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HRF modelling

 (21) and (25) have previously used a Bayesian framework to model epochs of the haemodynamic 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. To allow modelling of BOLD responses to general stimulation types, (18) introduced the use of convolution models assuming a linear time invariant system. (5), (7), (2) and (30) 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 (15). An additional assumption is that the stimulus represents the underlying neural activity. The stimulus (or neural activity) is then convolved with the assumed or modelled HRF to give the assumed BOLD response. In (11) and (33) HRF models, which are allowed to vary spatially, are considered within the framework of the linear model. Straightforward attempts to allow variation in parameterised forms would be nonlinear, preventing the use of the convenient linear modelling approach. To avoid this problem, variability in the HRF is introduced via basis sets. In (9) an interesting empirical Bayes approach is taken to HRF modelling with basis functions, whereby the HRF (and other parameters) within a dataset are probabilistically constrained by datasets from multiple sessions and multiple subjects, by inferring on a hierarchical model which incorporates all of the datasets. Basis sets specify a subspace in which a particular HRF either lies or does not. This represents a hard constraint and often the extent of the constraint is difficult to control and/or interpret. (25) make the point that in this regard using Bayesian prior information is preferable to a basis set approach. Bayesian modelling offers the possibility of soft constraints. In addition, unlike basis functions, a nonlinear parameterised HRF approach (that the Bayesian framework makes possible), allows interpretation of the parameters in terms of HRF shape characteristics directly. Furthermore, null hypothesis testing in a frequentist framework with basis functions, requires the overall effect for an underlying stimulus of interested to be tested for using f contrasts. These mean that the directionality of the test is lost - something which is very often of interest in FMRI experiments. For these reasons we present a Bayesian approach to linear HRF modelling for general stimuli using a novel parameterisation of the HRF with interpretable parameters. Our proposed form for the HRF is based on observed BOLD responses (23). This consists of a main response corresponding to an increase in the signal, and a dip in signal before and after the larger increase in signal, possibly reflecting a temporary imbalance between the metabolic activity and blood flow. The dip after the main response is now widely supported, whereas the existence of the early dip as a general phenomenon is still debated. One possibility would be to use an addition of Gaussians (35). However, there are a couple of problems evident with a Gaussian HRF model. Firstly, the HRF is not forced to be zero at time $ t=0$. Clearly, this does not reflect what we know physically. This is usually overcome using Gamma functions instead of Gaussians.
Figure 1: 11 evenly spread samples from the prior of the HRF, using (a) two Gaussian model, and (b) the half-cosine HRF model. The prior mean HRF is plotted along with different HRFs each of which have one parameter varying at the $ \pm 85^{th}$ percentile of the prior, with the other parameters held at the mean prior values.
(a) & (b)
The second problem is illustrated by Figure 1(a). This shows an evenly spread 11 samples of the HRF, taken from a sensible 5-dimensional prior probability space. The problem is that there is dependence between some of the HRF characteristics. It is difficult to interpret characteristics when more than one distinct combinations of parameters can affect them. This would also be a problem with the two-parameter Gamma HRF. The clearest example of this problem is the size of the post-stimulus undershoot. It is clear that the post stimulus undershoot size could be affected by a number of different combinations of parameters. Hence, this makes any attempts to investigate undershoot difficult to perform. A solution to both of these problems is to use an alternative parameterisation of the HRF. The one we present here is simply the addition of four half-period cosines. There are six parameters; four are the periods of the four cosines, and the other two are the ratio of the height of the post-stimulus undershoot to the height of the main peak and the ratio of the height of the initial dip to the height of the main peak. Figure 2 shows a schematic of how the HRF is parameterised.
Figure 2: Parameterisation of the HRF into four half-period cosines. There are six parameters.

picture(3232,3069)(2048,-3665) (2291,-2796)(0,0)[lb]$ m_1$% (3331,-3544)(0,0)[lb]$ m_3$% (4214,-2797)(0,0)[lb]$ m_4$% (4798,-3189)(0,0)[lb]$ c_2$% (2725,-711)(0,0)[lb]$ m_2$% (2048,-2993)(0,0)[lb]$ c_1$% (2059,-1830)(0,0)[lb]$ 1$%
Figure 1(b) shows an evenly spread 11 samples of the HRF without an initial dip ($ c_{i1}=0$ and $ m_{i1}=0$), taken from the resulting 4 dimensional prior probability space using the half-cosine HRF model. A disadvantage with this parameterisation is that its second derivative is discontinuous. However, the range of the HRF parameters are such that sharp transitions in the second derivative are avoided. Hence, sensible looking HRF shapes predominate, as illustrated by figure 1(b). This parameterisation does clearly impose the constraint that the HRF is zero at $ t=0$. Furthermore, parameters relating to HRF characteristics are independent. As with the Gaussian HRF, another big advantage of the half-cosine HRF model is that it could be parameterised in the frequency domain, hence speeding up the convolution. When figure 1(b) is compared with figure 1(a), it can be seen how a characteristic of the HRF shape, such as the size of the undershoot, is now controlled by a single parameter.

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