$Q$-functions, synchronization, and Arnold tongues for coupled stochastic oscillators

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 mode 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.