R.D. Pascual-Marqui

The KEY Institute for Brain-Mind Research, University Hospital of Psychiatry

Lenggstr. 31, CH-8029 Zurich, Switzerland

Tel.:+41-1-3884934 ; Fax:+41-1-3803043

pascualm@key.unizh.ch ; http://www.keyinst.unizh.ch/loreta.htm

Scalp electric potentials
(EEG) and extracranial magnetic fields (MEG) are due to the primary (impressed)
current density distribution that arises from neuronal post-synaptic processes.
A solution to the inverse problem, i.e. the computation of images of electric
neuronal activity based on extracranial measurements, would provide important
information on the time course and localization of brain function. In general,
there is no unique solution to this problem. In particular, an instantaneous,
distributed, discrete, linear solution capable of exact localization of point
sources is of great interest, since the principles of linearity and superposition
would guarantee its trustworthiness as a functional imaging method, given that
brain activity occurs in the form of a finite number of distributed “hot spots”.
Despite all previous efforts, linear solutions at best produced images with
systematic non-zero localization errors. A solution is reported here, which
yields images of standardized current density with zero localization error.
The purpose of this paper is to present the technical details of the method,
allowing researchers to test, check, reproduce, and validate the new method
(*s*LORETA).

This study is strictly
limited to EEG/MEG inverse solutions of the type: instantaneous, distributed,
discrete, and linear. The generic form of the inverse problem follows. There
are _{} instantaneous
extracranial measurements. There are _{} voxels in the brain. Typically, the voxels
are determined by subdividing uniformly the solution space, which is usually
taken as the cortical grey matter volume or surface. At each voxel there is
a point source, which may be a vector with three unknown components (i.e., the
three dipole moments), or a scalar (unknown dipole amplitude, known orientation).
The cases considered here correspond to _{}.

In 1984, Hämäläinen and Ilmoniemi (1) were the first to report an instantaneous, distributed, discrete, linear solution to the EEG/MEG inverse problem: the well known minimum norm solution. However, the minimum norm solution is notorious for totally misplacing actual deep sources onto the outermost cortex, as demonstrated in (3), (2), and (8).

The problem of excessively large errors of localization remained unsolved until the introduction of the method known as LORETA (low resolution brain electromagnetic tomography) in 1994 (18). LORETA has fairly good accuracy in localizing test sources even when they are deep. The overall average localization error is smaller than one grid unit (see e.g. (3), (2), and (8)).

A series of papers published in 1998 and 1999 (see (19)-(23)) introduced for the first time the method of high time resolution statistical parametric mapping for tomographic images of electric neuronal activity. The idea was to adopt the methods of statistical inference for the localization of brain function as used in PET and fMRI studies. This methodology was applied to high time resolution, time varying LORETA images.

In the present study
a new tomographic method for electric neuronal activity is introduced, where
localization inference is based on images of standardized current density. The
method is denoted as standardized low resolution brain electromagnetic tomography
(*s*LORETA). Unlike the method recently
introduced by Dale et al. (6), which has systematic non-zero localization error,
*s*LORETA has zero localization
error.

The equation of interest takes the form:

_{}
Eq. 1

In Eq. 1, _{} is a vector containing
scalp electric potentials measured at _{} cephalic electrodes, with respect to
a common, arbitrary reference electrode located anywhere on the body.

The primary (impressed)
current density _{} is defined as:

_{}
Eq. 2

where _{} for _{}. At the *l*^{th} voxel, _{} contains the three unknown
dipole moments.

The superscript “*T*” denotes transpose.

The lead field _{} has the following
structure:

_{}
Eq. 3

with _{}, for _{}, and for _{}. Note that _{}, where _{} is the scalp electric
potential at the *i*^{th}
electrode, due to a unit strength *X*-oriented
dipole at the *l*^{th} voxel;
_{} is the
scalp electric potential at the *i*^{th}
electrode, due to a unit strength *Y*-
oriented dipole at the *l*^{th}
voxel; and _{} is
the scalp electric potential at the *i*^{th}
electrode, due to a unit strength *Z*-
oriented dipole at the *l*^{th}
voxel.

In Eq. 1, *c* is an arbitrary constant which embodies
the fact that the electric potential is determined up to an arbitrary constant;
and _{} is
a vector of ones. The parameter *c*
allows the use of any reference for the lead field and the measurements.

Hämäläinen, M.S., and Ilmoniemi (1) were the first to publish a particular solution to the instantaneous, distributed, discrete, linear EEG/MEG inverse problem. Their solution is known as the minimum norm inverse solution. However, the minimum norm solution is notorious for totally misplacing actual deep sources onto the outermost cortex (2).

Dale et al. (6) proposed a method in which localization inference is based on a standardization of the current density estimates. In particular, they employed the current density estimate given by the minimum norm solution, and they standardized it by using its expected standard deviation, which is hypothesized to be originated exclusively by measurement noise. The method of Dale et al. (6) produces systematic non-zero localization errors (8), even in the presence of negligible noise. This fact was not evaluated nor admitted in their original paper.

*s*LORETA
is similar to the Dale et al. (6) method: it employs the current density estimate
given by the minimum norm solution, and localization inference is based on standardized
values of the current density estimates. However, standardization in *s*LORETA
takes a completely different route (explained below). The consequence is that,
unlike the Dale et al. (6) method, *s*LORETA
is capable of exact (zero-error) localization.

The minimum norm inverse
solution is harmonic (2), which means that the Laplacian of the current density
is zero, i.e., _{}, where **r** denotes volume coordinates in the brain. Therefore, the minimum
norm inverse solution is very smooth. The concept of smoothness employed here
is discussed in greater detail in (8), with special emphasis on its electrophysiological
interpretation. However, as previously mentioned, the minimum norm inverse solution
is notorious for its incapability of correct localization of deep point sources
(2).

This problem is solved by standardization of the minimum norm inverse solution, and basing localization inference on these standardized estimates.

The functional of interest here is:

_{}
Eq. 4

where _{} is a regularization
parameter. This functional is to be minimized with respect to **J** and *c*,
for given **K**, **F**,
and *a*.
The explicit solution to this minimization problem is (see e.g. (3)):

_{}
Eq. 5

where:

_{}
Eq. 6

_{}
Eq. 7

with _{} denoting the centering
matrix; _{} the
identity matrix; and _{} is a vector of ones.

For any matrix _{}, _{} denotes its Moore-Penrose
pseudoinverse (see e.g. (9)).

The centering matrix
**H** in Eq. 7 is the average reference operator.

In what follows, for
all EEG cases, the symbols **F** and **K** will denote the average reference transforms
of the EEG measurements and the lead field, respectively. This simplifies the
notation. But most important of all, the correct solution to EEG problems is
based on these average reference transforms.

Therefore, when using
average reference transforms of **F** and **K**, Eq. 1 becomes:

_{}
Eq. 8

and the functional in Eq. 4 becomes:

_{}
Eq. 9

with minimum:

_{}
Eq. 10

where:

_{}
Eq. 11

Standardization of the
estimate _{} requires
an estimate of its variance.

Note that Eq. 9 can be
derived from a Bayesian formulation of the inverse problem (see e.g. (5), Eq.
1.88 therein). In this view, the *actual*
source variance (*prior*) _{} is equal to the identity
matrix, i.e.:

_{}
Eq. 12

In addition, from the Bayesian point of view, the electric potential variance is due to noisy measurements:

_{}
Eq. 13

Note that in Eq.
13, the average reference operator **H**
plays the role of the identity matrix in the subspace spanned by the measurement
space.

It is usually assumed that activity of the actual sources and the noise in the measurements are uncorrelated.

Based on the linear relation
in Eq. 8, making use of Eqs. 12 and 13, and taking into account the independence
of actual source activity and measurement noise, the electric potential variance
_{} then is:

_{}
Eq. 14

See e.g. (10), Eqs. 1.5.1-1.5.6 therein.

Due to the linear relation
in Eq. 10, and making use of Eq. 14, the variance of the *estimated* current density is:

_{}
Eq. 15

See e.g. (10), Eqs. 1.5.1-1.5.6 therein, and (9).

Note that the variance
of the *estimated* current density
is equivalent to the Backus and Gilbert (4) resolution matrix, which is obtained
by plugging Eqs. 8 and 11 into 10:

_{}
Eq. 16

with:

_{}
Eq. 17

where **R** is the resolution matrix.

Note that the variance
of the *estimated* current density
in Eqs. 15 and 17 is __not__ the *posterior*
variance in the Bayesian formulation (see e.g. (5), Eq. 1.94).

In contrast, according
to Dale et al. (6), the variance of the *estimated*
current density is based on the assumption that the only source of variation
is measurement noise. This means that Eq. 14 now is:

_{}
Eq. 18

and Eq. 15 now is:

_{}
Eq. 19

Note that unlike the
approach of Dale et al. (6), *s*LORETA
takes into account two sources of variation: mainly the variation of the actual
sources, and then finally, if any, the variation due to noisy measurements.

