Biomedical Engineering Reference
In-Depth Information
8
6
4
2
0
-2
-4
-6
-8
-8
-6
-4
-2
0
2
4
6
8
x
Fig. 3.36.
The
z
component of the magnetic field due to a current dipole only
gnetic measurements. It has been shown [23] that knowledge of the electro-
magnetic field outside a conductor is not sucient to uniquely determine the
current distribution inside the conductor. This means that there is no unique
solution to the inverse problem, so that additional constraints must be added
to restrict the number of possible solutions. Basically, one can distinguish two
approaches for the resolution of the inverse problem, depending on whether
distributed or localized sources are assumed.
Imaging methods [24-29] assume distributed sources and compute an esti-
mate of the full current distribution in the brain. Among the set of all current
distributions
J
, an “optimal” solution
J
opt
is computed by using Tikhonov
regularization and finding that
J
T
WW
T
−
1
2
+
λ
J
opt
=
arg
min
J
M
−
GJ
J
,
(3.81)
where
M
is the observed measurements,
G
is the lead field matrix,
λ
is
a regularization parameter, and
is a weight matrix that can take several
forms, such as the identity matrix or the Laplacian operator, depending on the
desired constraints. The drawbacks of imaging methods is that they require
manipulation and computation of huge matrices, and that they usually tend
to give oversmoothed estimates of the sources.
An alternative method is to assume that the measured electromagnetic
field results from the activity of one or a few highly localized neuronal sources
of a dipolar nature: this is called the equivalent current dipole model [27,30-
32]. In this case, for a single time slice, the inverse problem reduces to finding
q
opt
such that
W
2
.
q
opt
=
arg
min
q
M
−
Gq
(3.82)
Since
is a vector of
n
dipoles, and each dipole corresponds to six parameters
(position, direction, and intensity), the number of parameters to be estimated
q
