Functional Connectivity: PLI / PLV

Initialize NeuroAnalyzer

using NeuroAnalyzer
eeg = load("files/eeg.hdf")
e10 = epoch(eeg; ep_len = 10)

Phase Lag Index (PLI) measures asymmetry in phase differences between signals, focusing on non-zero phase lags.

Phase Locking Value (PLV): measures consistency of phase differences, regardless of lag.

Key Differences:

PLI filters out zero-lag (artificial) connections.
PLI is robust against volume conduction but may miss some interactions.
PLV includes all phase relationships.
PLV captures all synchronization but is sensitive to zero-lag artifacts.

Feature	PLV	PLI
Primary goal	Measures the stability of the phase difference between two signals	Measures the consistency of the sign of the phase difference
Sensitivity	All phase differences	Only non-zero phase lags
Volume Conduction	Affected by zero-lag artifacts	Less affected
Range	0 to 1	0 to 1
Sensitivity to noise	Less sensitive to noise	More sensitive to noise perturbations
Use Case	General phase synchronization	True interaction detection

Use PLI if you want to minimize the effect of volume conduction and focus on true interactions between sources.

Use PLV if you want to detect any phase synchronization, regardless of lag.

Phase Lag Index

Phase Lag Index (PLI) measures asymmetry in phase differences between two signals, ignoring zero-lag synchronization (which may be due to volume conduction).

Formula:

\[ \text{PLI} = \left \vert \frac{1}{N} \sum_{k=1}^N \text{sign}(\Delta \phi_{ij} (t_n)) \right \vert \]

where: $\text{sign}(\sin(\Delta\phi))$ returns 1 if the phase difference is positive, -1 if negative.

Range: 0 (no lag) to 1 (consistent non-zero lag).

Use Case: Ideal for identifying true interactions in EEG/MEG data.

Interpretation:

PLI = 0: No consistent phase lag (random or zero-lag synchronization).
PLI = 1: Perfect consistent phase lag (true directional connectivity).

Advantages:

Reduces volume conduction effects by ignoring zero-lag synchronization.
Use Case: Studying directional connectivity in brain networks.

Calculating PLI:

pli_data = pli(e10,
               e10;
               ch1 = "F3",
               ch2 = "F4",
               ep1 = 1,
               ep2 = 1)
println("PLI: $(pli_data.pv[1])")

PLI: 0.04453125

Weighted Phase Locking Index

Weighted Phase Locking Index (WPLI) is a weighted version of PLI that accounts for the magnitude of phase differences (Nolte et al. 2004).

Formula:

\[ \text{WPLI} = \frac{\left \vert \sum_{n=1}^{N} w_n e^{i \Delta \phi_{ij}(t_n)} \right \vert}{\sum_{n=1}^{N} \vert w_n \vert} \]

where: $w_n = \text{Im}(e^{i \Delta \phi_{ij}(t_n)})$ (the imaginary part of the complex phase difference).

Alternatively, WPLI can be calculated using Cross Power Spectrum Density (CPSD):

\[ \text{WPLI}(f) = \frac{\left \vert \left \langle \text{Im}(S_{xy}(f)) \right\rangle \right \vert}{\left \langle \vert S_{xy}(f) \vert \right\rangle} \]

where:
- $\text{Im}(S_{xy}(f))$ is the imaginary part of the CPSD,
- $\left \vert \cdot \right \vert$ denotes the magnitude.

Range: 0 (no phase locking) to 1 (perfect phase locking).

Advantage:

More robust to noise and volume conduction compared to PLV.
Provides a more accurate estimate of true brain connectivity.

Why Use WPLI?

Reduces false positives from weak or spurious connections.
Focuses on meaningful interactions by emphasizing strong phase relationships.
Less sensitive to zero-lag artifacts than PLV.

Use Cases

EEG/MEG studies where noise and volume conduction are concerns.
Network analysis to identify core hubs or critical connections.

The wPLI weights each phase difference by the magnitude of the imaginary component of the cross-spectrum, which helps to downplay the contribution of phase differences around $0$ or $\pi$ (which are more likely to be due to volume conduction).