Finally, *s*LORETA corresponds to the following estimates
of standardized current density power:

_{}
Eq. 20

where _{} is the current density
estimate at the *l*^{th}
voxel given by Eqs. 10 and 11 (for average reference transforms); and _{} is the *l*^{th} diagonal block of matrix _{} in Eqs. 15 or 17.

Note that the pseudo-statistic
in Eq. 20 has the form of an “*F*”
statistic.

Note that Eq. 20 is different in form from the Dale et al. (6) standardization (see (6), Eq. 7 therein). The Dale et al. (6) standardized estimates are:

_{}
Eq. 21

where _{} is the *l*^{th} diagonal block of matrix _{} in Eq. 19; and for
any symmetric matrix **M**, _{} is the diagonal matrix
formed by the diagonal elements of **M**.

This case usually corresponds
to the inverse problem when the cortical surface is completely known. Voxels
are now distributed along the cortical surface, and the dipoles at each voxel
have known orientation (perpendicular to the cortical surface). The unknowns
correspond to the amplitudes, which may take positive, zero, or negative values.
The dipole orientations (*defined as unit
length vectors with three components*) can be incorporated into the
lead field **K** in Eq. 8. Details
can be found in (3).

In this case Eq. 8 has
the same form, but now _{} since it only contains one unknown scalar
per voxel, and _{} since it includes the dipole orientation
at each voxel. Details can be found in (3). All the derivations employed in
Eqs. 8-17 remain formally identical.

However, *s*LORETA now corresponds to the following
estimates of standardized current density power:

_{}
Eq. 22

where the scalar
_{} is the
current density amplitude estimate at the *l*^{th}
voxel; and the scalar _{} is the *l*^{th} diagonal element of matrix _{}.

Note that the pseudo-statistic
in Eq. 22 has the form of an “*F*”
statistic.

The equations for the MEG case have identical form to Eqs. 8-17, 20, and 22, depending on the case of unknown dipole moments, or only unknown amplitudes.

Note that the average reference does not apply to MEG.

The only change corresponds to the equations for the MEG lead field, which are different to those for the EEG.

Simulations were carried out in a three-shell spherical head model registered to the Talairach human brain atlas (11), available as a digitized MRI from the Brain Imaging Centre, Montreal Neurological Institute. Registration between spherical and realistic head geometry used EEG electrode coordinates reported by Towle et al. (12).

In one set of practical, realistic, simulations, the solution space was restricted to cortical gray matter and hippocampus, as determined by the corresponding digitized Probability Atlas also available from the Brain Imaging Centre, Montreal Neurological Institute. A voxel was labeled as gray matter if it met the following three conditions: its probability of being gray matter was higher than that of being white matter, its probability of being gray matter was higher than that of being cerebrospinal fluid, and its probability of being gray matter was higher than 33%. Only gray matter voxels that belonged to cortical and hippocampal regions were used for the analysis. A total of 6430 voxels at 5mm spatial resolution were produced under these neuroanatomical constraints. At each voxel, three unknown values (the three dipole moments) were estimated, making a total of 6430x3=19290 unknowns. 25 electrodes (in EEG experiments), or 25 magnetometer sensors (in MEG experiments) were used. In both cases, sensors and electrodes were placed in the same locations.

In the second set of practical, realistic, simulations, the solution space was restricted to the cortical surface, represented as 12980 triangles (voxels) (13). This case corresponded to unknown current density amplitude (but with known orientation), making a total of 12980 unknowns. 101 electrodes (in EEG experiments), or 101 magnetometer sensors (in MEG experiments) were used. In both cases, sensors and electrodes were placed in the same locations.

The minimum norm solution,
the method of Dale et al. (6), and *s*LORETA
were compared in terms of localization errors and spatial spread. The methods
were tested with point sources located at the voxels. For the case corresponding
to 3 unknowns per voxel, an arbitrary (random) orientation of the test source
was employed. The test sources were used to generate the measurements (forward
equation (Eq. 8)), which were then given to the imaging methods. Simulations
included “noise free” and “noisy” measurements.

In the minimum norm solution
case, the imaging method is based on Eqs. 20 and 22, but without standardization,
which is achieved by setting the variance to the identity matrix, i.e., _{}.

