Comparing PLI, wPLI, and dPLI

This example demonstrates the different connectivity information captured by the phase lag index (PLI) [1], weighted phase lag index (wPLI) [2], and directed phase lag index (dPLI) [3] on simulated data.

# Authors: Kenji Marshall <kenji.marshall99@gmail.com>
#          Charlotte Maschke <charlotte.maschke@mail.mcgill.ca>
#          Stefanie Blain-Moraes <stefanie.blain-moraes@mcgill.ca>
#
# License: BSD (3-clause)


import matplotlib.pyplot as plt
import mne
import numpy as np
from mne.datasets import sample

from mne_connectivity import spectral_connectivity_epochs

Background

The formulae for PLI, wPLI, and dPLI are given below. In these equations, \(X_{ij}\) is the cross-spectral density (CSD) between two signals \(i, j\). Importantly, the imaginary component of the CSD is maximal when the signals have a phase difference given by \(k\pi+\frac{\pi}{2}\), and is \(0\) when the phase difference is given by \(k\pi\) (where \(k \in \mathbb{Z}\)). This property provides protection against recognizing volume conduction effects as connectivity, and is the backbone for these methods [2]. In the equations below, \(\mathcal{I}\) refers to the imaginary component, \(\mathcal{H}\) refers to the Heaviside step function, and \(sgn\) refers to the sign function.

\(PLI = |E[sgn(\mathcal{I}(X_{ij}))]|\) [1]

\(wPLI = \frac{|E[\mathcal{I}(X_{ij})]|}{E[|\mathcal{I}(X_{ij})|]}\) [2]

\(dPLI = E[\mathcal{H}(\mathcal{I}(X_{ij}))]\) [3]

All three of these metrics are bounded in the range \([0, 1]\).

• For PLI, \(0\) means that signal \(i\) leads and lags signal \(j\) equally often, while a value greater than \(0\) means that there is an imbalance in the likelihood for signal \(i\) to be leading or lagging. A value of \(1\) means that signal \(i\) only leads or only lags signal \(j\).

• For wPLI, \(0\) means that the total weight (not the quantity) of all leading relationships equals the total weight of lagging relationships, while a value greater than \(0\) means that there is an imbalance between these weights. A value of \(1\), just as in PLI, means that signal \(i\) only leads or only lags signal \(j\).

• With dPLI, we gain the ability to distinguish whether signal \(i\) is leading or lagging signal \(j\), complementing the information provided by PLI or wPLI. A value of \(0.5\) means that signal \(i\) leads and lags signal \(j\) equally often. A value in the range \((0.5, 1.0]\) means that signal \(i\) leads signal \(j\) more often than it lags, with a value of \(1\) meaning that signal \(i\) always leads signal \(j\). A value in the range \([0.0, 0.5)\) means that signal \(i\) lags signal \(j\) more often than it leads, with a value of \(0\) meaning that signal \(i\) always lags signal \(j\). The PLI can actually be extracted from the dPLI by the relationship \(PLI = 2|dPLI - 0.5|\), but this relationship is not invertible (dPLI can not be estimated from the PLI).

Overall, these different approaches are closely related but have subtle differences, as will be demonstrated throughout the rest of this example.

Capturing Leading/Lagging Phase Relationships with dPLI

The main advantage of dPLI is that it’s directed, meaning it can differentiate between phase relationships which are leading or lagging. To illustrate this, we generate sinusoids with Gaussian noise. In particular, we generate signals with phase differences of \([-\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi]\) relative to a reference signal. A negative difference means that the reference signal is lagging the other signal.

fs = 250  # sampling rate (Hz)
n_e = 300  # number of epochs
T = 10  # length of epochs (s)
f = 10  # frequency of sinusoids (Hz)
t = np.arange(0, T, 1 / fs)
A = 1  # noise amplitude
sigma = 0.5  # Gaussian noise variance

data = []

phase_differences = [0, -np.pi, -np.pi / 2, 0, np.pi / 2, np.pi]
for ps in zip(phase_differences):
    sig = []
    for _ in range(n_e):
        sig.append(
            np.sin(2 * np.pi * f * t - ps)
            + A * np.random.normal(0, sigma, size=t.shape)
        )
    data.append(sig)

data = np.swapaxes(np.array(data), 0, 1)  # make epochs the first dimension

A snippet of this simulated data is shown below. The blue signal is the reference signal.

