PLV Connectivity: Theory & Applications

• PLV Connectivity is a metric that quantifies the temporal stability of phase differences between oscillatory signals, facilitating network inference in neuroscience.
• Its computational implementations leverage matrix-based formulations and real-time hardware optimizations, significantly increasing efficiency in high-dimensional analyses.
• Advanced variants such as lagged, amplitude‐weighted, and warped phase coherence address challenges like spectral leakage and bias while enhancing interpretability.

Phase-Locking Value (PLV) Connectivity is a central metric in network neuroscience, nonlinear dynamics, and oscillator theory, quantifying the temporal stability of phase differences between oscillatory processes. It is widely applied in EEG, MEG, fMRI, systems neuroscience, BCI, and physical oscillator networks to infer the structure and dynamics of functional and effective connectivity. PLV connectivity provides an amplitude-agnostic, phase-specific measure that is robust and interpretable under controlled conditions, but its implementation, interpretation, and statistical handling present important subtleties and caveats.

1. Mathematical Foundations of PLV Connectivity

PLV quantifies the consistency of phase differences between two band-limited signals. Let $x_i(t)$ and $x_j(t)$ be (possibly filtered) time series from oscillatory systems. The analytic signals $$s_i(t) = x_i(t) + i \mathcal{H}[x_i(t)] = r_i(t) e^{i\phi_i(t)}$$ and $s_j(t)$ are computed using the Hilbert transform $\mathcal{H}[\cdot]$, yielding instantaneous phases $\phi_i(t)$ and $\phi_j(t)$. The classical PLV over $T$ samples is defined by

$$\mathrm{PLV}_{ij} = \left| \frac{1}{T} \sum_{t=1}^T \exp \Big( i[\phi_i(t) - \phi_j(t)] \Big) \right|~,$$

with $\mathrm{PLV}_{ij}\in[0,1]$; unity reflects perfectly stable phase difference, zero indicates random or uniform phase differences (Kovach, 2017). Alternative formulations average across trials, epochs, or repeated observations rather than time.

Table: Key PLV Definitions and Generalizations

Metric	Formula/Description	Reference
Standard PLV	$$\mathrm{PLV} = |(1/T)\sum_t e^{i[\phi_1(t)-\phi_2(t)]}|$$	(Kovach, 2017)
Lagged PLV	$$\mathrm{PLV}(\tau) = (1/(T-|\tau|))\sum_t e^{i[\phi_i(t)-\phi_j(t+\tau)]}$$ with max	(Leyva et al., 2020)
n:m PLV	$$| (1/K)\sum_k e^{i[m\phi_k(f_n,t)-n\phi_k(f_m,t)]}|$$	(Vasudeva et al., 2021)
Amplitude-weighted PLV	$$awPLV = | \sum_n A_x(n)A_y(n) e^{i\Delta\phi_n}|/ \sum_n A_x(n)A_y(n)$$	(Kovach, 2017)
Multi-PLV	$$| (1/K) \sum_k e^{i[\sum_\ell m_\ell\phi_k(f_\ell,t-\tau)-n\phi_k(f_\Sigma,t)]}|$$	(Vasudeva et al., 2021)

2. Computational Approaches and Algorithmic Optimizations

For high-dimensional datasets (e.g., EEG with hundreds or thousands of channels), efficient computation is crucial. The naive loop-based PLV is computationally expensive; matrix-based formulations dramatically increase efficiency:

• Compute all analytic signals $Z\in\mathbb{C}^{N_c \times N_s}$, normalize to unit modulus $U=Z./|Z|$.
• PLV matrix: $\mathrm{PLV} = |U U^H| / N_s$ (Bruña et al., 2017).
• This method yields a $\sim 100$-fold speedup over elementwise loops.

Hardware implementations (e.g., low-power SoCs) use LUT-based arctan phase extraction, sliding-window accumulators, and l∞-norm approximations for real-time PLV of neural signals. These designs achieve $>95\%$ correlation with exact software implementations at $<$10 μW DSP power; trade-offs include window size, approximation error, and latency (Shin et al., 2022).

