Collective synchronization in populations of globally coupled phase oscillators with drifting frequencies

Biology – Quantitative Biology – Quantitative Methods

Scientific paper

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5 pages, 5 figures, accepted in Phys. Rev. E

Scientific paper

10.1103/PhysRevE.73.011104

We generalize the Kuramoto model for coupled phase oscillators by allowing the frequencies to drift in time according to Ornstein-Uhlenbeck dynamics. Such drifting frequencies were recently measured in cellular populations of circadian oscillator and inspired our work. Linear stability analysis of the Fokker-Planck equation for an infinite population is amenable to exact solution and we show that the incoherent state is unstable passed a critical coupling strength $K_c(\ga, \sigf)$, where $\ga$ is the inverse characteristic drifting time and $\sigf$ the asymptotic frequency dispersion. Expectedly $K_c$ agrees with the noisy Kuramoto model in the large $\ga$ (Schmolukowski) limit but increases slower as $\ga$ decreases. Asymptotic expansion of the solution for $\ga\to 0$ shows that the noiseless Kuramoto model with Gaussian frequency distribution is recovered in that limit. Thus varying a single parameter allows to interpolate smoothly between two regimes: one dominated by the frequency dispersion and the other by phase diffusion.

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