Coherence is a commonly used measure in EEG analysis. What is Coherence and what are the practical considerations, limitations and interpretations of this measure when applied to EEG?
How electrical activity flows in the brain to create groups of cells that are functionally integrated (i.e. working together) versus functionally segregated (i.e. working independently or separate from each other) is considered to be fundamentally important for understanding how the brain drives cognitive function and behavior. This comes down to devising ways to identify which regions are interacting or independent at any period of time, and essentially to construct maps of functional connectivity.
Such functional integration or connectivity is typically defined as the statistical inter-dependencies between the functional properties (for example electromagnetic activation or BOLD response) of these brain areas. There are other types of connectivity, such as structural connectivity which refers to direct anatomical links between brain regions or effective connectivity which gives information about the causality or directed influence (region A caused the activation of region B).
Another important approach to estimating functional connectivity is to look at spatial relationships in the EEG. Neuronal oscillations and their inter-regional synchronization is believed to be one of the mechanisms that enables interaction between various brain regions. Thus, quantifying synchrony using EEG data has become a popular approach to study functional brain connectivity. One of the widely used methods to study synchrony is coherence. Like many such methods, this measure of functional connectivity simply tells us that region A is linked to region B and says nothing about the direction of influence. It is therefore symmetric.
The Coherence Measure
We can think of coherence as a measure of similarity between two signals. When operating in the time domain, correlation is typically used to asses similarity between a pair of signals. One can think of coherence in a similar fashion, but instead if assessing similarity in the time-domain, coherence can be used to asses similarity in the frequency domain. Loosely speaking, coherence answers the following question – given a pair of signals, how similar are the power spectra of these two signals ? So, two completely independent signals will have a coherence value of 0 while two identical signals will have a coherence of 1.
Mathematically, given two time series x_i(t) and y_i(t) recorded from two channels , coherence C can be defined as
where Γ_ij ( f ) is the cross-spectrum (in practice, averaged over many epochs of EEG data) and Γ_ii (f) and Γ_jj (f) are the power spectrums of signals in channel i and j respectively.
Finally, the absolute value of coherency, |C_ij (f)| is defined as coherence. (Note that rather than using the square root in the denominator and doing this in two steps, some folks represent coherence as the square of the absolute value of the numerator and square of the denominator of the above equation. In essence these are similar.).
Thus computing coherence involves first computing the power spectra of the two signals at hand using for example, FFT or the Welch method. Then computing cross-spectra, normalizing by the square root of the power spectra of both the signals and finding the absolute value. Cross-spectra can be obtained as the product of the FFT of two signals which provides a measure of the energy in the frequency band that is common to both the signals.
Problem of volume conduction
Volume conduction is an important issue to be dealt with when measuring interactions between different brain regions, particularly in the EEG where the intervening tissue (skull, skin etc.) conducts charge. Volume conduction refers to the fact that, the activity of a source can be picked up by all the EEG electrodes, with zero or near zero phase delays i.e., the effects of volume conduction are instantaneous. Such an assumption is valid under the quasi-static approximations of the Maxwell’s equations, where the time derivatives are ignored. If you want to dig deep into this, the article by Plonsey and Heppner is an excellent reference.
Volume conduction results in high coherence values, especially for electrodes that are close to each other. One way to mitigate this issue is to first solve the EEG inverse problem and then estimate coherency between the estimated source activity. This approach also has its challenges and must be carefully interpreted (see related post The Inverse Problem in EEG – Assumptions and Pitfalls ).
Another work around is to use the imaginary part of coherence , rather than looking at its absolute value, to overcome the problem of volume conduction. The rationale here is that, volume conduction, as mentioned above, is an instantaneous , zero-phase delay phenomena and this maps onto the real part. Thus by looking at only the imaginary part of coherence, one can exclude the possible effects of such spurious, zero-phase delay correlations and look at interactions that have a phase delay. It is important to understand here that a phase lag in frequency domain is equivalent to having a time-lag between two signals in temporal domain and volume conduction cannot explain such delayed interactions. Thus, the other possible explanation for delayed interaction is that the two regions are, in fact, functionally connected.
Issues with coherence
Since coherence essentially depends on the power spectra, all the issues related with the computation of power spectra apply to coherence. Primary among these is the data length and non-stationarity. For instance, if the epochs are shorter, then analyzing interactions at lower frequencies can be biased due to unreliable estimation of the power spectra. Similarly, if the signal is nonstationary (i.e., composed of frequencies that vary with time), the Fourier transform cannot give reliable results. Thus, choosing an appropriate data length such that the signal is stationary and at the same time covers your frequencies of interest is not trivial. Without careful consideration of these aspects, you may end up with biased results.
see related post Factors that Impact Power Spectral Density Estimation
Finally, Lachaux and colleagues argued that coherence does not strictly reflect the phase relationship, but can be affected by the amplitude covariation of the signals . Thus, one can obtain high values of coherence as the amplitude covariance increases. Thus, whether it is the phase covariance or amplitude covariance that contributes to the coherence value cannot be easily untangled.
Inspite of these shortcomings, coherence and particularly, imaginary part of coherence has found widespread applications in EEG connectivity analysis. Several extensions to the coherence measure have been proposed and these include Partial coherence (multivariate extension of coherence) and partial directed coherence (which also gives the direction of influence). It is important to carefully interpret coherence in the context of these factors as results can be different based on the choice of underlying algorithms and methods.
1. Nolte, G., O. Bai, L. Wheaton, Z. Mari, S. Vorbach, and M. Hallett 2004. Identifying true brain interaction from eeg data using the imaginary part of coherency. Clinical neurophysiology, 115(10):2292–230
2. Lachaux, J.-P., E. Rodriguez, J. Martinerie, F. J. Varela, et al. 1999. Measuring phase synchrony in brain signals. Human brain mapping, 8(4):194–208