The largest Liapunov exponent for random matrices and directed polymers in a random environment , 1989 .

[63]  Charles M. Newman,et al.  Dynamic Basis of Food Web Organization , 1988 .

[64]  A. Crisanti,et al.  Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices , 1988 .

[65]  Kapral,et al.  Domain growth and nucleation in a discrete bistable system. , 1987, Physical review. A, General physics.

[66]  Eric S. Key,et al.  Computable examples of the maximal Lyapunov exponent , 1987 .

[67]  C. W. Li,et al.  Almost sure stability of linear stochastic systems with Poisson process coefficients , 1986 .

[68]  S. Geman.  THE SPECTRAL RADIUS OF LARGE RANDOM MATRICES , 1986 .

[69]  Volker Wihstutz,et al.  Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and application , 1986 .

[70]  Charles M. Newman,et al.  The distribution of Lyapunov exponents: Exact results for random matrices , 1986 .

[71]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[72]  Charles M. Newman,et al.  When Will a Large Complex System Be Stable , 1985 .

[73]  J. Hüsler.  Extremes and related properties of random sequences and processes , 1984 .

[74]  Arkady Pikovsky,et al.  On the interaction of strange attractors , 1984 .

[75]  Charles M. Newman,et al.  The Stability of Large Random Matrices and Their Products , 1984 .

[76]  K. Cooke,et al.  Discrete delay, distributed delay and stability switches , 1982 .

[77]  P. A. Mello,et al.  Random matrix physics: Spectrum and strength fluctuations , 1981 .

[78]  R. Datko.  A procedure for determination of the exponential stability of certain differential-difference equations , 1978 .

[79]  N. Macdonald.  Time lags in biological models , 1978 .

[80]  Jim M Cushing,et al.  Integrodifferential Equations and Delay Models in Population Dynamics , 1977 .

[81]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[82]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[83]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[84]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[85]  W. Feller,et al.  An Introduction to Probability Theory and its Applications , 1958 .

[86]  W. Feller.  An Introduction to Probability Theory and Its Applications , 1959 .

[87]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .