Global dynamics of oscillator populations under common noise

Common noise acting on a population of identical oscillators can synchronize them. We develop a description of this process which is not limited to the states close to synchrony, but provides a global picture of the evolution of the ensembles. The theory is based on the WatanabeStrogatz transformation, allowing us to obtain closed stochastic equations for the global variables. We show that at the initial stage, the order parameter grows linearly in time, while at the later stages the convergence to synchrony is exponentially fast. Furthermore, we extend the theory to nonidentical ensembles with the Lorentzian distribution of natural frequencies and determine the stationary values of the order parameter in dependence on driving noise and mismatch. Copyright c © EPLA, 2012 Introduction. – Synchronization of oscillations by a periodic forcing is a general phenomenon observed in numerous experiments. In this setup the system follows the driving and has, in particular, the same frequency, so one often speaks on frequency entrainment. Much more nontrivial is the effect of synchronization by an external noise. Here one can also distinguish between the cases when the driven system is entrained by the noise (synchrony) and the situations when the noise is not followed (asynchrony). While the difference between these regimes can be hardly seen by observing just one responding oscillator, it becomes evident if an ensemble of identical systems driven by the same noise is observed: in the case of synchronization all the oscillators in the ensemble follow the forcing and their states thus coincide, while in the asynchronous state the states of systems remain different. This effect is therefore called synchronization by common noise [1–6]. An interesting realization of this type of synchronization is the effect of reliability of neurons [7]. Here one does not use an ensemble of identical neurons, but takes one neuron and applies the same pre-recorded noise to it several times. The synchronous case appears as a reliable response to the forcing where all the noise-induced spikes are at the same position at all runs, while for asynchrony (antireliability) the same noise produces different spike patterns [8,9]. Synchronization by (a)E-mail: [email protected] common noise was also observed in physical experiments with phase-locked loop [10] and noise-driven lasers [11]. Synchronization by common noise can be characterized by the largest Lyapunov exponent of the noise-driven dynamics. This exponent governs the growth/decay of small perturbations of a synchronous state; a negative exponent corresponds to synchrony while a positive one to asynchrony [1–6] (notice that here the largest Lyapunov exponent can be interpreted as a “transverse” one, determining the growth/decay of the difference between oscillators in the ensemble). For periodic oscillators, which in the noise-free case have a zero maximal Lyapunov exponent, small noise generally leads to a negative exponent (while large noise can desynchronize); for chaotic systems with a positive Lyapunov exponent, strong noise can synchronize (see examples of the synchronization-desynchronization transition in [1,2,12–14]). Calculation of the Lyapunov exponent is a relatively easy numerical task, and in many cases it can be obtained analytically [4–6]. This theory is, however, restricted to the linear analysis of a stability of the synchronous state, and does not allow one to follow the evolution starting from a broad distribution of the phases. The goal of this letter is to present a global analytic theory of the synchronization by common noise, i.e. a theory describing the evolution toward synchrony of the population starting from the distributed, asynchronous state. Our theory is based on the Watanabe-Strogatz ansatz [15] that allows one to describe a population