EEG spatial placement and enhancement method

An improved brain wave electroencephalograph (EEG) method measures the physical positions of EEG electrodes applied to the subject's scalp. A set of standard cross-directional lines are obtained to provide measurements of the subject's head and a computer system automatically defines the subject's head shape according to predetermined head shape classes. The position of the electrode on the subject's scalp is then determined by scaling the measurements of the subject's head to a head model of the same head shape class.

BACKGROUND OF THE INVENTION 
1. Field of Invention 
The invention relates to brain wave medical systems and methods and more 
particularly to an electroencephalograph (EEG) system and method having 
improved spatial resolution. 
2. Related Art 
It is presently known that the brain waves of a human subject, at the 
microvolt level, may be amplified and analyzed using electroencephalograph 
(EEG) equipment. Generally 19 electrodes are electrically connected to the 
scalp of the subject and the brain waves are amplified and displayed in 
analog form. The U.S. Pat. No. 4,736,751, incorporated by reference, 
describes a system using a larger number of electrodes and various digital 
computer based methods to obtain more information from the brain wave 
signals. 
However, when many electrodes are used, for example, 256 electrodes, it is 
time consuming to obtain their location on the subject's scalp using the 
conventional technique of tape measurements. If the subject is a child, 
senile or infirm, the subject may not be able to hold still for the 15-30 
minutes which may be required for such measurements. The physical location 
of the electrodes is required to compare the EEG recordings, at each 
electrode, with the information obtainable concerning the head of the 
subject from other medical image systems, such as MRI (Magnetic Resonance 
Imaging). 
The information produced from the EEG recordings may be made more clear and 
meaningful when the brain wave data is enhanced and analyzed by various 
methods executed in a digital computer system. However, the presently 
available methods do not provide sufficient spatial resolution for some 
purposes. 
SUMMARY 
A functional-anatomical brain scanner with a temporal resolution of less 
than a hundred milliseconds measures the neural substrate of higher 
cognitive functions, as well as diagnosing seizure disorders. 
Electrophysiological techniques, such as electroencephalograms (EEGs), 
have the requisite temporal resolution but their potential spatial 
resolution has not yet been realized. Progress in increasing the spatial 
detail of scalp-recorded EEGs and in registering their functional 
information with anatomical models of a patient's brain has been inhibited 
by the lack of a convenient means of rapidly placing many electrodes on 
the scalp, determining their location, determining the local thickness and 
conductivity of the scalp and skull, incorporating this information into a 
mathematical model, and using the mathematical model to enhance the 
spatial detail of the EEG signals. The three-dimensional positions of each 
electrode and the shape of the head are rapidly determined, the local 
thickness and conductivity of the skull and scalp are determined, and this 
information is used to reduce blur distortion of scalp-recorded EEGs, in 
effect mathematically placing the electrode just above the surface of the 
brain. 
In accordance with one aspect of the present invention, a method is 
presented to improve the spatial resolution of electroencephalograph (EEG) 
medical images. The physical location of each of the electrodes in a large 
set of electrodes placed on the head of a subject is measured. A flexible 
hat is placed on the subject's head and piezoelectric ribbons, embedded in 
the hat are stretched. The ribbons are directed in cross-lines and the 
extent the ribbons are stretched (tensioned) determines the size of the 
subject's head in the direction along which the ribbon lies. The head 
measurements are entered into a pattern recognition computer system, 
preferably a neural network computer, and the subject's head shape is 
classified as falling into one class of head shapes. The electrode 
locations are mathematically derived in a computer system based on the 
known locations of the corresponding electrodes on a head model in the 
same head shape class and the size of the subject's head. 
In another aspect of the present invention, the EEG spatial resolution is 
improved using information about the geometry and conductivity of the 
skull to remove distortion due to transmission from the brain to the 
scalp. This requires 1) Measuring the size and shape of the subject's 
head; 2) Measuring the local skull thickness using an ultrasonic source 
and analyzing the reflected waves from the skull; 3) Estimating the local 
conductivity of the skull; and 4) Using these measurements to correct the 
EEG signals for distortion.

