Combining EEG and MRI Techniques - Quantitative EEG Analysis Methods and Clinical Applications - page 344

is not similar to the standardized HRF, used for most fMRI analysis, and is the
subject of numerous investigations [79-82].
One of the first objectives in using EEG-fMRI for the study of epileptic activity is
to find which areas of the brain are activated by the physiological spikes in the EEG.
It should be remembered that fMRI does not directly measure electrical activity of
the neurons but the changes in blood oxygenation indirectly caused by this activity.
The fMRI responses to EEG spikes are delayed and are dispersed by about 4 to 6 sec-
onds. The responses can be modeled by convolution of the filtered EEG spike with
an HRF. A simple model at a point s in D -dimensional Euclidean space ( D = 3 here)

is a linear model

() () ()()

Ys = X s β + σ s ξ s (12.10)

where Y ( s ) is a column vector of n observations at point s. X is a design matrix incor-
porating the response to the neural excitations. In MATLAB, the matrix X is
obtained by convolving a column vector of 1s and 0s with the standard HRF. The 1s
are placed at scan locations where the EEG data manifests a neural excitation (spike)
and 0s elsewhere. β ( s ) is an unknown coefficient, σ ( s ) is a scalar standard deviation,

and ( s ) is a column vector of temporally correlated Gaussian errors. The HRF can

be modeled as a gamma function or as the difference between two gamma functions,
whose parameters may be estimated as well, creating a nonlinear model [83-85].
The steps outlined above seem straightforward. However, the difficulty arises
because: (1) each observation Y ( s ) is an entire three-dimensional image, rather than
a single value, and neighboring voxels tend to be correlated, and (2) all activations
corresponding to EEG spikes may not have a direct correlation.
The analyses are generally carried out on a voxel-by-voxel basis. The parameter
estimates are therefore suboptimal. The correlation from neighboring voxels is mod-
eled by the variance in the term

ξ

ξ ( s ). The variance is reduced by spatial smoothing of

the parametrized image data.
An advantage of using the GLM is that the design matrix can be extended to
include columns that model effects of correlated noise in the fMRI data. The
regressors added may be a vector that correlates the cardiac activity or other con-
founding effects such as low-frequency rhythms of the EEG. A primary confound
used in most EEG-fMRI models incorporates the six rigid body transformation
parameters (obtained while realigning the spatial data to standard stereotaxic coor-
dinate space) [86].
The parameters of primary importance are the effects that are correlated to the
physiological spikes in the EEG (i.e., corresponding to the first column of the design
matrix) and their standard deviations. The key quantity for activation detection is
their ratio, or T statistic, T ( s ). These parameters are not smoothed, because smooth-
ing always increases bias as the cost for reducing noise. Smoothing is perhaps best
reserved for parameters of secondary importance, such as temporal correlations or
other ratios of variances and covariances.
The coefficient vector varies from voxel to voxel, and they are estimated sepa-

rately for each voxel. The method of least-squares estimates

β

β by minimizing the

sum of the squared residuals: