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MEG Forward Problem Formulation Using Equivalent Surface Current Densities Nicolás von Ellenrieder*, Student Member, IEEE, Carlos H. Muravchik, Senior Member, IEEE, and Arye Nehorai, Fellow, IEEE
Abstract—We present a formulation for the magnetoencephalography (MEG) forward problem with a layered head model. Traditionally the magnetic field is computed based on the electric potential on the interfaces between the layers. We propose to express the effect of the volumetric currents in terms of an equivalent surface current density on each interface, and obtain the magnetic field based on them. The boundary elements method is used to compute the equivalent current density and the magnetic field for a realistic head geometry. We present numerical results showing that the MEG forward problem is solved correctly with this formulation, and compare it with the performance of the traditional formulation. We conclude that the traditional formulation generally performs better, but still the new formulation is useful in certain situations. Index Terms—BEM, current density, forward problem, MEG.
I. INTRODUCTION
I
N this paper (see also [1]) we introduce a new formulation for the magnetoencephalography (MEG) forward problem with a layered head model. Traditionally the magnetic field is computed based on the electric potential on the interfaces between the layers [2]–[5]. In the proposed formulation the effect of the volumetric currents is assigned to an equivalent surface current density on the interfaces between the layers, for which an integral equation can be stated. The magnetic field is easily obtained from the equivalent surface current density. The equations are solved by the boundary elements method (BEM), with the integrals over the elements solved analytically. The proposed method is useful for fundamental understanding of MEG related problems and has mostly theoretical interest. Our numerical examples show that the discretized implementation of our formulation has a generally poorer performance than the traditional formulation based on electric potential, hence its practical use in solving the inverse problem
Manuscript received June 16, 2004; revised December 23, 2004. The work of N. von Ellenrieder was supported by Consejo Nacional de Investigaciones Científicas y Técnicas. The work of C. Muravchik was supported by Comisión de Investigaciones Científicas de la provincia de Buenos Aires. The work of A. Nehorai was supported by the National Science Foundation (NSF) under Grant CCR-0105334 and Grant CCR-0330342. Asterisk indicates corresponding author. *N. von Ellenrieder is with the Laboratorio de Electrónica Industrial, Control e Instrumentación, Departamento de Electrotecnia, Facultad de Ingeniería, Universidad Nacional de La Plata, Argentina (e-mail:
[email protected], ellenrie@ ing.unlp.edu.ar). C. H. Muravchik is with the Laboratorio de Electrónica Industrial, Control e Instrumentación, Departamento de Electrotecnia, Facultad de Ingeniería, Universidad Nacional de La Plata, Argentina. A. Nehorai is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA. Digital Object Identifier 10.1109/TBME.2005.847549
is limited. However it can be applied, for example, to compute the electric potential gradient on the surface of a realistically shaped head. This electric potential gradient is related to the equivalent current density, and could be directly measured and used as an alternative way to solve the EEG inverse problem, as suggested in [6]. Since the gradient is the first spatial derivative of the electric potential, its use on EEG problems has some points in common with the surface laplacian, i.e., the second spatial derivative of the potential, which has been proposed for use ECG and EEG problems [7], [8]. Related ideas for the lead field formulation have also been proposed in [9]. In Section II we derive the integral equations for an equivalent surface current density and explain how the MEG forward problem is solved based on it. In Section III we present some numerical examples and a comparison with the traditional formulation of the problem using the electric potential where we show that the performance of the new formulation is poorer than the performance of the traditional formulation. Some conclusions regarding the method are presented in the Fourth Section. II. FORWARD PROBLEM We model the head as an arbitrarily shaped body, with layers of different constant electric conduc(with ), and smooth interfaces betivities is defined on these surtween them. The normal vector faces as the outward pointing unit vector normal to the surface at point . The groups of active neurons that are the source of electric activity in the brain are characterized by a primary cur. rent density We adopt the quasi-static formulation of the Maxwell equations, valid for typical values of the electromagnetic properties of the tissues [3]. Under this approximation the curl of the elec, and the contric field is null at any point , i.e., tinuity of the electric charge imposes , is the volumetric current density. Then we can write where the following differential equations for the volumetric current density within each constant conductivity layer (1) A. Equivalent Surface Current Density In this section, we obtain an integral equation for an equivalent surface current density which describes the problem without resorting to the electric potential. The tangential components of the current density have a discontinuity at the interfaces between layers of different electric