DETAILED DESCRIPTION 
I. EEG Recording Method 
A. Nomenclature for 256 Electrode Placements 
A nomenclature is defined for the extra equidistant coronal rows added 
between the original rows of the conventional International 10-20 System 
as well as the extra equidistant electrodes added to fill in the spaces in 
each row (FIGS. 1A-1C). For example, the preferred number of electrodes is 
64-512 and most preferred are 256 scalp electrodes. The letter "a" is 
prefixed to the conventional 10-20 row name to indicate a position 
anterior to the existing one. For example aO is anterior occipital, aP is 
anterior parietal, aC is anterior central and aF is anterior frontal. Two 
additional numbers appear after the location number (1-8) of the 10-20 
positions. These two numbers (0-9) indicate the proportional distance to 
the next anterior and medial 10% electrode position, respectively. For 
example, an electrode halfway between P3 and aP1 would be named P355, 
while an electrode halfway between P3 and P1 would be named P305. With the 
original nineteen electrodes of the 10-20 System, the typical distance 
between electrodes on an average adult male head is about 6 cm; with 124 
electrodes, the typical distance is about 2.25 cm; with 256 electrodes, 
the typical distance is about 1 cm. 
B. Smart EEG Hat 
In most routine clinical EEGS, electrodes are prepared and positioned on 
the head individually, while in some labs commercially available electrode 
caps with up to 32 built-in electrodes are used. Extending this idea, we 
developed a more efficient system for recording up to 124 or more EEG 
channels with little or no preparation of the scalp, as described in U.S. 
Pat. No. 5,038,782, incorporated by reference herein. 
C. Measuring Electrode Positions 
Traditionally, individual electrodes are placed according to measurements 
taken with a tape measure. While this has been sufficient for qualitative 
examination of strip chart tracings with up to 21 electrodes, greater 
precision and a method using less time are needed when many more 
electrodes are used and when one wishes to relate a recording position on 
the scalp to the underlying brain anatomy. 
In accordance with the present invention, the position of each electrode on 
a subject's head is measured with a probe 12, such as the 3-Space Isotrak 
digitizer by Polhemus Navigation Sciences, that has coils for sensing the 
three-dimensional position of the probe tip with respect to a magnetic 
field source attached to the head support 10 (FIG. 2). Adjustable guides 
11 built into the head support hold the subject's head comfortably in 
place while the measurements are made. A menu-driven program on a control 
digital computer 13 is used to select electrodes to be measured and 
display the digitized position of each electrode on a two-dimensional 
projection display. Position measurement is accurate to better than 3 mm 
[RMS]. While this procedure provides greatly increased precision over the 
use of a tape, it can take fifteen minutes to measure the positions of 124 
electrodes during which time the patient must keep his head still in the 
head support device. Such lack of motion for that time period may be 
difficult or impossible for very young or infirm patients. 
An alternative procedure is to attache the transmitter for the Isotrak 
digitizer to the head with an adjustable elastic band, eliminating the 
need for the head support device. This alternative is more comfortable for 
the patient, but does not reduce the time required to measure each 
electrode. 
In accordance with the present invention, in another alternate procedure, 
electrode positions are determined by classifying a subject's head shape 
as one of seven canonical head shapes (FIG. 3), determining a scale factor 
which is the size of the subject's head relative to an average sized head 
of that canonical shape, and estimating electrode positions as scaled 
values of the standard electrode positions for that particular canonical 
head. The patient's head shape is classified by: (1) measuring hat stretch 
of stretch hat 20 having elastic bands 21 at several points over the head 
using piezo-electric sensors 22 as ribbons embedded within the hat (see 
FIG. 6; (2) scaling the vector of stretch measurements, termed the 
"stretch vector," to have unit magnitude; and (3) feeding this vector to a 
neural network or other mathematical computer pattern classifier that has 
been trained to make the desired classification. The scaling factor is the 
ratio between the stretch vector magnitudes of the subject and that 
determined from a canonical head of average size. For each electrode 
montage provided with the hat and each canonical head shape, accurate 
measurements of electrode position are made on an average sized model of 
that head. Appropriately scaling these electrode positions gives a 
reasonably accurate estimate of actual electrode position and has the 
advantage that no additional time is added to the EEG recording session. 
Another alternative procedure is to place a flexible grid of electric field 
sources in close proximity to the head and then measure the potential 
induced in each electrode when each node in the grid is activated. For 
example the nodes (electric field sources) may be resonant circuits or 
switch points which form a grid on a flexible cap. The relative position 
of each node is calculated by interpolation from the pattern of induced 
voltages. The grid may be embedded in a hat which fits over the hat 
containing the recording electrodes or the recording electrodes and field 
source may be combined. This has the advantage of increased precision with 
no significant increase in recording time. 
D. Software System for EEG Analysis 
Although commercial EEG computer systems have improved during the past few 
years, some important limitations still restrict their utility, including: 
(1) system capacity that is inadequate for recording large numbers of 
channels (124 or more) and collecting and storing large databases; and (2) 
spatial sampling and analyses that are inadequate, with too few channels, 
a lack of effective means to reduce volume conduction distortion, a lack 
of cross-channel analyses (e.g., crosscovariance and crosspower, 
correlation and coherence,), and a lack of means to investigate the 
relationship between neuroelectric data and cortical anatomy and 
physiology revealed by MRI and PET imaging technologies. 
