Time-varying coupling functions: Dynamical inference and cause of synchronization transitions.

Interactions in nature can be described by their coupling strength, direction of coupling, and coupling function. The coupling strength and directionality are relatively well understood and studied, at least for two interacting systems; however, there can be a complexity in the interactions uniquely dependent on the coupling functions. Such a special case is studied here: synchronization transition occurs only due to the time variability of the coupling functions, while the net coupling strength is constant throughout the observation time. To motivate the investigation, an example is used to present an analysis of cross-frequency coupling functions between delta and alpha brain waves extracted from the electroencephalography recording of a healthy human subject in a free-running resting state. The results indicate that time-varying coupling functions are a reality for biological interactions. A model of phase oscillators is used to demonstrate and detect the synchronization transition caused by the varying coupling functions during an invariant coupling strength. The ability to detect this phenomenon is discussed with the method of dynamical Bayesian inference, which was able to infer the time-varying coupling functions. The form of the coupling function acts as an additional dimension for the interactions, and it should be taken into account when detecting biological or other interactions from data.