A amygdala probability mask was first obtained from the Harvard-Oxford subcortical structural atlas provided in FSL in the MNI 152 standard space. The mask was then warped to the subject diffusion space through the transformation achieved by aligning T1W of each subject to the T1W in MNI space using a nonlinear registration algorithm (fnirt, FSL). A threshold of 50% was then applied to the probability mask to exclude extraneous tissue. The resulting masks were conservatively away from the edge to avoid alignment errors and partial voluming effects.
Amygdala Parcellation: K-mean Spectral Clustering
For each voxel, 28 SH coefficient pairs (lmax = 6, even orders) were extracted to describe the diffusion characteristics that reflect the underlying tissue microstructure. Voxels within the mask of the amygdala would be grouped together according to the similarity of their SH coefficients.
To prepare for the subsequent Laplacian transformation of Spectral Clustering, the graph similarities (Sj between two voxels i, j were computed by converting the weighted pair-wise Pearson’s correlation coefficient, i.e., Cj of the SH coefficients according to their spherical distance . The weighting, Wij, is to adjust physical (Euclidean) distance between voxels i, j. The dimension of the S matrix, M x M, equals to the number of voxels within the segmented amygdala.
Sigma is a threshold parameter that deems the importance of cells in C where values below sigma are penalized. Therefore, S is a sparser matrix than C, as higher similarity Sij is achieved only when the two voxels i , j have similar SH coefficients and are physically close to each other. The value of sigma was optimized by iteratively incrementing sigma until minimum Fiedler Value of the Laplacian matrix (see below or ) was achieved.
The graph similarity matrix (S) of each subject was then transformed into a normalized symmetric graph Laplacian matrix, on which eigen decomposition was performed. According to spectral clustering theory , the first few ordered eigen values contain critical structural information regarding the data. To determine the number of eigen values that best reflect the underlying structure, we tested the eigen values against those generated from unstructured data. The unstructured data were generated by randomizing the SH coefficients. The randomization process was bootstrapped for 1200 iterations to create a null distribution of eigenvalues of the unstructured Laplacian matrices. For each subject, eigenvalues of original “structured” Laplacian matrix were tested against the null distribution using z- scoring, and the number of significant eigenvalues were determined as the number of clusters, denoted as N.
To perform k-mean clustering to classify the voxels within the amygdala, we picked the N eigenvectors corresponding to the N eigenvalues starting from Fiedler Value. Each eigen-vector has M elements that equals to the dimension of the Laplacian and S matrix. Note that M also denotes the number of voxels within the segmented amygdala. The N eigen-vectors were stacked up to form a N x M matrix. Thus, the N x M matrix described N distinct features for M voxels. K-mean clustering was performed across M voxels to yield a cluster label for each voxel. The clustering would then be complete and yield N amygdala subfields for each subject in the native space. In order to check the inter-subject variability, the individual results were transformed to the template brain. Individual clusters were averaged across subjects to generate a consistency map.