Field spread refers to the phenomenon that the magnetic fields that are associated with electrical currents (of neural and non-neural origin) are not confined to the vicinity of the current generators, but are measurable far away from their source. For any neuronal source this leads to a widespread representation at the level of the sensor array. As a matter of fact, thanks to this feature, it is possible in the first place to measure MEG extracranially and to build models of the underlying neural sources. Yet, electromagnetic field spread also has important consequences for the interpretability of connectivity measures estimated between pairs or sensors. The reason for this is that any single source of neural (or non-neural) electric activity is visible to many sensors at once. This is illustrated by Fig. 1a. The spatial topography shows the correlation between one channel and the rest, from simulated data containing one single dipole plus uncorrelated sensor noise. Obviously, MEG sensor recordings represent the superposition of the activity of multiple sources, which are either or not functionally connected. Also with multiple sources present, even if the underlying sources are ‘unconnected’, many connectivity measures estimated between pairs of sensors will yield spurious estimates due to the instantaneous mixing process. To illustrate this we simulated the activity of 821 temporally uncorrelated dipoles, with an orientation parallel to the axis between the nasion and the midpoint of the interauricular line, and that were randomly distributed on the cortical sheet. Clearly, the orientations chosen are physiologically not meaningful, but are appropriate to demonstrate the effect of field spread on connectivity analysis. Using FieldTrip (Oostenveld et al. 2011), we simulated 50 s of data for a 275-channel CTF axial gradiometer system, by using a single shell volume conductor model (Nolte 2003) and uncorrelated sensor noise. Figure 1b shows the absolute value of the correlation coefficient between all MEG sensor pairs as a function of their distance. Even though the underlying source activities are temporally uncorrelated, many sensor pairs show high values of correlation.

In order to reduce the interpretational difficulties caused by field spread, one potential strategy could be to analyze changes in connectivity caused by an experimental manipulation, rather than the strength of the connectivity as such. The rationale for using experimental contrasts in this context is based on the assumption that the effects of electromagnetic field spread are identical across the experimental conditions and therefore subtract out. Unfortunately, the spatial structure of field spread is highly dependent on changes in the signals, and on changes in the noise. As a consequence, estimated modulations in connectivity do not necessarily always reflect modulations in actual connectivity between relevant neuronal sources. Experimental manipulations will most likely always lead to changes in activity of the underlying sources, or in the activation of different sources. Also, in studies that involve the comparison between different groups of subjects (e.g. patients versus controls), it is not unlikely that difference in the distribution and activity of the underlying sources exist. These potential confounds in the interpretation of estimated differences in connectivity should therefore always be taken into account. This is illustrated in Fig. 2. Here, we simulated two dipoles oscillating at 20 Hz in left and right ‘motor cortex’, at a phase difference of 90°, against a background of 821 uncorrelated dipoles evenly distributed across the cortical sheet. We generated 2 conditions of data where the amplitude of the motor cortex dipoles was twice as large in condition 1 compared to condition 2 (Fig. 2a). We computed coherence and the imaginary part of coherency (see Sect. 3.3) across all channel pairs and display the sensor pairs in which the difference in connectivity across the conditions exceeded a certain threshold. Clearly there is interesting spatial structure in the differential coherence and imaginary coherency maps, that cannot be accounted for by a change in actual connectivity (which as a matter of fact in both conditions was simulated to be equal to 1).

For the reasons outlined above, field spread is problematic in the interpretation of sensor-level connectivity estimates, and an important motivation to perform the connectivity analysis at the source level. Also, contrasting connectivity between two experimental conditions in sensor space will likely reduce (but not abolish) negative effects of field spread (Schoffelen and Gross 2009). In addition to this, there are other important motivations to perform the analysis on the source level. First of all, there is a more direct indication of the anatomical location of the interacting brain regions. Secondly, source level analysis facilitates subsequent group analysis because the data can be averaged in a meaningful standardized space.

Another strategy to address interpretation problems associated with electromagnetic field spread is to use connectivity measures that are insensitive to this effect. In general, this type of measure can be divided into two categories. The first category consists of measures that are derived from conventional linear measures. The imaginary part of coherency (Nolte et al. 2004) is a well-known example of this type of measure. Another example is amplitude-envelope correlation with the zero time-lag correlation removed (Hipp et al. 2012) (see Sect. 5). The other category consists of measures that are derived from the non-linear dynamics framework. This type of measure includes the phase lag index (Stam et al. 2007) synchronization likelihood (Stam and Van Dijk 2002) and transfer entropy (Vicente et al. 2011).

Measures that are insensitive to field spread usually exploit the fact that field spread caused by point sources has an instantaneous effect on the sensors. In other words, field spread causes cross-correlation effects between sensors at a time lag of 0 ms, or equivalently at a phase difference of 0 or 180°. Explicitly removing the zero ms time lag contribution to the estimate of connectivity reduces the risk of falsely interpreting the estimate as true interaction. In the case of the imaginary part of coherency the removal of the 0° phase difference contribution is achieved by projection of the vector representation of the complex-valued coherency onto the imaginary axis.

An important caveat needs to be raised here, which is related to the fact that spurious connectivity is addressed only when contributing sources can be modeled as single point sources (equivalent current dipoles). This is illustrated in Fig. 3 where we present results of an analysis of the weighted phase lag index (WPLI) at 10 Hz (Vinck et al. 2011). For each channel, we computed the average WPLI between that channel and the rest of the channels and represented this in a spatial topography (panel A). Red here means that the underlying channels on average have a positive phase difference with the other channels, blue means that the underlying channels on average have a negative phase difference with the other channels. Thus the picture suggests a fronto-occipital gradient of time-lagged neural oscillations at 10 Hz, where the frontal channels ‘lead’ the occipital channels. However, the data that was used to generate this topography was constructed by back-projecting two independent components that were estimated from a few minutes of resting state MEG data. The time course of these components and their corresponding spatial topographies are shown in panel B. Clearly, these two components mainly represent cardiac activity. The slight time lag between the individual components in combination with the different topographies leads to a non-trivial mixing with significant interaction at non-zero time lag.

For the reasons outlined above it is increasingly acknowledged that the functional interactions should be studied at the level of the neuronal sources.