Abstract
A large variety of methods exist to estimate brain coupling in the frequency domain from electrophysiological data measured, e.g., by EEG and MEG. Those data are to reasonable approximation, though certainly not perfectly, Gaussian distributed. This work is based on the well-known fact that for Gaussian distributed data, the cross-spectrum completely determines all statistical properties. In particular, for an infinite number of data, all normalized coupling measures at a given frequency are a function of complex coherency. However, it is largely unknown what the functional relations are. We here present those functional relations for six different measures: the weighted phase lag index, the phase lag index, the absolute value and imaginary part of the phase locking value (PLV), power envelope correlation, and power envelope correlation with correction for artifacts of volume conduction. With the exception of PLV, the final results are simple closed form formulas. In an excursion we also discuss differences between short time Fourier transformation and Hilbert transformation for estimations in the frequency domain. We tested in simulations of linear and non-linear dynamical systems and for empirical resting state EEG on sensor level to what extent a model, namely the respective function of coherency, can explain the observed couplings. For empirical data we found that for measures of phase-phase coupling deviations from the model are in general minor, while power envelope correlations systematically deviate from the model for all frequencies. For power envelope correlation with correction for artifacts of volume conduction the model cannot explain the observed couplings at all. We also analyzed power envelope correlation as a function of time and frequency in an event related experiment using a stroop reaction task and found significant event related deviations mostly in the alpha range.
Highlights
Electrophysiological recordings like electroencephalography (EEG) and magnetoencephalography (MEG) have a high temporal resolution, but are non-invasive measurements with a low spatial resolution
In this paper we presented mathematical relations between linear and non-linear measures of brain coupling assuming Gaussian distributed data
We considered four different non-linear measures of phase-phase coupling: weighted phase lag index (wPLI), phase lag index (PLI), and absolute value and imaginary part of phase locking value (PLV)
Summary
Electrophysiological recordings like electroencephalography (EEG) and magnetoencephalography (MEG) have a high temporal resolution, but are non-invasive measurements with a low spatial resolution. The question to be addressed here is whether the corresponding measures really describe different phenomena as it is at least mathematically, conceivable that, e.g., phasephase coupling determines amplitude-amplitude coupling even if the actual values are not identical, which is the case for Gaussian distributed data as will be shown below. In such a case the latter would be a function of the former; the estimation of the latter would not add information on the brain dynamics and our measures would be essentially redundant. We tried to keep the main body of the paper as simple as possible, and we moved all mathematical derivations, which are technically quite involved, to an Supplementary Material
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