Abstract
Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the spatial priors used have several less appealing properties, such as being improper and having infinite spatial range. We propose a statistical inference framework for whole-brain fMRI analysis based on the class of Matérn covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of possibly anisotropic spatial Matérn fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditionally on the inferred hyperparameters, we make a fully Bayesian treatment of the brain activity. The Matérn prior is applied to both simulated and experimental task-fMRI data and clearly demonstrates that it is a more reasonable choice than the previously used priors, using comparisons of activity maps, prior simulation and cross-validation.
Highlights
We show how the spatial priors used in these previous articles can be seen as special cases of the Gaussian Markov random field (GMRF) representation of Gaussian fields of the Matern class, using the stochastic partial differential equation (SPDE) approach presented in Lindgren et al (2011)
Our model for Functional magnetic resonance imaging (fMRI) data can be divided into three parts: (i) the measurement model, which consists of a regression model that relates the observed blood oxygen level dependent (BOLD) signal in each voxel to the experimental paradigm and nuisance regressors, and a temporal noise model (Section 2.1), (ii) the spatial prior that models the dependence of the regression parameters between voxels (Sections 2.2 and 2.3), and (iii) the priors on the spatial hyperparameters and noise model parameters (Sections 2.4 and 2.5)
We start by analysing simulated fMRI data, to demonstrate the empirical Bayes (EB) method’s capability to estimate the true parameters, and to visualise the differences between the spatial priors in a controlled setting
Summary
Functional magnetic resonance imaging (fMRI) is a noninvasive technique for making inferences about the location and magnitude of neuronal activity in the living human. An alternative to the mass-univariate approach is to use Bayesian spatial smoothing priors for the brain activity, and an early example of this is the two-dimensional prior in slice-wise fMRI analysis proposed by Penny et al (2005). (Groves et al, 2009) use a spatial GP prior with Gaussian covariance and do the analysis slice-wise, due to the computational cost For this reason, much work on spatial modeling of fMRI data has been using GMRFs instead, see for example Gossl et al (2001); Woolrich et al (2004); Penny et al (2005); Harrison and Green (2010); Siden et al (2017). We develop a fast Bayesian inference algorithm that allows us to use spatial three-dimensional whole-brain priors of the Matern class on the activity coefficients, for which previous MCMC and VB approaches are not computationally feasible. The new methods in this article have been implemented and added to the BFAST3D extension to the SPM software, available at http://www.fil.ion.ucl.ac.uk/spm/ ext/#BFAST3D
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