Prediction of Typical Features

Because this model theoretically represents an efficient and over-complete 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 trained on the data from half of the HVs and tested on the other half.  

Outlier Detection

Using our normative model of cortical variability, we 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.  This radius was selected to be small enough to identify changes within specific gyri/sulci, and large enough to minimize local noise and artifacts, and is similar to the smoothing size used in similar studies, for example \cite{Adler2017}. 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.  To create an average FCD vector, we computed the average across  patches within each MRI+ patient's FCD mask (n=11), then averaged across patients to give each lesion equal weighting regardless of size. 
In this feature space, outliers can be defined by the Mahalanobis distance, MD, or the distance form the origin to the patch's center, as the probability of finding a patch with a given average feature vector \(\mathbf{m}_p\) depends only on the magnitude of the feature vector \(\|\mathbf{m}_{p}\|\) 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 then wished to explore whether some cortical regions reproducibly appeared as relative outliers across healthy volunteers.  To explore this, we arbitrarily defined all patches with an average distance from the center (MD) of > 2.7 as outliers, corresponding to the top ~5% of patches (p = 0.043).  We retained all resulting clusters with > 30 vertices in order to eliminate isolated atypical patches, as we were seeking to identify atypical regions at larger spatial scales as described by cortical cyto- and myeloarchitectonic maps.  Of the resulting clusters, we selected two of the three most outlying regions of interest (ROI) as exemplars, and compared their average MD to those of randomly selected normal and FCD patches.