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

TL;DR

The paper introduces a generalized multi-phase locking value (M-PLV) method to quantify complex multi-frequency phase coupling, including delayed interactions, in coupled oscillatory systems, surpassing limitations of existing metrics.

Contribution

It proposes a novel M-PLV approach capable of detecting and quantifying multi-frequency and non-integer frequency coupling, including delayed phase interactions.

Findings

M-PLV reliably detects significant phase coupling across multiple frequencies.

It accurately estimates time lag in delayed coupling scenarios.

The method performs well on synthetic and real oscillator systems.

Abstract

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.