Electroencephalographic (EEG) signals recorded from the scalp surface are generally highly correlated. Each channel is a linear mixture of concurrently active brain and non-brain electrical sources whose activities are volume conducted to the scalp electrodes with broadly overlapping patterns (Nunez et al., 1997). This property is particularly relevant to connectivity analyses, which seek to detect and characterize active interactions between brain regions. Therefore, meaningful connectivity patterns can be derived only from measures of cortical source activities and not directly from EEG channel activities (Michel and Murray, 2012). However, this poses a serious problem, since estimating the nature, number, brain (or non-brain) locations, and time courses of the active sources contributing to the scalp EEG is not straightforward (Baillet et al., 2001). Several methods have been proposed to estimate source activities from multi-channel EEG recordings, thereby removing the confounding effects of volume conduction. These methods can be grouped into three categories: (a) simple spatial filters that seek to reduce correlations between scalp channels based on idealized assumptions; (b) more complex spatial filters that seek to estimate net activities within ROIs based on detailed neurophysiological head models; and (c) blind spatial source separation methods that seek to separately identify source signals by exploiting source signal information differences. Whereas simple spatial filters such as bipolar derivations and Laplacian filters can reduce, to some extent, correlations among scalp-recorded channels (Fisch, 2012), more sophisticated spatial filtering methods use inverse imaging methods to estimate the time courses of cortical sources in given or estimated region of interest (ROI) (Baillet et al., 2001). Blind source separation techniques, in particular independent component analysis, by contrast, learn spatial filters from the EEG time courses that separate the data into constituent independent source activities. Their corresponding brain (or non-brain) locations can then be estimated using neurophysiological inverse imaging methods (Makeig et al., 1996; Jung et al., 2001; Delorme et al., 2012). Vector autoregressive (VAR) models are versatile tools for analyzing multivariate time series, including multi-channel EEG or multivariate source activities. VAR models predict current values of time series from their recent past (Lutkepohl, 2005). Importantly, they can be used to derive various electrophysiological connectivity measures (Schlogl and Supp, 2006). Volume conduction in biological tissue can be modeled as instantaneous propagation of activity from sources to recording channels. The resulting zero-phase connectivity may be treated as noise added to lagged connectivity patterns of interest. Although some measures, including the imaginary part of the coherency (Nolte et al., 2004), are insensitive to zero-phase connectivity, measures derived from VAR model coefficients do not include such zero-phase terms. Thus, volume conduction effects are not accounted for by the model and affect the correlation structure of the model residuals, which are normally assumed to be uncorrelated. Popular connectivity measures derived from VAR models include the Directed Transfer Function (DTF) (Kaminski and Blinowska, 1991) and the Partial Directed Coherence (PDC) (Baccala and Sameshima, 2001). Whereas the PDC is defined in terms of the system matrix (a frequency domain representation of the VAR model), the DTF is based on the inverse of the system matrix. Viewing lagged dependencies between source signals as information flow, the DTF may be said to be normalized by the inflow of information to some sink, while the PDC is normalized by the outflow of information from some source. As we will demonstrate below, both the DTF and the PDC are indeed adversely affected by volume conduction from multiple sources to the scalp electrodes, in contrast to the claim of Kaminski and Blinowska in their recent opinion article (Kaminski and Blinowska, 2014). Thus, in general direct application of connectivity measures to scalp EEG signals produces less than accurate results and also does not allow their clear interpretation in terms of underlying source dynamics.
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