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VON ELLENRIEDER et al.: MEG FORWARD PROBLEM FORMULATION USING EQUIVALENT SURFACE CURRENT DENSITIES
conductivity. This discontinuity is equivalent to a surface current density. We define this equivalent surface current density , where and are as the current densities at a point as seen from outside and inside of an interface, respectively. The following lemma, proved in Appendix I, gives the solution of the equivalent current density forward problem. Lemma 1: For the layered head model given above and a source or primary current density , the equivalent surface current density satisfies
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dipole and its orientation and intensity. This model is valid for some epileptic sources and responses to sensorial stimulus [11]. With the dipolar model we can solve (3) and (5) analytically (7) (8) More complex sources can be considered, although in some cases (3) and (5) may not be analytically solvable (e.g., general distributed sources). D. Numerical Implementation
(2) for , and . The term is equivalent to the volumetric current density generated by the primary current density in a three-dimensional (3-D) infinite media, denoted by (3) where density.
Equations (2) and (6) are solved for and , respectively with the BEM. The surfaces are tesselated and a simple expression is assumed for in each element, transforming the integral equations in linear systems. In this work we choose constant over triangular elements denoted by , with the nodes placed at the center of the triangles. Since the equivalent surface current density is a vectorial quantity tangential to the surface we can write for each triangular element
is any region including the primary current (9)
B. Magnetic Field An expression for the magnetic field generated by the volumetric currents of the th layer can be obtained from the Ampère–Maxwell equation [10]
where and are two orthogonal unit vectors tangent to the triangle surface so that , and and are the components of the equivalent surface current density in the directions of and , respectively. The integral equation (2) over the interfaces is rewritten as a sum of integrals over each triangle, see Appendix II. The resulting equations form a linear system with two unknowns ( and ) per triangle (10)
(4) where
is the magnetic permeability of free space and the term is the magnetic field generated by the primary current density in an infinite media (5) The last term of (4) is zero because for every layer . Then, we can add the contribution of each layer and obtain (6) for
any point in
and
.
C. Source Terms The expressions (3) and (5) allow us to compute the current density and magnetic field generated by the primary current in an infinite media. In this work we adopt for simplicity a dipolar , where is the position of the source model
where the matrix is a diagonal matrix related to the conducis a matrix related to the shape of tivity of the layers, and the head model. The matrices and are both square matrices , where is the total number of triangles of of size the tesselated surfaces. The vector has both tangential components of the equivalent surface current density over the triangles, and is a vector related to the source of electric activity. If the source parameters change only this vector must be recomputed. Using the traditional formulation, the forward problem is a Neuman boundary conditions problem, and its solution is unique up to a constant. When discretized using the BEM, this results in a singular linear system. This is not the case with our formulation, where the solution to the integral problem is unique, and the linear system (10) is solved without resorting to deflation techniques. Using the same surface discretization for (6) a linear system is obtained for the magnetic field (11) where is a vector with the magnetic field components of ina vector depending on the source parameters and terest, the matrix related to the geometry of the head model and the positions of magnetic field sensors, see Appendix II.
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E. Isolated Skull Approach The shielding effect of the low conductivity layer representing the skull causes numerical instabilities in the forward problem solution based on the electric potential. The same effect deteriorates the forward problem solution in the present formulation, so to obtain a better solution a procedure similar to the isolated skull approach (ISA) [12] should be applied, just as when solving the electric potential forward problem. In this case, the surface current density is written as the sum , where is the surof two terms face current density that would be present if the conductivity of the layer representing the skull was zero. Let the skull be the th layer, then for , since there is no current in the outer layers if the conductivity is zero. from (2) we get For
where if placing expression for the term
, and
Fig. 1. Current density and electric potential error on the innermost surface versus the normalized source distance to the center of the head. The figure shows the mean NRDM of the equivalent current density components solved with and without ISA, and the NRDM of the electric potential solved with and without ISA.
