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We generalize the Kuramoto model for coupled phase oscillators by allowingthe frequencies to drift in time according to Ornstein-Uhlenbeck dynamics. Suchdrifting frequencies were recently measured in cellular populations ofcircadian oscillator and inspired our work. Linear stability analysis of theFokker-Planck equation for an infinite population is amenable to exact solutionand we show that the incoherent state is unstable passed a critical couplingstrength $K_c(\ga, \sigf)$, where $\ga$ is the inverse characteristic driftingtime and $\sigf$ the asymptotic frequency dispersion. Expectedly $K_c$ agreeswith the noisy Kuramoto model in the large $\ga$ (Schmolukowski) limit butincreases slower as $\ga$ decreases. Asymptotic expansion of the solution for$\ga\to 0$ shows that the noiseless Kuramoto model with Gaussian frequencydistribution is recovered in that limit. Thus varying a single parameter allowsto interpolate smoothly between two regimes: one dominated by the frequencydispersion and the other by phase diffusion.Comment: 5 pages, 5 figures, accepted in Phys. Rev.

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