# Graph Laplacian Priors for fMRI

Users of spatial priors for fMRI now have additional flexibility as to which voxels they analyze. Previously, either the full volume or specified slices could be selected, whereas now sub-volumes, e.g. using a mask generated from an effect of interest measured using smoothed data and a standard mass-univariate SPM analysis, can be analyzed. The options available in “Bayesian 1st-Level”, under “Analysis space” are now; “Volume”, “Slices” or “Clusters”. In addition, after this, the user can choose how these volumes are divided into smaller blocks, which is necessary for computational reasons, c.f. in spm_spm a slice is also divided into blocks. These blocks can be either slices (by selecting “Slices”) or 3D segments (“Partitions”), whose extent is computed using a graph partitioning algorithm. The latter option means that the spatial prior is truly 3D, instead of 2D spatial priors stacked one on another.

Two additional spatial precision matrices have been included; unweighted graph Laplacian, ‘UGL’, and weighted graph-Laplacian, ‘WGL’. This makes appreciating differences between the previous priors ‘GMRF’ and ‘LORETA’ clearer as they are functions of the ‘UGL’, i.e. normalized and squared respectively. The ‘WGL’ empirical prior uses the ordinary least squares estimate of regression coefficients to inform the precision matrix, which has the advantage of preserving edges of activations compared to ‘UGL’, ‘GMRF’ and ‘LORETA’. Diffusion kernels (see Harrison et al 2007-2008), i.e. matrix exponential of ‘UGL’ and ‘WGL’ will be included in the near future. The benefit of these is that the spatial extent of the spatial precision matrix can also be optimized, which is not possible using the graph Laplacian alone. This is achieved by controlling the dispersion of the diffusion kernel through an additional parameter, tau, i.e. spatial precision matrix = expm(L*tau), where L is the graph Laplacian.

L. Harrison, W.D. Penny, J. Ashburner, N. Trujillo-Bareto, and K.J. Friston. Diffusion-based spatial priors for imaging. NeuroImage, 38:677-695, 2007.

L. Harrison, W.D. Penny, J. Daunizeau, and K.J. Friston. Diffusion-based spatial priors for functional magnetic resonance images. NeuroImage, 41(2):408-423, 2008.

These descriptions of the new features are taken from the SPM8 Release Notes

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