Synchronize the random!

Synchronization, the coordinated behavior of coupled oscillators, is a fundamental phenomenon observed across various scientific disciplines, from biological systems (e.g. circadian rhythms) to physical networks. While well-established theoretical tools exist for analyzing synchronization in deterministic oscillators, the inherent randomness in stochastic oscillators presents significant challenges. The Theory of Complex Systems and Neurophysics Group (Benjamin Lindner) have introduced a novel approach to this problem by employing the Q-function, a mathematical tool derived from the stochastic Koopman operator. Their work offers new insights into defining and analyzing synchronization in coupled stochastic oscillators. The paper addresses the limitations of traditional methods, like phase reduction, when applied to systems where noise complicates the very notion of a well-defined phase. By focusing on the Q-function, the slowest decaying complex mode of the system, the authors propose a new perspective: a synchronized system of coupled stochastic oscillators can be viewed as a single, higher-dimensional stochastic oscillator. This allows for a fresh examination of phenomena like Arnold tongues, which map regions of synchronization. When you are interested about the details, read this article about coupled stochastic oscillators.

Abstract

Phase reduction is an effective theoretical and numerical tool for studying synchronization of coupled deterministic oscillators. Stochastic oscillators require new definitions of asymptotic phase. The Q-function, i.e. the slowest decaying complex model of the stochastic Koopman operator (SKO), was proposed as a means of phase reduction for stochastic oscillators. In this paper, we show that the Q-function approach also leads to a novel definition of “synchronization" for coupled stochastic oscillators. A system of coupled oscillators in the synchronous regime may be viewed as a single (higher-dimensional) oscillator. Therefore, we investigate the relation between the Q-functions of the uncoupled oscillators and the higher-dimensional Q-function for the coupled system. We propose a definition of synchronization between coupled stochastic oscillators in terms of the eigenvalue spectrum of Kolmogorov’s backward operator (the generator of the Markov process, or the SKO) of the higher dimensional coupled system. We observe a novel type of bifurcation reflecting (i) the relationship between the leading eigenvalues of the SKO for the coupled system and (ii) qualitative changes in the cross-spectral density of the coupled oscillators. Using our proposed definition, we observe synchronization domains for symmetrically-coupled stochastic oscillators that are analogous to Arnold tongues for coupled deterministic oscillators.