Collective phase dynamics of globally coupled oscillators: Noise-induced anti-phase synchronization ✩

Abstract We formulate a theory for the collective phase description of globally coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms. Collective phase equations describing such macroscopic rhythms are derived by means of a two-step phase reduction. The collective phase sensitivity and collective phase coupling functions, which quantitatively characterize the macroscopic rhythms, are illustrated using three representative models of limit-cycle oscillators. As an important result of the theory, we demonstrate noise-induced anti-phase synchronization between macroscopic rhythms by direct numerical simulations of the three models.

[1]  Y. Kuramoto,et al.  Collective phase description of globally coupled excitable elements. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Huzihiro Araki,et al.  International Symposium on Mathematical Problems in Theoretical Physics , 1975 .

[3]  Eric Shea-Brown,et al.  On the Phase Reduction and Response Dynamics of Neural Oscillator Populations , 2004, Neural Computation.

[4]  Khashayar Pakdaman,et al.  Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model , 2009, 0911.1499.

[5]  S. Strogatz,et al.  Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action. , 2009, Chaos.

[6]  M C Cross,et al.  Frequency precision of two-dimensional lattices of coupled oscillators with spiral patterns. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Arkady Pikovsky,et al.  Self-emerging and turbulent chimeras in oscillator chains. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  J. Kurths,et al.  Synchronization in Oscillatory Networks , 2007 .

[9]  Yoji Kawamura,et al.  Phase Description of Stable Limit-cycle Solutions in Reaction-diffusion Systems , 2012 .

[10]  Ralf Tönjes,et al.  Perturbation analysis of complete synchronization in networks of phase oscillators. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[12]  G Bard Ermentrout,et al.  Stochastic phase reduction for a general class of noisy limit cycle oscillators. , 2009, Physical review letters.

[13]  Steven H. Strogatz,et al.  The Spectrum of the Partially Locked State for the Kuramoto Model , 2007, J. Nonlinear Sci..

[14]  G Bard Ermentrout,et al.  Dynamics of limit-cycle oscillators subject to general noise. , 2009, Physical review letters.

[15]  A. Edwards,et al.  Sync-how order emerges from chaos in the universe, nature, and daily life , 2005 .

[16]  Khashayar Pakdaman,et al.  Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators , 2011, 1107.4501.

[17]  Edward Ott,et al.  Comment on "Long time evolution of phase oscillator systems" [Chaos 19, 023117 (2009)]. , 2010, Chaos.

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

[19]  M. Bär,et al.  Front propagation in one-dimensional spatially periodic bistable media. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  S. Swain.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences , 1984 .

[21]  Julia Kluge,et al.  Emergence Of Dynamical Order Synchronization Phenomena In Complex Systems , 2016 .

[22]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[23]  Yoshiki Kuramoto,et al.  Cooperative Dynamics of Oscillator Community : A Study Based on Lattice of Rings , 1984 .

[24]  István Z Kiss,et al.  Resonance clustering in globally coupled electrochemical oscillators with external forcing. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  M. Rosenblum,et al.  Partially integrable dynamics of hierarchical populations of coupled oscillators. , 2008, Physical review letters.

[26]  Per Sebastian Skardal,et al.  Hierarchical synchrony of phase oscillators in modular networks. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  M C Cross.  Improving the frequency precision of oscillators by synchronization. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Yoji Kawamura,et al.  Noise-induced turbulence in nonlocally coupled oscillators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  K. Showalter,et al.  Dynamical Quorum Sensing and Synchronization in Large Populations of Chemical Oscillators , 2009, Science.

[30]  Naoki Masuda,et al.  Analysis of relative influence of nodes in directed networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Yoji Kawamura,et al.  Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case. , 2010, Chaos.

[32]  J. Crawford,et al.  Scaling and singularities in the entrainment of globally coupled oscillators. , 1995, Physical review letters.

[33]  Naoki Masuda,et al.  Collective fluctuations in networks of noisy components , 2009, 0911.5013.

[34]  S. Strogatz,et al.  Stability of incoherence in a population of coupled oscillators , 1991 .

[35]  J. Crawford,et al.  Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings , 1997 .

[36]  Carlo R. Laing,et al.  The dynamics of chimera states in heterogeneous Kuramoto networks , 2009 .

[37]  Y. Kuramoto,et al.  A Soluble Active Rotater Model Showing Phase Transitions via Mutual Entertainment , 1986 .

[38]  Yoji Kawamura,et al.  Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case. , 2010, Chaos.

[39]  Monika Sharma,et al.  Chemical oscillations , 2006 .

[40]  Jun-nosuke Teramae,et al.  Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise. , 2010, Chaos.

[41]  Hiroshi Kori,et al.  Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization , 2007, Science.

[42]  Arkady Pikovsky,et al.  Dynamics of heterogeneous oscillator ensembles in terms of collective variables , 2011 .

