Multi-phase locking value: A generalized method for determining instantaneous multi-frequency phase coupling

Abstract

BackgroundMany physical, biological and neural systems behave as coupled oscillators, with characteristic phase coupling across different frequencies. Methods such as n:m phase locking value (where two coupling frequencies are linked as: mf1=nf2) and bi-phase locking value have previously been proposed to quantify phase coupling between two resonant frequencies (e.g. f,2f/3) and across three frequencies (e.g. f1,f2,f1+f2), respectively. However, the existing phase coupling metrics have their limitations and limited applications. They cannot be used to detect or quantify phase coupling across multiple frequencies (e.g. f1,f2,f3,f4,f1+f2+f3-f4), or coupling that involves non-integer multiples of the frequencies (e.g. f1,f2,2f1/3+f2/3). New methodsTo address the gap, this paper proposes a generalized approach, named multi-phase locking value (M-PLV), for the quantification of various types of instantaneous multi-frequency phase coupling. Different from most instantaneous phase coupling metrics that measure the simultaneous phase coupling, the proposed M-PLV method also allows the detection of delayed phase coupling and the associated time lag between coupled oscillators. ResultsThe M-PLV has been tested on cases where synthetic coupled signals are generated using white Gaussian signals, and a system comprised of multiple coupled Rössler oscillators, as well as a human subject dataset. Results indicate that the M-PLV can provide a reliable estimation of the time window and frequency combination where the phase coupling is significant, as well as a precise determination of time lag in the case of delayed coupling. This method has the potential to become a powerful new tool for exploring phase coupling in complex nonlinear dynamic systems.