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by projecting the data onto a magnetic signal subspace obtained from a truncated
expansion of spherical harmonic basis functions of the scalar magnetic potential.
Regardless of any transformation or averaging of the original measurements, the
data to be simultaneously solved can be represented as a
d
b
by
d
v
real or complex
matrix
B
, which contains
d
v
measurement column vectors. For example, if
B
is a
time series matrix, its
i
th column vector is the
d
b
-dimensional measurement vector
at the
i
th time. But
B
needs not be a time series, it can also be any given set of vectors
obtained from the data that benefit from simultaneous source localization (e.g., a
subspace spanned by the data). When
d
v
=
1, the single measurement problem is
recovered. This case is also used for localizing individual sensor maps obtained
from a decomposition of the data (e.g., ICA).
8.4 Overview of Modeling Steps
A quick overview of several aspects of modeling, which directly affect source es-
timation is presented in this section. Throughout this chapter, uppercase Latin and
Greek letters will be used to represent matrices, and Latin lowercase letters will be
used for vectors, except when in italics, which will be used for scalars. Lowercase
Greek letters will be used for vectors, scalars, and functions depending on the con-
text. The
i
th element of a vector will be specified by
a
i
, and the notation
A
:
i
and
A
i
:
will be used to refer, respectively, to the
i
th column vector and
i
th row vector of
A
.
Also,
A
:
i
and
A
i
:
will represent matrices made out, respectively, of the column and
row vectors specified by the vector of indices
i
.
8.4.1 Modeling of Neural Generators
The source model refers to the mathematical model used to approximate the pri-
mary current density within a cellular assembly. A popular source model for surface
and volume distributions of neural currents is the equivalent current dipole (ECD),
which approximates the current density as concentrating to a single point in space
r
q
=(
T
as expressed by
j
p
, where
j
p
x
,
y
,
z
)
(
r
)=
q
δ
(
r
−
r
q
)
(
r
)
is the 3D primary cur-
rent density vector at 3D spatial coordinates
r
, and
δ
is the Dirac delta distribution
j
p
with dipole moment
q
dr
flowing along a preferred direction as derived by
the average morphology of cells within the neural ensemble [31, 6]. The popularity
of the ECD source model stems from its compact description of distributed current
flows using a limited number of parameters: 3 for position, 2 for orientation, and
1 for amplitude. Higher dimensional parametric source models have been proposed
to describe more complex geometries of the primary current through most notably,
multipolar expansions [39].
EM recordings are dependent on the total flow of currents generated by neural
activity. We have so far described the modeling of the primary currents generated
=
(
r
)
