Functional brain imaging: Dipole localization and laplacian methods

0042-6989/93 $6.00 + 0.00 Copyright 0 1993 Pcrgamon Press Ltd

Vision Res. Vol. 33, No. 17, pp. 2413-2419, 1993 Printed in Great Britain. All rights reserved

Functional Brain Imaging: Dipole Localization and Laplacian Methods R. SREBRO,* R. M. OGUZ,* K. HUGHLETT,*

P. D. PURDY?

Received 30 October 199.2; in revised form 25 January l!W3; in finat form 15 April 1993

The perfomance of two methods, used to localize brain activity from evoked potential fields measured on the scalp, was assessed In a tank model of tbe human bead. This pkysical model contained a human skull encased in a polymer simulating the resistivity and geometry of brain and scalp. The dipole locution method misla the positions of known dipole c#urces by several centimeters. The . axon was systematic. The d.ipoles were iocalized too deepiy in the kead. The Laphian method yielded a field resembling the brain smface fieId (epiu&ical potential field) provided that the iso-potential contours of the s&p field closed witbin the measurement range. Clipping @ted in a serious mislocalization of the position -of the peak of the epicortieal potential field. Visually evoked potential

Dipole localization

Laplacian analysis

most. The MRI method has a temporal resolution of about 2 set, much too slow to be useful for this purpose. Moreover, the temporal resolution is set not by technical features of the method but rather by the dynamics of brain blood flow. Rather than supplanting the older technologies, MRI studies underscore the need to make the older t~hnolo~es more useful than they have so far been. VEPs have played and continue to play a significant role in exploring issues ranging from the development of the visual system to the physiological basis of hyperacuity (Regan, 1989). In the last few years, multi-electrode recording techniques have added an important new dimension to VEP studies. Scalp topography is often used to infer info~ation about the location of brain regions activated by carefully constructed visual stimuli. The vision researcher is increasingly called upon to evaluate procedures used to localize human brain activity from recorded scalp fields. Thus the vision research community holds a proprietary interest in the further development of VEP techniques. The excellent temporal resolution of the VEP (< 1 msec) is unfortunately coupled to di~c~ties in using scalp fieids to localize brain activity. The problem is not so much one of spatial resolution, typically about 1 cm (Srebro, 1990), as it is the issue of how to make best use of the localizing information provided by a scalp field, Two methods are currently in use. The first, called the dipole localization method (DL), is a parameter estimation ~h~que which attempts to account for a scalp field by the activity of one or more dipole current *Departmentof Ophthalmology, University of Texas Southwestern sources in the brain (Scherg & Von Cramon, 1985,1986; Medical Center at Dallas, 5323 Harry Hines Blvd, Dallas, A&m, Richer & Saint-Hilaire, 1988; DeMunck, Van TX 75235-8592, U.S.A. Dijk & Spekreijse, 1988). The second, called the LaplaPepartment of Radiology, Neurology and Neurosurgery,University cian (or current source density) method, is a deblurring of Texas SouthwesternMedicalCenter at Dallas, 5323 Harry Hines Blvd, Dallas, TX 75235-8896, U.S.A. technique that attempts to deconvolve the low-pass

This year has witnessed a remarkable advance in the use of magnetic resonance imaging (MRI) to measure regional blood flow, a marker for brain activity in humans (Belliveau, Kennedy, McKinstry, Buchbinder, Weisskoff, Cohen, Vevea, Brady & Rosen, 1991; Ogawa, Tank, Menon, Ellermann, Kim, Merkle & Ugurbil, 1992). But this advance does not render obsolete older technologies, such as evoked potentials and evoked magnetic fields. In fact quite the contrary is true. The advances reveal the limitations of MRI and make it clear that the older technologies must fill a vital gap. Consider a relatively simple visual discrimination. (We choose one that we have been studying as an example.) A subject is presented six rectangles in hexagonal array centered at fixation as an 80 msec flash, The subject is required to respond by pressing a button whenever one of the rectangles differs in orientation from the remaining five. The reaction time for this pop-out discrimination (T&man, 1991) is about 400 msec. During this short epoch the brain has processed the sensory information, formed a cognitive model, made a decision about whether to respond or not, and executed a motor action. Reelecting the complexity of the brain processing, the visually evoked potential (VEP) has multiple temporal peaks and a singular value decomposition of the VEPs from 31 electrodes shows the existence of at least 4 topographically distinct components. The temporal resolution required to separate these components and to study their interaction is a few tens of milliseconds at

