Bayesian Inference on Constrained Linear Basis Sets for HRF Models

FMRI analysis ideally requires flexible haemodynamic response function (HRF) modelling, with the HRF being allowed to vary spatially and between subjects. To achieve this flexibility, in [42] we proposed voxel-wise HRF modelling using a parameterised HRF consisting of a number of joined half-cosines. This was embedded in a fully Bayesian framework, incorporating non-separable spatio-temporal noise modelling. A fully Bayesian approach allows for the uncertainties in the noise and signal modelling to be incorporated together to provide full posterior distributions of the HRF parameters.

However, inference on such parameterised HRF models is slow. A more practical approach to voxel-wise HRF modelling is to use basis functions. This allows us to proceed in the more manageable GLM framework. The problem with this is that, as shown in fig. 3a, a large amount of the subspace spanned by the basis functions produces nonsensical HRF shapes. Therefore, in [41] we proposed a technique for choosing a basis set, and then, importantly, the means to constrain the subspace spanned by the basis set to only include sensible HRF shapes. The choice of the basis set can be driven by a standard parametric HRF, or a physiologically informed model such as the balloon model [11]. Using the GLM in a Bayesian framework we can then use priors on the basis function regression parameters to constrain the linear combinations of HRFs to sensible HRF shapes, as shown in fig. 3b. This was augmented with spatially regularised autoregressive noise modelling.

Inference was carried out using Variational Bayes (an efficient alternative to MCMC for posterior distribution estimation, making simplifying assumptions about the form of the posterior). As shown in fig. 3, constraining the subspace spanned by the basis set allows for superior separation of activating voxels from non-activating voxels in FMRI data. This approach will be included in future versions of FEAT.

Figure 3: Samples from the HRF basis set: (a) unconstrained (b) constrained. Below is the difference in voxel classification between the constrained HRF model and the unconstrained HRF model for a single-event pain stimulus dataset. [red] voxels are active for both models, [yellow] voxels are active for just the constrained HRF model, and [blue] voxels are active for just the unconstrained HRF model.

(a) Unconstrained (b) Constrained