IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008
1103
Array Response Kernels for EEG and MEG in
Multilayer Ellipsoidal Geometry
David Gutiérrez*, Member, IEEE, and Arye Nehorai, Fellow, IEEE
Abstract—We present forward modeling solutions in the form
of array response kernels for electroencephalography (EEG) and
magnetoencephalography (MEG), assuming that a multilayer ellipsoidal geometry approximates the anatomy of the head and a
dipole current models the source. The use of an ellipsoidal geometry is useful in cases for which incorporating the anisotropy of
the head is important but a better model cannot be defined. The
structure of our forward solutions facilitates the analysis of the
inverse problem by factoring the lead field into a product of the
current dipole source and a kernel containing the information corresponding to the head geometry and location of the source and
sensors. This factorization allows the inverse problem to be approached as an explicit function of just the location parameters,
which reduces the complexity of the estimation solution search.
Our forward solutions have the potential of facilitating the solution
of the inverse problem, as they provide algebraic representations
suitable for numerical implementation. The applicability of our
models is illustrated with numerical examples on real EEG/MEG
data of N20 responses. Our results show that the residual data after
modeling the N20 response using a dipole for the source and an ellipsoidal geometry for the head is in average lower than the residual
remaining when a spherical geometry is used for the same estimated dipole.
Index Terms—Dipole source signal, electroencephalography
(EEG), ellipsoidal head model, magnetoencephalography (MEG),
N20 response, sensor array processing.
I. INTRODUCTION
A
RRAY processing methods have been developed to solve
problems related to the localization of brain activity
sources using electroencephalography (EEG) and magnetoencephalography (MEG) arrays, and the solutions are useful in
neurosciences and clinical applications [1], [2]. Solutions to the
forward modeling problem in EEG/MEG consist of computing
the electric potentials over the scalp and the magnetic field
outside the head, respectively, given a current source within
the brain. The forward model is necessary for solving the
inverse problem (i.e., finding the current distributions using
EEG/MEG measurements). Since solving the inverse problem
often involves an iterative solution of the forward problem, it
Manuscript received February 19, 2007; revised June 11, 2007. Asterisk indicates corresponding author
*D. Gutiérrez is with Centro de Investigación y Estudios Avanzados
(CINVESTAV), Unidad Monterrey, Apodaca, NL 66600, México (e-mail:
[email protected]).
A. Nehorai is with the Department of Electrical and Computer Engineering,
Washington University, St. Louis, MO 63139 USA (e-mail: [email protected].
edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TBME.2007.906493
is important to have an efficient form for the analytical and numerical solutions of the forward problem in order to minimize
the computational burden [3].
The use of an ellipsoidal geometry to model the head is useful
in cases for which incorporating the anisotropy of the head is
important but a better model cannot be defined. This is the case
of fetal MEG studies [4], where the inaccessibility to the fetal
head as well as health issues do not permit the use of tomographic techniques to obtain more realistic head models. Furthermore, the ellipsoidal model is useful in MEG studies in
adults, as it decouples not only the source location from the
dipole moment, but also the source location from the sensor location, allowing for further simplification in the computation.
Recently, second-order approximations for the electric potentials and magnetic fields in multilayer ellipsoidal geometry have
been developed [5], [6]. However, the mathematical expressions
for those approximations are not suitable for direct use in the inverse neuroelectromagnetic problem.
In this paper (see also [7] and [8]), we present forward modeling solutions in the form of array response kernels for EEG/
MEG, assuming that a multilayer ellipsoidal geometry approximates the anatomy of the head and a dipole current models
the source. The structure of our solution facilitates the analysis
of the inverse problem by decoupling the dipole source signal
(linear parameter) from the source location (nonlinear parameter). We factor the lead field into a product of the current dipole
source and kernel. This factorization allows the inverse problem
to be approached as an explicit function of just the location parameters, which reduces the complexity of the estimation solution search [9].
