Biomedical Engineering Reference
In-Depth Information
Substituting of
(
8.3) in
(
8.1) leads to a linear system of equa-
tions. The only choice to be made is for the spatial dependence
of
ω
MN
(
r
). This choice is usually made to smooth over biases in
the sensitivity of the sensors to different parts of the source space
while “projecting out” source configurations that are already
present in some baseline measurement. Linearity allows such tasks
to be performed at the level of signal properties as described for
example in the covariance matrices of active and control condi-
tions. Although the (w)MN choice seems natural, it can not be
justified a priori. It puts an enormous load on the weight fac-
tor,
, demanding that just its spatial dependence can recover the
strength and location of generators. Effectively, the simplicity and
computational advantage of the linear models is bought at the
expense of using only a small amount of the information in the
data.
Generalized MFT admits a power expansion of
ω
ω
(
r
,
J
(
r
)) in
the modulus of the current density,
1
p
+
J(r)
=|
J(r)
|
A
m
ϕ
m
(
r
)
ω
p
(r)
(8.4)
m
Leading to a family of methods
(10, 21)
: MN, wMN and
LORETTA
(19)
for p
1, and a version of the FOCUSS
(20)
algorithm (corrected for gauge invariance) for p
=−
1. Stan-
dard MFT, as was initially selected via simulations corresponds
to p
=+
0
(9, 10)
.
The way standard MFT draws on the data makes the method
computationally demanding but it also confers two contrasting
and highly desirable properties that are necessary for accurate and
unbiased localization. First, only for standard MFT (with p
=
0)
the right hand side of
(
8.4) depends on the modulus of the cur-
rent density, just like the left hand side does, thus allowing sharp
discontinuities in the current density vector with small values of
the expansion coefficients. Second, standard MFT satisfies the
principle of least sensitivity to both variations of the data and iter-
ations of the non-linear norm constraints
(10)
. On the practical
side, MFT allows only part of the function,
=
, the a priori weight,
ω
0
(
r
), to be computed in advance from simulations with com-
puter generated data
(9)
or in more general ways
(10)
.Thefull
current density must be obtained from a highly non-linear system
of equations for each snapshot of data. Specifically, the strength
must be determined more explicitly from the MEG signal itself. A
more detailed discussion about the theoretical basis and algorith-
mic implications of different choices of p can be found in
(10,21)
,
and a discussion of the pitfalls of choosing values other than p
ω
=
0
in
(32)
.
