GUÍA DE EJERCICIOS PARA ESTIMULAR LA ATENCIÓN

For fMRI being measured on a metric scale, it is assumed thati follows a multivariate

normal distribution

i ∼ N(0, σ2iV)

Usually fMRI data exhibit short-range serial or temporal correlations (Friston et al., 2008, p. 121). Long-range correlations are assumed to be removed by the applied highpass filter, cf. Section 2.1.1. If ignoring serial correlations, inappropriate estimates for degrees of freedom

are obtained, which enterT−or F−statistics in classical inference approaches. Hence,

V is typically chosen to model serial correlations. A popular choice of V consists of a

AR(1)plus white noise process. Classical estimation forVcan be accomplished by ReML

procedures. We refer to Friston et al. (2008) for a detailed discussion of modeling serial correlations in fMRI.

If the baseline captures small-range trend changes as in Gössl et al. (2000, 2001a), an uncorrelated error structure seems adequate. Within their Bayesian modeling framework, the authors have observed that temporal trends are sufficiently captured by their time- varying trend effect, confirming the assumption of white error noise. In their experience no gain in precision is obtained by considering correlated error terms. Note, however, that the application of this small-scale trend component is restricted to the analysis of block experiments. In event-related designs, this trend component might confound stimulus effects.

As a starting point, we derive the model proposed in Chapter 3 on the assumption of independent error terms:

i ∼ NT(0, σ2iI),

whereIis the identity matrix of sizeT. Though, our chosen highpass filter removes only

long-range frequencies, the derivation of a combined fMRI-EEG model is substantially complicated when adding serial correlations. For testing whether the model exhibits a sufficient usefulness, we revert to independent noise.

The main focus of an fMRI study lies on the identification of activated voxels. Thereby, an voxel is said to be activated if its fMRI time series correlates with the transformed stimulus time series. In the classical fMRI regression framework, activation detection is implemented by voxelwise statistical hypothesis tests evaluating whether stimulus effect parameters equal zero (see for example Friston et al., 1994, 2002). Applying a significance threshold (derived by spatial multiple test procedures) to corresponding T- or F-maps results in a so-called statistical parametric map (SPM). In analogy to this, activation maps of Bayesian models are derived by thresholding posterior probability maps (PPMs) of effect parameters (see for example Gössl et al., 2000; Friston et al., 2002; Woolrich et al., 2004b). However, neither the classical approach nor the mentioned Bayesian approach constitutes a direct measure of activation. In the classical approach, a voxel is declared to be active when a corresponding test statistic is larger than a critical value. This critical value represents a limit for test statistic values that are acceptable under the null hypothesis of the true effect beinh zero. PPMs constitute a more direct measure for being defined as the posterior probability that the effect is larger than some threshold. The threshold, though, rests upon a definition from which effect size on a voxel can be declared activated—which can be quite arbitrary.

To overcome these problems, Smith et al. (2003) resp. Smith and Fahrmeir (2007) proposed to use a Bayesian activation detection scheme estimating voxelwise probabilities of ac- tivation directly. For this, they applied the theory of Bayesian variable selection (Smith and Kohn, 1996; George and McCulloch, 1997) introducing a binary variable being 1 if the stimulus regressor is selected, i.e. if there is a relationship between stimulus and signal time series, and 0 otherwise. Estimation of these binary indicators was spatially regularized by an Ising prior (Hurn et al., 2003). The posterior probabilities of voxelwise indicators being 1 constitute the posterior activation probabilities.

For our combined fMRI and EEG model, we choose to extend their approach to incorporate EEG information in an fMRI regression model. For this, we exchange the Ising prior with

a prior based on a binary regression model. EEG information is included as predictor with an—optionally—spatially-varying coefficient allowing for adaption to local brain response. Estimation of spatially-varying coefficients is regularized by Gaussian Markov random fields (GMRF). We consider two alternative regularization schemes either based on an intrinsic GMRF (Lang and Brezger, 2004; Rue and Held, 2005) or on a Gaussian conditional autoregression (CAR) (Weir and Pettitt, 2000; Pettitt et al., 2002; Smith and Smith, 2006).

With this spatial probit model, a strong EEG effect at voxelishould increase its activation

probability.

This chapter is organized as follows. After briefly reviewing our choice for the fMRI regression model for readers who skipped Chapter 2, we introduce the activation detection scheme based on Bayesian variable selection in Section 3.2. Prior specifications for all parameters, including the priors for incorporating EEG information in a probit hierarchy, are discussed in Section 3.3. Posterior inference based on a Markov Chain Monte Carlo (MCMC) scheme is derived in Section 3.4. A minor model extension is described in Section 3.5.

3.1 The fMRI regression model

The design of the predictor we use for modeling the fMRI signal is discussed in depth in Chapter 2. For ease of reading, we shortly summarize its form.

Letyi denote the vectoryi = (yi,t, t = 1, . . . , T)0 of the fMRI signal time series at voxeli,

i= 1, . . . , N. We include three predictor components which can be linearized with respect to unknown effect parameters. Hence, the following multiple regression model results

yi =Wδi+Cνi+Zζi+i,

whereWis theT ×p1 design matrix for the baseline trend,Cis theT ×p2 design matrix

of covariates andZis theT ×p3design matrix for modeling the hemodynamic response to

stimuli. The vectori is the vector of random errorsi = (i,t, t = 1, . . . , T)0. Note that all

three design matrices do not vary over voxels, whereas voxel dependent effect estimates allow for adaption to local brain response.

Simplifying the notation, we combine all linear regression parts into model

where X = [W|C|Z] is the overall T × p design matrix (with p = p1 + p2 + p3) and

β_i = (δ0_i,ν0_i,ζ0_i)0 is an according parameter vector of unknown weights and effects.

The likelihood of our model depends on the assumptioni ∼ NT(0, σi2I), whereIis the

identity matrix of sizeT.