1. Data Collection 
One or two computers are used for stimulus presentation and data 
collection. One computer is used to present stimuli and gather behavior 
response data from the subject while the other collects physiological data 
and controls the first computer. When a very fast computer is used, or not 
many channels are required, a single computer can be used to perform all 
operations. A control software system runs the two computers, presents a 
variety of visual, auditory and somatosensory stimuli to the subject 
according to flexible task protocols, and digitizes up to 256 channels of 
evoked brain wave physiological data. Up to 128 EEG traces can be 
monitored in real time as data are collected. Most parameters of an 
experiment can be altered via a menu-driven interactive display. A 
calibration module numerically adjusts the gain of all channels according 
to the magnitude of a calibration signal which is injected, under computer 
control, into each electrode. Another module detects gross artifacts and 
color-codes contaminated data such as eye movements, gross head and body 
movements and bad electrode contacts. Stimulus, behavior and physiological 
data are stored on hard disk according to a self-recording data 
description language, and archived to optical disk or magnetic tape. The 
data are immediately available to researchers at their desks using either 
remote terminals, or UNIX workstations via Ethernet and Network File 
System. 
2. Data Analysis 
For subsequent data analysis the system also has a number of other 
functional improvements over its predecessors including: (1) least-squares 
and 3-D spline Laplacian derivation estimation and spatial deblurring 
using finite element or integral methods to reduce volume conduction 
distortion; (2) digital filters with user-specified characteristics; (3) 
time series analysis including spectral analysis, Wigner Distributions 
(Morgan and Gevins, 1986), and event-related covariance analysis (Gevins 
et al., 1987, 1989; Gevins and Bressler, 1988); (4) neural-network-driven 
pattern recognition to extract optimal or near-optimal subsets of features 
for recognizing different experimental or clinical categories (Gevins, 
1980, 1987; Gevins and Morgan, 1988); and (5) anatomical modeling to 
construct 3-D mathematical models of the brain and head from MRI to CT 
brain scans. Four on-line, interactive subsystems are used to examine and 
edit data for residual artifacts (on an individual channel basis if 
desired), sort data according to stimulus, response or other categories, 
perform exploratory data analysis, and produce three-dimensional graphics 
representations of the brain and head. 
II. Brain Modeling Methods 
Since commercial magnetic resonance image (MRI) analysis packages lack 
essential features for functional-anatomical integration studies, there is 
described below algorithms and a softward outline to produce 3-D brain 
models suitable for functional localization studies. Visualization 
softward permits construction of 3-D composites of multiple 2-D image 
planes, as well as 3-D surface rendering based on surface contours. Since 
generating surface contours manually is laborious and subject to error, we 
have largely automated the procedure. We have also automated the alignment 
of the digitized EEG electrode positions with the scalp surface contours, 
which is a critical first step in a functional-anatomical analysis. The 
MRI contour information is used to produce mathematical finite element or 
surface integral models of the brain and head suitable for equivalent 
dipole source localization and scalp EEG deblurring procedures. 
A. EEG-MRI Alignment Procedure 
In order to visualize the brain areas underlying EEG electrodes, a 
procedure is needed for aligning scalp electrode positions with the MR 
Images. In our procedure, x, y, z translation and x, y, z axis rotations 
are computed iteratively to align the digitized positions of the EEG 
electrodes with the MRI data. This is done for each electrode by finding 
the distance to the closest point on the scalp surface MRI contours and 
minimizing the mean distance for all electrodes. With MR images that have 
a 3 mm inter-slice spacing, the mean error distance is usually better than 
2 mm. This is more accurate and less subjective than alignment procedures 
that use skull landmarks such as the nasion and pre-auricular points 
located visually in the MR images. FIG. 1A shows the electrodes displayed 
schematically on a scalp surface reconstructed from horizontal MR scalp 
contours as described below. 
B. 3-D Composite MRI Displays 
In order to visualize how the individual MRI slices fit together to form a 
three-dimensional object, a means is needed to juxtapose an arbitrary 
number of slices from sagittal, coronal and horizontal scanning planes 
into a composite image, and view the juxtaposed slices from an arbitrary 
viewing angle. 
