## 2 Materials and Methods
### Directional Connectivity
Volume conduction is a serious problem for directional connectivity computations of EEG signals [30] due to the relatively poor spatial resolution of scalp EEG. A single source on the brain, through volume conduction, can reach adjacent channels leading to erroneous inferences about connectivity [18]. In [31], a method was proposed to orthogonalize the power envelope of the received signals to suppress volume conduction effects. By orthogonalizing power signals at each channel, adjacent channels do not share the trivial co-variability made by common sources. Reference [18] proves that the aforementioned orthogonalizing procedure can be reflected in the imaginary part of cross-spectral density between two channels using the MVAR modeling of the received signals. This method (using the time scaling technique) is called general orthogonal partial directed coherence (gOPDC). If we assume that at most K independent sources affect each EEG channel and the sources add linearly, such a relationship based on Equation (1) in the frequency domain is:  
\[Y_{i}(f)=\sum_{k=1}^{K}v_{ik}S_{k}(f) \tag{5}\]  
Equation (5) can be written in its matrix form:  
\[Y(f)=VS(f) \tag{6}\]  
\(Y(f)\in C^{M}\) is a multi-channel EEG collected datum in the frequency domain, \(S(f)\in C^{M}\) is the multivariate source signal in the frequency domain, and \(V\in\mathbb{R}^{M\times M}\) includes all source weights. \[V=\left[ \begin{array}{ccc}v_{11}&\cdots&v_{1k}\\ \vdots&\ddots&\vdots\\ v_{M1}&\cdots&v_{Mk}\\ \end{array} \right] \tag{7}\]  
The cross-spectral density function \(C_{ij}(f)\) between \(Y_{i}(f)\) and \(Y_{j}(f)\) is defined by Equation (8):  
\[C_{ij}(f)=\left\langle Y_{i}(f)Y_{j}^{*}(f)\right\rangle=\sum_{k=1}^{K}V_{ik} V_{jk}\langle|S_{k}|\rangle^{2} \tag{8}\]  
The MVAR model in Equation (8) aids in checking the directional connectivity in the cross-spectral density. The innovative idea behind [18] is that instead of having two orthogonal signals, they search for orthogonal MVAR representations. The imaginary part of \(Imag\big{\{}C_{ij}(f)\big{\}}\), in effect, subtracts the volume conduction effects from the received signal. It can be shown that this value can be computed by Equation (9) [27]. \(\Sigma_{w}^{-1}\) is the inverse covariance matrix of \(l\)-th column of \(A(n,f)\), and \(a_{l}(n,f)\) is the \(l\)-th column of \(A(n,f)\). The inverse covariance value is used to ensure scale invariance:  
\[\Psi_{kl}(n,f)=\frac{|Real\{A_{kl}(n,f)\}|}{\sqrt{a_{l}^{H}(n,f)a_{l}(n,f)}} \times\frac{|Imag\{A_{kl}(n,f)\}|}{\sqrt{a_{l}^{H}(n,f)a_{l}(n,f)}}\text{ if }k\
eq 1 \tag{9}\]Analogous to the definition of gPDC, OPDC can be extended to gOPDC, \(\widetilde{\Psi}_{kl}(n,f)\) by taking the effect of time series scaling into consideration:  
\[\widetilde{\Psi}_{kl}(n,f)=\frac{1}{\lambda_{kl}^{2}}\times\frac{|Real\{A_{kl}(n,f)\}|}{\sqrt{a_{l}^{H}(n,f)\sum_{w}^{-1}a_{l}(n,f)}}\times\frac{|Imag\{A_{kl}( n,f)\}|}{\sqrt{a_{l}^{H}(n,f)\sum_{w}^{-1}a_{l}(n,f)}}\text{ if }k\
eq 1 \tag{10}\]  
We computed DC from EEG signals during rest using the model in Equation (1) and the formula of Equation (10). Each element of DC matrix is the mean value of time components and frequency bins in the regarded band derived from Equation (10). After computation of gOPDC via Equation (10), we analyzed the resulting directional connectivity to see how strong the connections were. Directional connectivity (DC) in this paper is computed by averaging over time and frequency bins' (in each of the 4 bands) component values of any computed gOPDC connection. Because the number of variables is 2808 (4 \(\times\) 2 \(\times\) \(\left( \begin{array}{c}27\\ 2\end{array} \right)\) - 4 frequency bands, 2 directions of DC, and \(\left( \begin{array}{c}27\\ 2\end{array} \right)\) related to all possible existing connections based on choosing 2 combinations of 27 numbers of EEG channels) (based on 4 frequency bands and 2 directions), the DC matrix is 33 \(\times\) 2808, including 2808 for each of 18 HC and 15 PD subjects.