Functional similarities and distance properties

The analysis of functional and effective brain connectivity forms an important tool for unraveling structure-function relationships from neurophysiological data. It has clinical applications, supports the formulation of hypotheses regarding the role and localization of functional processes, and is often an initial step in modeling. However, only a few of the commonly applied connectivity measures respect metric properties: reflexivity, symmetry, and the triangle inequality. This may hamper interpretation of findings and subsequent analysis. Connectivity indices obtained by metric measures can be seen as functional distances, and may be represented in Euclidean space by the methods of multidimensional scaling. We sketch some classes of measures that do allow for such a reconstruction, in particular the class of Wasserstein distances, and discuss their merits for interpreting cortical activity assessed by magnetoencephalography. In an application to magnetoencephalographic recordings during the execution of a bimanual task, the Wasserstein distances between relative circular variances indicated cortico-muscular synchrony as well as cross-talk between bilateral primary motor areas in the β-band.