Collective phase description of globally coupled excitable elements

Abstract

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type-I for the saddle-node bifurcation and type-II for the Hopf bifurcation.

18 pages, 6 figures

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• Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems
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• Sparse optimization of mutual synchronization in collectively oscillating networks
• Collective excitability in highly diluted networks of rotators
• Paradoxical phase response of gamma rhythms facilitates their entrainment in heterogeneous networks
• Phase reduction analysis of traveling breathers in reaction--diffusion systems
• Phase reduction of reaction-diffusion systems with delay