(12)
The error measure used is the normalized relative difference measure (NRDM) [15], defined as
. With (12), and rein (2), we obtain the following
(14)
(13)
where is the Kronecher delta, i.e., the last term is different . from zero only when The ISA consists then in the use of the solution of a reduced size problem as the source term of the full size problem. Note that in (13) only the vector associated with the source is different than in the original formulation. III. NUMERICAL EXAMPLES In this section we show some numerical results obtained with the new formulation, and compare them with the results obtained when using the traditional formulation based on the electric potential, solved with constant potential over the triangles for the same tesselation of the interfaces. The results shown in this section correspond to a three layered spherical head model. This geometry is chosen because its analytical solution can be computed [13] and used for comparisons. The three layers have radii 87, 92, and 100 mm, each surface is tesselated in 320 triangular elements, and the conductivity of the first and third layers is 80 times larger than the second’s. This relationship among conductivities is supposed to approximate average measured values [14]. The magnetic field is sensed at 180 points regularly spaced on the upper hemisphere of a spherical surface 30 mm over the head.
where is the analytically computed quantity, the same quantity computed numerically, and the sum is over all the nodes , i.e., the center of the triangles for the current density or electric potential, or the sensor positions for the magnetic field. For vector quantities we use the mean NRDM of the components. The NRDM measures the error in the “shape” of the solution, i.e., it is related to the position and orientation parameters of the source, and independent of the intensity parameter. An analysis of the error in the magnitude of the solution is not presented here but leads to similar conclusions. Each point shown in the figures corresponds to the mean error of 84 tangentially oriented dipolar sources of current density, distributed regularly on the upper hemisphere of a sphere at a given depth below the brain surface. For radially oriented dipoles the errors are similar and slightly higher, just as when computed using potentials. A. Equivalent Surface Current Density Figs. 1 and 2 show the error of the equivalent surface current density as a function of the source distance to the sphere’s center, normalized to the radius of the layer representing the brain. The figures show the mean NRDM of the current density components and the NRDM obtained for the electric potential with the same nodes. Fig. 1 corresponds to the error on the innermost surface and Fig. 2 to the error on the outer surface of the head. We can see that the error for the current density is higher than for the potential. The reason is that since the current density is proportional to the electric potential gradient, it has higher spatial frequencies. Hence, the approximation of the current density for a given surface discretization will have higher errors.
VON ELLENRIEDER et al.: MEG FORWARD PROBLEM FORMULATION USING EQUIVALENT SURFACE CURRENT DENSITIES
Fig. 2. Current density and electric potential error on the outer surface versus the normalized source distance to the center of the head. The figure shows the mean NRDM of the equivalent current density components solved with and without ISA, and the NRDM of the electric potential solved with and without ISA.
Fig. 3. Convergence of the current density and electric potential numerical solutions on the outer surface. The figure shows the mean NRDM of the equivalent current density components and the NRDM of the potential. In both cases solved using ISA. The error corresponds to sources located at a distance to the center equal to 0.9 times the inner sphere radius. The surface discretizations are of 80, 160, 320, 640, and 1280 triangles per surface.
In the figures we also show the performance of the method when using ISA. We can see in Fig. 1 that on the inner surface the use of ISA has little effect, but on the outer surface (Fig. 2) this approach is necessary to obtain a good solution of the equivalent surface current density forward problem. We also tested the convergence of the numerical solution of the forward problem using the equivalent current density. In Fig. 3 we show how the error decreases when the number of triangles per surface increases. The figure shows the mean error on the outer surface. It corresponds to sources near the brain surface, at a depth equal to the 10% of the inner sphere radius. The error of the electric potential numerical solution is also plotted in the figure. We can see that although the error in the current density is approximately four times higher than the error in the electric potential, the convergence rate is similar in both cases. Although not shown in this paper, studies of the convergence of the magnetic field solution showed similar results.
