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In previous work Friman et al. (2003) also looked to constrain the
possible linear combinations of the basis set, but within the
canonical correlation analysis (CCA) framework. However, they only
looked to constrain the linear combination coefficients to be
positive. In this work we apply a more complete constraint by
fitting a multivariate Normal distribution to describe the desired
constrained space probabilistically. A big limitation of the work
in Friman et al. (2003) is that they did not address the issue of how
to threshold the resulting correlations of the CCA. In contrast,
the framework in this paper is the Variational Bayesian GLM
framework first used in FMRI by Penny et al. (2003). This framework
has the advantage that it takes into account important issues such
as temporal autocorrelation in FMRI and at the same time
intrinsically produces approximate probability distributions from
which inference can take place.
The HRF modelling in this paper all assumes linearity of the HRF.
Friston et al. (1998b) produced compelling work addressing the use of
basis functions for non-linear HRF modelling using Volterra series.
They model the first and second order kernels using Gamma basis
functions in a frequentist inferred GLM.
In Friston et al. (2000) they derive Volterra kernels from the
balloon model (Buxton et al., 1998) and fit them to empirically found Volterra
kernels from the frequentist inferred GLM of Friston et al. (1998b). In
Friston (2002) they infer on the balloon model parameters
directly from the FMRI data in a Bayesian framework. Within the
Bayesian framework they can incorporate priors on the balloon model parameters
deduced empirically in Friston et al. (2000). This incorporation of the
prior information from previous empirical evidence will constrain the
balloon model parameters in the same way we
constrain the HRF shape basis function parameters
in this work. One area of future
work is to extend the Variational Bayesian inference in this paper
to deal with second order Volterra kernel
basis functions and nonlinearities in ways related to the
work of Friston et al. (1998b).
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