Multi-Phase Locking Value: A Generalized Method for Determining Instantaneous Multi-frequency Phase Coupling

Many physical, biological and neural systems behave as coupled oscillators, with characteristic phase coupling across different frequencies. Methods such as $n:m$ phase locking value 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. $f_1$, $f_2$, $f_1+f_2$), 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. $f_1$, $f_2$, $f_3$, $f_4$, $f_1+f_2+f_3-f_4$), or coupling that involves non-integer multiples of the frequencies (e.g. $f_1$, $f_2$, $2f_1/3+f_2/3$). To 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. The M-PLV has been tested on cases where synthetic coupled signals are generated using white Gaussian signals, and a system comprised of multiple coupled R\"ossler oscillators. 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.