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Fig. 4. Magnetic field radial component error versus the normalized source distance to the center of the head. The figure shows the NRDM of the radial magnetic field computed based on the equivalent surface current density (with and without ISA) and of the radial magnetic field computed based on the electric potential (with and without ISA).
Fig. 5. Magnetic field tangential components error versus the normalized source distance to the center of the head. The figure shows the mean NRDM of the tangential components of the magnetic field computed based on the equivalent surface current density (with and without ISA) and of the tangential components of the magnetic field computed based on the electric potential (with and without ISA).
B. Magnetic Field In Figs. 4 and 5, we show the results obtained for the magnetic field solution. The NRDM of the radial component of the magnetic field is shown in Fig. 4, and Fig. 5 shows the mean NRDM of the tangential components of the magnetic field. We see that the magnetic field solved using the current density has a poorer performance than when solved based on the electric potential. This is specially true for sources near the brain surface, i.e., for cortical sources, as is the case in many neurological situations. We can see in Figs. 4 and 5 that the use of the ISA in the equivalent current density forward problem deteriorates the magnetic field solution. The effect is more noticeable for the tangential components of the magnetic field (Fig. 5), and for these components the effect is also seen, to a lesser degree, for the magnetic field computed based on the potential.
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Fig. 6. Magnetic field error versus normalized source distance to the center of a realistically shaped brain model. The figure shows the mean NRDM of the “tangential” components and the NRDM of the “radial” component of the magnetic field computed based on the equivalent surface current density.
Note that the error of the radial component of the magnetic field is more than one order of magnitude lower than for the tangential components. This is so because for a spherically symmetric head model the radial magnetic field depends only on the primary current and not on the volumetric currents [3]. The error in the radial component appears only because the tesselated head model is not exactly spherical. For the tangential components the error reflects also the approximation of the current density (or potential) as a piecewise constant function over the triangles. For a realistically shaped head model there are no true radial or tangential components, thus this effect is not so noticeable, as we can see in Fig. 6. The figure shows the NRDM of the “radial” and “tangential” components of the magnetic field for a realistically shaped constant conductivity brain model. The surface of the brain model was tesselated in over 2000 plane triangles and the magnetic field computed based on the current density. The errors were measured comparing the results to the magnetic field forward problem solution computed based on the electric potential for the same surface tesselated in over 8000 plane triangles. As expected, the difference between the error in the “radial” and “tangential” components is much smaller.
Since the primary and volumetric currents are confined to the bounded region in space that models the head, the far magnetic field should behave as a dipolar field [10]. We imposed this condition on the solution of the forward problem to see if it improved the solution of the field near the head model. For a spherical head model this was true in some cases, but for realistically shaped head models there was no improvement since one important assumption is that the head model should be centered at the origin, and the center is not well defined for an arbitrary shape. We also tested the performance of the method in solving the MEG inverse problem. We found that the error in the forward problem solution induces important errors in the source parameter estimation, even in the absence of noise. This is a drawback of the method, but it still can be used to gain a deeper understanding, or a conceptual alternative solution of the MEG forward problem. As we mentioned earlier, it is possible to exploit the relationship between the surface current density and the electric potential gradient on the interfaces. Indeed, the electric potential gradient can be directly measured by a special kind of sensors as proposed in [6] and then a new inverse problem can be established. Note that in (10) the electric conductivities of the layers appear only on the diagonal matrix and in (11) do not appear at all. This simple dependence with the conductivities makes our formulation appealing for their estimation. However, the estimation of conductivities from MEG measurements is not practical, because the head shape is approximately spherical, and for such a geometry the magnetic field is independent of the electric conductivity profile [3]. Thus, to use our formulation for estimating the conductivities, it would be necessary to measure the electric potential gradient, as mentioned in the previous paragraph. APPENDIX I PROOF OF LEMMA 1 Proof: We start with the vector form of the Green second identity [16] for two vectors fields and in a bounded region
IV. DISCUSSION We presented an alternative formulation for the MEG forward problem. The magnetic field can be computed with the presented method, but numerical examples show that the error is higher than when using the traditional formulation of the MEG forward problem. From the computational point of view, the proposed formulation is again outperformed by the traditional one, since a linear system of twice the size must be solved. However, in the proposed method the matrix associated with the linear system is not singular, thus there is no need to use deflation when solving the system as in the traditional formulation. We studied some modifications to improve the present method with limited success. One of the modifications consisted of adopting a linear variation of the current density over the triangles. In this case the tangential direction on the vertices is not well defined and the method failed to converge. Another modification consisted of correcting the behavior of the far magnetic field.