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R. SREBRO et al.

spatial filter effect of volume conduction in the head (Hjorth, 197.5; Srebro, 1985; Per&, Bertrand & Pernier, 1987; Gevins, 1989; Gevins, Brickett, Costales, Le & Reutter, 1990; Nunez, Pilgreen, Westdorp, Law & Nelson, 199 1). Dipole localization requires an explicit model of the passive electrical properties of the human head. Most often the spherical 3 shell model of Rush and Driscoll (1969) is used. The main virtue of this model is that it leads to a relatively simple analytical expression for the potential at a surface point due to one or more dipoles inside. Since the human head does not conform to this cartoon rendition, it is necessary to use a transformation that projects the electrode locations to the surface of the spherical model. Both this transformation and the fact that the 3 shell model is only a rough approximation to volume conduction in a real head can be expected to affect the parameter estimation. The Laplacian method does not require an explicit model of volume conduction in the head, but it does require a spline approximation of the scalp potential field. It is claimed that the Laplacian estimates the potential field that exists on the surface of the brain, in other words, the epicortical potential field. And it is presumed that the epicortical potential field is more useful than the scalp field for localizing brain activity. This paper addresses the performance of both the DL and Laplacian methods. To evaluate performance, we built a tank model of the human head in which a human skull is embedded in a polymer matrix mimicking both the resistivity and geometry of the human head. We inserted dipole current sources at measured locations inside the skull, measured both the “scalp” and “brain surface fields”, and localized the dipole positions from the measured scalp fields.

GOODS Tank model of the human head A human skull in good condition was encased in a polymer matrix with resistivity simulating scalp and brain. The polymer was made from a mixture of dry agar, polyvinyl chloride, sodium chloride, and sodium azide (Kato, Hiraoka & Ishida, 1986; Kato & Ishida, 1987). They were mixed with water and gently heated to produce a viscous fluid which was poured into the prepared skull (see below) and into a mold surrounding the skull so that the skull was completely filled and encased. When cooled, the polymer was firm, moist, and rubbery. The resistivity was measured using a 4 electrode chamber and the formula adjusted to yield a conductivity of about 1 S/m. The calvarium of the skull was removed and 26 gold disk electrodes, each 4mm dia, were attached to the inner surface. Also, four electrode pairs, for passing electrical current, were fixed 1.5 cm deep to the inner surface at several locations. Each pair consisted of 2 mm diameter silver chloride beads set 1 cm apart and lead off with insulated silver wires. We inserted each pair from

