Modelling the fMRI response using smooth FIR lters

F. A. Nielsen, C. Goutte and L. K. Hansen Dept. of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark Introduction We describe a exible semi-parametric linear time-invariant convolution model for the fMRI response. It extends the FIR model [1] with a Gaussian process prior over the parameters (the FIR coe cients). Linear models with Poisson, Gamma and Gaussian kernels have few parameters and are not very exible, e.g., they are not able to model post-activation undershoot. A exible model such as the FIR lter is able to model more complex linear responses but su ers from unstable (i.e., large variance) estimation in the case where long lters are necessary due to short TRs. Ridge regularization for the FIR lter estimation does not help su ciently and principal component regression introduces biases that are dependent on the input function (the paradigm). The smooth FIR overcomes the disadvantages of the FIR lter by introducing a regularization that ties the parameters together, enabling a more stable estimation. This is done by modelling the smoothness of the lter by a correlation between the parameters and use that information as a prior in a Bayesian setting. Method For a fMRI signal y(t) corresponding to activation x, and a standard assumption of additive Gaussian noise, the linear lter of parameters w models the fMRI signal as y(t) = wTx + . Using the Gaussian process prior on the parameters w, p(w) / exp wT 1w=2 , the posterior probability distribution is tractable and the maximum a posteriori (MAP) estimate becomes: wMAP = XTX+ 2 1 XTy with ij = exp 1 2l2 (i j)2 (1) where we model the correlation by a decreasing Gaussian function. If we are only concerned about the MAP estimate, only two hyperparameters need to be set: The ratio 2= determines the amount of regularization and is equivalent to the ridge parameter in ridge regression; l corresponds to a \characteristic length", i.e. the typical length in which the lter varies. When l goes to zero, we recover ridge regression, while large values of l give a constant lter. There are several options to set the hyperparameters. Strong prior knowledge can be used in some cases, e.g., setting the characteristic length to a value corresponding to 7sec. It is also possible to use a data oriented adaptive method, such as optimization of the likelihood or posterior of the hyperparameters, the evidence framework [3] or integrate over hyperparameters using Markov Chain Monte Carlo sampling [4]. Finally, the signi cance of activation can be assessed by calculating the probability that w = 0 lies in a credible region: the \highest posterior density" (HPD) region [5, 6]. Our full approach is described in more detail in [2]. Result and Discussion