To allow for more straightforward estimations of outlierness and similarity of vertices across the cortical sheet, we transformed the data into a latent representation with a known probability density function (PDF).  We sampled our 39 feature vector onto the the FreeSurfer generated smooth white matter surface and each feature was standardized within each subject.  This was followed by dimensionality reduction and whitening using principal component analysis (PCA) of all cortical vertices on the selected surface across all HVs implemented in scikit-learn in Python \cite{scikit-learn}, retaining 14 components explaining 90% of the variance.  Next, we applied a rotation-based iterative Gaussianization (RBIG) procedure, consisting of 10 iterations of a pair of sequential transformations: 1) a non-linear univariate Gaussianization transformation applied to each of the data matrix columns (marginals)  that converts percentile scores computed using the rank transformation to standard scores (scikit-learn's QuantileTransformer); and 2) a linear orthogonal transformation applied to the entire data matrix using PCA, retaining all components after each iteration \cite{Laparra_2011}.  Source code for generating the features and model is publicly available on GitHub as part of the JEM python package {http://github.com/InatiLab/jem}.

Feature Prediction

Because this model theoretically represents an efficient and overcomplete representation of the local features of each image, we hypothesized that we should be able to predict the values of other commonly used local features such as curvature, sulcal depth, cortical thickness, and gray/white contrast (as calculated using FreeSurfer) or measures of myelination, here calculated by dividing the T1  by the T2 intensity, as in  \cite{Glasser_2011}, sampled onto the gray-white junction surface.  Using scikit-learn, our feature set served as the input to ordinary least-squares regression models (polynomial order 2), predicting each target feature. We used a 10-fold cross validation procedure; each fold was trained on 90% of the cortical vertices from all HVs and tested on the remaining 10% of cortical vertices.  

FCDs as Global Outliers

In our latent representation of normal cortical variability, we first wished to identify whether vertices within FCD lesions appeared as global outliers compared to non-lesional vertices in healthy volunteers.  To smooth the data and facilitate comparisons across subjects, we created patches centered at every vertex, including all neighboring vertices within a 5 mm radius, and used this smoothed data for all subsequent analyses.  For each patch \(p\), the center \(\mathbf{m}_{p}\) was computed by averaging the feature vectors of the vertices within the patch: \(\mathbf{m}_{p}=\sum_{\mathbf{x}\in p}\mathbf{f}(\mathbf{x})\). The direction \(\mathbf{\hat{u}}_{p}\), the unit vector pointing in the direction of the patch’s center, was computed by dividing the patch’s mean feature vector by its length \(\hat{u}_{p}=\mathbf{m}_{p}/\|\mathbf{m}_{p}\|\).  For each normal or pathological ROI, we defined the center and direction as the average across patches within that ROI.  For the FCDs, we computed the average across  patches within each MRI+ FCD mask (n=11) then averaged across patients to give each lesion equal weighting regardless of size. 
In this feature space, the probability of finding a patch with a given average feature vector \(\mathbf{m}_p\) depends only on the magnitude of the feature vector \(\|\mathbf{m}_{p}\|\), i.e. the Mahalanobis distance, \(\)MD, or the distance from the origin to the patch’s center, and not on its direction, with MD2  following a cumulative chi-squared distribution.  We measured the Mahalanobis distance (MD) of 1000 randomly sampled patches from normal cortex across HVs and 1000 patches selected from FCD lesion masks.  As there was significant variability in the MD of normal cortical vertices, we identified reproducibly outlying regions by plotting the average MD across HVs for each vertex, thresholding resulting maps at MD > 2.7 and retaining the top ~5% of patches (p = 0.043) occurring in clusters with > 30 vertices.  We selected two of the three most outlying regions of interest (ROI) as exemplars, and compared their average MD to those of the randomly selected normal and FCD patches.