the outside through a small burr hole in the calvarium and sealed the burr hole with waterproof epoxy. Each pair simulated a radial dipole current source at vertex, mid-occipital, frontal and left lateral sites. The locations of all electrodes, including the dipole pairs, were measured using a three-dimensional digitizer with an accuracy of 1 mm. All measurements were converted to a skull based coordinate system using the tips of three wooden dowel rods cemented to the skull as reference points. The electrodes were lead out of the foramen magnum. A plaster of par-is cast was made of the inside of the skull. Contours of the cast were digitized with the three-dimensional digitizer. The resulting wire frame model of the inner surface (brain surface) was tiled with triangular elements. The calvarium was cemented to the base of the skull with waterproof epoxy. The skull was thoroughly hydrated by soaking in 0.9 N sodium chloride before it was encased in the polymer. An X-ray was taken after encasing it to ensure that the electrodes were not displaced. The polymer block was then sculpted as closely as possible to human dimensions based on MRI data. During the sculpting process, a very fine probe was used to measure polymer thickness. Several preliminary measurements were done to verify the model. (1) Electrical current was passed between two scalp points and the field measured over the entire scalp. The field spread on the scalp was found to be similar to that seen in humans (Burger & Van Milaan, 1943). (2) The properties of the dipole pairs were examined in a hemispherical bowl of saline. Fields were measured on the surface of the saline around the circumference of the bowl and compared to predictions for a dipole source based on bioelectric theory (Hosek, Sances, Jodet & Larson, 1978). Thus, although the current sources consisted of 2 mm beads separated by 1 cm, the potential fields produced by passing current between them were virtually identical in shape to those due to a theoretical dipole located at the midpoint between the two beads and having the same directional moment as the line between the two beads. (3) The stability of the electrode pair was assured by repeated measurements over several days. (4) Finally, the resistive properties of the polymer were checked by replacing the saline with polymer and repeating the measurements in the hemispherical bowl to assure that the polymer had the correct isotropic resistivity. Measurements were made in the tank model by passing electrical current from an isolated current source through a dipole pair as square waves at 10, 20 and 100 HZ. (The fields were identical.) Scalp potentials were measured at 152 discrete locations whose coordinates were obtained with the three-dimensional digitizer. (We will call these locations scalp electrodes.) The measurements were made from oscilloscope traces and using an RMS meter. The error of the measurement was approx. 1% of the maximum scalp field value. Current was continuously monitored. We carefully explored all scalp

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regions near seals in the skull for evidence of field distortion but found none at all. The center of the head coordinate system is somewhat arbitrary r&cting the positions of the wooden dowel rods. The x-axis “runs” laterally from left (approx. -7 cm) to right (8 cm), the y-axis runs anteriorposterior from front (-2 cm) to back (17 cm) and the z-axis runs superior-inferior from top (-4 cm) to neck (18 cm).

(1)

C(C)’

’

N

where V; = calculated field value at electrode i VF = measured field value at electrode i N = number of electrodes. Values of RE < 0.2 are indicative of excellent agreement between the 2 fields. Laplacian method

Dipole parameter estimation

The Laplacian of the scalp field was calculated using the method of Gevins (1989). This method provides an estimate of the Laplacian at every scalp measurement point. (We use the expression “Laplacian field” for this result and represent it by iso-Laplacian contours.) The Laplacian fields on the scalp were projected onto the brain surface representation and interpolated to the nodes of this surface with a high resolution cubic spline (Keys, 1981). Similarly the voltages measured at the 26 inside electrodes were used to construct the observed brain fields. This procedure allows the 2 fields to be compared on equal footing. It proved useful to compare the 2 fields by estimating how widely they spread on the brain surface. This estimate was made by first identifying the location of the node corresponding to the peak voltage. Then the half maximum voltage iso-potential contour was constructed. This contour intersected some of the triangles representing the brain surface. Vectors were constructed from these intersections to the location of the node corresponding to the peak voltage and field spread estimated as twice the average magnitude of these vectors.

The scalp electrode locations were fit to a sphere and projected to the surface of the sphere. The center of the sphere was at (0.1, 7.1, 5.5 cm) and the radius of the sphere was 9.5 cm which is slightly larger than the value 9.2cm used by Rush and Driscoll (1969). A gradient search technique was used to estimate the dipole location from its corresponding scalp field (Revington, 1969). The starting location for each dipole was taken as the actual location and the gradient probe increment was set initially at 1 cm. The search was allowed to continue as long as the error decreased. Then the probe increment was shortened and the search continued. This procedure was iterated until shortening the gradient prohe increment did not reveal any improvement in the fit. Additionally, we used several different starting locations and several different initial gradient probe increments, but the final dipole positions varied little. The scalp fields predicted from the best fitting dipole was compared to the measured scalp field using the relative error measure (RE) as described by Rudy and Messinger-Rapport (1988),

TABLE 1. Performance of the dipole localization method Actual location of dipole (cm) X Occipital Vertex Frontal Lateral

2.1 1.5 1.1 -1.4

CaIculated location of dipole (cm)

Y

Z

R

X

Y

Z

R

Difference (cm) (Euclidean)

12.8 7.5 1.8 7.6

-0.2 -1.7 0.5 -0.5

8.3 7.4 7.4 4.3

0.0 -0.3 -0.2 -5.2

9.7 7.2 3.0 5.8

0.2 1.7 1.4 2.2

5.9 3.9 5.8 6.2

3.8 3.9 2.0 5.0

Relative error 0.16 0.12 0.16 0.16

R, radial coordinate from center of best fit sphere.

