Structural vs. Chemical: Discovering Brain Map Coupling

The Spatial Spin Test

To combat this, we utilize the Alexander-Bloch Spin Test. Instead of destroying the data's inherent spatial structure by randomly shuffling vertices, we project the maps onto a sphere and rotate them relative to one another.

This preserves the intrinsic spatial topology of the original data while randomizing the alignment between the two maps. When combined with Max-T permutation correction, this guarantees that observed correlations represent true biological coupling rather than random geometric coincidence.

Pairwise Density Scatter & Ranking

Given the high dimensionality of our data (~32,000 vertices per map), standard scatter plots become unreadable. We implemented hexbin density scatter plots to visualize the concentration of data points across map pairs.

Because interactions in the brain are rarely perfectly linear, we applied rank transformations to the data. This linearization of monotonic relationships allowed us to evaluate spatial associations rigorously using the Spearman rank correlation coefficient (\(\rho\)), converting curved trends into linear diagonal associations.

Principal Component Analysis (PCA)

To characterize the dominant low-dimensional structure underlying the multimodal covariance between brain maps, we performed Principal Component Analysis (PCA) on the multimodal correlation matrix.

Because the \(8\times8\) Spearman correlation matrix \(R\) is symmetric and positive semi-definite by construction, PCA was implemented directly via eigendecomposition:

\[ R = V \Lambda V^T \]

Here, \(\Lambda\) contains the eigenvalues (sorted in descending order to calculate variance explained), and \(V\) contains the corresponding orthonormal eigenvectors. Component loadings, which quantify the contribution of each original brain map to each principal component, were computed as:

\[ L = V \Lambda^{1/2} \]

This mathematical property allows us to precisely isolate the orthogonal axes capturing the maximum shared spatial variance across structural and functional modalities.

Subgroup Clustering

Finally, we applied agglomerative hierarchical clustering to systematically test our "Segregation Hypothesis". Because clustering algorithms operate on distances rather than similarities, we converted the Spearman correlation matrix \(R\) into a dissimilarity matrix \(D\):

\[ D = 1 - R \]

We set the diagonal to zero (\(D_{ii} = 0\)) to enforce zero self-distance. This transformation maps highly positively correlated map pairs to small distances. We then used average linkage (UPGMA), where the distance between two clusters \(A\) and \(B\) is defined as the mean pairwise distance between all elements across clusters:

\[ d(A,B) = \frac{1}{|A||B|} \sum_{i \in A} \sum_{j \in B} D_{ij} \]

We cut the resulting dendrogram into two distinct clusters (\(k=2\)) to measure whether maps couple more strongly within their biological domains (e.g., structure-to-structure) than between them.