6 Methods for Functional Connectivity Analysis
137
6.4.3 FC Measures Robust to Volume Conduction Effect
Another solution is to use the FC metrics which is inherently robust to the volume
conduction [34, 36, 38, 52, 58]. Nunez et al. [38] proposed a modified version of
coherence, called the reduced coherence. It is calculated by subtracting the random coherence from the measured coherence. Alternatively, partial coherence was
introduced, which removes the linear effect of the third time-series (considered as
common source) from the coherence calculated from a pair of time-series [34].
Nolte et al. [36] suggested imaginary coherence (ImC). It is based on the hypothesis that the imaginary part of coherence (i.e., the non-zero phase difference) cannot
be affected by the volume conduction [36]. In the same vein, Stam et al. [52] proposed the PLI based on how much the non-zero phase differences are distributed to
negative (phase lag) or positive (phase lead) sides of the x axis on the complex plane
[52]. More recently, an extended version of PLI, called the weighted PLI (WPLI),
was suggested to take into account the magnitude as well as the distribution of the
phase differences [58].
Figure 6.5 illustrates the advantage of PLI to mitigate the volume conduction
effects, as compared to the PLV. The phase differences between EEGs of two nearby
channels Fpz and Fp1 are distributed symmetrically around zero degree, which
resulted in much lower value of PLI as compared to the PLV (0.09 vs. 0.86). In
addition, the correlation between the FC and the inter-electrode distance became
drastically reduced to −0.174, from −0.787 in the case of PLV (Fig. 6.5b). The
correlation with the spectral power became insignificant as well (Fig. 6.5c). All the
results in Fig. 6.5 shows that spurious FC due to the volume connection can be
alleviated by using the PLI. Vinck et al. [58] pointed out the problems of the PLI.
Temporal discontinuity may occur when there exist small perturbations which lead
to phase lags from phase leads (and vice versa) between two time-series. Also, the
estimation of PLI is statistically biased, hence its calculation may suffer from the
small sample size [52, 58]. Modified versions of the PLI (weighted PLI and debiased
weighted PLI) have been proposed to overcome these limitations [58].
6.5 EEG FC Analysis on Source Space
Spatial filters such as surface Laplacian can be applied before the FCA to lessen the
effect of volume conduction as shown above. More recently, the reconstruction of
cortical current sources is performed prior to the FCA by solving an inverse problem
[4, 33, 40]. This provides time-series of cortical current densities at numerous vertices
on cortical surface, which enables the calculation of FC measures among cortical
regions. The FCA at the cortical source space is advantageous also for the better
explanation of the obtained results, since each pair of connection has anatomical
interpretation [46].
137
6.4.3 FC Measures Robust to Volume Conduction Effect
Another solution is to use the FC metrics which is inherently robust to the volume
conduction [34, 36, 38, 52, 58]. Nunez et al. [38] proposed a modified version of
coherence, called the reduced coherence. It is calculated by subtracting the random coherence from the measured coherence. Alternatively, partial coherence was
introduced, which removes the linear effect of the third time-series (considered as
common source) from the coherence calculated from a pair of time-series [34].
Nolte et al. [36] suggested imaginary coherence (ImC). It is based on the hypothesis that the imaginary part of coherence (i.e., the non-zero phase difference) cannot
be affected by the volume conduction [36]. In the same vein, Stam et al. [52] proposed the PLI based on how much the non-zero phase differences are distributed to
negative (phase lag) or positive (phase lead) sides of the x axis on the complex plane
[52]. More recently, an extended version of PLI, called the weighted PLI (WPLI),
was suggested to take into account the magnitude as well as the distribution of the
phase differences [58].
Figure 6.5 illustrates the advantage of PLI to mitigate the volume conduction
effects, as compared to the PLV. The phase differences between EEGs of two nearby
channels Fpz and Fp1 are distributed symmetrically around zero degree, which
resulted in much lower value of PLI as compared to the PLV (0.09 vs. 0.86). In
addition, the correlation between the FC and the inter-electrode distance became
drastically reduced to −0.174, from −0.787 in the case of PLV (Fig. 6.5b). The
correlation with the spectral power became insignificant as well (Fig. 6.5c). All the
results in Fig. 6.5 shows that spurious FC due to the volume connection can be
alleviated by using the PLI. Vinck et al. [58] pointed out the problems of the PLI.
Temporal discontinuity may occur when there exist small perturbations which lead
to phase lags from phase leads (and vice versa) between two time-series. Also, the
estimation of PLI is statistically biased, hence its calculation may suffer from the
small sample size [52, 58]. Modified versions of the PLI (weighted PLI and debiased
weighted PLI) have been proposed to overcome these limitations [58].
6.5 EEG FC Analysis on Source Space
Spatial filters such as surface Laplacian can be applied before the FCA to lessen the
effect of volume conduction as shown above. More recently, the reconstruction of
cortical current sources is performed prior to the FCA by solving an inverse problem
[4, 33, 40]. This provides time-series of cortical current densities at numerous vertices
on cortical surface, which enables the calculation of FC measures among cortical
regions. The FCA at the cortical source space is advantageous also for the better
explanation of the obtained results, since each pair of connection has anatomical
interpretation [46].