fig, ax = plt.subplots(1, 2, figsize=(12, 5), sharey=True)
ax[0].plot(t[:fs], data[0, 0, :fs], label="Reference")
ax[0].plot(t[:fs], data[0, 2, :fs])

ax[0].set_title(r"Phase Lagging ($-\pi/2$ Phase Difference)")
ax[0].set_xlabel("Time (s)")
ax[0].set_ylabel("Signal")
ax[0].legend()

ax[1].plot(t[:fs], data[0, 0, :fs], label="Reference")
ax[1].plot(t[:fs], data[0, 4, :fs])
ax[1].set_title(r"Phase Leading ($\pi/2$ Phase Difference)")
ax[1].set_xlabel("Time (s)")

plt.show()

We will now compute PLI, wPLI, and dPLI for each phase relationship.

conn = []
indices = ([0, 0, 0, 0, 0], [1, 2, 3, 4, 5])
for method in ["pli", "wpli", "dpli"]:
    conn.append(
        spectral_connectivity_epochs(
            data,
            method=method,
            sfreq=fs,
            indices=indices,
            fmin=9,
            fmax=11,
            faverage=True,
        ).get_data()[:, 0]
    )
conn = np.array(conn)

Connectivity computation...
    using t=0.000s..9.996s for estimation (2500 points)
    frequencies: 9.0Hz..11.0Hz (21 points)
    connectivity scores will be averaged for each band
    computing connectivity for 5 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: PLI
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    frequencies: 9.0Hz..11.0Hz (21 points)
    connectivity scores will be averaged for each band
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    the following metrics will be computed: WPLI
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[Connectivity computation done]
Connectivity computation...
    using t=0.000s..9.996s for estimation (2500 points)
    frequencies: 9.0Hz..11.0Hz (21 points)
    connectivity scores will be averaged for each band
    computing connectivity for 5 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: DPLI
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[Connectivity computation done]

The estimated connectivites are shown in the figure below, which provides insight into the differences between PLI/wPLI, and dPLI.

Similarities Of All Measures

• Capture presence of connectivity in same situations (phase difference of \(\pm\frac{\pi}{2}\))

• Do not predict connectivity when phase difference is a multiple of \(\pi\)

• Bounded between \(0\) and \(1\)

How dPLI is Different Than PLI/wPLI

• Null connectivity is \(0\) for PLI and wPLI, but \(0.5\) for dPLI

• dPLI differentiates whether the reference signal is leading or lagging the other signal (lagging if \(0 <= dPlI < 0.5\), leading if \(0.5 < dPLI <= 1.0\))

x = np.arange(5)

plt.figure()
plt.bar(x - 0.2, conn[0], 0.2, align="center", label="PLI")
plt.bar(x, conn[1], 0.2, align="center", label="wPLI")
plt.bar(x + 0.2, conn[2], 0.2, align="center", label="dPLI")

plt.title("Connectivity Estimation Comparison")
plt.xticks(x, (r"$-\pi$", r"$-\pi/2$", r"$0$", r"$\pi/2$", r"$\pi$"))
plt.legend()
plt.xlabel("Phase Difference")
plt.ylabel("Estimated Connectivity")

plt.show()

Robustness to Outliers and Noise with wPLI

The previous experiment illustrated the advantages conferred by dPLI when differentiating leading and lagging phase relationships. This experiment will now focus on understanding the advantages of wPLI, and explore how it extends upon PLI.

The main difference between PLI and wPLI is in how different phase relationships are weighted. In PLI, phase differences are weighted as \(-1\) or \(1\) according to their sign. In wPLI, phase differences are weighted based on their value, meaning that phase differences closer to \(\pm\frac{\pi}{2}\) are weighted more heavily than those close to \(0\) or any other multiple of \(\pi\).