TABLE 2. Performance of Laplacian method Actual location of

Calculated location of

peak @=4

peak (cm)

X Occipital Vertex Frontal Lateral

2.1 0.3 -0.9 -4.0

Y 11.3 8.0 0.9 8.3

Z -1.0 -2.1 0.3 -0.3

X 1.0 1.5 0.2 -2.4

Y

Z

Difference bn) (Euclidean)

11.1 8.3 5.0 7.7

-1.4 -2.0 -2.2 -1.4

1.2 I.2 4.9 1.9

Half peak amplitude width Actual (cm) Calculated (cm) 5.5 3.8 3.2 3.6

7.6 6.0 5.4 11.@

*This estimate is large because of excessive ripple in the Laplacian field. Ripple is inherent in the operation of a high-pass spatial filter.

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R. SREBRO et rd. Vertex

Occipital

Lateral

Frontal

Observed scalp

Lapkcian scalp

Laplacian brain

Observed Brain

FIGURE I. Fields in the tank model. Each field is shown as 15 iso-potential contours plotted in the three-dimensional reconstruction of the appropriate surface. Fields are shown for each of four radial dipole sources located at the level of the cortex in the occipital, vertex, lateral and frontal regions as marked for each column. Row 1 (top): the potential tields on the surface of the model (scalp fields) and row 2 (next down): the Laplacian of each scalp field. The dots represent discrete measurement sites. Row 3: the Laplacian field projected to the “brain” surface (epicortical potential field) and row 4 (bottom row): the observed epicortical potential field. The dots represent internal electrodes. The “brain surface” was constructed from a plaster of paris cast of the inner skull.

RESULTS Tables I and 2 and Fig. 1 show the main results of this study. The DL method resulted in serious misiocations of the dipoles ranging from 2 to 5cm. The Laplacian method produced estimates of the brain surface fields with relatively complex structure. Each Lapiacian field consists of a more or less prominent peak surrounded by several valleys. This field structure is anticipated from the application of a second-order spatial derivative operator and the valleys can be thought of as unavoidable ripple. In the regions surrounding the peak, each Lapiacian field resembles its corresponding observed brain surface field. However, for the lateral dipole the ripple is so prominent that we could easily have mistaken the peak location had we not known its sign in advance. The locations of the peaks in the Laplacian fields are

close to the observed peaks in the brain surface fields only for the occipital and vertex dipoles, Serious mislocations (2-5 cm) occur for the lateral and frontal dipole sources. Examination of the scalp potential fields for the lateral and frontal dipole sources reveals that the isopotential contours are clipped; i.e. many of them do not close within the measurement space. All the Laplacian fields spread more widely on the brain surface than the observed fields by 4040%.

DISCUSSION We examined the dipole localization method in a tank model of the human head containing implanted radial dipole sources. We find that the error of the dipole localization is substantial. The measured scalp fields are

FUN~IONAL

virtually free of noise and the sampling of the fieid at 152 points widely scattered over the scalp far exceeds the sampling used in human experiments. The actual location of the dipole is known to an accuracy of 1 mm and we took some pains to assure that the current passing electrodes did in fact behave as dipole sources. The locations of the dipoles were such as to simulate sources in the cortex. Our method of least squares minimization is virtually identical to that used by previous workers (e.g. Achim et al., 1988). It is essentially the ~~~~ewton method. We consider only the simplest case, namely 1 dipole source. These errors in dipole localization could reflect inaccuracies associated with either the approximation of the scalp as a sphere or the use of the 3 shell model to describe volume conduction. Figure 2 shows the actual locations of the scalp electrodes and their spherical projections. The two representations are in good overall agreement. The average error, that is the average distance between the actual electrode position on the scalp and its spherical projection is 0.28 cm. The largest error was 0.69 cm. Table 1 shows that for each dipole source the scalp field predicted by the best fitting dipole is a very good approximation to the observed scalp field, relative errors range from 0.12 to 0.16. The high quality of the predicted fields implies that the spherical approximation of the scalp was

