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fMRI-to-EEG approaches target at enhancing the performance of the EEG source recon- struction problem. EEG source reconstruction techniques try to find the three-dimensional location of neuronal generators of the corresponding (two-dimensional) signals measured on the scalp surface. Two different classes of models have been developed to solve the problem (Michel et al., 2004; Daunizeau et al., 2010): equivalent current dipole (ECD) mod- els (Mosher et al., 1992; Kiebel et al., 2008) and distributed source models (Hämäläinen and Ilmoniemi, 1994).

For ECD models, an assumption must be made beforehand on how many activation foci are to be found. This assumption is generally content-driven. Algorithms then estimate the location and orientation of dipoles via a nonlinear optimization procedure, because the used forward model is nonlinear in dipole locations. The spatial distribution of the activated area involved, however, cannot be inferred. ECD models are no imaging tech- nique in a narrower sense, because they do not calculate a three-dimensional voxel image, but only a limited number of dipole locations. fMRI information can be incorporated as a prior constraint for the position of dipoles, which are supposed to lie in the center of fMRI activation regions (cf. e.g. Wagner and Fuchs, 2001).

Distributed source models do not suffer from mentioned disadvantages of ECD models like yielding position estimates for only a small, specified number of dipoles without information about their spatial extend. In the following, we give some more details on the methodology behind distributed source models based solely on EEG, because we pursue to incorporate these in the combined fMRI-EEG model proposed later in this thesis. Afterwards we give some information on fMRI-constrained distributed source models proposed so far in the literature.

Distributed source models calculate the strength and direction of dipoles on a dense (voxel) grid typically distributed all over the cortical sheet producing a three-dimensional image of current density values. This unknown current density distribution is the solution of

an under-determined system of equations, in which the number of parameterspto be

estimated exceeds the number of observationsNeby far (Ne p). Hence, restrictions have

to be introduced to constrain the solution space by prior information. The distributed source model is generally written as a multivariate linear model

with M being the Ne×T observation matrix of scalp EEG measurements recorded at

time pointst = 1, . . . , T, Kis a knownNe×ptransition matrix (i.e. a lead field matrix

as described in Fuchs et al., 2002) containing the mapping of cortical sources to the scalp

surface by an underlying physical model, andJ is ap×T matrix of unknown current

density parameters of p dipoles (located on the pre-specified voxel grid) at each time

pointt. Note that, the number of parameterspcoincides with the number of grid voxels,

which we generally denote asN in this thesis. It is assumed that the model is correctly

specified except for an additive white noise termE. Proposed distributed source models

use different regularization techniques to yield solutions with certain properties (Mattout et al., 2006). Estimation is generally based on minimizing a quadratic loss function in the form of a penalized least squares criterion (PLS):

P LS(λ) = ||M−KJ||2

WNe +λ||J||

2

Wp

where||.||2

W is theL2norm with respect to metricWandλis a regularization parameter

controlling the relative weight of both terms, which should be minimized. Given a normally distributed error, this criterion corresponds to a Bayesian linear regression model

with Gaussian error prior, E ∼ N(0,CNe)with CNe = (W

0

NeWNe)

−1_{, andJ ∼ N(0,C}

p)

withCp = (λW0pWp)−1as prior forJ. ChoosingWp =Ip, withIp being thep-dimensional

unity matrix, leads to the minimum-norm approach of Hämäläinen and Ilmoniemi (1994).

IfWpequals a spatial Laplacian matrix the smooth LORETA solution is calculated (Pascual-

Marqui et al., 1994). Mattout et al. (2006) modelCNe andCp as a linear combination of

variance component matrices within a two-stage hierarchical model to adapt variance structure estimates to more correlated correlation structures. The BASTA method of Daunizeau et al. (2006) can as well be derived from a multi-stage hierarchical model. Here, in contrast to Mattout et al. (2006) and related to ECD models, the current density distribution estimation is based on a brain parcelling into homogeneous clusters.

Source reconstruction of raw EEG trajectories is of minor practical interest, because single- stimulus profiles suffer from a low signal-to-noise ratio. More commonly, several EEG responses to a selected stimulus type are averaged in a defined time window, with this average being referred to as event-related potential (ERP). The improved signal-to-noise ratio reveals the typical response profile evoked by this stimulus type. For selected time points of this mean ERP, a source reconstruction method can then be used to infer the location of neuronal generators. In Figure 1.9, the process of a source reconstruction of two selected time points of an averaged ERP signal in response to acoustic stimuli is visualized. Data stem from the acoustic oddball experiment analyzed in Chapter 7.

Figure 1.9 depicts the mean ERP response to the so-called odd tone stimulus type. The shown source distribution solution has been calculated by the sLORETA software—an

extension to the LORETA software (http://www.uzh.ch/keyinst/loreta.htm).

Instead of prior information influencing solely the structure of the solution, like its smooth- ness, fMRI information (or other external information) can be included to constrain the solution space (Dale and Sereno, 1993). Generally, fMRI activation information in the

form of a binary activation mapZcan be incorporated into the prior variance-covariance

matrix ofJby choosingCp ∝f(Z)wheref(.)is an appropriate transformation function

(Daunizeau et al., 2006). The function f(Z)is often chosen as a linear combination of

variance-component matrices with one variance-covariance component being proportional

toλZDZ withλZ being an additional, positive weight parameter andDZ the diagonal

matrix of the vectorizedZentries (Liu et al., 1998; Phillips et al., 2002; Daunizeau et al.,

2006). Hence, if voxeliis activated, weightλZ is added to its element-specific variance

putting higher weight resp. probability toJi values farther away from 0. This approach

is related to the concept of spike-and-slab priors well-known in the Bayesian variable selection literature (George and McCulloch, 1993).

To classify fMRI-to-EEG approaches into the introduced neuronal source model, we assert that the high spatial resolution of the fMRI is used to add information to the ill-posed source construction problem to find more reliable source locations. Noise and modeling bias is intended to be reduced to detect neuronal sources that could not be detected on the basis of the EEG alone. Hence, fMRI-to-EEG approaches cover a larger area of neuronal sources. This is visualized in Figure 1.7d. The gain in knowledge about neuronal sources is visualized by the plain yellow area, whereas the dashed yellow area indicates sources that both the unimodal and multimodal method can infer. Having said this, we have to admit that this is a rather optimistic view. Incorporating fMRI information as prior information into EEG source reconstruction models can also introduce bias when some sort of decoupling occurs (as discussed in Section 1.1.4). For more details on this, we refer to Ahlfors and Simpson (2004), Ritter and Villringer (2006) and Daunizeau et al. (2010), and references therein.