Biomedical Engineering Reference
In-Depth Information
oretical MEG measurements were computed, which were then used in (5.25) and
(5.26) to obtain the estimated minimum norm current density, which showed maxi-
mum activity at the correct location, but with some spatial dispersion. These first
results were very encouraging. However, there was one essential omission: The
method does not localize deep sources. In a three-dimensional cortex, if the actual
source is deep, the method misplaces it to the outermost cortex. The reason for this
behavior was explained in Pascual-Marqui [26], where it was noted that the
EEG/MEG minimum norm solution is a harmonic function [27] that can only attain
extreme values (maximum activation) at the boundary of the solution space, that is,
at the outermost cortex.
5.4.3 Low-Resolution Brain Electromagnetic Tomography
The discrete, three-dimensional distributed, linear inverse solution that achieved
low localization errors (in the sense defined earlier by Hämäläinen and Ilmoniemi
[23]) even for deep sources was the method known as low-resolution electromag-
netic tomography (LORETA) [28].
Informally, the basic property of this particular solution is that the current den-
sity at any given point on the cortex be maximally similar to the average current
density of its neighbors. This “smoothness” property (see, e.g., [29, 30]) must hold
throughout the entire cortex. Note that the smoothness property approximates the
electrophysiological constraint under which the EEG is generated: Large spatial
clusters of cortical pyramidal cells must undergo simultaneously and synchronously
the same type of postsynaptic potentials.
The general inverse problem that includes LORETA as a particular case is
stated as
min
J
F
W
(5.27)
with
2
F
W
=− +
Φ
KJ
α
J
T
WJ
(5.28)
The solution is
J
W
=Φ
(5.29)
with the pseudoinverse given by
(
)
+
−
1
T
−
1
T
TWK WK H
W
=
+
α
(5.30)
R
(
3
NN
V
) (
×
)
where the matrix
W
∈
can be tailored to endow the inverse solution with
V
a particular property.
In the case of LORETA, the matrix
W
implements the squared spatial Laplacian
operator discretely. In this way, maximally synchronized PSPs at a relatively large
macroscopic scale will be enforced. For the sake of simplicity, lead field normaliza-
tion has not been mentioned in this description, although it is an integral part of the
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