[43]  Steven H. Strogatz,et al.  Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life , 2004 .

[44]  Dynamical quorum sensing and synchronization in collections of excitable and oscillatory catalytic particles , 2010 .

[45]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[46]  Yoji Kawamura,et al.  Collective-phase description of coupled oscillators with general network structure. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Carlo R Laing,et al.  Chimera states in heterogeneous networks. , 2008, Chaos.

[48]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[49]  E. Ott,et al.  Long time evolution of phase oscillator systems. , 2009, Chaos.

[50]  Yoji Kawamura,et al.  Collective phase sensitivity. , 2008, Physical review letters.

[51]  Moshe Sheintuch,et al.  Modeling periodic and chaotic dynamics in anodic nickel dissolution , 1992 .

[52]  Naoki Masuda,et al.  Structure of cell networks critically determines oscillation regularity. , 2011, Journal of theoretical biology.

[53]  G Bard Ermentrout,et al.  Phase-response curves of coupled oscillators. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  John L Hudson,et al.  Emerging Coherence in a Population of Chemical Oscillators , 2002, Science.

[55]  Yoshiki Kuramoto,et al.  Mutual Entrainment between Populations of Coupled Oscillators , 1991 .

[56]  S. Strogatz,et al.  Integrability of a globally coupled oscillator array. , 1993, Physical Review Letters.

[57]  Carlo R. Laing,et al.  Fronts and bumps in spatially extended Kuramoto networks , 2011 .

[58]  Bard Ermentrout,et al.  Stimulus-Driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate Function , 2010, SIAM J. Appl. Math..

[59]  Michelle Girvan,et al.  Multiscale dynamics in communities of phase oscillators. , 2012, Chaos.

[60]  S Yanchuk,et al.  Spectral properties of chimera states. , 2011, Chaos.

[61]  Aneta Stefanovska,et al.  Asymmetry-induced effects in coupled phase-oscillator ensembles: Routes to synchronization. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  R. Mirollo.  The asymptotic behavior of the order parameter for the infinite-N Kuramoto model. , 2012, Chaos.

[63]  J. Teramae,et al.  Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. , 2004, Physical review letters.

[64]  J. Elgin.  The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[65]  Aneta Stefanovska,et al.  Routes to synchrony between asymmetrically interacting oscillator ensembles. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  Edward Ott,et al.  Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times. , 2011, Chaos.

[67]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[68]  Kenneth Showalter,et al.  Chimera States in populations of nonlocally coupled chemical oscillators. , 2013, Physical review letters.

[69]  Juan P. Torres,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[70]  Alexander S. Mikhailov,et al.  From Cells to Societies: Models of Complex Coherent Action. Authorized translation from the English edition published by Springer-Verlag , 2006 .

[71]  E. Ott,et al.  Synchronization in networks of networks: the onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[72]  E. Ott,et al.  Low dimensional behavior of large systems of globally coupled oscillators. , 2008, Chaos.

[73]  J. Kurths,et al.  Synchronization of two interacting populations of oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[75]  Kenichi Arai,et al.  Phase reduction of stochastic limit cycle oscillators. , 2007, Physical review letters.

[76]  A. Winfree.  The geometry of biological time , 1991 .

[77]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[78]  Alexander S. Mikhailov,et al.  From Cells to Societies: Models of Complex Coherent Action. Authorized translation from the English edition published by Springer-Verlag , 2006 .

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

[80]  S. Strogatz,et al.  Constants of motion for superconducting Josephson arrays , 1994 .

[81]  T. Ikeguchi,et al.  Colored noise induces synchronization of limit cycle oscillators , 2012, 1402.1839.

[82]  Hayato Chiba,et al.  Center manifold reduction for large populations of globally coupled phase oscillators. , 2011, Chaos.

[83]  K. Showalter,et al.  Chimera and phase-cluster states in populations of coupled chemical oscillators , 2012, Nature Physics.

[84]  O. Omel'chenko,et al.  Coherence–incoherence patterns in a ring of non-locally coupled phase oscillators , 2013 .

[85]  S. Strogatz,et al.  Solvable model for chimera states of coupled oscillators. , 2008, Physical review letters.

[86]  B. Ermentrout,et al.  Response of traveling waves to transient inputs in neural fields. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[87]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[88]  D. S. Goldobina,et al.  Synchronization of self-sustained oscillators by common white noise , 2004 .

[89]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[90]  Yoji Kawamura,et al.  Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators. , 2007, Physical review letters.

[91]  Carlo R Laing,et al.  Disorder-induced dynamics in a pair of coupled heterogeneous phase oscillator networks. , 2012, Chaos.

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

[93]  David Terman,et al.  Mathematical foundations of neuroscience , 2010 .

[94]  Yoshiki Kuramoto,et al.  Rhythms and turbulence in populations of chemical oscillators , 1981 .