Collective phase description of globally coupled excitable elements.

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.