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

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 past a critical coupling strength K(c)(gamma,sigma(f)), where gamma is the inverse characteristic drifting time and sigma(f) the asymptotic frequency dispersion. Expectedly K(c)agrees with the noisy Kuramoto model in the large gamma (Schmolukowski) limit but increases slower as gamma decreases. Asymptotic expansion of the solution for gamma-->0 shows that the noiseless Kuramoto model with Gaussian frequency distribution is recovered in that limit. Thus varying a single parameter allows us to interpolate smoothly between two regimes: one dominated by the frequency dispersion and the other by phase diffusion.

[1]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[2]  Wiesenfeld,et al.  Synchronization transitions in a disordered Josephson series array. , 1996, Physical review letters.

[3]  S. Leibler,et al.  Mechanisms of noise-resistance in genetic oscillators , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Felix Naef,et al.  Cellular oscillators: rhythmic gene expression and metabolism. , 2005, Current opinion in cell biology.

[5]  Y. Kuramoto,et al.  A Soluble Active Rotater Model Showing Phase Transitions via Mutual Entertainment , 1986 .

[6]  J. Crawford,et al.  Amplitude expansions for instabilities in populations of globally-coupled oscillators , 1993, patt-sol/9310005.

[7]  Steven H. Strogatz,et al.  Cellular Construction of a Circadian Clock: Period Determination in the Suprachiasmatic Nuclei , 1997, Cell.

[8]  A. Goldbeter,et al.  Toward a detailed computational model for the mammalian circadian clock , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Steve A. Kay,et al.  Bioluminescence Imaging of Individual Fibroblasts Reveals Persistent, Independently Phased Circadian Rhythms of Clock Gene Expression , 2004, Current Biology.

[10]  D. Whitmore,et al.  Imaging of single light-responsive clock cells reveals fluctuating free-running periods , 2005, Nature Cell Biology.

[11]  A. Winfree.  Biological rhythms and the behavior of populations of coupled oscillators. , 1967, Journal of theoretical biology.

[12]  Felix Naef,et al.  Circadian Gene Expression in Individual Fibroblasts Cell-Autonomous and Self-Sustained Oscillators Pass Time to Daughter Cells , 2004, Cell.

[13]  John L Hudson,et al.  Emerging Coherence in a Population of Chemical Oscillators , 2002, Science.

[14]  S. Strogatz,et al.  Frequency locking in Josephson arrays: Connection with the Kuramoto model , 1998 .

[15]  S. Strogatz,et al.  Stability of incoherence in a population of coupled oscillators , 1991 .

[16]  J. Crawford,et al.  Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings , 1997 .

[17]  Y. Kuramoto,et al.  Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities , 1987 .

[18]  Yoshiki Kuramoto,et al.  Cooperative Dynamics of Oscillator Community : A Study Based on Lattice of Rings , 1984 .

[19]  J. Crawford,et al.  Scaling and singularities in the entrainment of globally coupled oscillators. , 1995, Physical review letters.

[20]  S. Strogatz,et al.  Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping. , 1992, Physical review letters.