This avoids a discontinuity at the transition between positive and negative phase, treating all phase differences near this transition in a similar way. This provides some robustness against outliers and noise when estimating connectivity. For instance, volume conduction can distort EEG/MEG recordings, wherein signals emanating from the same neural source will be picked up by multiple sensors on the scalp. This can effect connectivity estimations, bringing the relative phase differences between two signals close to \(0\). wPLI minimizes the contribution of phase relationships that are small but non-zero (and may thus be attributed to volume conduciton), while PLI weighs these in the same way as phase relationships of \(\pm\frac{\pi}{2}\).

To demonstrate this, we recreate a result from (Vinck et al, 2011) [2]. Two sinusoids are simulated, where the phase difference for half of the epochs is \(\frac{\pi}{2}\), and is \(-\frac{\pi}{100}\) for the others. We also explore the effect of applying uniform noise to this phase difference.

n_noise = 41  # amount of noise amplitude samples in [0, 4]
data = [[]]

# Generate reference
for _ in range(n_e):
    data[0].append(np.sin(2 * np.pi * f * t))

A_list = np.linspace(0, 4, n_noise)

for A in A_list:
    sig = []
    # Generate other signal
    for _ in range(int(n_e / 2)):  # phase difference -pi/100
        sig.append(
            np.sin(2 * np.pi * f * t + np.pi / 100 + A * np.random.uniform(-1, 1))
        )
    for _ in range(int(n_e / 2), n_e):  # phase difference pi/2
        sig.append(np.sin(2 * np.pi * f * t - np.pi / 2 + A * np.random.uniform(-1, 1)))
    data.append(sig)

data = np.swapaxes(np.array(data), 0, 1)

# Visualize the data
fig, ax = plt.subplots(1, 2, figsize=(12, 5), sharey=True)
ax[0].plot(t[:10], data[0, 0, :10], label="Reference")
ax[0].plot(t[:10], data[1, 1, :10])

ax[0].set_title(r"Phase Lagging ($-\pi/100$ Phase Difference)")
ax[0].set_xlabel("Time (s)")
ax[0].set_ylabel("Signal")
ax[0].legend()

ax[1].plot(t[:fs], data[0, 0, :fs], label="Reference")
ax[1].plot(t[:fs], data[-1, 1, :fs])
ax[1].set_title(r"Phase Leading ($\pi/2$ Phase Difference)")
ax[1].set_xlabel("Time (s)")

plt.show()

We can now compute PLI and wPLI

conn = []
indices = ([0] * n_noise, np.arange(1, n_noise + 1))
for method in ["pli", "wpli"]:
    conn.append(
        spectral_connectivity_epochs(
            data,
            method=method,
            sfreq=fs,
            indices=indices,
            fmin=9,
            fmax=11,
            faverage=True,
        ).get_data()[:, 0]
    )
conn = np.array(conn)

Connectivity computation...
    using t=0.000s..9.996s for estimation (2500 points)
    frequencies: 9.0Hz..11.0Hz (21 points)
    connectivity scores will be averaged for each band
    computing connectivity for 41 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: PLI
    computing cross-spectral density for epoch 1
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[Connectivity computation done]
Connectivity computation...
    using t=0.000s..9.996s for estimation (2500 points)
    frequencies: 9.0Hz..11.0Hz (21 points)
    connectivity scores will be averaged for each band
    computing connectivity for 41 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: WPLI
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[Connectivity computation done]

The results from the simulation are shown in the figure below. In the case without noise, the difference between wPLI and PLI is made obvious. In PLI, no connectivity is detected, as the \(-\frac{\pi}{100}\) phase differences are weighted in the exact same way as the \(\frac{\pi}{2}\) relationships. wPLI is able to avoid the cancellation of the \(\frac{\pi}{2}\) relationships.

