Coherence and phase synchronization: generalization to pairs of multivariate time series, and removal of zero-lag contributions

Coherence and phase synchronization between time series corresponding to different spatial locations are usually interpreted as indicators of the connectivity between locations. In neurophysiology, time series of electric neuronal activity are essential for studying brain interconnectivity. Such signals can either be invasively measured from depth electrodes, or computed from very high time resolution, non-invasive, extracranial recordings of scalp electric potential differences (EEG: electroencephalogram) and magnetic fields (MEG: magnetoencephalogram) by means of a tomography such as sLORETA (standardized low resolution brain electromagnetic tomography). There are two problems in this case. First, in the usual situation of unknown cortical geometry, the estimated signal at each brain location is a vector with three components (i.e. a current density vector), which means that coherence and phase synchronization must be generalized to pairs of multivariate time series. Second, the inherent low spatial resolution of the EEG/MEG tomography introduces artificially high zero-lag coherence and phase synchronization. In this report, solutions to both problems are presented. Two additional generalizations are briefly mentioned: (1) conditional coherence and phase synchronization; and (2) non-stationary time-frequency analysis. Finally, a non-parametric randomization method for connectivity significance testing is outlined. The new connectivity measures proposed here can be applied to pairs of univariate EEG/MEG signals, as is traditional in the published literature. However, these calculations cannot be interpreted as connectivity, since it is in general incorrect to associate an extracranial electrode or sensor to the underlying cortex.

💡 Research Summary

The paper addresses two fundamental challenges in estimating functional connectivity from EEG and MEG recordings that have been projected into source space using techniques such as sLORETA. First, each cortical location is represented by a three‑dimensional current density vector, so traditional coherence and phase‑synchronization measures, which are defined for scalar time series, must be extended to handle multivariate data. The authors propose a framework based on the complex cross‑spectral matrix: the covariance between two vector‑valued series is computed for each frequency, then normalized to yield a multivariate coherence matrix that captures linear coupling across all vector components. Phase synchronization is similarly generalized by extracting the complex phase of each vector and averaging the phase‑difference across components.

Second, the low spatial resolution of EEG/MEG source imaging introduces spurious zero‑lag correlations caused by volume conduction and field spread. To eliminate these artefacts, two complementary strategies are introduced. The first isolates the imaginary part of the cross‑spectral density (imaginary coherence), which by construction excludes instantaneous (zero‑lag) interactions. The second orthogonalizes one vector with respect to the other before recomputing coherence, thereby removing any shared zero‑lag component while preserving genuine delayed coupling. Both approaches aim to retain only physiologically meaningful, non‑instantaneous connectivity.

The paper also briefly discusses extensions: conditional coherence and phase synchronization, which control for a third time series to isolate direct connections; and non‑stationary time‑frequency analysis, where the multivariate measures are computed within sliding windows to track dynamic changes. Finally, a non‑parametric randomization test is described: surrogate datasets are generated by shuffling phases or trial labels, and the empirical multivariate connectivity values are compared against the surrogate distribution to obtain significance thresholds, thereby addressing multiple‑comparison issues inherent in high‑dimensional data.

Overall, the authors deliver a comprehensive methodological toolkit that (1) generalizes coherence and phase synchronization to multivariate source‑level signals, (2) removes artificial zero‑lag coupling, (3) allows conditional and time‑varying analyses, and (4) provides rigorous statistical testing. This framework enables more accurate inference of true neural interactions from EEG/MEG source reconstructions, moving beyond the simplistic interpretation of sensor‑level correlations.