(15) where is the outward pointing unit normal vector of the surface , boundary of . Let and where is any constant vector and , with . This is the Green function for a point source in 3-D, or the distribution of potential for a point source in 3-D in an , where homogeneous medium, i.e., is the Laplace operator applied to the variable , and the Dirac delta. We can write then (16) (17) (18) (19)
VON ELLENRIEDER et al.: MEG FORWARD PROBLEM FORMULATION USING EQUIVALENT SURFACE CURRENT DENSITIES
and with (15) we get for the
th constant conductivity layer
on the inner surface of normal defined on
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has opposite direction to the unit , then we can rewrite (24) as
(20) We can work with each term of (20) using some vector identities and the divergence theorem [10]. For the first term we get
(25) and the current density at point If we call from the outside and inside of an interface, respectively, the boundary condition for the normal current density at an interface between different conductivity regions is given by . Then if we add (25) for all the layers, with in one of the interfaces, we get
(26)
(21)
is confined to the region , Since the primary current the term is equivalent to the volumetric current density generated by the primary current density in an infinite media, given by (27)
and for the second term
(22) at a point where (assuming smooth surfaces), and term on the right side of (20)
where is any region including the primary source. The boundary condition for the tangential electric field on . This leads to the an interface is following condition for the current density
for otherwise. For the
(28) which holds even if the conductivity of a layer is zero. We call , and combine (26) and (28) to get
(29) (23) Since is any vector, the previous equations are valid for each direction, and we can assemble the expression
for
and
. APPENDIX II LINEAR SYSTEMS
A. Equivalent Surface Current Density
(24) Note that the boundary of the region is formed by , with . The normal unit vector the interfaces
be tesselated into a set Let the interfaces . Let the equivalent surface of plane triangles current density be constant over each triangle , for and the unit vectors defining a local coordinate system over the th triangle. Let be the and , and center of the th triangle,
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let belong to the tesselation of surface . From (2) we can write for the first tangential component of the equivalent surface current density
Combining (30), (31), and (33) we can form a linear system with one equation for each current density component over each triangle (34)
(30) and for the second component
where • the vectors and , each with elements, are formed by the current density components and ; , is a diagonal matrix re• the matrix , of size lated to the electric conductivity of the layers. Its diagonal elements are given by
(31) Next, we approximate the integral over the interfaces between layers by a sum of integrals over the plane triangles resulting from the surfaces tesselation, and since the current density and the normal direction are constant on each triangle we can write
•
•
(32)
where is the number of the interface that includes the triangle related to the th row; and , all of size , are the matrices related to the shape of the head model, and their elements are given by
the vector , with elements, is related to the source parameters, and given by
B. Magnetic Field sensors, Let the magnetic field be measured with a set of and with the orientation given by the each located at point . The measured components of unit vector the magnetic field are , and from (6) we can write
Let
these integrals can be computed analytically, see [17] and [18]. Using some vector identities we get (35) can be computed analytiwhere cally [5]. With one equation for each magnetic field sensor we can assemble the following linear system: (36)
(33)
where is formed by the measurements and , of size , have elements matrices and , and the elements of vector .
