# An eLORETA Longitudinal Analysis of Resting State EEG Rhythms in Alzheimer's Disease
## 2 Materials and Methods
### Loreta  
LORETA was the first linear distributed method for solving the EEG inverse problem which provided a valid solution from the neurophysiological point of view [25]. LORETA, in fact, solved the matter of the large source localization error of the minimum norm method, which misplaced deep source onto the outermost cortex [37]. LORETA employs a three-shell spherical head model (MNI305) registered to the [PERSON] human brain atlas [38; 39]. The head model is made up of the scalp, skull, and brain compartment. The brain compartment, limited to the gray matter and hippocampus, is discretized in 2394 voxels at 7 mm spatial resolution. Each source is placed on a voxel and is characterized by a current density vector with unknown components. Therefore, the potential recorded on the scalp derives from the electrical activity at each point. LORETA seeks the smoothest of all possible infinite solutions by means of a discrete spatial Laplacian operator. Therefore, the LORETA images represent the electrical activity at each voxel as power of the computed current density. Because of the smoothing, the location of the maximal activity is maintained but the images are \"blurred\". The average localization error is one voxel. The expression of the general inverse problem is [40]  
\[\min_{J}F_{W} \tag{1}\]  
with  
\[F_{W}=\|\mathbf{\Phi}-\mathbf{KJ}\|^{2}+\alpha\mathbf{J}^{T}\mathbf{WJ} \tag{2}\]  
where \(\mathbf{\Phi}\in R^{N_{E}x1}\) is the vector of the electric potential differences measured at \(N_{E}\) electrodes with respect to a single common reference electrode; \(\mathbf{K}\in R^{N_{E}x(3N_{V})}\) is the lead field matrix corresponding to \(N_{v}\) voxels, \(\mathbf{J}\in R^{(3N_{V})x1}\) is the current density, and \(\alpha>0\) is the Tikhonov regularization parameter [41]. The solution is  
\[\mathbf{\hat{J}_{W}}=\mathbf{T_{W}}\mathbf{\Phi} \tag{3}\]  
here \(\mathbf{T_{W}}\) is the pseudoinverse, given by  
\[\mathbf{T_{W}}=\mathbf{W}^{-1}\mathbf{K}^{T}(\mathbf{K}\mathbf{W}^{-1}\mathbf{ K}^{T}+\alpha\mathbf{H})^{+} \tag{4}\]  
The matrix \(\mathbf{H}\) is the average reference operator. For LORETA, the matrix \(\mathbf{W}\) represents the squared spatial discrete Laplacian operator. After a few years, the sLORETA algorithm was introduced [42]. Unlike LORETA, the brain volume is made up of 6239 voxels at 5 mm spatial resolution. The electric potential lead field used for the inverse solution is computed with the boundary element method applied to the MNI152 template [43; 44]. Moreover, sLORETA abandons the use of the laplacian operator and is based on the standardization of the current density. Differently from the Dale method [45], which takes into account only the variation due to measurement noise, in sLORETA the estimated current density considersthe variation of the actual sources and, if any, the variation due to noisy measurements. The electric potential variance is given by  
\[\mathbf{S_{\Phi}}=\mathbf{K}\mathbf{S_{j}}\mathbf{K}^{T}+\mathbf{S_{\Phi}}^{ Noise} \tag{5}\]  
where \(\mathbf{S_{\Phi}}^{Noise}\) is the variance of the noisy measurements and \(\mathbf{S_{j}}\) is the actual source variance (prior). The latter, according to the Bayesian formulation, is equal to the identity matrix. Therefore, the variance of the estimated current density can be expressed as  
\[\mathbf{S_{j}}=\mathbf{T}\mathbf{S_{\Phi}}\mathbf{T}^{T}=\mathbf{T}(\mathbf{ K}\mathbf{K}^{T}+\alpha\mathbf{H})\mathbf{T}^{T}=\mathbf{K}^{T}[\mathbf{K} \mathbf{K}^{T}+\alpha\mathbf{H}]^{+}\mathbf{K} \tag{6}\]  
Finally, sLORETA is given by the estimates of standardized current density power:  
\[\sigma_{v}^{T}\sigma_{v}=\hat{J}_{v}^{T}[S_{j}]_{v}^{-1}\hat{J}_{v} \tag{7}\]  
where \([S_{j}]_{v}\) is the \(v\)th 3 \(\times\) 3 diagonal matrix in \([S_{j}]\). As for the localization accuracy, sLORETA has zero localization error in ideal (no-noise) conditions and a lower localization error as compared with both the minimum norm and the Dale methods in the case of noisy measurements [37; 45].