Next: Overview
Up: tr04mw2
Previous: tr04mw2
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.
Subsections
Next: Overview
Up: tr04mw2
Previous: tr04mw2