Single Subject and Groupwise Parcelling Methodologies
In the previous section we hypothesised that the cortex is divided in clusters with homogeneous extrinsic connectivity, alongside intra-cluster and across-subject variability. In using the previous hypothesis, it is important to remark that we don’t have a priori knowledge of the cluster’s location or their variability. But, thanks to the proposed logistic random effects model, we formulated the problem of finding these clusters as a well-known clustering problem. This is because, after transforming the tractograms with the logit function as in Eq. (4) they will be in a Euclidean space . Even more, Eq. (5) states that the transformed tractograms come from a mixture of Gaussian distributions. This is known as a Gaussian mixture model.
To solve the Gaussian mixture model and find the clusters, we use a modified Agglomerative Hierarchical Clustering (AHC) algorithm. This was inspired by the method of Moreno-Dominguez et al. . To enforce the local coherence criterion we also modify the algorithm to accept one parameter: the minimum size of the resulting clusters. Clusters smaller than this size are merged with neighbors, i.e. physically close clusters in the cortex. As we are working in a Euclidean space, we use the Euclidean distance and the centroid as similarity and linkage functions, improving performance. Our technique’s time complexity is O(n2log(n)), with n the number of tractograms to cluster . AHC creates a dendrogram: a structure that comprises different levels of granularity for the same parcellation. This allows us to explore different parcellation granularities by choosing cutting criteria, without the need of recomputing each time.
The main advantage of the model we proposed in the previous section is that it allows us to create a groupwise parcellation using linear operations. Assuming direct seed correspondence across subjects, as in the HCP data set, our model lets us remove the subject variability of each seed’s tractogram by calculating the expected value across subjects:
where the last equality is due to Es(es) = 0 [Eq. (5)]. This allows us to create population-representative tractograms for each seed free of across-subject variability, which then can be clustered to create a groupwise parcellation.