TOP

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adequate. In addition the error of localization is least for the frontal dipole, not the vertex dipole where the spherical approximation is best. These findings suggest that the approximation of the scalp as a sphere was not an important source of the localization error. On the other hand, Table 1 shows that the radial coordinate of the estimated dipole location is smaller than the actual one by as much as 3.5 cm (for the vertex dipole). Thus, the estimated dipole locations lie deeper in the head than the actual ones. The deeper a dipole source lies in a 3 shell model, the wider its field spreads on the scalp. This suggests that the 3 shell model may underestimate the spread on the scalp so that a good fit to the observed scalp field can only be obtained by displacing the estimated dipole location deeper than its actual position, Figure 3 compares the observed scalp potential fields to those calculated from the (actual) measured dipole parameters using the 3 shell model. The half peak amplitude widths of the observed fields are consistently larger than those calculated from the 3 shell model. The observed half widths range from 5.6 cm (~ipital) to 6.9cm (lateral). The calculated half widths range from 3.9 cm (occipital) to 5.2 cm (lateral). Thus, the calculated scalp fields underestimate the spread by about 300/b. In a recent study Meijs, ten Voorde, Peters, Stok and Lopes

Back

Front

FIGURE 2. Scalp electrodes and their spherical projections. Solid circles, actual scaip electrodes. Open circles, spherical projections. Five views of the array of scalp electrodes are shown and labelled. In the top view the back of the head is at the bottom. In the back and front views the right side of the head is at the right. In the right and let? views (below) the front of the head is at the left.

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Occipital

Occipital

(OBS)

(CALC)

Vertex

Vertex

(OBS)

(CALC)

et al.

Frontal

Frontal

(OBS)

(CALC)

Lateral

Lateral

(OBS)

(CALC)

FIGURE 3. Observed and calculated scalp potential fields in a tank model of the human head. The tank model consists of a human skull encased in a polymer with resistivity and shape emulating human dimensions. Dipoles were implanted deep to the skull at known position and with known orientation. The observed (OBS) scalp fields are shown in the top row for four dipole locations as marked. The bottom row shows the scalp potential fields calculated from the 3 shell model (CALC). The view is looking directly down toward the vertex. Dots indicate points where the potentials were measured and calculated (electrodes). Scalp fields are represented by 10 equipotential contours. The relative errors of the calculated scalp fields are 0.42, 0.52, 0.45 and 0.44 for the occipital, vertex, frontal and lateral dipoles respectively.

da Silva (1988) compared the accuracy of dipole localization in several spherical models. A realistic head model based on MRI data and solved using the boundary element method (BEM) served as the standard. Dipole localizations were based on simulated magnetic fields. The localization error ranged from 2 to 8 mm for occipital sources when a 4 shell model was used. The error is likely to be substantially larger for localizations based on scalp potential fields rather than magnetic fields (Rose, Sucla-Soares & Sato, 1989; Stok, 1987). These studies still leave open the question of how well “realistic” head models represent either a real human head or the tank model used in our study. The BEM needed to solve the forward problem demands the assumption that all tissue resistivities are homogeneous and isotropic. This is not likely to be a good approximation to the properties of the skull either in a living human head or our tank model. The Laplacian method has its roots in Freeman’s “software lens” (1980). Its descendants include the “cortical imaging technique” (Kearfott, Sidman, Major & Hill, 1991) finite element methods (Gevins, 1989) and

