Biomedical Engineering Reference
In-Depth Information
the spatial correlation of sources. The necessary methods are described in the next
section.
12.1.4 Source Localization Techniques
The basics of source analysis were introduced in Chapter 5, and most of the tech-
niques described there can easily be adapted to make use of cortical locations and
orientations. In this section, some less obvious extensions that become available
with the advent of cortical triangle meshes are described, both dealing with the
incorporation of local neighborhood information.
12.1.4.1 Spatial Coupling
The LORETA method uses MNLS fitting with spatial coupling between source
locations. The spatial Laplacian (second derivative) of the source distribution is
used in the model term, rather than the source strengths as in standard MNLS [25].
The effect is that the model term demands minimum curvature of the source distri-
bution rather than minimum norm. Spatial coupling is achieved via a nondiagonal
Laplacian weighting matrix
B
, while the diagonal weighting matrix
D
is responsible
for removing the depth dependency [24, 26], so that
TT
WDBBD
=
(12.1)
In the case of locations on a regular three-dimensional grid, for any location
i
and its
N
i
neighbors
j
(with
N
i
≤
6) [27],
B
=−
1
ii
,
(12.2)
(
)
B
=+
6
N
12
N
ij
,
i
i
In the case of cortical sources [28], where
d
i,j
is the distance between locations
i
and
j
and the sums loop over all
N
i
neighbors
j
of location
i
,
(
)
B
=−
Σ
1
d
Σ
d
ii
,
i
i
(12.3)
(
)
B
=
1
d
Σ
d
ij
,
ij
,
i
Because
B
is nondiagonal but sparse, it can be favorable to minimize [see (12.1)]
J
=
arg min
Φ
−
KJ
+
λ
J
T
WJ
=
arg min
Φ
−
KJ
+
λ
J
T
D
T
B
T
BDJ
(12.4)
directly [24], instead of computing J as
(
)
−
1
JWKKWK
=
−
1
T
−
1
T
+
λ
1
Φ
(12.5)
12.1.4.2 Extended Sources
The methods described until now have used a lead field matrix
K
that encodes the
impact of point sources onto the measured data and a model term measuring prop-
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