Phase Lag Index is a tool to estimate connectivity in EEG in a way that eliminates volume conduction effects. However, it is also likely to discard a significant component of genuine interactions.
In the previous blog posts I discussed coherence and phase locking value as measures of functional connectivity. While the phase locking value only depends on the phase difference between the signals and disregards the amplitudes, coherence can increase with amplitude. We also saw how volume conduction can confound functional connectivity estimation.
In this blog post we will look at measures that are insensitive to volume conduction.
Let us briefly review the concept of phase synchronization. Borrowed from the field of physics, the concept of phase synchronization is widely used in neurosciences. Given two EEG signals in time-domain, we can transform them to be represented in the complex plane to reflect their amplitude and phase as shown in the figure below (see left).
Figure from 
If we compute the cross-spectrum (by multiplying the complex representation of one signal with the conjugate of the other), we obtain a vector whose length is the product of the two amplitudes and the angle between the vector and real axis represents the phase difference between the signals. A statistical definition can then be given to phase synchronization, which is the stability or consistency of phase differences across EEG time segments, or trials or epochs. Thus measures such as PLV reflect phase synchronization in true sense as it only looks at phase difference.
If one observes a phase difference of 0, this could mean that a neuronal source is seen by two sensors at the same instant and the phase observed at these sensors is the same (resulting in the phase difference of 0) . This could be due to volume conduction, although other alternatives cannot be ruled out purely on this basis (see section on Pitfalls). ue to such volume conduction, the individual phases observed at these two sensors can also have a difference of 180 degrees or π radians if one of the sensors sees the other end of the source dipole (i.e., records a negative voltage) !
Phase Lag Index
Phase lag index (PLI) was introduced by Stam and colleagues . The fundamental idea here is to disregard phase locking that is centered around 0 phase difference as a means of excluding volume conduction effects(at the risk of ignoring true instantaneous interactions). This also applies to phase locking at π, 2π and so on, i.e. repeating at every π , also given as
0 mod π. Mathematically this can be expressed as,
Where sign is the signum function that discards phase difference of 0 mod π. The PLI ranges between 0 and 1, with 0 indicating no coupling of instantaneous coupling due to volume conduction and 1 indicating true, lagged interaction.
Another way of looking at PLI is to think of the proportion of phase difference between signals over many trials above or below the 0 degree (i.e. the real axis) line. Asymmetry in this proportion reflects a consistent phase difference, while a flat distribution centered around 0 or π reflects no coupling of instantaneous coupling.
Figure from 
In the figure above, the distribution of phase differences is uniform around π (see left) and will thus give a PLI of 0. Where as, the distribution of the phase differences lying between 0 and π, in this case centered around π/2 (see right), will give a PLI of 1, indicating a consistent phase difference of π/2.
An extension to PLI, weighted PLI (wPLI) was introduced to de-weight vectors that are closer to the real axis (thus reflecting instantaneous mixing), such that those vectors have a smaller influence on the final PLI value .
Imaginary part of coherency
Coherency is a complex-valued quantity that is obtained using the normalized cross-spectrum of two signals. In other words, given N trials of a pair of EEG signals, coherency is the average of the phase difference between two signals over N trials, weighted by product of their normalized amplitudes. Thus Coherency can be expressed as ,
Where Sij represents cross-spectrum obtained by averaging over N trials and Sii is the power spectrum. It can be shown that coherency will be real-valued if the signals observed at two electrodes are purely due to independent sources. Thus discarding the real part of this quantity removes instantaneous interactions that are potentially spurious due to volume conduction and any non-instantaneous interaction between the two signals is reflected in the imaginary part of coherency.
Cohen  illustrated some pitfalls with phase-based measures that are insensitive to volume conduction. PLI underestimates the connectivity at small time lags and low signal-to-noise ratio. Also, frequency nonstationarities is a common issue for measures such as PLI or imaginary part of coherency . What this means is that, if there is a mismatch in the frequency of oscillation between two sources, then the distribution of the phase differences will fluctuate around 0 or π , which will naturally cause PLI or imaginary part of coherency to give a value of 0. This is illustrated in the figure below, where PLI or wPLI drops to zeros when the distribution of phase differences fluctuate around π, where as PLV, also referred to as inter-site phase clustering (IPSC) in , remains constant as it does not disregard interactions at phase difference of 0 or π.
Figure from 
Thus it is important to keep in mind that phase-based measures that are insensitive to volume conduction can reduce Type I errors , i.e., they will not falsely identify connections due to volume conduction but can give high Type II errors, meaning they may often miss true connections, for example due to case of small lags or frequency nonstationarities . While the reverse is true for measures such as PLV. In Local Field Potentials recorded in cortical tissue in vitro the phenomenology of neuronal avalanches and coherence potentials  demonstrate robust zero phase lag activity in cortical tissue that is not due to volume conduction (i.e. it ‘jumps’ across electrodes rather than spreading spatially contiguously, and can be disrupted with inhibitors of GABAergic inhibitory transmission) suggesting that the Type II errors may be exceptionally high. Consequently, additional methods that complement PLV are likely required in order to minimize these errors.
It also has to be kept in mind that PLV, PLI or imaginary part of coherency do not give any information about the direction of interaction. Measures such as phase slope index , which is also insensitive to volume conduction, can be used for this purpose. PSI can be calculated from imaginary part of coherency and it estimates the consistency of the direction of the change in the phase difference across frequencies. However the problems that apply to PLI or imaginary part of coherency, also apply to PSI (for example, high Type II errors).
- Bastos, André M., and Jan-Mathijs Schoffelen. “A tutorial review of functional connectivity analysis methods and their interpretational pitfalls.” Frontiers in systems neuroscience 9 (2016): 175.
- Stam, Cornelis J., 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 (2007): 1178-1193.
- Vinck M, Oostenveld R, van Wingerden M, Battaglia F, Pennartz CMA. An improved
index of phase-synchronization for electrophysiological data in the presence
of volume-conduction, noise and sample-size bias. Neuroimage 2011;55:
- Nolte, Guido, et al. “Identifying true brain interaction from EEG data using the imaginary part of coherency.” Clinical neurophysiology 115.10 (2004): 2292-2307.
- Cohen, Michael X. “Effects of time lag and frequency matching on phase-based connectivity.” Journal of neuroscience methods 250 (2015): 137-146.
- Thiagarajan, TC et al. Coherence Potentials: Loss-less all or none network events in the Cortex, PLOS Biology (2010)
- Nolte, Guido, et al. “Robustly estimating the flow direction of information in complex physical systems.” Physical review letters 100.23 (2008): 234101.