In Section II, we introduce the original formulations of
the forward solutions for both EEG and MEG. In Section III,
we present the algebraic steps necessary to manipulate the
EEG/MEG forward solutions and take them to their factored
forms, while in Section IV, we extend those solutions to an
array response representation for the case in which measurements are obtained from an array of detectors. In Section V,
experiments with real data are used to demonstrate the applicability of our methods to the solution of practical EEG/MEG
forward and inverse problems. Finally, our results and future
work are discussed in Section VI.
II. FORWARD MODELING SOLUTIONS
In this section, we present the derivation of the solution to
the forward problem of computing the magnetic field outside an
ellipsoidal conductor and the electric potential over the surface
due to a current dipole source.
0018-9294/$25.00 © 2008 IEEE
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008
A. Biot-Savart-Maxwell Solution
We model both the neuroelectric and neuromagnetic phenomena using the quasi-static approximation of Maxwell’s
equations given that the time-derivatives of the associated
electric and magnetic fields are sufficiently small to be ignored
[10]. Under this condition, the static electromagnetic field
equations can be written as
(1)
(2)
(3)
(4)
is the magnetic field,
is the electric field,
is the observation point,
is the magnetic permeability (assumed to be the same inside and outside
the brain), and is the current density. Since is irrotational,
it can be represented in terms of the electric potential as
where
Furthermore, assume that the source is modeled by an equivalent current dipole (ECD); i.e.,
(10)
is the dipole moment, and
is the source location. The ECD model is
a common simplification whose use is justified in cases where
the source dimensions are relatively small compared with the
distances from the source to measurement sensors [12], as is
often true for evoked response and event-related experiments.
Hence, substituting the ECD model in (9) and (8), and through
simple vector identities, we can rewrite the volume integrals as
a sum of surface integrals. For the case of the magnetic field,
we have [13]
where
(11)
(5)
The current density can be divided into passive and primary
result from the macrocomponents. The passive currents
scopic electric field in the conducting medium and are described
by the following expression:
and
are the conductivities on the inner and outer
where
sides of , respectively, and is the outward unit vector normal
to the surface at a point . Meanwhile, the electric potential
, is given by
on the boundary of , where
[14]
(6)
is the electric conductivity at . The primary currents
where
can be considered as the sum of the impressed neural current
and the microscopic passive cellular currents and are given by
(12)
(7)
Under these conditions, the equation that relates and
the integral form of the Biot–Savart–Maxwell law [11]:
where is the source point and
to the volume. Similarly, and
is
(8)
indicates the space interior
are related by
(9)
In the typical head model, we assume that the head may be
represented by different regions (typically four for the brain,
cerebrospinal fluid, skull, and scalp, or three when the cerebrospinal fluid is not considered). Another common assumpis constant and isotropic within
tion is that the conductivity
these regions. Therefore, the gradient of the conductivity is zero
except at the surfaces between regions, which allows the volume
integrals to be reworked into surface integrals. Under these conditions, we can assume the regions of our head model are
, going from the inner
bounded by surfaces , for
to the outer region, each with conductivity .
B. Forward Solutions for an Ellipsoidal Volume
Assume that our regions are bounded by concentric ellipsoids
, each defined by the following equation:
(13)
are the semi-axes of the th ellipsoid. Supwhere
pose, without loss of generality, that
Then, (13) defines an ellipsoidal system [15] with coordinates
such that
,
,
and
, where “ ” indicates
that any of the ellipsoids can be considered, as all
are
confocal. The equations connecting the ellipsoidal coordinates
to the Cartesian coordinates
are given in the
Appendix A.
GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY
1) Multilayer MEG Forward Solution: A multilayer model
is assumed so that different layers composing the head are included. Under this condition, using (13) in the evaluation of
(11), the solution for the magnetic field in Cartesian coordinates
becomes [5]
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are the conductivity constants of order and deIn (15),
gree . These conductivity constants reflect the effects of the
different paths that currents could possibly take within the various conductivity layers, as well as the effect of the geometry in
the volumetric currents. The conductivity constants in (15) for
layers can be computed by
(14)
refer to the components of in the ellipwhere
and
are the second-degree
soidal coordinate system;
exterior solid ellipsoidal harmonics of orders 1 and 2, respecis the second-degree elliptical integral of order
tively;
;
is the 3-D vector with “1” in the th position and
zero elsewhere;
and
are the roots of the quadratic equawith
;
denotes
tion
ellipsoidal terms of degrees greater than or equal to three; and
is the dipole moment modified by the spatial
and conductivity effects of the anisotropy imposed by the multilayer ellipsoidal geometry.
layers, is derived in [5] and it is given
For the case of
by
(16)
(17)
a
(18)
(19)
where , , , and
to outer layer), and
are the layer conductivities (from inner
(20)
(15)
where
is the interior Lamé function of order and degree
, and is a nuisance variable. Note that the elliptical integrals ,
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008
exterior solid ellipsoidal harmonics , and interior Lamé functions
used in (14)–(20) are defined in the Appendix B.
layers
In a similar way, we can write for the case of
by
and
(24)
2) Multilayer EEG Forward Solution: We evaluate (12) for
to obtain the electric potential measured
the case when
at the outer layer. The forward solution under those conditions
has been derived in [6] and it is given by (25), which is shown
and
are the
at the bottom of the page. In (25),
second-degree interior solid ellipsoidal harmonics (defined in
the Appendix B) of orders 1 and 2, respectively. For the case of
, constants
are given by (16), while for the case of
, constants
are defined in (22).
III. RESPONSE KERNELS
Clearly, (14) and (25) are not suitable for a numerical solution
of the inverse problem in EEG/MEG. Therefore, in this section
we develop novel reformulations to the forward solutions based
on algebraic factorizations. Our goal is then to represent the
magnetic field as
(26)
(21)
is the 3
where
electric potential as
3 kernel matrix for MEG; and the
and its corresponding conductivity constants will be given by
(27)
where
is the 3
1 kernel vector for EEG.
A. MEG Kernel Matrix
In order to reach the form of (26), we first need to factor (15)
and (21) in a matrix form. We start by defining the following
auxiliary matrices:
(22)
(28)
(23)
(29)
(25)
GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY
and
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then
(39)
(30)
Using (28)–(30), we can factor
as
This result has the advantage of decoupling not only and ,
but also and , while the effects of the geometry and conductivities of the layers are contained in matrix .
B. EEG Kernel Vector
(31)
where (16)–(19) are used in the calculation of (29) for the case
defined by (15), and (22)–(24) are used for the case of
of
defined by (21). Equation (31) can be further simplified
as
by defining matrix
In a similar way as in Section III-A, we define the auxiliary
, and the auxiliary matrices
, and
as
vector
(40)–(42), shown at the bottom of the page, where (16) is used to
for the case of
, and (22) is used to compute
compute
for the case of
, respectively.
Therefore, using (40)–(42)– in (25), and discarding the higher
is expressed as
order terms,
(32)
Then, (31) reduces to
(33)
,
Similarly, let us define the support matrices
as
and
,
(43)
where
. From (43) we note that decoupling
and in the EEG forward solution is not possible.
Finally, a summary of the solution kernels developed in this
section is presented in Table I.
IV. ARRAY RESPONSE MATRIX
(34)
(35)
In this section, we consider the case in which EEG/MEG measurements are obtained from an array of detectors located at
. Then, we can extend (26) and (27) to an array
representation that contains all kernel solutions.
A. MEG Array Response
and
and
. Then, we can
represent the forward solution of all the array as
Let
(36)
Using (33)–(36) in (14), and discarding the higher order
as
terms, we can express
(37)
(44)
where is the
be written as
3 array response matrix. Note that
with
can
(45)
We can further simplify (37) by defining
as
(38)
This last representation is especially useful for the computer
implementation of the forward model.