As noise gets added, PLI increases since the \(-\frac{\pi}{100}\) relationships are made positive more often than the \(\frac{\pi}{2}\) relationships are made negative. However, wPLI maintains an advantage in its ability to distinguish the underlying structure. Beyond a certain point, the noise dominates any pre-defined structure, and both methods behave similarly, tending toward \(0\). For a more detailed analysis of this result and the properties of wPLI, please refer to (Vinck et al, 2011) [2].

plt.figure()
plt.plot(A_list, conn[0], "o-", label="PLI")
plt.plot(A_list, conn[1], "o-", label="wPLI")
plt.legend()
plt.xlabel("Noise Amplitude")
plt.ylabel("Connectivity Measure")
plt.title("wPLI and PLI Under Increasing Noise")
plt.show()

Demo On MEG Data

To finish this example, we also quickly demonstrate these methods on some sample MEG data recorded during visual stimulation.

data_path = sample.data_path()
raw_fname = data_path / "MEG/sample/sample_audvis_filt-0-40_raw.fif"
event_fname = data_path / "MEG/sample/sample_audvis_filt-0-40_raw-eve.fif"
raw = mne.io.read_raw_fif(raw_fname)
events = mne.read_events(event_fname)


# Select gradiometers
picks = mne.pick_types(
    raw.info, meg="grad", eeg=False, stim=False, eog=True, exclude="bads"
)

# Create epochs
event_id, tmin, tmax = 3, -0.2, 1.5  # need a long enough epoch for 5 cycles
epochs = mne.Epochs(
    raw,
    events,
    event_id,
    tmin,
    tmax,
    picks=picks,
    baseline=(None, 0),
    reject=dict(grad=4000e-13, eog=150e-6),
)
epochs.load_data().pick_types(meg="grad")  # just keep MEG and no EOG now

fmin, fmax = 4.0, 9.0  # compute connectivity within 4-9 Hz
sfreq = raw.info["sfreq"]  # the sampling frequency
tmin = 0.0  # exclude the baseline period

# Compute PLI, wPLI, and dPLI
con_pli = spectral_connectivity_epochs(
    epochs,
    method="pli",
    mode="multitaper",
    sfreq=sfreq,
    fmin=fmin,
    fmax=fmax,
    faverage=True,
    tmin=tmin,
    mt_adaptive=False,
    n_jobs=1,
)

con_wpli = spectral_connectivity_epochs(
    epochs,
    method="wpli",
    mode="multitaper",
    sfreq=sfreq,
    fmin=fmin,
    fmax=fmax,
    faverage=True,
    tmin=tmin,
    mt_adaptive=False,
    n_jobs=1,
)

con_dpli = spectral_connectivity_epochs(
    epochs,
    method="dpli",
    mode="multitaper",
    sfreq=sfreq,
    fmin=fmin,
    fmax=fmax,
    faverage=True,
    tmin=tmin,
    mt_adaptive=False,
    n_jobs=1,
)

