Introduction

An important issue for detecting areas of brain activity in FMRI is the forward modelling of the temporal BOLD response. This forward model predicts what the BOLD response would be if we knew the underlying neural activity. To allow modelling of BOLD responses to general stimulation types, Friston et al. (1994) introduced the use of convolution models which assume a linear time invariant system. Cohen (1997), Dale and Buckner (1997) and Boynton et al. (1996) provide some evidence that the BOLD response possesses linear characteristics with respect to the stimulation. However, non-linearities are predominant when there are short separations (less than approximately 3 seconds) between stimuli (Friston et al., 1998b). An additional assumption is that the stimulus represents the underlying neural activity. The stimulus (or neural activity) is then convolved with an assumed or modelled impulse response function, known as haemodynamic response function (HRF), to give the assumed BOLD response (Friston et al., 1994). Genovese (2000) and Gössl et al. (2001) have previously used a Bayesian framework to model the voxel-wise BOLD response to a sustained period of stimulation. The advantage of a Bayesian approach is most obvious in the use of prior experience to justify the prior distributions used for these haemodynamic response parameters. However, the work of Genovese (2000) and Gössl et al. (2001) was restricted to just modelling the response to an epoch of fixed size. Woolrich et al. (2004b) generalised this to general stimulations via parametric modelling of the HRF assuming a linear time invariant system. However, the problem with parametric modelling of the HRF is that the model is difficult to infer upon without slow techniques such as Markov Chain Monte Carlo (MCMC). Friston et al. (1995) and Josephs et al. (1997) consider voxel-wise linear time invariant system HRF models within the framework of the General Linear Model (GLM). Flexibility to model the HRF is introduced via basis sets. However, a large amount of the subspace spanned by the basis functions produces nonsensical HRF shapes (see figure 4(a)). This is because the conventional GLM will indiscriminately allow all possible linear combinations of the basis set. In this work we propose a technique for using soft constraints to weight the subspace spanned by the basis set to only include sensible HRF shapes within a linear time-invariant system. The choice of the basis set can be driven by a standard parametric HRF, or a physiologically informed model such as the Balloon Model (Buxton et al., 1998). 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. Penny et al. (2003) showed how Variational Bayes can be used to infer on the GLM for FMRI. Here we extend this work to give inference on the GLM with the constrained HRF basis functions. We also extend the work of Penny et al. (2003) to spatially regularise the autoregressive noise parameters using a Markov Random Field (MRF). We demonstrate that the constrained basis function approach allows for far superior sensitivity, when compared with traditional unconstrained basis function approaches.