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Introduction

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.
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Next: Modelling Framework Up: tr03mw2 Previous: tr03mw2