2.3. Calculation of functional density and edge maps

Given a cortical mesh P with N vertices, in order to calculate a density value for each vertex, we first constructed a weighted graph by connecting all neighboring vertices of the mesh and computing the Pearson distance between each pair of neighboring vertices i and j:

where y_i and y_j are the normalized fMRI time series and 〈 ⋅ ,  ⋅ 〉 denotes the standard inner product. The shortest path between each pair of vertices through the weighted graph was then computed and defined as their geodesic distance (Honnorat et al., 2015). The geodesic distance between two vertices reflects the dissimilarity of their time series through the surface. However, it should be mentioned that the geodesic distance is also affected by the spatial distance since it is the accumulated sum along the shortest path. Tow vertices with a long spatial distance are more likely to have a large geodesic distance. The density value at vertex i can then be computed using a Gaussian kernel (Rodriguez and Laio, 2014):

where d¯ik is the geodesic distance between vertex i and vertex k, N is the total number of vertices, and d_c is a free parameter. The parameter a was set to 1 and the parameter b was set to zero as suggested in Rodriguez and Laio (2014). It can be seen that a vertex which is functionally similar to other vertices (i.e., has short geodesic distances to other vertices) tends to have a larger density value. In the present study, for each subject, the parameter d_c was set to the top 0.1% smallest geodesic distance for each subject since we found that with this parameter setting the functional density map can capture fine details in the functional architecture. Figure 1 shows examples of the functional density map. Vertices within each functionally homogeneous region have high density while vertices near the boundary of functional regions have low density values. Although the top 0.1% smallest geodesic distance appears to be a good choice for d_c when calculating the functional density map, functional organization at different scales can be observed when other values of d_c are used, which may provide additional information for functional analysis. Therefore, we calculate a functional edge map in order to utilize the multi-scale information in the functional density map. Specifically, a watershed segmentation method (Beucher and Lantuéjoul, 1979; Gordon et al., 2014) was applied to the functional density maps computed at a range of d_c values, generating binary edges at different scales. A cross-scale functional edge map can then be obtained by averaging the functional edges across scales.

Calculation of the functional edge map using multi-scale information from the functional density maps. When calculating the functional density map, the parameter d_c was set such that the effective number of neighboring vertices corresponds to 0.1% of the total number of vertices on the surface. When calculating the functional edge map, the parameter d_c was set at the following values: {0.05%, 0.1%, 0.2%, 0.3%, 0.4%, 0.5%, 0.6%, 0.7%, 0.8%, 0.9%, 1%}. These values were used to extract fine to coarse scale information from the functional connectivity patterns. The second row shows the binary edge map derived from each functional density map with different values of d_c. A multi-scale functional edge map can be obtained by averaging the binary edge maps across scales.