Calculating wPLI:

wpli_data = wpli(e10,
                 e10;
                 ch1 = "F3",
                 ch2 = "F4",
                 ep1 = 1,
                 ep2 = 1)
println("wPLI: $(wpli_data.pv[1])")

wPLI: 2.0229218207816276e-17

Additionally, a debiased wPLI estimator can be calculated:

wpli_data = wpli(e10,
                 e10;
                 ch1 = "F3",
                 ch2 = "F4",
                 ep1 = 1,
                 ep2 = 1,
                 debiased = true)
println("WPLI: $(wpli_data.pv[1])")

WPLI: -0.006158630306069087

Directed Phase Lag Index

Directed Phase Lag Index (dPLI) was introduced in by Stam et al. (2012) to capture the phase and lag relationship as a measure of directed functional connectivity.

dPLI is defined as the probability that the instantaneous phase of $x$ was smaller than the phase of $y$ (modulo $\pi$).

if $0.5 < dPLI_{xy} \leq 1.0$, $x$ is leading $y$
if $0.0 \leq dPLI_{xy} = 0.5$, $y$ is leading $x$
if $dPLI_{xy} = 0.5$, neither $x$ nor $y$ is leading or lagging

dpli_data = dpli(e10,
                 e10;
                 ch1 = "F3",
                 ch2 = "F4",
                 ep1 = 1,
                 ep2 = 1)
println("dPLI: $(dpli_data.pv[1])")

dPLI: 0.417578125

Null connectivity is 0 for PLI and wPLI, but 0.5 for dPLI.

wPLI is less sensitive to noise and is the method of choice for studying an underlying phase relationship.

To study directionality, use wPLI or dPLI.

Phase Locking Value

Phase Locking Value (PLV) measures consistency of phase differences, including zero and non-zero lags.

Formula:

\[ PLV = \frac{1}{N} \left \vert \sum_{n=1}^{N} e^{i \Delta \phi_{ij}(t_n)} \right \vert \]

Range: 0 (random phases) to 1 (perfect phase locking).

Use Case: Useful for detecting transient synchronization.

PLI reflects the proportion of phase angle differences above or below the real (horizontal) axis.

Advantage:

PLI is not sensitive to volume conduction and is robust common noise sources.

It only measures phase synchronization, not amplitude correlation.

For multiple channels, you can compute PLI between all pairs to create a connectivity matrix.

PLI is insensitive to instantaneous (zero-lag) synchronization, which helps reduce the influence of volume conduction artifacts.

Interpretation: PLI = 0: no consistent phase locking (random phase differences), PLI = 1: perfect phase locking (constant phase difference).

High PLI: Indicates strong phase synchronization between the two signals, suggesting true interaction between sources.

Low PLI: Indicates weak or no phase synchronization.

Phase Locking Value

Phase Locking Value (PLV)

Definition: A normalized measure of phase synchronization between two signals, ranging from 0 (no synchronization) to 1 (perfect synchronization).
Formula:
$
= | _{k=1}^N e^{i (_1(k) - _2(k))} |
$
- Same as ISPC, but often used interchangeably.
Advantages:
- Phase-invariant: Does not depend on the absolute phase of oscillations.
- Robust to noise: Less sensitive to amplitude variations.
Use Case: Studying functional connectivity in cognitive tasks or clinical conditions.

Phase Locking Value (PLV): reflects the proportion of phase angle differences above or below the real (horizontal) axis.

It is used for measuring synchrony (especially changes in synchrony) across brain regions/electrodes.

\[ PLV = \frac{1}{N} \left \vert \sum{e^{1i \times (\varphi_1 - \varphi_2)}} \right \vert \] $e^{\varphi_1 - \varphi_2}$ gives the randomness of $\Delta \varphi$

$N$: number of segments to average. In practice it is best if you have a large number of trials.

Interpretation: PLV $\approx$ 0: phases are randomly distributed (no phase locking), PLV ≈ 1: phases are highly synchronized (perfect phase locking).

PLV uses the Hilbert transform to calculate instantaneous phases, hence the signal should be narrowband.

Calculate PLV:

plv_data = plv(e10,
               e10;
               ch1 = "F3",
               ch2 = "F4",
               ep1 = 1,
               ep2 = 1)
println("PLV: $(plv_data.pv[1])")

PLV: 0.14681551787660263

Since we might be interested whether PLV changed after a stimulus, we need to normalize (using z score) using the the pre-stimulus period as a baseline (subtract z score of the baseline PLV values from PLV).

Imaginary Phase Locking Value (IPLV) was proposed to resolve PLV’s sensitivity to volume conduction and common reference effects.

\[ IPLV = \frac{1}{N} \left \vert \mathfrak{I} \left ( \sum{e^{1i \times (\varphi_1 - \varphi_2)}} \right ) \right \vert \]

iplv_data = iplv(e10,
                 e10;
                 ch1 = "F3",
                 ch2 = "F4",
                 ep1 = 1,
                 ep2 = 1)
println("IPLV: $(iplv_data.ipl[1])")

IPLV: 0.062038174364113366