3. Variants, Extensions, and Limitations

Phase-Locking with Time-Shift/Lag:

Time-shifted PLV quantifies delayed interactions. For each lag $\tau$ in $[-\tau_{max},\tau_{max}]$,

$$\mathrm{PLV}_{ij}(\tau) = \frac{1}{T-|\tau|} \sum_{t=1}^{T-|\tau|} e^{i[\phi_i(t)-\phi_j(t+\tau)]}$$

and similarity $S_{ij}(\tau) = |\mathrm{PLV}_{ij}(\tau)|$ is maximized to extract optimal lag $\tau_{ij}$ (Leyva et al., 2020).

Frequency-generalized and Cross-frequency PLV:

Multi-Phase Locking Value (M-PLV) captures phase coupling involving arbitrary frequency ratios and delayed cross-frequency relationships: $$\Psi(f_1, ..., f_L; m_1, ..., m_L, n; t, \tau) = \left| \frac{1}{K} \sum_{k=1}^K \exp \{ i [\sum_\ell m_\ell\phi_k(f_\ell, t-\tau)-n\phi_k(f_\Sigma,t)] \} \right|$$ enabling systematic probing of nonlinear and delayed couplings (Vasudeva et al., 2021).

Amplitude-weighted PLV (awPLV):

Standard PLV discards amplitude, potentially introducing bias through envelope normalization, spectral leakage, and ambiguous interpretation under correlated amplitude modulations.

$$\mathrm{awPLV} = \left| \sum_{n=1}^N A_x(n)A_y(n) e^{i\Delta\phi_n} \right|/ \sum_{n=1}^N A_x(n)A_y(n)$$

awPLV mitigates the nonlinear bias and confines spectral content to the analysis band (Kovach, 2017).

Warped Phase Coherence (WPC):

Addition of a complex constant $c$ to the analytic signal warps the phase, incorporating amplitude fluctuations into the PLV measure and potentially improving discrimination of true structural connections, especially in noisy or partially synchronized regimes. Optimal $|c|$ is typically 1–4 times the mean analytic amplitude (Minati et al., 2019).

Zero-lag Insensitive Metrics:

iPLV and ciPLV discard the real component and correct for normalization, mitigating spurious synchrony from volume conduction:

• iPLV: $$\operatorname{Im}\big\{\frac{1}{T}\sum_t e^{i[\phi_j(t)-\phi_k(t)]}\big\}$$
• ciPLV: normalization corrects bias when the mean phase difference is not $\pi/2$ (Bruña et al., 2017).

4. Application Domains and Empirical Findings

Network Inference and Synchronization Detection

• PLV and lagged PLV enable robust reconstruction of sparse network topologies among coupled nonlinear oscillators such as Kuramoto or Rössler systems. Inclusion of lag statistics improves true/false positive separation: AND-thresholding on both magnitude and lag achieves AUC$\sim$0.8–0.9 for network detection (Leyva et al., 2020).
• M-PLV captures multi-frequency and delayed synchronization, capable of resolving conduction delays in EEG–EMG coupling (e.g., corticospinal lag $\sim$25ms) (Vasudeva et al., 2021).

Brain Functional Connectivity and Biometrics

• PLV-based matrices serve as features for BCI, mild cognitive impairment detection, and EEG biometrics. In EEG biometrics, gamma-band PLV yields 97.4% cross-subject accuracy (56 ch), robust to removal of channels (21 ch: 95%), and optimally exploits short (4s) epochs (G et al., 2022).
• In EEG-based AD research, PLV connectivity in the lower alpha band is reduced in patients; clustering and global efficiency also decrease, correlating with cognitive impairment (MMSE $R\sim0.8$) (Kabbara et al., 2017). MHSA fMRI-PLV networks, augmented by temporal attention, improve mild cognitive impairment detection beyond single-scale approaches (Yuan et al., 11 Dec 2024).

