  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.
