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Many physical, biological and neural systems behave as coupled oscillators,with characteristic phase coupling across different frequencies. Methods suchas $n:m$ phase locking value and bi-phase locking value have previously beenproposed 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 theirlimitations and limited applications. They cannot be used to detect or quantifyphase 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 thefrequencies (e.g. $f_1$, $f_2$, $2f_1/3+f_2/3$). To address the gap, this paperproposes a generalized approach, named multi-phase locking value (M-PLV), forthe quantification of various types of instantaneous multi-frequency phasecoupling. Different from most instantaneous phase coupling metrics that measurethe simultaneous phase coupling, the proposed M-PLV method also allows thedetection of delayed phase coupling and the associated time lag between coupledoscillators. The M-PLV has been tested on cases where synthetic coupled signalsare generated using white Gaussian signals, and a system comprised of multiplecoupled R\"ossler oscillators. Results indicate that the M-PLV can provide areliable estimation of the time window and frequency combination where thephase coupling is significant, as well as a precise determination of time lagin the case of delayed coupling. This method has the potential to become apowerful new tool for exploring phase coupling in complex nonlinear dynamicsystems.

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