Reliability of the Clustering Algorithm for the Model

In the previous sections we presented a model for the cortical extrinsic connectivity and a clustering technique to parcellate the brain. Our technique allows us to create single subject and groupwise parcellations, encoded with different levels of granularity in a dendrogram. However, is not immediate that the chosen clustering algorithm (AHC) solves a Gaussian mixture model [Eq. 5] since it was not designed for this particular case [13]. That is, it’s not immediate that the algorithm finds the clusters if they are stated as in the model [Eq. 5]. Now we show that the modified version of the algorithm (Sect. 2.3) to enforce the local coherence criterion (Sect. 2.2), solves the model for reasonable levels of variability. Moreover, it retrieves the right clusters using one of the simplest criterion to cut the dendrogram: the horizontal cut, i.e. cutting the dendrogram just by choosing the cut’s height.

To test the technique, we started by creating synthetic data from the model [Eq. (5)]. We randomly took ten subjects from our chosen set alongside their extant Desikan parcellations. Then, we created synthetic connectivity fingerprints representing the connections between their Desikan areas. Next, for each vertex in their cortical surface we: replicated those fingerprints; transformed them with the logit function and added cluster-specific variability and across-subject variability as in our model. Finally, we grouped the vertex based on their connectivity using our clustering technique.

If our parcelling technique is able to solve the model, then the Desikan Areas should be encoded in the resulting dendrogram. To show that the Desikan areas were encoded in the resulting dendrogram, we calculated the best obtainable overlap between each Desikan area and the clusters in the dendrogram using the Dice coefficient. To evince the accuracy of the horizontal cut criterion, we compared every obtainable parcellation through cutting our dendrogram in the average subject case against the Desikan atlas using the corrected Rand index [8]. A Rand index of 1 means that the two parcellations were equal.

Figure 2-left shows the best dice coefficients obtained for every Desikan region under different levels of variability. The Signal-to-Noise-Ratio (SNR) in the figure represents the amount of variability added respect to the original variability of the synthetic connectivity fingerprint. The result in Fig. 2-left shows that parcels were retrieved and well-encoded inside the dendrogram for reasonable levels of variability, specially in the average case (dark blue line) were we get ride of the across-subject variability by averaging. Figure 2-right shows the best obtainable rand index using horizontal cut on the dendrogram under different levels of variability. The high Rand indices obtained show that we can solve our model by simply using the horizontal cut criterion.