(40)
(41)
(42)
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008
TABLE I
SUMMARY OF THE SOLUTION KERNELS FOR THE MULTILAYER ELLIPSOIDAL HEAD MODEL
In practical situations only some of the components of the full
magnetic field are available. Note that each component of can
be expressed as
(46)
(47)
(48)
where denotes the Kronecker product and
is an
identity matrix. For any of these components, we can express
the array response matrix as
(49)
Finally, we can easily extend (44) to a spatio-temporal model
.
if we allow to change in time; i.e.,
B. EEG Array Response
In
a
similar
the
array
as
in
response
Section
IV-A,
and
let
de-
as
. Then, we
can extend (27) to an array representation that contains all
kernel solutions; i.e.,
fine
3
way
matrix
(50)
while the spatio-temporal model is given by
.
V. NUMERICAL EXAMPLES
We conducted a series of experiments using real EEG/MEG
data. The goal of our experiments is to show the applicability
of our methods by determining the goodness-of-fit in modeling
the data using the proposed multilayer ellipsoidal geometry. We
will use a current dipole to model the source. Then, we will
compare the unexplained (residual) data obtained from using
the ellipsoidal geometry against the residuals produced when a
classical multilayer spherical geometry is used.
A. Measurements and Models
The measurements used in our experiments correspond to real
EEG/MEG data of the N20 response from six healthy subjects.
The EEG and MEG data were recorded simultaneously over the
contralateral somatosensory cortex using a bimodal array with
32 EEG and 31 MEG channels (Philips, Hamburg, Germany).
Constant current of 0.2 ms square-wave pulses were delivered
to the right or left wrist at a stimulation rate of 4 Hz. The data
were sampled at 5000 Hz with a 1500 Hz antialiasing lowpass
filter, resulting in 250 time samples for each subject.
Even though N20 response is more extended in one direction
(from superior-posterior of the wall of the central sulcus to inferior-anterior) and then is a good example where line-source
models can be applied [16], [17], we decided to use a single
dipole to model the source in order to compare the head models,
which is the main interest in this paper.
For the head, we used two different models: The
first model corresponds to a three-layer ellipsoid where
of the outer layer were
the semi-axes
chosen to fit the scalp of each subject as closely as possible. The dimensions of the inner layers were chosen
to be
and
. The second
head model corresponds to a three-layer sphere with radii
, where
is the radius of
the outer sphere that provides the best fit to each subject’s
scalp. Finally, for the layer conductivities, we used the values
of
in both head
geometries.
B. Experiments and Results
We applied the spatial filtering method described in [18] to estimate the source location from the measured EEG/MEG data.
The search of the source was constrained to the region over the
central sulcus, where the N20 generator is know to be located
[19]. For the ellipsoidal model we used the forward models proposed in Sections IV-A and IV-B, while the forward solutions
for EEG/MEG in [3] were used for the spherical model.
Then, we computed the residual data as
(51)
GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY
Fig. 1. Interpolated iso-contour maps of the original, fitted, and residual potentials over the scalp for one subject at t = 20 ms. The black dots indicate the position of the electrodes. Maps (a)–(c) represent the original potential, the fitted
potential using a spherical model, and the fitted potential using an ellipsoidal
model, respectively. Maps (d) and (e) represent the residual data for spherical
and ellipsoidal model, respectively.
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Fig. 3. Standard deviation of the residual data over all channels as a function
of time for the EEG data of one subject. The dashed–dotted line indicates the
average value.
Fig. 4. Standard deviation of the residual data over all channels as a function
of time for the MEG data of one subject. The dashed–dotted line indicates the
average value.