, the are
VON ELLENRIEDER et al.: MEG FORWARD PROBLEM FORMULATION USING EQUIVALENT SURFACE CURRENT DENSITIES
ACKNOWLEDGMENT The authors wish to thank Dr. M. Wagner from Neuroscan Labs, for providing them with several tesselated human heads used to test some of the presented procedures. REFERENCES [1] N. von Ellenrieder, C. H. Muravchik, and A. Nehorai, “MEG forward problem solution avoiding the electric potential,” presented at the Proc. 13th Int. Conf. Biomagnetism (BIOMAG’02), Jena, Germany, Aug. 2002. [2] R. J. Ilmoniemi, M. S. Hämäläinen, and J. Knuutila, “The forward and inverse problems in the spherical model,” in Biomagnetism: Applications & Theory, H. Weimberg, G. Stroink, and T. Katila, Eds. New York: Pergamon, pp. 278–282. [3] M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa, “Magnetoencephalography—Theory, instrumentation, and applications to noninvasive studies of the working human brain,” Rev. Mod. Phys., vol. 65, no. 2, pp. 413–497, Apr. 1993. [4] J. Sarvas, “Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,” Phys. Med. Biol., vol. 32, pp. 11–22, 1987. [5] A. S. Ferguson, X. Zhang, and G. Stroink, “A complete linear discretization for calculating the magnetic field using the boundary element method,” IEEE Trans. Biomed. Eng., vol. 41, no. 5, pp. 455–460, May 1994. [6] N. von Ellenrieder, C. H. Muravchik, E. Spinelli, A. Nehorai, J. Roitman, W. Silva, and S. Kochen, “Performance of the electroencephalography inverse problem using electric potential gradient measurements,” in Proc. 25th IEEE Engineering in Medicine and Biology Conf., Sep. 2003, pp. 2814–2817. [7] B. He, “Theory and applications of body-surface Laplacian ECG mapping,” IEEE Eng. Med. Biol. Mag., vol. 17, no. 5, pp. 102–109, 1998. [8] D. B. Geselowitz and J. E. Ferrara, “Is accurate recording of the ECG surface laplatian feasible?,” IEEE Trans. Biomed. Eng., vol. 46, no. 4, pp. 377–381, Apr. 1999. [9] J. J. Riera and M. E. Fuentes, “Electric lead field for a piecewise homogeneous volume conductor model of the head,” IEEE Trans. Biomed. Eng., vol. 45, no. 6, pp. 746–753, Jun. 1998. [10] J. D. Jackson, Classical Electrodynamics, 2nd ed. New York: Wiley, 1975. [11] J. C. de Munck, B. W. van Dijk, and H. Spekreijse, “Mathematical dipoles are adequate to describe realistic generators of human brain activity,” IEEE Trans. Biomed. Eng., vol. 35, no. 11, pp. 960–966, Nov. 1988. [12] M. S. Hämäläinen and J. Sarvas, “Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data,” IEEE Trans. Biomed. Eng., vol. 36, no. 2, pp. 165–171, Feb. 1989. [13] J. C. de Munck and M. J. Peters, “A fast method to compute surface potentials in the multisphere model,” IEEE Trans. Biomed. Eng., vol. 40, no. 11, pp. 1166–1174, Nov. 1993. [14] L. A. Geddes and L. E. Baker, “The specific resistance of biological material, a compendium of data for the biomedical engineer and physiologist,” Med. Biol. Eng., no. 5, pp. 271–293, 1967. [15] J. W. H. Meijs, O. W. Weier, M. J. Peters, and A. van Oosterom, “On the numerical accuracy of the boundary element method,” IEEE Trans. Biomed. Eng., vol. 36, no. 10, pp. 1038–1049, Oct. 1989. [16] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [17] A. van Oosterom and J. Strackee, “The solid angle of a plane triangle,” IEEE Trans. Biomed. Eng., vol. BME–30, no. 2, pp. 125–126, Feb. 1983. [18] J. C. de Munck, “A linear discretization of the volume conductor boundary integral equation using analytically integrated elements,” IEEE Trans. Biomed. Eng., vol. 39, no. 9, pp. 986–990, Sep. 1992.
