We recently outlined a Bayesian scheme for analyzing fMRI data using diffusion-based spatial priors [Harrison, L.M., Penny, W., Ashburner, J., Trujillo-Barreto, N., Friston, K.J., 2007. Diffusion-based spatial priors for imaging. NeuroImage 38, 677-695]. The current paper continues this theme, applying it to a single-subject functional magnetic resonance imaging (fMRI) study of the auditory system. We show that spatial priors on functional activations, based on diffusion, can be formulated in terms of the eigenmodes of a graph Laplacian. This allows one to discard eigenmodes with small eigenvalues, to provide a computationally efficient scheme. Furthermore, this formulation shows that diffusion-based priors are a generalization of conventional Laplacian priors [Penny, W.D., Trujillo-Barreto, N.J., Friston, K.J., 2005. Bayesian fMRI time series analysis with spatial priors. NeuroImage 24, 350-362]. Finally, we show how diffusion-based priors are a special case of Gaussian process models that can be inverted using classical covariance component estimation techniques like restricted maximum likelihood [Patterson, H.D., Thompson, R., 1974. Maximum likelihood estimation of components of variance. Paper presented at: 8th International Biometrics Conference (Constanta, Romania)]. The convention in SPM is to smooth data with a fixed isotropic Gaussian kernel before inverting a mass-univariate statistical model. This entails the strong assumption that data are generated smoothly throughout the brain. However, there is no way to determine if this assumption is supported by the data, because data are smoothed before statistical modeling. In contrast, if a spatial prior is used, smoothness is estimated given non-smoothed data. Explicit spatial priors enable formal model comparison of different prior assumptions, e.g., that data are generated from a stationary (i.e., fixed throughout the brain) or non-stationary spatial process. Indeed, for the auditory data we provide strong evidence for a non-stationary process, which concurs with a qualitative comparison of predicted activations at the boundary of functionally selective regions.