Fig. 2. Interpolated iso-contour maps of the original, fitted, and residual tangential components of the magnetic field outside the head for one subject at
t = 20 ms. Maps (a)–(c) represent the original field, the fitted field using a
spherical model, and the fitted field using an ellipsoidal model, respectively.
Maps (d) and (e) represent the residual data for spherical and ellipsoidal model,
respectively.
where is the vector containing the measured data in all channels at a single time, and is the fitted data using either the ellipsoidal or spherical head model. Note that and are used instead of
and
, respectively, for notational convenience.
Examples of and for one subject at the activation of the N20
response are shown in Figs. 1 and 2 for the EEG and MEG data,
respectively. The quantity (51) is essentially the data which is
orthogonal to the space spanned by the electric potential or magnetic field induced by a dipole.
Once we have computed the residual, we calculated the standard deviation of this residual over all channels and plotted it as
a function of time. An example of such plot for one subject is
shown in Figs. 3 and 4 for the EEG and MEG data, respectively.
In those figures we can note that the standard deviation is, in average over time, lower when we use an ellipsoidal head model
than the case when a spherical model is used, and this is true for
both the EEG and MEG cases. We repeated this experiment for
all six subjects and the average value of the standard deviation,
, are shown in Tables II and
as well as the value at
III for the EEG and MEG data, respectively. Again, our results
show that a better fit (indicated by the lower average standard
deviation) is achieved in all cases when an ellipsoidal model is
used, with an average improvement of up to 45.63% less standard deviation in the residual data (Table II, subject 2).
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008
TABLE II
COMPARISON OF GOODNESS-OF-FIT OF THE EEG DATA
USING SPHERICAL AND ELLIPSOIDAL HEAD MODELS
APPENDIX A
ELLIPSOID COORDINATE SYSTEM
The ellipsoidal coordinates
the Cartesian coordinates
relationships:
are connected to
through the following
(A.1)
(A.2)
(A.3)
TABLE III
COMPARISON OF GOODNESS-OF-FIT OF THE MEG DATA
USING SPHERICAL AND ELLIPSOIDAL HEAD MODELS
where the subscript of the semi-axes that indicates the layer has
been dropped for notational convenience as all ellipsoids are
assumed confocal.
Most of the time we want to go from the Cartesian to the Ellipsoidal coordinates. Therefore, we need to solve the nonlinear
system formed by (A.1)–(A.3) constrained to the range in values
of , , and as described in Section II-B.
APPENDIX B
ELLIPSOIDAL HARMONICS
VI. CONCLUSION
We presented a solution to the EEG/MEG forward problem
for a multilayer ellipsoidal head model in the form of an array
response kernel. This matrix structure has the potential of facilitating the solution of the inverse problem as it provides an algebraic representation suitable for numerical implementations.
The simplification is greater in the case of MEG where all the
signal and location parameters are decoupled from each other.
A series of numerical examples with real EEG/MEG data
showed the applicability of our forward solutions in the inverse
problem of estimating the source location, and in the direct
problem of modeling the data. Our experiments showed that a
better fit of the original data may be achieved by using the proposed multilayer ellipsoidal model instead of the classical spherical model. Still, many other factors should be considered in the
modeling problem, like the model used for the source and the
accuracy of the source localization. However, under similar conditions, the anisotropy introduced by the ellipsoidal model provides advantageous conditions for more accurate data modeling.
In future work we will evaluate the accuracy of higher order
approximations to the full ellipsoidal solutions, such as those
described in [20], in order to determine the optimal expansion
size required for the EEG/MEG inverse problem without compromising the overall accuracy of the computation. Future work
will also focus on solving this issue by approximations similar
to those described in [21] for the spherical head model, as well
as the application of our models to the estimation of conductivities in ellipsoidal geometries.
In order to evaluate (14) and subsequent equations, we first
need to compute the values of the elliptic integrals , the interior
solid ellipsoidal harmonics , and the exterior solid ellipsoidal
harmonics of different orders and degrees.
