Functional coordinates: Modeling interactions between brain regions as points in a function space

Here, we propose a novel technique to investigate nonlinear interactions between brain regions that captures both the strength and type of the functional relationship. Inspired by the field of functional analysis, we propose that the relationship between activity in separate brain areas can be viewed as a point in function space, identified by coordinates along an infinite set of basis functions. Using Hermite polynomials as bases, we estimate a subset of these values that serve as "functional coordinates," characterizing the interaction between BOLD activity across brain areas. We provide a proof of the convergence of the estimates in the limit, and we validate the method with simulations in which the ground truth is known, additionally showing that functional coordinates detect statistical dependence even when correlations ("functional connectivity") approach zero. We then use functional coordinates to examine neural interactions with a chosen seed region: the fusiform face area (FFA). Using k-means clustering across each voxel's functional coordinates, we illustrate that adding nonlinear basis functions allows for the discrimination of interregional interactions that are otherwise grouped together when using only linear dependence. Finally, we show that regions in V5 and medial occipital and temporal lobes exhibit significant nonlinear interactions with the FFA.