Next: Modelling Framework
Up: tr03mw2
Previous: tr03mw2
Functional magnetic resonance imaging (FMRI) uses fast MRI
techniques to enable studies of dynamic physiological processes at
a time scale of seconds. This can be used for spatially localising
dynamic processes in the brain, such as neuronal activity.
However, to achieve this we need to be able to infer on models of
4-dimensional data. Predominantly, for statistical and
computational simplicity, analysis of FMRI data is performed in
two-stages. Firstly, the purely temporal nature of the FMRI data
is modelled at each voxel independently, before considering
spatial modelling on summary statistics from the purely temporal
analysis (14,18). Clearly, it would be
preferable to incorporate the spatial and temporal modelling into
one all encompassing model. This would allow for correct
propagation of uncertainty between temporal and spatial model
parameters. Previous work (29,8,26,1) has
taken such a combined spatio-temporal approach. In this
work we look to propose an alternative spatio-temporal approach.
There is a wealth of possibility when considering
spatio-temporal models for FMRI. A systematic approach is required
to determine the most appropriate model. We break down the work
into two main areas in this paper. They are spatio-temporal noise
modelling and haemodynamic response (HRF) signal modelling.
Previous FMRI spatio-temporal noise models have either been
separable (1) or modelling deterministic
trends (26). As we shall see, the assumptions
underlying a separable noise model are not well met by FMRI data.
In this work, we focus not on
longer-term (spatially and temporally) deterministic trends, but
on shorter-term correlated noise processes. This results in the
proposed use of a novel space-time non-separable autoregressive
process to FMRI data.
In the signal modelling we focus on HRF modelling. It is now widely
accepted that FMRI modelling requires flexible HRF modelling,
with the HRF varying spatially and between
subjects. Flexibility in linear
modelling has been introduced with the use of basis functions
(33,11). However, basis functions suffer
from a number of limitations. They impose a hard constraint on the
allowed HRF shape and often the extent of the constraint is
difficult to control and/or interpret. To overcome these problems
we use a parameterised HRF
approach (21,25).
Unlike previous work (21,25) we do not
limit ourselves to fixed epoch designs and we propose a novel
half-cosine parameterisation to separately represent different
shape characteristics of the HRF.
Importantly, and for the first time in FMRI,
with such highly parameterised models we introduce
the use of Automatic Relevance Determination (ARD) priors, to
force parameters with insufficient evidence in the data to support
them to be zero with high precision. This adaptively avoids
over-fitting at the time of inference on the data.
The use of a Bayesian framework provide us with the
ability to probabilistically incorporate prior
information (for example, about the expected HRF shape).
However, this is not the only reason for adapting the Bayesian
approach.
To correctly infer on parameters of interest, for example, the HRF
parameters or activation height, is not an easy task with the
complexity of model proposed here. Indeed, there are no solutions
available in the frequentist literature, precluding the use of
frequentist null hypothesis testing. However, as
in previous work (29,26,21,25) we are
able to infer on these complex models by using a fully
Bayesian framework. Whilst the fully Bayesian framework
is not the only method for dealing with such complex models,
it does provide us with a systematic framework for doing full
inference by
incorporating all of the uncertainty in the model parameters via
marginalisation. Hence this approach is of great use, even if
non-informative priors were to be assumed for all parameters.
Next: Modelling Framework
Up: tr03mw2
Previous: tr03mw2