Extracting phase coupling functions between networks of dynamical elements that exhibit collective oscillations: Direct extraction from time-series data

Many real-world systems are often regarded as weakly coupled limit-cycle oscillators, in which each oscillator corresponds to a dynamical system with a large number of degrees of freedom exhibiting collective oscillations. One of the most practical methods for investigating the synchronization properties of such a rhythmic system is to statistically extract phase coupling functions between limit-cycle oscillators directly from observed time-series data. Particularly, using the method that combines phase reduction theory and Bayesian inference, the phase coupling functions can be extracted even from the time-series data of only one variable in each oscillatory dynamical system with many degrees of freedom. However, it remains unclear how the choice of the observed variables affects the statistical inference for the phase coupling functions. In this study, we examine the influence of observed variable types on the extraction of phase coupling functions using some typical dynamical elements under various conditions. Thus, we demonstrate that our method can consistently extract the macroscopic phase coupling functions between two phases representing collective oscillations regardless of the observed variable types, e.g., even when using one variable of any element in one system and the mean-field value over all the elements in another system.