Multi-Phase Locking Value (M-PLV)

• M-PLV is a generalized metric that detects both instantaneous and delayed phase coupling among multiple oscillatory components using integer and rational frequency relationships.
• It unifies classical measures such as n:m PLV, bPLV, and MSPC through a comprehensive mathematical formulation and a sliding-window averaging approach.
• Empirical validations using synthetic signals, Rössler networks, and EEG-EMG data demonstrate M-PLV’s high accuracy in capturing complex, multi-frequency interactions.

The Multi-Phase Locking Value (M-PLV) is a generalized quantitative metric for detecting and characterizing instantaneous and delayed phase coupling among arbitrarily many oscillatory components, including both integer and non-integer (rational) frequency relationships. Unlike prior metrics such as the $n:m$ Phase Locking Value (PLV), bi-phase locking value (bPLV), or Multi-Spectral Phase Coupling (MSPC), M-PLV encompasses high-order, non-integer, and delayed couplings, providing a unified analytic tool for phase relationships in physical, biological, and neural oscillatory systems (Vasudeva et al., 2021).

1. Motivation and Context

The study of phase coupling in oscillatory systems underlies analysis across neuroscience, physics, and engineering. Classic phase locking metrics such as the $n:m$ PLV (e.g., $mf_1 ≈ nf_2$ with $m,n \in \mathbb{Z}$) are confined to two-component integer relationships, while bPLV detects quadratic three-frequency couplings of the form $f_3 = f_1 + f_2$ or $f_1 - f_2$. Extensions like MSPC can generalize to more frequencies but are inherently limited to integer combinations and cannot detect non-integer or rational couplings or resolve phase relationships with explicit time delay.

Complex systems such as neural populations, cardiovascular-respiratory dynamics, and networked oscillators often exhibit higher-order and nonlinearly delayed coupling not adequately captured by these measures. M-PLV was developed to address this methodological gap, providing capacity to measure:

• Multi-frequency (order $>3$) interactions, e.g., $f_1$, $f_2$, $f_3$, $n:m$0, and combinations like $n:m$1.
• Non-integer/rational-ratio couplings, e.g., $n:m$2, $n:m$3, $n:m$4.
• Delayed interactions between input and output frequency combinations.

2. Mathematical Formulation

Let $n:m$5 denote the instantaneous phase of signal $n:m$6 at frequency $n:m$7 and time $n:m$8, extracted from the analytic signal after band-pass filtering. For $n:m$9 input oscillations with frequencies $mf_1 ≈ nf_2$0 and an output at $mf_1 ≈ nf_2$1, a generalized resonance satisfies: $mf_1 ≈ nf_2$2 A delay $mf_1 ≈ nf_2$3 is allowed between the inputs and output. The phase difference for trial $mf_1 ≈ nf_2$4 is: $mf_1 ≈ nf_2$5 The M-PLV metric is then: $mf_1 ≈ nf_2$6 Key special cases include:

Special Case	Coefficient Choices	Recovers
$mf_1 ≈ nf_2$7 PLV	$mf_1 ≈ nf_2$8, $mf_1 ≈ nf_2$9, $m,n \in \mathbb{Z}$0, $m,n \in \mathbb{Z}$1, $m,n \in \mathbb{Z}$2	Two-frequency PLV
bPLV	$m,n \in \mathbb{Z}$3, $m,n \in \mathbb{Z}$4, $m,n \in \mathbb{Z}$5, $m,n \in \mathbb{Z}$6, $m,n \in \mathbb{Z}$7	Quadratic three-frequency PLV
Pairwise/1:1 PLV	$m,n \in \mathbb{Z}$8, $m,n \in \mathbb{Z}$9, $f_3 = f_1 + f_2$0, $f_3 = f_1 + f_2$1, $f_3 = f_1 + f_2$2 unused	Classical pairwise PLV
MSPC	$f_3 = f_1 + f_2$3, integer $f_3 = f_1 + f_2$4, $f_3 = f_1 + f_2$5	Integer multi-frequency phase coupling

This formulation subsumes all previously established PLV family metrics.