Nicolás von Ellenrieder (S’01) was born in Argentina, April 27, 1975. He graduated in 1998 as an Electronics Engineer from the National University of La Plata, La Plata, Argentina, where he is working towards the Ph.D. degree. He is an Assistant Lecturer at the Department of the Electrical Engineering of the National University of La Plata and a member of its Industrial Electronics, Control and Instrumentation Laboratory (LEICI). His research interests include statistical processing of biomedical signals.
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Carlos H. Muravchik (S’81–M’83–SM’99) was born in Argentina, June 11, 1951. He graduated as an Electronics Engineer from the National University of La Plata, La Plata, Argentina, in 1973. He received the M.Sc. in statistics (1983) and the M.Sc. (1980) and Ph.D. (1983) degrees in electrical engineering, from Stanford University, Stanford, CA. He is a Professor at the Department of the Electrical Engineering of the National University of La Plata and a member of its Industrial Electronics, Control and Instrumentation Laboratory (LEICI). He is also a member of the Comision de Investigaciones Cientificas de la Pcia. de Buenos Aires. He was a Visiting Professor at Yale University, New Haven, CT, in 1983 and 1994, and at the University of Illinois at Chicago in 1996, 1997, 1999, and 2003. His research interests are in the area of statistical signal and array processing with biomedical, control, and communications applications, and nonlinear control systems. Since 1999, Dr. Muravchik is a member of the Advisory Board of the journal Latin American Applied Research and is currently an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING.
Arye Nehorai (S’80–M’83–SM’90–F’94) received the 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. After graduation he worked as a Research Engineer for Systems Control Technology, Inc., in Palo Alto, CA. From 1985 to 1989, he was Assistant Professor and from 1989 to 1995 he was Associate Professor with the Department of Electrical Engineering at Yale University, New Haven, CT. In 1995, he joined the Department of Electrical Engineering and Computer Science at The University of Illinois at Chicago (UIC), as a Full Professor. From 2000 to 2001, he was Chair of the department’s Electrical and Computer Engineering (ECE) Division, which is now a new department. In 2001, he was named University Scholar of the University of Illinois. He holds a joint professorship with the ECE and Bioengineering Departments at UIC. His research interests are in signal processing, communications, and biomedicine. Dr. Nehorai is Vice President-Publications and Chair of the Publications Board of the IEEE Signal Processing Society. He is also a member of the Board of Governors and of the Executive Committee of this Society. He was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from January 2000 to December 2002, and is currently a Member of the Editorial Board of Signal Processing, the IEEE Signal Processing Magazine, and The Journal of the Franklin Institute. He is the founder and Guest Editor of the special columns on Leadership Reflections in the IEEE Signal Processing Magazine. He has previously been an Associate Editor of IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, IEEE SIGNAL PROCESSING LETTERS, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE JOURNAL OF OCEANIC ENGINEERING, and Circuits, Systems, and Signal Processing. He served as Chairman of the Connecticut IEEE Signal Processing Chapter from 1986 to 1995, and a Founding Member, Vice-Chair, and later Chair of the IEEE Signal Processing Society’s Technical Committee on Sensor Array and Multichannel (SAM) Processing from 1998 to 2002. He was the co-General Chair of the First and Second IEEE SAM Signal Processing Workshops held in 2000 and 2002. He was co-recipient, with P. Stoica, of the 1989 IEEE Signal Processing Society’s Senior Award for Best Paper, and coauthor of the 2003 Young Author Best Paper Award of this Society, with A. Dogandzic. He received the Faculty Research Award from the UIC College of Engineering in 1999 and was Adviser of the UIC Outstanding Ph.D. Thesis Award in 2001. He was elected Distinguished Lecturer of the IEEE Signal Processing Society for the term 2004 to 2005. He has been a Fellow of the Royal Statistical Society since 1996.
