Biomedical Engineering Reference
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weight matrix used in LORETA. The technical details of the LORETA method can
be found in [26, 28].
When LORETA is tested with point sources, low-resolution images with very
low localization errors are obtained. These results were shown in a
nonpeer-reviewed publication [31] that included discussions with M. S.
Hämäläinen, R. J. Ilmoniemi, and P. L. Nunez. The mean localization error of
LORETA with EEG was, on average, only one grid unit, which happened to be three
times smaller than that of the minimum norm solution. These results were later
reproduced and validated by an independent group [32].
It is important to take great care when implementing the Laplacian operator.
For instance, Daunizeau and Friston [33] implemented the Laplacian operator on a
cortical surface consisting of 500 vertices, which are very irregularly sampled, as can
be unambiguously appreciated from their Figure 2 in [33]. Obviously, the Laplacian
operator is numerically worthless, and yet they conclude rather abusively that “the
LORETA method gave the worst results.” Because their Laplacian is numerically
worthless, it is incapable of correctly implementing the smoothness requirement of
LORETA. When this is done properly with a regularly sampled solution space, as in
[31, 32], LORETA localizes with a very low localization error.
At the time of this writing, LORETA has been extensively validated, such as in
studies combining LORETA with fMRI [34, 35], with structural MRI [36], and with
PET [37]. Further LORETA validation has been based on accepting as ground truth
localization findings obtained from invasive implanted depth electrodes, in which
case there are several studies in epilepsy [38-41] and cognitive ERPs [42].
5.4.4 Dynamic Statistical Parametric Maps
The inverse solutions previously described correspond to methods that estimate the
electric neuronal activity directly as current density. An alternative approach within
the family of discrete, three-dimensional distributed, linear imaging methods is to
estimate activity as statistically standardized current density.
This approach was introduced by Dale et al. in 2000 [43], and is referred to as
the dynamic statistical parametric map (dSPM) approach or the noise-normalized
current density approach. The method uses the ordinary minimum norm solution
for estimating the current density, as given by (5.25) and (5.26). The standard devia-
tion of the minimum norm current density is computed by assuming that its variabil-
ity is exclusively due to noise in the measured EEG.
Let
S
Φ
R
N
E
×
denote the EEG noise covariance matrix. Then the corre-
sponding current density covariance is
Noise
∈
Noise
Noise
T
S
=
TS
T
(5.31)
Φ
J
with
T
given by (5.26). This result is based on the quadratic nature of the covariance
in (5.31), as derived from the linear transform in (5.19) (see, e.g., Mardia et al. [44]).
From (5.31), let
[]
S
J
Noise
∈
R
33
denote the covariance matrix at voxel
v
. Note that
×
v
this is the
v
th 3
×
3 diagonal block matrix in
S
J
Noise
, and it contains current density
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