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straints can take many forms but are generally handled by making assumptions about
the nature of the sources (e.g., number of brain areas involved, constraints on their
spatial extent and the relative smoothness or sparsity of the current density, priors
on source parameters and hyperparameters, priors on their anatomical locations -
e.g., on the cortex - and from electrophysiology - e.g., the maximum expected am-
plitude of currents). Thus, the accuracy and validity of the source estimates depend
to some extent on the biological correctness of the assumptions and priors adopted
in the models. A recent trend in the domain of NSI research consists is considering
that such priors should - to some extent - be flexible and adaptive to the data under
study. The rest of this chapter focuses on presenting a variety of inverse modeling
approaches. We have identified three basic approaches that encompass most of the
methods that have been published so far: (1) parametric source model fitting, (2)
source imaging techniques explained within a general Bayesian framework, and (3)
spatial scanning and filtering through beamforming.
8.5 Parametric Dipole Modeling
One of the most common assumptions adopted to handle nonuniqueness is that the
measurements were generated within a small number of brain regions that can be
modeled using a limited number of ECDs. The associated estimation algorithms
minimize a data-fit cost function, defined typically in the least-squares sense, in the
multidimensional space of nonlinear parameters. Usually, algorithms estimate five
nonlinear parameters per dipole: the
x
,
y
, and
z
parameters that define the dipole
position, and the two angles necessary to define the dipole orientation in 3D space.
However, in the MEG spherically symmetric volume conductor model only one an-
gle (on the tangent space of the sphere) is necessary because the radial dipole com-
ponent is silent, thereby reducing the dimensionality to four dimensions per dipole.
The dipole amplitudes are linear parameters estimated directly from data. The di-
mension of the space where the cost function is minimized can be reduced further
to three dimensions per dipole if the dipole orientations are allowed to be obtained
linearly from the data. Technically, parametric dipole modeling is performed in the
sense of a least-squares fit of a model of the data, which writes differently depending
on the model of noise statistics under consideration as we shall now describe.
8.5.1 Uncorrelated Noise Model
Parametric dipole fitting algorithms, minimize a data-fit cost function such as the
square of the Frobenius norm of the residual,
B
L
s
J
s
||
F
F
L
s
L
s
)
F
P
L
s
B
F
min
s
||
B
−
||
=
||
B
−
=
||
(
I
−
B
)
||
=
||
||
,
(8.1)
