Warped Phase Coherence (WPC) Overview

• Warped Phase Coherence (WPC) is an advanced synchrony measure that extends traditional phase locking value by incorporating amplitude sensitivity and average phase offset detection.
• WPC employs a warping transformation on analytic signals, enabling better discrimination of coupling regimes in both simulated oscillator networks and experimental setups.
• Empirical studies show that WPC improves connectivity inference and EEG classification performance by capturing hybrid phase–amplitude synchrony beyond conventional methods.

Warped Phase Coherence (WPC) is an empirical synchronization measure that generalizes the conventional phase locking value (PLV) by integrating amplitude sensitivity and average relative phase detection into the quantification of signal synchrony. WPC was introduced to address limitations of phase-only coherence metrics, especially in physiological recordings and nonlinear oscillator networks, by exploiting a warping transformation in the analytic signal domain. Through analytic, simulation, and experimental studies, WPC has demonstrated utility in enhancing inference of structural couplings and improving classification accuracy in brain–computer interface (BCI) paradigms (Minati et al., 2019).

1. Mathematical Foundation and Definition

Let $x(t)$ denote a real-valued time series. Its analytic signal is expressed as

$$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$

where $\mathcal{H}[\cdot]$ is the Hilbert transform, $r(t)$ the instantaneous amplitude, and $\alpha(t)$ the instantaneous phase.

For two signals $x_j(t)$ and $x_k(t)$, traditional PLV considers the phase difference

$$\Delta\alpha_{jk}(t) = \arg[\psi_j(t)\,\overline{\psi_k(t)}] = \alpha_j(t) - \alpha_k(t),$$

with the PLV defined as

$$\mathrm{PLV}_{jk} = \left| \left\langle e^{i\,[\alpha_j(t) - \alpha_k(t)]} \right\rangle_t \right|.$$

WPC introduces a “warping” step by adding a complex constant $c$ to the analytic signal: $$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$0 resulting in the unnormalized WPC,

$$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$1

with $$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$2. For normalization and removal of trivial trends at large $$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$3, a baseline is estimated by random temporal shuffling: $$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$4 where $$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$5 denotes the incoherent baseline. Closed-form partial derivatives,

$$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$6

illustrate how sensitivity to amplitude ($$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$7) and phase ($$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$8) is introduced for $$\psi(t) = x(t) + i\,\mathcal{H}[x(t)] = r(t) e^{i \alpha(t)},$$9.

2. Relationship with Phase Locking Value (PLV)

WPC reduces to PLV in the limit $\mathcal{H}[\cdot]$0. The mapping $\mathcal{H}[\cdot]$1 in WPC injects amplitude and mean phase offset sensitivity, distinguishing it from the phase-only PLV. For $\mathcal{H}[\cdot]$2, $\mathcal{H}[\cdot]$3 and WPC resembles PLV. For $\mathcal{H}[\cdot]$4, $\mathcal{H}[\cdot]$5 and the measure transitions toward correlation-like behavior, dominated by high-amplitude fluctuations. Critically, WPC is sensitive both to amplitude correlations and to nonzero average phase shifts, properties to which PLV is insensitive. These distinctions make WPC particularly informative in scenarios where amplitude fluctuations and phase offsets encode meaningful information.

3. Algorithmic Implementation

The WPC computation proceeds as follows for $\mathcal{H}[\cdot]$6 real-valued time series $\mathcal{H}[\cdot]$7 and a chosen warping constant $\mathcal{H}[\cdot]$8:

1. Compute the analytic signal $\mathcal{H}[\cdot]$9 for each $r(t)$0 using the Hilbert transform.
2. Calculate the warped angle $r(t)$1 for each series.
3. For each pair $r(t)$2, compute phase differences $r(t)$3.
4. Evaluate $r(t)$4.
5. If normalization is required, shuffle $r(t)$5, compute $r(t)$6, and apply the normalization formula to obtain $r(t)$7.

Empirically, choosing $r(t)$8 relative to the mean amplitude ($r(t)$9) balances amplitude sensitivity with avoidance of trivial collapse to unity. The phase of $\alpha(t)$0 can be tuned to emphasize average phase offsets or large-amplitude events, depending on the application.