Brain-Computer Interfaces and Cognitive Paradigms

• PLV is foundational for BCI feature extraction. End-to-end deep learning (PSNet) with learned spatial filters finds PSC pairs with PLV$>$0.87 for tongue motor imagery—previously unreported strong phase synchrony, marking a new data-driven approach to connectivity-based decoding (Niu et al., 2023).
• In imagined speech and visual imagery, PLV reveals paradigm-specific synchrony in language and visual networks, supporting hybrid and individualized BCI paradigms (Lee et al., 14 Nov 2024); both modalities tend to reduce global PLV (desynchronization) compared to rest, consistent with task-related network reconfiguration (Lee et al., 2020).

Real-Time Clinical and Engineering Systems

• Low-power neuromorphic hardware employs in situ PLV extraction, enabling closed-loop stimulation contingent on ongoing neural connectivity. This real-time approach is validated in rat models, with design trade-offs in window size and signal quantization (Shin et al., 2022).

5. Statistical Handling, Biases, and Limitations

Spectral Leakage and Bias

• Phase extraction via envelope normalization (analytic phase) introduces spectral leakage—energy from amplitude modulations is spread across spectral bands, potentially leading to spurious phase locking when amplitude envelopes are correlated (Kovach, 2017).
• PLV is subject to decentering (nonzero circular mean) bias; two independent signals with nonuniform phase distributions yield nonzero PLV by chance, exacerbated with amplitude covariations.

Remedies and Caveats

• Recentering and phase-uniformization only remove the DC component; they do not correct for amplitude-modulation sidebands and fail when shared envelopes are present.
• Amplitude-weighted PLV and coherence better separate phase and amplitude dependencies.
• iPLV/ciPLV, while mitigating zero-lag artifacts (e.g., from volume conduction), may underestimate high coupling unless the true phase difference is near $\pi/2$ (Bruña et al., 2017).
• PLV assumes narrowband oscillations and stationarity within analysis windows; violations degrade interpretability and statistical sensitivity.
• Small effective sample sizes inflate PLV estimates; correction factors such as $1/\sqrt{N}$ (PLV) or $1/\sqrt{\nu}$ (awPLV) are advised (Kovach, 2017).
• Recent Bayesian extensions estimate the latent phase as a smooth function, yielding noise-robust, uncertainty-quantified model-based PLV with well-calibrated credible intervals (Sugasawa et al., 8 Sep 2025).

6. Practical Implementation Guidelines

• Bandpass filter to the frequency band of interest (e.g., 0.1–45 Hz for EEG, 0.01–0.1 Hz for fMRI).
• Compute analytic signals via Hilbert transform; extract phase by argument.
• For PLV, multiply and average unit phasors across time, trials, or windows.
• For lagged PLV and M-PLV, scan candidate time-shifts or delays, maximizing synchrony or estimating physiological delays (Leyva et al., 2020, Vasudeva et al., 2021).
• For group analysis, threshold PLV matrices at a fixed density for graph construction; compute clustering, efficiency, etc. (Kabbara et al., 2017).
• For robust inference, use surrogate data or permutation testing to assess statistical significance; report effective sample size and correction methods (Kovach, 2017).
• In BCI/biometric applications, select frequency band (gamma frequently optimal), epoch length (balance resolution and data requirements), and features (PLV vector, subnetwork, or PSC pairs by data-driven methods) (G et al., 2022, Niu et al., 2023).

7. Contemporary Developments and Outlook

Phase-Locking Value Connectivity continues to evolve with innovations in

• data-driven filter learning and component selection (e.g., deep networks for phase synchrony extraction (Niu et al., 2023)),
• multi-modal integration (EEG–fNIRS, EEG–fMRI for cross-modal PLV connectivity (Qaremohammadlou et al., 4 Nov 2025, Yuan et al., 11 Dec 2024)),
• advanced statistical modeling (Bayesian circular-functional PLV for credible-interval estimation under high noise (Sugasawa et al., 8 Sep 2025)),
• and generalizations to the detection of nonlinear, delayed, and cross-frequency couplings (M-PLV) in complex, multiscale systems (Vasudeva et al., 2021).

Consensus on artifact rejection, bias correction, and null-hypothesis testing is developing. Challenges remain in disentangling true physiological synchrony from indirect coupling, signal leakage, and amplitude confounds, but comprehensive methodological developments and careful implementation enable PLV connectivity to serve as a robust metric across diverse research domains.