direct calculation of the epicortical potential field (Srebro, Oguz, Hughlett & Purdy, 1993). We tested the assertion that the Laplacian of the scalp field estimates the epicortical potential field. The results show that provided the scalp field is not clipped (i.e. that most contour lines close within the range of measurement), the Laplacian field resembles the epicortical potential field. In this case, the peaks are within about 1.5 cm of the observed peaks but the field spread is about 4&80% greater than the observed field. However, substantial mislocalization of the peak of the epicortical potential field occurs when the scalp field is clipped. In human application, it is often not possible to measure potentials widely enough on the head to avoid clipping and it is all too easy to forget the severity of the penalty thus incurred. Perusal of Fig. 1 makes it clear that the Laplacian is not a satisfactory estimate of the epicortical potential field. And it is possible to estimate the epicortical potential field directly from the scalp field without using the Laplacian at all (Gevins et al., 1990; Srebro et al., 1993). In this paper we consider only the accuracy of the

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Laplacian and dipole localization methods and we find both wanting. However, there is another important point to consider. The problem of localizing brain activity from bio-electric fields is important. It will not just evaporate because of advances in MRI technology. It is likely that both brain blood flow and bio-electric field studies wiI1 be required to explore human brain function in a satisfactory way. Neither the Laplacian method nor the dipole localization method provides a side-by-side comparison to the tomographic results of MRI or PET. Recent studies suggest that tomographic inversion may be possible (Clarke & Janday, 1989; Ioannides, Bolton & Clarke, 1990; Wang, Williamson & Kaufman, 1992). In the meantime it would be prudent to view with some caution conclusions based on Laplacian or dipole localization methods. REFERENCES Achim, A., Richer, F. & Saint-Hilaire, J. M. (1988). Methods for separating temporally overlapping sources of neuroelectric data. Brain Topography, 1, 22-28. Belliveau, J. W., Kennedy, D. N., McKinstry, R. C., Buchbinder, B. R., Weisskoff, R. M., Cohen, M. S., Vevea, J. M., Brady, T. J. & Rosen, B. R. (1991). Functional mapping of the human visual cortex by magnetic resonance imaging. Science, 254, 716719. Bevington, P. R. (1969). Data reduction and error analysis for the physical sciences. New York: McGraw-Hill. Burger, H. C. & Van Milaan, J. B. (1943). Measurements of the specific resistance of the human body to direct current. Acta Medica Scandivanica, 114, 584607.

Clarke, C. J. S. & Janday, B. S. (1989). The solution of the biomagnetic inverse problem by maximum statistical entropy. Inverse Problems, 5, 483-500.

De Munck, J. C., Van Dijk, B. & Spekreijse, H. (1988). Mathematical dipoles are adequate to describe realistic generators of human brain activity. IEEE Transactions on Biomedical Engineering, 35,96&-9&i. Freeman, W. J. (1980). Use of spatial deconvolution to compensate for distortion of EEG by volume conduction. IEEE Transactions on Biomedical Engineering, 27, 421429.

Gevins, A. (1989). Dynamic functional topography of cognitive tasks, Brain Topography, 2, 37-56.

Gevins, A., Brickett, P., Costales, B., Le, J. & Reutter, B. (1990). Beyond topographic mapping: towards functional-anatomical imaging with 124channel EEGs and 3-D MRIs. Bruin Topography, 3, 53-64.

Hjorth, B. (1975). On-line transformation of the EEG scalp potentials into source derivations. Efectroencephalography and Chnical Neurophysiology, 39, 526530.

Hosek, R. S., Sances, A. Jr, Jodet, R. W. % Larson, S. J. (1978). The contributions of intracerebral currents to the EEG and evoked potentials. IEEE Transactions on Biomedical Engineering, 2.5, 405413.

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Kearfott, R. B., Sidman, R. D., Major, D. J. & Hill, C. D. (1991). Numerical tests of a method for simulating electrical potentials on the cortical surface. IEEE Transactions on Biomedical Engineering, 38, 294-299. Keys, R. G. (1981). Cubic convolution interpolation for digital image processing. IEEE Transactions on Acoustics, Speech and Signal Processing, 29, 1153-l 160.