4. Empirical Results: Simulation and Physical Experiments

A. Coupled Rössler Oscillator Networks

In simulated Rössler systems, WPC effectively distinguishes different coupling regimes:

• For moderate coupling, $\alpha(t)$1 rises from 0.42 at $\alpha(t)$2 (PLV) to approximately 0.55 at $\alpha(t)$3, indicating amplitude entrainment (effect vanishes if amplitudes are clamped).
• In uncoupled cases, $\alpha(t)$4 but $\alpha(t)$5 for $\alpha(t)$6, improving discrimination of asynchrony.
• In $\alpha(t)$7 node Erdős–Rényi networks ($\alpha(t)$8), structural-link inference accuracy increases from $\alpha(t)$9 (PLV) to $x_j(t)$0 ($x_j(t)$1). Removing amplitude or average-phase information degrades this accuracy.
• Distinct $x_j(t)$2 profiles are observed across phase-synchronization transitions, reflecting sensitivity to underlying dynamics.

B. Chaotic Oscillator Hardware Networks

In a network of 90 single-transistor Hartley-like oscillators with 9 strong long-range links, WPC focused connectivity inference sharply on the actual strong links and suppressed peripheral spurious synchrony:

• Link inference accuracy increased nonmonotonically from 0.89 (PLV) to 1.00 at $x_j(t)$3.
• WPC with $x_j(t)$4 produces connectivity maps similar to amplitude-sensitive metrics such as linear correlation, mutual information, and generalized synchronization (L-index), in contrast to PLV.

5. Application to Brain Signals and Decoding Performance

WPC has been applied to electroencephalography (EEG) data for motor imagery tasks:

• Dataset: 106 subjects, 64 channels, 4 s epochs at 160 Hz, motor imagery and control conditions, band-pass 7–30 Hz, common-average reference.
• For each epoch, the upper triangular part of the WPC($x_j(t)$5) matrix is vectorized as feature input for sparse logistic regression with leave-one-out subject-wise classification.
• Active vs idle discrimination improves from $x_j(t)$6 (PLV) to $x_j(t)$7 (p < 0.001).
• Left vs right imagery classification rises from $x_j(t)$8 to $x_j(t)$9.
• Improvements extend to actual hand movement and fists-vs-feet tasks.
• Warping amplifies contributions from large-amplitude, event-related synchrony (μ/β volleys) and average phase structure, which PLV underestimates.

6. Practical Considerations and Limitations

Recommended practice is to select $x_k(t)$0 approximately twice the mean amplitude, e.g., $x_k(t)$1 for normalized data. The direction (phase) of $x_k(t)$2 may be set real positive to accentuate positive amplitude events. Computational complexity is dominated by the Hilbert transform (FFT-based) and $x_k(t)$3 pairwise combinations, allowing feasibility for moderate channel counts ($x_k(t)$4) and short epochs ($x_k(t)$5 s) in BCI applications.

Care must be exercised in interpretation: for $x_k(t)$6, the warped angle $x_k(t)$7 may lose monotonicity over time and the metric transitions from pure phase concordance toward hybrid phase–amplitude or even correlation-like behavior. WPC is particularly informative in partial synchronization regimes with informative amplitude fluctuations, such as physiological dynamics, engineered chaotic systems, and BCI.

Table: Effect of Warping Constant $x_k(t)$8 in WPC

$x_k(t)$9 Regime	WPC Behavior	Interpretation
$$\Delta\alpha_{jk}(t) = \arg[\psi_j(t)\,\overline{\psi_k(t)}] = \alpha_j(t) - \alpha_k(t),$$0	Reduces to PLV	Pure phase synchrony assessment
$$\Delta\alpha_{jk}(t) = \arg[\psi_j(t)\,\overline{\psi_k(t)}] = \alpha_j(t) - \alpha_k(t),$$1	Maximal amplitude-phase sensitivity	Hybrid phase–amplitude synchrony
$$\Delta\alpha_{jk}(t) = \arg[\psi_j(t)\,\overline{\psi_k(t)}] = \alpha_j(t) - \alpha_k(t),$$2	Approaches trivial unity (unnormalized); normalized WPC→0 under asynchrony	Correlation-dominated regime

Further research directions include systematic comparison with amplitude-weighted and debiased PLV variants, evaluation of noise and volume conduction effects (notably in EEG), and application to denser topologies. Warped Phase Coherence constitutes a rapid, robust index for quantifying synchrony with enhanced sensitivity to amplitude-modulated interactions (Minati et al., 2019).