## 2 Materials and Methods  
### Simulation Procedure  
The simulation framework developed by [1] was used to generate simulated EEG data originating from three dipoles. This data generating process, as well as the forward and inverse problems, were implemented in MATLAB (2020). Figure 1 shows the data generation procedure. The standard length of each generated series was 1500 time steps for Ground Truth 1 and 2. In Ground Truth 1, three fixed dipoles were used with the directionality of the connections as well as their strength being imposed (Figure 2), a strategy used before [20; 21; 22]:  
\[ \begin{array}{c}X_{1}(t)=0.5X_{1}(t-1)-0.7X_{1}(t-2)+c12(t)X_{2}+\in_{1}(t) \\ X_{2}(t)=0.7X_{2}(t-1)-0.5X_{2}(t-2)+0.2X_{1}+c23(t)X_{3}(t-1)+\in_{2}(t)\\ X_{3}(t)=0.8X_{3}(t-1)+\in_{3}(t)\end{array}  \tag{1}\]  
with \(X_{1}\), \(X_{2}\) and \(X_{3}\), three electrical sources contributing to the simulated scalp-EEG signals and with:  
\[ \begin{array}{c}c12(t)=0.5\frac{t}{L}\leavevmode\
obreak\ if\leavevmode \
obreak\ t\leavevmode\
obreak\ \leq\leavevmode\
obreak\ \frac{t}{2}\leavevmode\
obreak\ \leavevmode\
obreak\,\leavevmode\
obreak\ c12(t)=0.5\frac{L-t}{L}\leavevmode\
obreak\ if\leavevmode \
obreak\ t>\frac{L}{2}\\ c23(t)=0.4\leavevmode\
obreak\ if\leavevmode\
obreak\ t<0.7L,\leavevmode \
obreak\ c23(t)=0\leavevmode\
obreak\ if\leavevmode\
obreak\ t\geq 0.7L\end{array}  \tag{2}\]  
Figure 1: Simulation Procedure followed by Connectivity Estimation. TCDF = Depthwise Separable 1D Temporal Causal Discovery Framework, LSTM-NUE = Long Short-Term Memory with Non-Uniform Embedding, Conv2D = 2D Convolutional Network, TRGC = Time-Reversed Granger Causality. Coordinates in the Ground Truths denote MNI-coordinates. \(L=\) length of the generated time series (\(L=\) 1500), \(t=\) the current time step and \(\epsilon=\) uncorrelated white noise, varying with time. We further assume an EEG cap with 108 electrodes. For Ground Truth 2, we considered two fixed, one moving dipole and only one true connection (Figure 3) and focused on the presence or absence of this connectivity as well as its directionality:  
\[\left[ \begin{array}{c}X_{s}(t)\\ X_{r}(t)\\ X_{n}(t)\end{array} \right]=\sum_{p=1}^{P}\left[ \begin{array}{ccc}a_{11}(p)&0 &0\\ a_{21}(p)&a_{22}(p)&0\\ 0&0&a_{33}(p)\end{array} \right]\left[ \begin{array}{c}X_{s}(t-p)\\ X_{r}(t-p)\\ X_{n}(t-p)\end{array} \right]+\left[ \begin{array}{c}\epsilon_{1}(t)\\ \epsilon_{2}(t)\\ \epsilon_{3}(t)\end{array} \right] \tag{3}\]  
with \(X_{s}\), the moving dipole, as a sender, and two fixed dipoles, with \(X_{r}\) the receiver and \(X_{n}\) the fixed non-interactive dipole, and \(a_{\rm ij}\) (\(p\)), \(i,j\in\{1,2,3\}\) and \(p\in\{1,\ldots,P\}\) the coefficients with \(a_{21}\) the coupling strength between sender and receiver. All \(a_{\rm ij}\) are randomly picked from the interval [0.3, 1]. Finally, \(\epsilon\) is uncorrelated, biological, white noise. The moving dipole (the sender) changes location (far, deep, close, superficial) at every iteration, with a total of 1004 iterations. The maximum time lag t is two. The reason for this ground truth is that the sender can be located at really challenging locations (too close to one of the other dipoles or very deep in the brain). Two conditions were created for both ground truths: one condition consisted of three superficial dipoles far away from each other, while the other consisted of three dipoles located \"deep\" in the brain, but each dipole was still positioned far away from the other dipoles. The corresponding MNI-coordinates of the two fixed dipoles that Ground Truth 1 and 2 have in common are depicted in Figure 4. The full set of coordinates of Ground Truth 1 (including the coordinates of the third fixed dipole) is denoted in Figure 1.  
In Ground Truth 2, the first two coordinates are the same as in Ground Truth 1, for each dipole condition, while the third dipole moves throughout the brain as described above.