Bayesian fMRI time series analysis with spatial priors

Abstract

We describe a Bayesian estimation and inference procedure for fMRI time series based on the use of General Linear Models (GLMs). Importantly, we use a spatial prior on regression coefficients which embodies our prior knowledge that evoked responses are spatially contiguous and locally homogeneous. Further, using a computationally efficient Variational Bayes framework, we are able to let the data determine the optimal amount of smoothing. We assume an arbitrary order Auto-Regressive (AR) model for the errors. Our model generalizes earlier work on voxel-wise estimation of GLM-AR models and inference in GLMs using Posterior Probability Maps (PPMs). Results are shown on simulated data and on data from an event-related fMRI experiment. D 2004 Elsevier Inc. All rights reserved.

Variational free energy and the Laplace approximation

“…Bayesian inversion using VB is ubiquitous in neuroimaging (e.g., Penny et al, 2005). Its use ranges from spatial segmentation and normalisation of images during pre-processing (e.g., Ashburner and Friston, 2005) to the inversion of complicated dynamical casual models of functional integration in the brain .…”

Bayesian spatiotemporal model of fMRI data using transfer functions

“…to set up a spatiotemporal formulation. Some of the spatiotemporal models proposed in the literature are Bowman (2007), Penny et al (2005), Katanoda et al (2002) and Woolrich et al (2004a). Although all these models are based on convolution, the four of them present a di↵erent modelisation of the spatiotemporal correlation structure between voxels.…”

“…Bowman (2007) incorporates a functionally defined distance metric into a parametric structure for spatial correlations within a ROI (with the di culty of choosing the ROI and the functional distance between voxels in it) and includes temporal correlations between scans. Penny et al (2005) propose a fully Bayesian model with spatial priors defined over regression coe cients of a GLM, using Gaussian Markov random fields (GMRF), and the errors are modelled as an autoregressive process. Katanoda et al (2002) propose a spatiotemporal regression model for each voxel that involves the time series of the neighbouring voxels together with its own.…”

“…Gossl et al (2001) use GMRF to spatially regularise regression coe cients and Woolrich et al (2004b) to spatially regularise AR coe cients. Moreover, several choices of the precision matrix of the GMRF prior on regression coe cients of a GLM-AR model have been considered in the Bayesian literature, these include uninformative priors , global-shrinkage priors Friston and Penny (2003) and Laplacian priors (Penny et al, 2005) among others. An interesting comparison of these priors can also be found in Penny et al (2005).…”

“…Moreover, several choices of the precision matrix of the GMRF prior on regression coe cients of a GLM-AR model have been considered in the Bayesian literature, these include uninformative priors , global-shrinkage priors Friston and Penny (2003) and Laplacian priors (Penny et al, 2005) among others. An interesting comparison of these priors can also be found in Penny et al (2005). Quirós et al (2010) propose a Bayesian spatiotemporal model to determine active areas into the brain by merging the two stages of the model first proposed by Kornak (2000) and extended by Quirós et al (2006).…”