Choosing a Basis Set

Basis sets used previously range from a single canonical HRF plus its temporal derivative to a set of Gamma functions Friston et al. (1998a)). These basis functions are then separately convolved with the known stimulus to give the same number of regressors as there are basis functions for use in the linear model. Hossein-Zadeh and Ardekani (2002) and Friman et al. (2003) have previously shown how we can generate a basis set using singular value decomposition (SVD). This produces a basis set from samples of the HRF or regressors resulting from a parametric forward model of the haemodynamics. This is the approach we take in this paper. In this paper we base the basis set on a parameterised model of the HRF. The HRF is parameterised by four half-cosines, requiring six parameters, as illustrated in figure 1.

Figure 1: Parameterisation of the HRF into four half-period cosines. There are six parameters.

To complete the HRF parameterised model we need to specify probabilities for the parameter values in the HRF model from which we generate physiologically plausible HRF shapes. With this information we can then draw HRF samples. Figure 2 shows 20 HRF samples drawn from the half-cosine parameterisation using the HRF parameter value probabilities:

$$h_1 \sim (0s,2s)$$

$$h_2 \sim (2s,6s)$$

$$h_3 \sim (2s,6s)$$

$$h_4 \sim (2s,8s)$$

$$f_1 =$$ 0

$$f_2 \sim (0,0.5)$$ (14)

We strongly emphasise that this is one of many possible choices of obtaining HRF samples. An alternative, attractive approach would be to use a physiologically based model, such as the Balloon model (Friston et al., 2000), with physiologically meaningful parameters which can be given sensible ranges. Using a probabilistic model of choice, one can obtain a set of samples of the HRF, which represents the space of HRFs expected. The $N_H$ HRF samples, each of length $N_T=\rho T$, can be placed into a $N_T\times N_H$ matrix, $W$. To obtain a basis set that spans this space of HRFs we perform a Singular Value Decomposition (SVD) on this matrix. This gives us $N_H$ eigenHRFs (of length $N_T$) and $N_H$ corresponding eigenvalues (describing the power each corresponding eigenHRF explains). Figure 3 shows the four eigenHRFs (and their corresponding eigenvalues) with the largest eigenvalues obtained from an SVD on $N_H=1000$ HRF samples of length $N_T=512$ (resolution of $0.5$ seconds) from the half-cosine parameterisation. It is worth noting that the first three eigenHRFs/basis functions look remarkably like the commonly used canonical, delay (temporal) derivative and width (dispersion) derivative of a Gamma/Gaussian parameterised HRF (in that order). To form our basis set we need to decide how many of the largest eigenHRFs we want to include. For the rest of this paper we use the $N_b=3$ largest eigenHRFs as our basis set.

Figure 2: 20 HRF samples drawn from the half-cosine parameterisation

Figure 3: Four eigenHRFs (and their corresponding eigenvalues) with the largest eigenvalues from $N_H=1000$ HRF samples of length $N_T=512$ (resolution of $0.5$ seconds) from the half-cosine parameterisation.