Phase-based connectivity

To compute phased-based connectivity metric between $S_1$ and $S_2$, we need to estimate a phase. Here we will use the instantaneous phase given by their analytic signal ($\widehat{S_1}$ and $\widehat{S_2}$). Thanks to the Hilbert transform ($\mathcal{H}(.)$) and Euler's formula, we obtain: $$\widehat{S_1}(t)= S_1(t)+i\mathcal{H}(S_1(t)) = Ae^{i(2\pi f_1t + \theta_1)} + \widehat{\epsilon_1}(t),$$ $$\widehat{S_2}(t)= Ae^{i(2\pi f_2t + \theta_2)} + \widehat{\epsilon_2}(t).$$ The instantaneous phase is defined as: $$\phi_i(t)= 2\pi f_it + \theta_i,$$ $$\textrm{so }\widehat{S_1}(t)= Ae^{i\phi_1(t)} + \widehat{\epsilon_1}(t), \textrm{ and } \widehat{S_2}(t)= Ae^{i\phi_2(t)} + \widehat{\epsilon_2}(t),$$ as $\widehat{\epsilon_i}$ is also a complex, we can write again thanks to Euler's formula: $$\widehat{S_i}(t)= \tilde{A}e^{i\tilde{\phi_i}(t)}$$ where $\tilde{A}$ and $\tilde{\phi_i}$ are the (noisy) estimation of $A$ and $\phi_i$. In other word, when the noise becomes null, $\\lim_{\sigma\to0} \tilde{A} = A$ and $\\lim_{\sigma\to0} \tilde{\phi_i} = \phi_i$. For simplicity of the notation we will only use $A$ and $\phi_i$.

The Phase Locking Value (PLV) is a measure of phase synchrony between two signals, independent of the amplitude of the signal (Lachaux et al., 1999). $$\text{PLV}(S_1,S_2) = \left| \frac{1}{N} \sum_{t=1}^{N} e^{i (\phi_1(t) - \phi_2(t))} \right|$$ $$\text{or, PLV}(S_1,S_2) = \left| \frac{1}{N} \sum_{t=1}^{N} e^{i \Delta\phi_{1,2}(t)} \right|$$
$N$ is the total number of time points, $\Delta\phi_{1,2}$ the phase difference, and $\left| . \right|$the absolute value. From the definition it is clear that the coherence is symmetrical or undirected, i.e. $\text{PLV}(S_1,S_2) = \text{PLV}(S_2,S_1)$
This measure quantifies how consistent the phase difference between two signals is over time, ranging from 0 (no phase synchrony) to 1 (perfect synchrony). Noteworthy, $A$ is not used in the defition of the $\text{PLV}$ rendering this metric independent of the amplitude. Beware, that it is still dependent on the signal-to-noise ratio, but not the amplitude of the signal per se. $$\Phi_{\text{PLV}}(S_1,S_2) = \arg \left( \frac{1}{N} \sum_{t=1}^{N} e^{i \Delta\phi_{1,2}(t)} \right)$$ $\arg(.)$ is the argument function, which gives the phase angle of the complex mean phase difference.
$\Phi_{\text{PLV}}$ represents the preferred phase difference. Note that $\Phi_{\text{PLV}}(S_1,S_2) = -\Phi_{\text{PLV}}(S_2,S_1)$
One can represent both information as a phasor in a polar plot, where its radius would be the $\text{PLV}$ and $\Phi_{\text{PLV}}$ its angle, that is: $$\mathcal{P}_{\text{PLV}}(S_1,S_2) = \frac{1}{N} \sum_{t=1}^{N} e^{i \Delta\phi_{1,2}(t)} = \text{PLV}e^{i \Phi_{\text{PLV}}}$$

Case 1: constant amplitude

As a first example, let us define $A_1=A_2=A=1$. The plot below let you modify $f_2$, $\theta_2$ and $\sigma$ and see their effect on the $\text{PLV}$.
First, you could test that when $\sigma = 0$ and $f_1 = f_2$, $\Phi_{\text{PLV}} = \theta_1 - \theta_2 = - \theta_2$ whatever the value of $\theta_2$ and $\text{PLV} = 1$.

Case 2: amplitude modulated signal

Let us now investigate the PLV metric when the amplitude is not constant, but varies over time. $$A_1(t) = \frac{1}{2}(a\cos(2\pi f_{A_1}t + \theta_{A_1}) + 1)$$ $f_{A_1}$ the frequency of the cosine amplitude modulation, $\theta_{A_1}$ the phase shift of the cosine amplitude modulation and $a$ the amount of modulation. Let us set $f_{A_1} = 2\textrm{ Hz}$, $\theta_{A_1} = 0\textrm{ rad}$ and $a = \frac{4}{5}$.

Interestingly, when $\sigma = 0$, $\text{PLV} = 1$, even if the amplitude/envelope of signal 1 is not constant anymore. This is because the computation of the $\text{PLV}$ does not take the amplitude into account (and because the envelope is never 0 in our case, otherwise the phase could not be determine correctly).

If $\sigma \ne 0$, one can see that the $\text{PLV}$ from case 2 is lower than the $\text{PLV}$ from case 1 for the same amount of noise. This is due to the fact that at the trough of the envelope, the level of is relatively high compared to the amplitude of the signal. The instantaneous phase become more dependent to the noise than to the signal of interest. This is why, even though the $\text{PLV}$ is not sensitive to the amplitude it is sensitive to the signal to noise ratio.

Version of the Phase Locking Value robust to spatial leakage.

In case of spatial leakage (signal mixing), such in Electrical Source Imaging, $S_1$ and $S_2$ can be spuriously correlated. However, this spurious correlation is instantaneous and thus occurs at 0-lag. One way to compensate for that is by removing the contribution to the 0-lag in the computation of the PLV. Importantly, 0-lag means that the phase difference equals 0. Any contribution at 0-lag happens on the real axis. Therefore the imaginary part of the PLV ($\text{iPLV}$) is defined as (Bruña et al., 2018): $$\text{iPLV} = \left|\Im\left( \mathcal{P}_{\text{PLV}} \right)\right|$$
$\Im$ is the imaginary part. This metric is now robust to spurious 0-lag but also removes any genuine 0-lag. We cannot, however, estimate the true preferred phase due to the mixing. Intuitively, $\text{iPLV}$ is the projection of the $\text{PLV}$ phasor onto the imaginary axis. Check the $\text{iPLV}$ box, and see how this metric behaves according to different changes in phase and noise.

Unfortunately, $\text{iPLV}$ is sensitive to the preferred phase difference $\Phi_{\text{PLV}}$. A corrected version of $\text{iPLV}$, named corrected iPLV, mitigates this effect and is defined as (Bruña et al., 2018): $$\text{ciPLV} = \frac{\Im\left( \mathcal{P}_{\text{PLV}}\right)}{\sqrt{1-\Re\left( \mathcal{P}_{\text{PLV}} \right)^2}}$$
$\Re$ is the real part.
Check the $\text{ciPLV}$ box, and see how this metric behaves according to different changes in phase and noise.

Phase-based connectivity

Introduction

Definition

Key concept

Example: Computing the Phase-Locking Value (PLV)

Case 1: constant amplitude

Case 2: amplitude modulated signal

Version of the Phase Locking Value robust to spatial leakage.