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Collective phase description of globally coupled excitable elements.

Yoji Kawamura1, Hiroya Nakao, Yoshiki Kuramoto

  • 1Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan. ykawamura@jamstec.go.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 21, 2011
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Summary

We present a new theory for collective phase description in noisy excitable systems. This framework extends phase reduction methods to infinite-dimensional systems, revealing bifurcation-dependent phase sensitivity functions for macroscopic rhythms.

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Area of Science:

  • Dynamical Systems
  • Statistical Physics
  • Nonlinear Science

Background:

  • Globally coupled noisy excitable elements can exhibit macroscopic oscillations.
  • Understanding collective dynamics in such systems is crucial for various scientific fields.
  • Existing phase reduction methods are typically limited to finite-dimensional systems.

Purpose of the Study:

  • To develop a theory for collective phase description of globally coupled noisy excitable elements.
  • To extend conventional phase reduction methods to infinite-dimensional dynamical systems.
  • To analyze the dependence of collective phase sensitivity on bifurcation types.

Main Methods:

  • Derivation of collective phase equations from Langevin-type equations using a nonlinear Fokker-Planck equation.
  • Extension of the phase reduction method to time-periodic solutions of nonlinear Fokker-Planck equations.
  • Analysis of collective phase sensitivity functions near the onset of collective oscillations.

Main Results:

  • A theory for collective phase description of macroscopic rhythms in noisy excitable systems is established.
  • The nonlinear Fokker-Planck equation is used to represent macroscopic rhythms in infinite-dimensional systems.
  • The collective phase sensitivity function's type (I or II) is shown to depend on the bifurcation type (saddle-node or Hopf).

Conclusions:

  • The developed theory provides a framework for analyzing collective dynamics in complex oscillatory systems.
  • The findings highlight the critical role of bifurcation types in shaping collective phase dynamics.
  • This work advances the understanding of macroscopic rhythms in infinite-dimensional nonlinear systems.