3. Algorithmic Implementation

The practical application of M-PLV proceeds via the following steps:

1. Band-pass Filtering: For each frequency $f_3 = f_1 + f_2$6, extract narrowband signals $f_3 = f_1 + f_2$7 with, e.g., zero-phase Butterworth filters; a typical bandwidth is 1–3 Hz.
2. Instantaneous Phase Extraction: Compute analytic signal $f_3 = f_1 + f_2$8 using the Hilbert transform $f_3 = f_1 + f_2$9; obtain $f_1 - f_2$0.
3. Combined Phase Argument Formation: Define integer/rational coefficients $f_1 - f_2$1 for the hypothesized coupling, compute $f_1 - f_2$2 for all trials and desired time shifts.
4. Sliding-Window Averaging: Over window $f_1 - f_2$3, often 5–10 cycles of the lowest frequency, calculate:

$f_1 - f_2$4

1. Coefficient and Delay Search: For exploratory analyses, scan over small $f_1 - f_2$5 and $f_1 - f_2$6, as well as appropriate delays $f_1 - f_2$7.
2. Statistical Significance: Assess significance using surrogate distribution of M-PLV maxima; e.g., phase-shuffling or trial-shuffling to establish a null, reporting if observed M-PLV exceeds the 95th percentile of the surrogate.

Efficiency is improved by precomputing analytic phases, vectorizing computations, and using FFT-based Hilbert transforms for long-duration data.

4. Delayed Coupling and Parameter Estimation

M-PLV natively incorporates delay analysis. By evaluating $f_1 - f_2$8 over a delay grid $f_1 - f_2$9, the time lag $>3$0 maximizing mean or peak M-PLV over the analysis window is identified as the estimated interaction lag. This approach produces high accuracy for both synthetic and experimental data, with delay estimates in model systems recovering the ground-truth lag to within $>3$1 error (Vasudeva et al., 2021).

Key parameter selection guidelines:

• Window length $>3$2: Covers several cycles of lowest frequency for a stable mean.
• Bandwidth $>3$3: Wide enough to accommodate frequency variability but sufficiently narrowband for phase estimation.
• Number of trials/epochs $>3$4: Larger $>3$5 reduces estimator variance $>3$6.
• Coefficient choice: Guided by theoretical predictions or preliminary spectral analysis.

5. Validation and Empirical Results

Empirical validation incorporates synthetic signals and real neurophysiological recordings:

• Coupled White-Gaussian Signals: Synthetic outputs constructed from phase-combined band-limited noise demonstrate that M-PLV peaks accurately only at the expected summed frequency ($>3$7) and only during the imposed coupling window. Detection window accuracy is $>3$8–$>3$9 error for moderate $f_1$0 ($f_1$1–$f_1$2).
• Rössler Oscillator Networks: Both integer and non-integer frequency couplings, with and without delay, were precisely identified in chaotic multi-input/single-output Rössler systems. Recovered coupling-window errors were under $f_1$3 for $f_1$4; injected delays of $f_1$5~s were measured with mean error less than $f_1$6.
• Real EEG–EMG Data: Analysis of motor cortex EEG (32-channel, C3/C4), and deltoid EMG during sustained contraction identified significant 1:1 $f_1$7-band coupling with peak delay ($f_1$8 ms) consistent with corticospinal conduction. 2:1 nonlinear coupling was also intermittently detected, in line with known neural pathway complexities.

6. Implementation Notes and Pseudocode

A single M-PLV analysis for parameters $f_1$9 follows:

$f_2$1

Efficiency is maximized by precomputing phases, using vectorized evaluation, and discarding filter transients. Significance is established via surrogate procedures (e.g., shuffling) whereby the M-PLV distribution under the null is built and observed statistics compared against its upper quantiles.

7. Applications and Scope

M-PLV supports analysis of phase coupling in:

• Neuroscience: Cross-frequency EEG/MEG coupling, cortex–muscle coherence, Dirichlet–Neumann boundary oscillations, spike–LFP interactions.
• Physiology: Cardio-respiratory and autonomic dynamics, sleep-state transitions.
• Climate Science: Interactions among oscillatory climate modes (seasonal, interannual, decadal).
• Engineering: Detection of vibrational coupling in machinery, grid oscillatory stability.

Any system exhibiting simultaneous or lagged multi-frequency phase relationships is amenable to this analysis. M-PLV offers a unified framework subsuming $f_2$0 PLV, bPLV, and MSPC, also extending to non-integer, higher-order, and time-shifted couplings with demonstrated empirical efficacy (Vasudeva et al., 2021).