Opening raw data file /home/circleci/mne_data/MNE-sample-data/MEG/sample/sample_audvis_filt-0-40_raw.fif...
    Read a total of 4 projection items:
        PCA-v1 (1 x 102)  idle
        PCA-v2 (1 x 102)  idle
        PCA-v3 (1 x 102)  idle
        Average EEG reference (1 x 60)  idle
    Range : 6450 ... 48149 =     42.956 ...   320.665 secs
Ready.
Not setting metadata
73 matching events found
Setting baseline interval to [-0.19979521315838786, 0.0] s
Applying baseline correction (mode: mean)
0 projection items activated
Loading data for 73 events and 256 original time points ...
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
    Rejecting  epoch based on EOG : ['EOG 061']
24 bad epochs dropped
NOTE: pick_types() is a legacy function. New code should use inst.pick(...).
Adding metadata with 3 columns
Connectivity computation...
    using t=0.000s..1.498s for estimation (226 points)
    frequencies: 4.7Hz..8.6Hz (7 points)
    connectivity scores will be averaged for each band
only using indices for lower-triangular matrix
    computing connectivity for 20503 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: PLI
    computing cross-spectral density for epoch 1
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    assembling connectivity matrix
[Connectivity computation done]
Replacing existing metadata with 3 columns
Connectivity computation...
    using t=0.000s..1.498s for estimation (226 points)
    frequencies: 4.7Hz..8.6Hz (7 points)
    connectivity scores will be averaged for each band
only using indices for lower-triangular matrix
    computing connectivity for 20503 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: WPLI
    computing cross-spectral density for epoch 1
    computing cross-spectral density for epoch 2
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    computing cross-spectral density for epoch 48
    computing cross-spectral density for epoch 49
    assembling connectivity matrix
[Connectivity computation done]
Replacing existing metadata with 3 columns
Connectivity computation...
    using t=0.000s..1.498s for estimation (226 points)
    frequencies: 4.7Hz..8.6Hz (7 points)
    connectivity scores will be averaged for each band
only using indices for lower-triangular matrix
    computing connectivity for 20503 connections
    Using multitaper spectrum estimation with 7 DPSS windows
    the following metrics will be computed: DPLI
    computing cross-spectral density for epoch 1
    computing cross-spectral density for epoch 2
    computing cross-spectral density for epoch 3
    computing cross-spectral density for epoch 4
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    computing cross-spectral density for epoch 48
    computing cross-spectral density for epoch 49
    assembling connectivity matrix
[Connectivity computation done]

In this example, there is strong connectivity between sensors 190-200 and sensors 110-160.

Moreover, after observing the presence of connectivity, dPLI can be used to ascertain the direction of the phase relationship. Here, it appears that the dPLI connectivity in this area is less than \(0.5\), and thus sensors 190-200 are lagging sensors 110-160.

In keeping with the previous simulation, we can see that wPLI identifies stronger connectivity relationships than PLI. This is due to its robustness against volume conduction effects decreasing the detected connectivity strength, as was mentioned earlier.

fig, axs = plt.subplots(1, 3, figsize=(14, 5), sharey=True)
axs[0].imshow(con_pli.get_data("dense"), vmin=0, vmax=1)
axs[0].set_title("PLI")
axs[0].set_ylabel("Sensor 1")
axs[0].set_xlabel("Sensor 2")

axs[1].imshow(con_wpli.get_data("dense"), vmin=0, vmax=1)
axs[1].set_title("wPLI")
axs[1].set_xlabel("Sensor 2")

im = axs[2].imshow(con_dpli.get_data("dense"), vmin=0, vmax=1)
axs[2].set_title("dPLI")
axs[2].set_xlabel("Sensor 2")

fig.colorbar(im, ax=axs.ravel())
plt.show()

Conclusions

Both wPLI and dPLI are extensions upon the original PLI method, and provide complementary information about underlying connectivity.

• To identify the presence of an underlying phase relationship, wPLI is the method of choice for most researchers as it provides an improvement in robustness over the original PLI method

• To know the directionality of the connectivity identified by wPLI, dPLI should be used

Ultimately, these methods work great together, providing a comprehensive estimate of phase-based connectivity.

References

[1] (1,2)

Cornelis J. Stam, Guido Nolte, and Andreas Daffertshofer. Phase lag index: assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources. Human Brain Mapping, 28(11):1178–1193, 2007. doi:10.1002/hbm.20346.

[2] (1,2,3,4,5)

Martin Vinck, Robert Oostenveld, Marijn van Wingerden, Franscesco Battaglia, and Cyriel M.A. Pennartz. An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias. NeuroImage, 55(4):1548–1565, 2011. doi:10.1016/j.neuroimage.2011.01.055.

[3] (1,2)

C. J. Stam and E. C. W. van Straaten. Go with the flow: use of a directed phase lag index (dpli) to characterize patterns of phase relations in a large-scale model of brain dynamics. NeuroImage, 62(3):1415–1428, Sep 2012. doi:10.1016/j.neuroimage.2012.05.050.