Meijs, J. W. H., ten Voorde, 3. J., Peters, M. J., Stok, C. J. & Lopes da Silva, F. H. (1988). The influence of various head models on EEGs and MEGs. In Pfurtscheller, G. & Lopes da Silva, F. H. (Eds), Functional brain imaging. Lewiston, N.Y.: Hans Huber. Nunez, P., Pilgreen, K. L., Westdorp, A. F., Law, S. K. & Nelson, A. V. (1991). A visual study of surface potentials and Laplacians due to distributed neocortical sources: Computer simulations and evoked potentials. Brain Topography, 4, 151-168. Ogawa, S., Tank, S. W., Menon, R., Ellermann, J. M., Kim, S. G., Merkle, H. & Ugurbil, K. (1992). Intrinsic signal changes accompanying sensory stimulation: Functional brain mapping with magnetic resonance imaging. Proceedings of the National Academy of Sciences, U.S.A., 89, 5951-5955.

Perrin, F., Bertrand, 0. & Pemier, J. (1987). Scalp current density mapping: Value and estimation from potential data. IEEE Transactions on Biomedical Engineering, 34, 283-288.

Regan, D. (1989). Human brain eIectrophysiology. New York: Elsevier. Rose, F. D., Sucla-Soares, E. & Sato, S. (1989). Improved accuracy of MEG localization in the temporal region with inclusion of volume current efforts. Brain Topography, I, 1755181. Rudy, Y. & Messinger-Rapport, B. J. (1988). The inverse problems in electrocardiography: Solutions in terms of epicardial potentials. CRC Critical Reviews in Bioengineering,

16, 215-268.

Rush, S. & Driscoll, D. D. (1969). EEG electrode sensitivity-An application of reciprocity. IEEE Transactions on Biomedical Engineering, 16, 15-22. Scherg, M. & Von Cramon, D. (1985). Two bilateral sources of the late AEP as identified by a spatio-temporal dipole model. Electroencephalography and Clinical Neurophysioiogy, 62, 32-44.

Scherg, M. & Von Cramon, D. (1986). Evoked dipole source potentials of the human auditory cortex. Electroencephalography and Clinical Neurophysiology, 65, 344-360.

Srebro, R. (1985). Localization of visually evoked cortical activity in humans. Journal of Physiology, London, 360, 233-246. Srebro, R. (1990) Realistic modeling of VEP topography. Vision Research, 30, 1001-1009.

Srebro, R. Oguz, R. M., Hughlett, K. & Purdy, P. D. (1993). Estimating regional brain activity from evoked potential fields on the scalp. IEEE Transactions on Biomedical Engineering. In press. Stok, C. J. (1987). The influence of model parameters on EEG/MEG single dipole source estimation. IEEE Transactions on Biomedical Engineering, 34, 289-296.

Treisman, A. (1991). Search, similarity and integration of features between and within dimensions. Journal of Experimental Psychology: Human Perception and Performance,

17, 652476.

Wang, J. Z., Williamson, S. J. & Kaufman, L. (1992). Magnetic source images determined by a lead-field analysis: The unique minimumnorm least-squares estimation. IEEE Transactions on Biomedical Engineering, 39, 665675.

Ioannides, A. A., Bolton, J. P. R. & Clarke, C. J. S. (1990). Continuous probablistic solutions to the biomagnetic inverse problem. Inverse Problems, 6, 523-542. ato, H. & Ishida, T. (1987). Development

of an agar phantom adaptable for simulation of various tissues in the range 540 MHz.

Physics in Medicine and Biology, 32, 221-226.

Kate, H., Hiraoka, M. & Ishida, T. (1986). An agar phantom for hyperthermia. Medical Physics, 13, 396-398.

Acknowledgements-This

work was supported in part by an unrestricted research grant from Research to Prevent Blindness, Inc. New York, and by grant ROI-EY09041 from the National Eye Institute. The first author is a Research to Prevent Blindness Senior Scientific Investigator. Computing resources for this work were provided by the University of Texas System Center for High Performance Computing.