94
Y. Zhang
To obtain the time-course of activity of the current source at r di p , one can apply the
spatial filter matrix W to each of the measurement vectors v(t) for all t 1, . . . , T .
Furthermore, this approach can reconstruct the dipole activity at any location by
simply changing the r di p , so long as the new source position is anatomically realistic.
Beamforming techniques may, however, struggle in cases where highly correlated,
spatially distinct sources are active [70, 80]. A straightforward strategy to handle
this problem is the introduction of longer time windows, as the high correlation
between sources is less likely to persist across the longer period. Brookes et al.
proposed a modified source model in a dual source beamformer technique that could
successfully reconstruct correlated sources in a simulation study [12], while other
simulation studies have suggested that the LCMV method is robust against moderate
level of source correlation [76, 80].
5.2.2 Non-parametric Optimization Methods
In contrast to ECD models, which assume that underlying sources can be represented by a single or small group of equivalent dipoles that explain the observed
measurements, non-parametric optimization methods make use of cortically distributed source (CDS) models. The CDS models base on the assumption that the
primary current sources are the cortical pyramidal neurons that span the cortex and
orient normally to the surface. Thus, the source space is constructed with a current
dipole assigned at each of the mesh element of the cortical layer, with dipole orientations either fixed to the local surface normal or left as unknown. In practice, cortical
surface models are usually extracted from brain anatomical images (e.g. MRI, CT)
through segmentation algorithms [17, 18, 24, 25]. In this setting, the dipole locations
are known and the parameter of interest is the dipole moment of each source location. The number of source points on the cortical mesh will vary depending on the
chosen model and mesh element properties, but is typically on the order of several
thousands. This again makes the inverse problem highly underdetermined (P N ).
The inverse solutions are thus obtained by applying some form of regularization to
the cost function. The formulation of the EEG inverse problem is presented again
below, with the various regularization schemes and their limitations described in
subsequent sections. Consider the following linear formulations of the forward and
inverse problems:
V G J + n
ˆ
J M V
where we assume a priori that statistical distributions of the dipole moment J and
sensor vector n exist such that J ∼ N (0, R) and n ∼ N (0, C). Matrix M is the linear
inverse operator that maps the EEG measurements V into the estimated source ˆ
J.
Y. Zhang
To obtain the time-course of activity of the current source at r di p , one can apply the
spatial filter matrix W to each of the measurement vectors v(t) for all t 1, . . . , T .
Furthermore, this approach can reconstruct the dipole activity at any location by
simply changing the r di p , so long as the new source position is anatomically realistic.
Beamforming techniques may, however, struggle in cases where highly correlated,
spatially distinct sources are active [70, 80]. A straightforward strategy to handle
this problem is the introduction of longer time windows, as the high correlation
between sources is less likely to persist across the longer period. Brookes et al.
proposed a modified source model in a dual source beamformer technique that could
successfully reconstruct correlated sources in a simulation study [12], while other
simulation studies have suggested that the LCMV method is robust against moderate
level of source correlation [76, 80].
5.2.2 Non-parametric Optimization Methods
In contrast to ECD models, which assume that underlying sources can be represented by a single or small group of equivalent dipoles that explain the observed
measurements, non-parametric optimization methods make use of cortically distributed source (CDS) models. The CDS models base on the assumption that the
primary current sources are the cortical pyramidal neurons that span the cortex and
orient normally to the surface. Thus, the source space is constructed with a current
dipole assigned at each of the mesh element of the cortical layer, with dipole orientations either fixed to the local surface normal or left as unknown. In practice, cortical
surface models are usually extracted from brain anatomical images (e.g. MRI, CT)
through segmentation algorithms [17, 18, 24, 25]. In this setting, the dipole locations
are known and the parameter of interest is the dipole moment of each source location. The number of source points on the cortical mesh will vary depending on the
chosen model and mesh element properties, but is typically on the order of several
thousands. This again makes the inverse problem highly underdetermined (P N ).
The inverse solutions are thus obtained by applying some form of regularization to
the cost function. The formulation of the EEG inverse problem is presented again
below, with the various regularization schemes and their limitations described in
subsequent sections. Consider the following linear formulations of the forward and
inverse problems:
V G J + n
ˆ
J M V
where we assume a priori that statistical distributions of the dipole moment J and
sensor vector n exist such that J ∼ N (0, R) and n ∼ N (0, C). Matrix M is the linear
inverse operator that maps the EEG measurements V into the estimated source ˆ
J.