# Phase Locking Value

Phase Locking Value (PLV) is a statistic that can be used to investigate task-induced changes in long range synchronization of neural activity from EEG data. This method is introduced in Lachaux et al., (1999). I have implemented the computation of this statistic in MATLAB and you can download it from the link below. In this post, I will talk a little bit about how PLV can be useful when analyzing EEG data. Note that this statistic may also be used to quantify LFP (local field potential) data recordings to study synchrony at a finer scale compared to EEG.

It is useful to keep in mind a few properties of the PLV statistic before diving in further.

1. PLV statistic is a time course. This means that for every time point in your EEG data, you can extract a measure of connectivity. Therefore, this quantity can be used to observe transient changes in connectivity without pre-defining a time window of analysis.
2. One PLV time course for every electrode pair. For example, if data is recorded from 28 EEG electrodes, there are 378 possible PLV time courses.
3. One PLV time course is computed over multiple trials. In practice, I have observed that this metric is suitable for datasets with a large number of trials. It is good to have a few hundred trials for each experimental condition for which long range synchronization is to be quantified using PLV.

The PLV statistic can be argued to be a proxy for connectivity. Intuitively, if the EEG signal in two channels (electrodes) during an experimental condition rises and falls together more than a baseline value, then there is more synchronization or loosely speaking, enhanced connectivity between these two electrodes. If it is less than the baseline value, there is desynchronization or loosely speaking, decreased connectivity between the two electrodes. Note that this metric does not care about the co-variation in the power of the EEG signal between two electrodes.

The schematic below illustrates how PLV is computed in this implementation. Explanation for each step follows.

Filtering. The EEG data is first filtered in the desired frequency band of interest, for example, gamma band, 35-45 Hz. In practice, it is best to use an FIR (finite impulse response) filter to filter such data in comparison to IIR (infinite impulse response) filters. Loosely speaking, FIR filters filter the signal in time domain and IIR filters operate in the frequency domain. Put simply, an FIR filters computes the value of the filtered signal at a time point from the values of the previous and future points. How far the FIR filter looks is knows as the order of the FIR filter.  For EEG signals, a useful rule of thumb is to ‘look at’ about 4 to 5 cycles of the desired EEG rhythm. As an example, consider gamma rhythm (35-45 Hz). Here, one cycle is about 25 ms and I would set the order of the filter to be 100 ms. If the sampling rate is 500 Hz, then the filter order would be 50 data points.

Hilbert transform. This step is used to quantify rising and falling of EEG data. Hilbert transform of a signal can be used to compute the instantaneous amplitude as well as the instantaneous phase. We ignore the former and use the phase (φ) for PLV computation. φ is a value between –π and π. Think of the filtered EEG signal as a series of crests and troughs. A value of π indicates the peak of a crest and a –π indicates the bottom of a trough.

PLV metric. Remember that there is a phase time course φ for every EEG electrode. Now consider the phase time courses of two electrodes. The difference between these two time courses (Δφ) quantifies locking between the phases of these two electrodes. If an experimental stimulus influences signal in two electrodes to rise and fall together or with a certain lag, then Δφ will be consistent between trials. If there is absolutely no relationship between when the signal in these two electrodes will rise and fall, then Δφ will be random. All we need to do now is to quantify the randomness in Δφ. This can be done using a little trick from complex number theory. Please see the equation in the analysis schematic to understand this trick.

Normalization. This step is performed to make the PLV metric useful in practice. Note that due to its definition, PLV is always a value between 0 and 1; 0 signifying purely random rise and fall whereas a value of 1 signifies that one signal perfectly follows the other. In practice, the value of a PLV obtained is going to vary very little over time and the absolute PLV is not what we are interested in. We are interested in knowing whether an experimental stimulus induced a change in PLV. We are interesting in answering questions such as “was there increased connectivity between the frontal and parietal electrodes when the subject performed a task?” . To find this, we should test if PLV after stimulus is significantly greater than the PLV before stimulus. To simplify things, we can use the pre-stimulus period as a baseline and perform a z-transform normalization.

Good luck!