Graph-partitioned spatial priors for functional magnetic resonance images

Spatial models of functional magnetic resonance imaging (fMRI) data allow one to estimate the spatial smoothness of general linear model (GLM) parameters and eschew pre-process smoothing of data entailed by conventional mass-univariate analyses. Recently diffusion-based spatial priors [Harrison, L.M., Penny, W., Daunizeau, J., and Friston, K.J. (2008). Diffusion-based spatial priors for functional magnetic resonance images. NeuroImage.] were proposed, which provide a way to formulate an adaptive spatial basis, where the diffusion kernel of a weighted graph-Laplacian (WGL) is used as the prior covariance matrix over GLM parameters. An advantage of these is that they can be used to relax the assumption of isotropy and stationarity implicit in smoothing data with a fixed Gaussian kernel. The limitation of diffusion-based models is purely computational, due to the large number of voxels in a brain volume. One solution is to partition a brain volume into slices, using a spatial model for each slice. This reduces computational burden by approximating the full WGL with a block diagonal form, where each block can be analysed separately. While fMRI data are collected in slices, the functional structures exhibiting spatial coherence and continuity are generally three-dimensional, calling for a more informed partition. We address this using the graph-Laplacian to divide a brain volume into sub-graphs, whose shape can be arbitrary. Their shape depends crucially on edge weights of the graph, which can be based on the Euclidean distance between voxels (isotropic) or on GLM parameters (anisotropic) encoding functional responses. The result is an approximation the full WGL that retains its 3D form and also has potential for parallelism. We applied the method to high-resolution (1 mm(3)) fMRI data and compared models where a volume was divided into either slices or graph-partitions. Models were optimized using Expectation-Maximization and the approximate log-evidence computed to compare these different ways to partition a spatial prior. The high-resolution fMRI data presented here had greatest evidence for the graph partitioned anisotropic model, which was best able to preserve fine functional detail.