Discussion

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).