The elliptic integral of order and degree is given by
(B.1)
is the interior Lamé function of order and degree
where
, and is a nuisance variable. In our computations, we are
only for degrees less or equal than 2.
required to evaluate
Under this condition, the interior Lamé functions [15] are given
by
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
Furthermore, the interior solid ellipsoidal harmonics are defined
as
(B.7)
Then, we can express the exterior solid ellipsoidal harmonics as
(B.8)
GUTIÉRREZ AND NEHORAI: ARRAY RESPONSE KERNELS FOR EEG AND MEG IN MULTILAYER ELLIPSOIDAL GEOMETRY
ACKNOWLEDGMENT
The authors are thankful to Prof. J. Haueisen of the Neurological University Hospital at Jena, Germany, for providing the
real EEG/MEG data used in the paper.
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[21] P. Berg and M. Scherg, “A fast method for forward computation of multiple-shell spherical head models,” Electroencephalography and Clinical Neurophysiology, vol. 90, no. 1, pp. 58–64, 1994.
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David Gutiérrez (M’05) received the B.Sc. degree
(hons) in electrical engineering from the National
Autonomous University of Mexico (UNAM),
Mexico, in 1997, the M.Sc. degree in electrical engineering and computer sciences, and the Ph.D. degree
in bioengineering from the University of Illinois at
Chicago (UIC), in 2000 and 2005, respectively.
From March 2005 to May 2006, he held a postdoctoral fellowship at the Department of Computer
Systems Engineering and Automation, Institute
of Research in Applied Mathematics and Systems
(IIMAS), UNAM. In June 2006, he joined the Center of Research and Advanced
Studies (CINVESTAV) at Monterrey, Mexico. There, he works as Researcher
and Academic in the area of medical sciences. His research interests are in
statistical signal processing and its applications to biomedicine. He is also
interested in image processing, neurosciences, and real-time algorithms.
Dr. Gutiérrez is a former Fulbright Scholar and he has been a fellow of the National Council for Science and Technology (CONACYT), Mexico, since 1998.
Arye Nehorai (S’80–M’83–SM’90–F’94) received
the B.Sc. and M.Sc. degrees in electrical engineering
from the Technion, Haifa, Israel, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA.
From 1985 to 1995, he was a faculty member
with the Department of Electrical Engineering, Yale
University, Hartford, CT. In 1995, he joined as Full
Professor the Department of Electrical Engineering
and Computer Science at The University of Illinois at
Chicago (UIC). From 2000 to 2001, he was Chair of
the department’s Electrical and Computer Engineering (ECE) Division, which
then became a new department. In 2001 he was named University Scholar of
the University of Illinois. In 2006, he became Chairman of the Department
of Electrical and Systems Engineering at Washington University in St. Louis.
He is the inaugural holder of the Eugene and Martha Lohman Professorship
and the Director of the Center for Sensor Signal and Information Processing
(CSSIP) at WUSTL since 2006.
Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL
PROCESSING during the years 2000 to 2002. In the years 2003 to 2005 he was
Vice President (Publications) of the IEEE Signal Processing Society, Chair
of the Publications Board, member of the Board of Governors, and member
of the Executive Committee of this Society. From 2003 to 2006 he was the
founding editor of the special columns on Leadership Reflections in the IEEE
Signal Processing Magazine. He was co-recipient of the IEEE SPS 1989 Senior
Award for Best Paper with P. Stoica, coauthor of the 2003 Young Author Best
Paper Award and co-recipient of the 2004 Magazine Paper Award with A.
Dogandzic. He was elected Distinguished Lecturer of the IEEE SPS for the
term 2004 to 2005 and received the 2006 IEEE SPS Technical Achievement
Award. He is the Principal Investigator of the new multidisciplinary university
research initiative (MURI) project entitled Adaptive Waveform Diversity for
Full Spectral Dominance. He has been a Fellow of the Royal Statistical Society
since 1996.