Signal Modelling

We now turn our attention to the modelling of $r_{it}$ in equation 1. Typically, the experimenter has knowledge of the timing of stimuli during the experiment and it is this, along with a model of the BOLD response, that can be used in a model-based approach. The approach taken in this work is to separate out the height of the response, $a_i$, by the normalisation of the assumed response $s_t \otimes h(t;\vec{\lambda_i})$. We also use a linear, time-invariant model:

$$r_{it} = a_i(s_t \otimes h(t;\vec{\lambda_{i}}))/std(s_t \otimes h(t;\vec{\lambda_{i}}))$$ (19)

where $\otimes$ represents convolution, $h(t;\vec{\lambda_{i}})$ is the parameterised impulse response function, or haemodynamic response function (HRF). The choice of models for the HRF $h(t;\vec{\lambda_i})$ and for the response size $a_i$ is now addressed.