In the minimum norm solution
and in *s*LORETA, the regularization
parameter *a* in the previous equations was estimated
by cross-validation. Exact details and equations for a practical implementation
of the cross-validation method can be found in (14).

In the Dale et al. (6)
method, the parameter *a* is interpreted
as the variance of the noise in the measurements, and this value was determined
by the simulation design. In the “noise free” case, a very small value of *a*
was used, typically in the order of 10^{-10} times the power of the
scalp field produced by the test source with lowest scalp field power.

Localization error was defined as the distance between the actual test source and the location of the maximum in the imaging method. The spatial spread was defined identically as in (3), which corresponds to a measure of spatial standard deviation of the imaging method centered at the actual test source, and not at the imaging method’s own maximum, since this would unjustifiably favor the method’s performance.

Figures 1a-1h summarize
localization error, spatial spread, and estimated activity values for the three
imaging methods (minimum norm, Dale, and *s*LORETA).

Note that the estimated
activity values at test source locations cannot be compared among the different
imaging methods, since these values are in different units for the different
imaging methods. However, this feature is very informative for comparing the
quality of the different methods. For example, from Fig 1a, the ratios of estimated
source activity (maximum to minimum) were 850, 103, and, 30, for minimum norm,
Dale, and *s*LORETA, respectively.
This means that with *s*LORETA,
some sources (especially deep ones) will be underestimated. However, *s*LORETA out-performs tremendously the minimum
norm and the Dale methods in this aspect.

In all noise free simulations,
only *s*LORETA has exact, zero error
localization. In all noisy simulations, *s*LORETA has by far the lowest localization
errors. In most cases, the spatial spread (i.e. “blurring”) of *s*LORETA is smaller than that of the Dale
method.

Figure 1: “a-h” summarize
in tabular form localization error, spatial spread, and estimated activity values
for the three imaging methods (minimum norm, Dale, and *s*LORETA). (a) EEG, 6430 voxels, 3 unknowns
per voxel, 25 electrodes, 6430 test sources with random orientation, no noise.
(b) Same as (a), but with additive random noise (noise scalp field standard
deviation equal to 0.12 times the test source with lowest scalp field standard
deviation). (c) MEG, 6430 voxels, 3 unknowns per voxel, 25 sensors, 6430 test
sources with random orientation, no noise. (d) Same as (c), but with additive
random noise (noise scalp field standard deviation equal to 7.21 times the test
source with lowest scalp field standard deviation). (e) EEG, 12980 voxels, 1
unknown per voxel, 101 electrodes, 100 randomly selected test sources, no noise.
(f) Same as (e), but with additive random noise (noise scalp field standard
deviation equal to 0.082 times the test source with lowest scalp field standard
deviation). (g) MEG, 12980 voxels, 1 unknown per voxel, 101 sensors, 100 randomly
selected test sources, no noise. (h) Same as (g), but with additive random noise
(noise scalp field standard deviation equal to 8.49 times the test source with
lowest scalp field standard deviation). MNE: minimum norm localization error
(mm); MNSSD: minimum norm spatial standard deviation (mm); MNMaxAbs: estimated
minimum norm activity value at test source location (arbitrary units); DaleE:
Dale localization error (mm); DaleSSD: Dale spatial standard deviation (mm);
DMaxAbs: estimated Dale activity value at test source location (arbitrary units);
LORE: *s*LORETA localization error
(mm); LORSSD: *s*LORETA spatial
standard deviation (mm); LORMaxAbs: estimated *s*LORETA activity value at test source location
(arbitrary units). Note that the estimated activity values at test source locations
cannot be compared among the different imaging methods (see text for explanation).

The main properties of
*s*LORETA, for both EEG and MEG,
based on estimates of activity given by Eq. 20 are:

1. Exact, zero error, localization for test dipoles located at voxel positions, in the absence of noisy measurements.

2. Exact, zero
error, localization of test dipoles *with
arbitrary orientation*, located at voxel positions, in the absence
of noisy measurements.

3. Exact, zero
error, localization of test dipoles *with
arbitrary orientation*, located at voxel positions, in the absence
of noisy measurements, even under regularization _{}.

4. Exact, zero
error, localization even for dipoles corresponding to a non-connected grid.
For example, *cortical and non-connected subcortical
grey matter can now be modeled as the solution space*. The error remains
zero.

The main properties of
*s*LORETA based on estimates of
activity given by Eq. 22 are:

1. Exact, zero error, localization of test dipoles located at voxel positions, in the absence of noisy measurements.

3. Exact, zero
error, localization of test dipoles located at voxel positions, in the absence
of noisy measurements, even under regularization _{}.

