# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming
## 2 Materials and Methods
### EEG Data
#### 2.6.1 Simulated Data  
Realistic EEG measurements were simulated using the four-layers geometrical model of the head shown in Figure 2. The measurements were obtained for an array of \(M=90\) sensors. Based on that, we generated \(K=100\) measurements with independent biological noise (i.e., background activity coming from other regions in the brain) and measurement noise realizations, such that (2) became extended to  
\[Y_{k}=A(\mathbf{\theta})Q+A(\mathbf{\theta}_{\Delta})Q_{\Delta}+V_{k},\ \ \text{for}\ k=1,2, \ldots,K, \tag{17}\]  
where \(\mathbf{\theta}_{\Delta}\) and \(Q_{\Delta}\) are the positions and dipole moments, respectively, of five hundred cortical and one hundred thalamic sources simulating the biological noise. The number of cortical and thalamic sources were chosen to approximate realistic spatially correlated noise and to provide enough power to simulate a desired range of noise conditions (for a discussion on random dipole modeling of spontaneous brain activity, see [25; 26]). For both cortical and thalamic sources, their \(\|Q_{\Delta}\|\) were modeled as independent Gaussian random variables with zero mean and variance \(\sigma_{\Delta}^{2}\). The orientation of each \(Q_{\Delta}\) remained always normal to surface of the triangle in which it was located in the tessellated thalamus. Using (17), and by adjusting the corresponding variances \(\sigma_{\text{v}}^{2}\) and \(\sigma_{\Delta}^{2}\), we created different scenarios based on the following ratios (in decibels):  
* signal-to-measurement-noise ratio, given by \[\eta_{m}=20\log\frac{\|A(\mathbf{\theta})Q\|_{\text{r}}}{\|V_{k}\|_{\text{r}}},\] (18) where \(\|\cdot\|_{\text{r}}\) denotes the Frobenius norm;
* signal-to-biological-noise ratio, given by \[\eta_{b}=20\log\frac{\|A(\mathbf{\theta})Q\|_{\text{r}}}{\|A(\mathbf{\theta}_{\Delta })Q_{\Delta}\|_{\text{r}}}.\] (19)  
Hence, we simulated different sets of measurements with a combination of \(\eta_{m}=0,\,5,\,10\) dB, and \(\eta_{b}=-5,\,0,\,5,\,10\) dB (average values over \(K=100\) independent noise realizations). Under those conditions, each trial \(k\) was created through (17) to have specific \((\eta_{m},\eta_{b})\) while keeping \(\mathbf{\theta}\) and \(Q\) unchanged. In all data, we considered the case of \(L=3\) sources located in the occipital lobe and in close proximity to each other in order to generate spatially correlated measurements. For \(Q\), their temporal evolution was chosen to model realistic evoked responses. Furthermore, two out of the three dipoles were independent between each other (they are shown in Figure 3). The third dipole was constructed as a linear combination of the other two, i.e.,  
\[\mathbf{q}_{3}=\left[ \begin{array}{ccc}0.4&0.1\\ 0.2&0.7\\ 0.5&0.9\end{array} \right]\left[ \begin{array}{c}\mathbf{q}_{1}\\ \mathbf{q}_{2}\end{array} \right]. \tag{20}\]  
This approach was taken in order to further enforce a situation of source correlation in which traditional beamformers are known to fail. Finally, \(A(\mathbf{\theta})\) and \(A(\mathbf{\theta}_{\Delta})\) in (17) were calculated by using the comprehensive BEM toolbox that was proposed in [11]. Once we solved the forward problem of data generation, we proceeded with the solution of the inverse problem of estimating \(\mathbf{\theta}\). For that, all neural indices were computed only within a region-of-interest (ROI) comprising the visual cortex. Therefore, the ROI contained 517 candidate positions out of the 8633 total positions that the tessellated model of the brain cortex had. Next, the pool of candidate positions was reduced by choosing those for which their \({}_{tLOW}(\mathbf{r})\) were above the 70-percentile. That process left out all but  
Figure 3: Temporal evolution of dipoles \(q_{1}\) and \(q_{2}\). Their Cartesian components \(x,y,\) and \(z\) are shown in red, blue, and green, respectively. Figure 2: Geometrical model with four layers (scalp, skull, brain and thalami). (**a**) Frontal. (**b**) Sagittal. 150 candidate source positions.