Similarity Estimation Across Subjects
To smooth the data and facilitate comparison across subjects, we created patches centered at every vertex, including all neighboring vertices within a 5 mm radius. Homotopic patches were defined as patches in the same location across subjects; heterotopic patches are in non-overlapping different locations. 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 \(\hat{u}_p = \mathbf{m}_p / \| \mathbf{m}_p \|\). 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 \(\| \mathbf{m}_p \|\), which is also the Mahalanobis distance, \(d\), defined as the distance from the origin to the patch's center with \(d^2\) following a cumulative chi-squared distribution. The similarity between two patches \(p_i\) and \(p_j\) can be assessed using simple metrics, such as 1) the Euclidean distance between their centers \(d_{ij} = \| \mathbf{m}_i - \mathbf{m}_j \|\), and 2) the cosine similarity between their directions \(s_{ij} = \mathbf{\hat{u}_i} \cdot \mathbf{\hat{u}_j}\).
Global Anomaly Detection
In our representation of cortical variability, we hypothesized that cortical lesions, but also possibly some normal cortical regions known to have atypical structural characteristics, would appear as global outliers in our feature space. To identify such regions of normal cortex, we calculated the average Mahalanobis distance for each cortical patch across all HVs (fig). Specific outlier regions were identified by thresholding the average distance map across HVs at a threshold of 2.7, retaining 4.3% of the patches (equivalent to \(p=0.043\)), and clustering with a minimum of 30 nodes. As exemplars, for further analysis we selected 2 of the resulting outlier regions of interest (ROI) in the anterior insula and primary motor cortex.
Directional Outliers and Similarity Maps
Although some brain regions appear to be consistent global outliers across HVs, this does not mean that they are similar to each other. In our representation, similarities or differences in combinations of features can be represented as differences in direction. We explored this using our 2 exemplar outlier ROIs, defining the center and direction of each ROI as the average of the patches within that ROI across all HVs. We similarly defined the average FCD ROI center and direction by averaging all of the patches within each MRI+ FCD mask (n=10), then averaging across the FCDs, to give each lesion equal weighting. Using the average direction of the insula ROI as the x-axis and the average direction of the motor ROI as the y-axis, we plotted the patches within each outlier ROI, the FCD ROI center, and 1000 randomly selected cortical patches for comparison (fig).