4. Exact, zero
error, localization even for dipoles corresponding to non-connected grids. For
example, *cortical and non-connected subcortical
grey matter can now be modeled as the solution space*. The error remains
zero.

5. These results
mean that the distribution of voxels can be quite arbitrary. For example, voxels
do not have to be uniformly distributed from the geometrical point of view,
although they should be uniformly distributed from the “*grey matter density*” point of view. Furthermore,
different types of voxels may exist, some with unknown current density vector,
and some with known current density orientation but unknown amplitude.

Suppose there exist reasons
to *believe* that the *actual* (*prior*)
current density variance is the diagonal, positive definite matrix **W**. This situation arises for example, in some
approaches that force fMRI hot spot locations onto the EEG/MEG inverse solution
(see for example (6)). In this case, Eq. 8 can be rewritten as:

_{}
Eq. 23

where the new
unknown variable _{} has
been “pre-standardized” to have the identity matrix as its variance. This transformed
variable plays the role of **J** in
all equations above, and _{} plays the role of the lead field in all
equations above. All else proceeds identically with these new formal substitutions.

Note that the final *s*LORETA image corresponds to standardized
estimates of activity (Eqs. 20 or 22) for the pre-standardized current density
_{}.

Note that this approach
can be applied to any *actual* (*prior*)
current density variance **W**, as long as it is positive definite, and
there exists a meaningful decomposition:

_{}
Eq. 24

for the square
root matrix _{}.
For example, this is the case of the classical LORETA method (2), where _{} embodies a discrete
spatial Laplacian operator that achieves smoothness between neighboring voxels.

The regularization parameter
*a* cannot be estimated from the functional
in Eq. 8. However, it can be estimated via the cross-validation functional.
This has been published in (14), in a reply to comments made to the paper in
(2), which includes the detailed derivation of the method, and a set of equations
that can be used efficiently in practice.

Although *s*LORETA calculations produce pseudo-statistics,
it is highly recommended to *not*
use these values as actual statistics in testing of hypotheses in experimental
designs.

Unlike the approach of
Dale et al. (6), which makes use of their statistics for hypothesis testing,
it is recommended to use *s*LORETA
pseudo-statistic values as *estimates of activity*,
and to apply standard techniques such as in statistical non-parametric mapping
(SnPM) (7) for the analysis of experimental designs.

Note that tests for absolute
activation with *s*LORETA can be
performed by using the modified pseudo-statistics:

_{}
Eq. 25

or:

_{}
Eq. 26

which correspond to Eqs. 20 and 22, respectively.

These pseudo-statistics
should be used in an experimental design where there are *N* independent *s*LORETA images. For example, in a visual event related potential
study with _{} subjects, consider the 10 *s*LORETA images at the P100 latency.

In Eq. 25, _{} denotes the unique
symmetric inverse square root matrix of _{}. The pseudo-statistic in Eq. 26 has the
form of a univariate Student’s *t*-statistic,
and the pseudo-statistic in Eq. 25 has the form of a Mahalanobis transform (10).

In the case of unknown amplitudes only, significant absolute activation is based on testing for zero mean of the pseudo-statistic in Eq. 26. SnPM can be used to correct for multiple comparisons and to bypass assumptions of Gaussianity.

In the case of unknown
current density vector, significant absolute activation is based on testing
for zero mean of the “*max-statistic*”
of the pseudo-statistic in Eq. 25. This corresponds to the maximum of the absolute
value among the three components. The “*max-statistic*”
reduces three numbers per voxel to a single number per voxel. This is then used
in SnPM to correct for multiple comparisons and to bypass assumptions of Gaussianity.

1. Localization error can not be improved beyond the present result. It is zero. Up to the present, no other instantaneous, distributed, discrete, imaging method for EEG/MEG has been published (to the best of the author’s knowledge) that achieved perfect localization. All other previously published methods at best produced systematic non-zero localization errors (see (2), (6), (15), (16), (17)).

2. If the aim
is localization, this new method, denoted as *s*LORETA,
at least has perfect first order localization.

3. A distributed imaging method capable of exact localization of point sources is of great interest, since the principles of linearity and superposition would guarantee its trustworthiness as a functional imaging method, given that brain activity occurs in the form of a finite number of distributed “hot spots”.

4. The detailed information provided here allows the reader to reproduce, check, test, and validate the previous claims.

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