C. Contour Extraction and External Surface Rendering 
In order to visualize the external three dimensional surface of the brain 
and head, a means is needed to compute the curved surfaces from the 
information contained in the set of two-dimensional MRI slices. Scalp and 
cortical surface contours are extracted from MRIs using two image analysis 
methods we developed. The first technique used intensity thresholding, 
which involves extracting contours along which the image intensity is 
equal to a defined threshold. This technique is useful for extracting the 
scalp surface contour which can be discerned easily from the black 
(approximately zero intensity) background of the image. The technique has 
six steps: 1) for each pair of adjacent contours, find the point on the 
second contour closest to the first; 2) calculate the two distances from 
each of these points to the next point on the adjacent contour; 3) make a 
triangle using the point with the shortest distance, advancing on that 
contour; 4) repeat steps two and three until all points are exhausted; 5) 
repeat steps two to four in the reverse direction along the contours; and 
6) piece together the "best" results. 
The second contour extraction method involves differential intensity 
analysis. Using this technique, contours that separate image regions with 
different local average intensities are extracted. Resulting contours pass 
through the points in the image at which the local average image intensity 
is changing most rapidly. This technique requires no a priori intensity 
threshold value, and is useful for extracting the cortical surface contour 
which has a less well-defined image intensity value throughout an image 
than the scalp surface contour. The first-order and second-order partial 
derivatives of the image are estimated using 2-D filters, and these 
derivatives are used to locate the local maxima in the gradient of the 
image intensity. Highly computationally efficient filtering techniques 
have been developed for the estimation of the partial derivatives of the 
image (Algazi et al., 1989). FIG. 6 shows an example of contours 
corresponding to locations of local maxima in the gradient of the image 
intensity, which trace sulci. 
D. 3-D Cortical Internal Surface Image Model 3-D surface models of the 
external convexity of the cerebral cortex are not sufficient to visualize 
and computationally model the cortical surface within fissures and sulci. 
As a result, we developed an algorithm to model cubic volume elements 
(voxels). The faces of the voxels lie in the horizontal, coronal, and 
sagittal planes for which MRI data have been obtained. The MRI data are 
mapped onto the faces of the voxels to obtain a 3-dimensional display of 
the image data. The image planes used are averaged horizontal planes that 
lie halfway between the horizontal planes in which MRI data have been 
acquired, and coronal and sagittal image planes (that can optionally be 
synthesized from the acquired horizontal images). The spacing between 
image planes is 3 mm, which yields voxels with dimensions of 3 mm by 3 mm 
by 3 mm. The images have pixel dimensions of approximately 1 mm by 1 mm. 
The initial set of voxels is the set bounded by those voxels that lie just 
inside the cortical surface. To view the MRI data at a slightly deeper 
level, a mathematical morphology erosion operation is used to remove the 
boundary layer of voxels, thereby exposing the faces of the vosels that 
lie one layer deeper. By eroding the model iteratively, anatomical 
structures can be tracked and 3-dimensional models of the structures can 
be made (see FIG. 7). 
E. 3-D Electrical Brain Model: Finite Element Method 
A mathematical 3-D representation of the brain and head are needed to most 
effectively deblur the EEG recorded at the scalp and to compute equivalent 
current dipole sources from the scalp EEG. We developed an efficient 
finite element method for this purpose. 
Maxwell's Equation, 
EQU .gradient..multidot.(.sigma..gradient.u)+.gradient..multidot.J=0 in 
.OMEGA.(1) 
is frequently used to study the electromagnetic field generated by 
populations of neurons. Here .sigma.(&gt;0) is the conductivity tensor at a 
point =(x,y,z) in .OMEGA.(e.g., a human head), u is the electric potential 
at R, J is the electric current density at R and .gradient. is the grad 
vector operator. When .OMEGA. represents a human head, the following 
boundary condition for equation (1) is obtained 
EQU .sigma..gradient.u.multidot.n=0 on S(headsurface) (2) 
since electric current does not flow out of the head in the direction 
normal to the surfaces. The limited scalp potential measurement specifies 
another boundary condition. 
EQU u=U(x,y,z) on S.sub.1. (3) 
Where S.sub.1 is a subsurface of S corresponding to the area covered by the 
electrodes. 
The Finite Element Method (FEM) we developed finds the numerical solution 
of equation (1) with the boundary condition as specified in equation (2) 
when the electromagnetic field activities of a head are modeled. Our 
implementation has the following special features: 
a) A procedure to generate a list of wedges within a head such that each 
wedge has an unique tissue type and the interfaces between two neighboring 
wedges are perfectly matched to guarantee a correct numerical solution to 
Eq. (1). 
b) A proper numbering scheme on a chosen set of nodes and the use of local 
supporting functions to produce a diagonally condensed block sparse 
coefficient matrix. 
c) The use of elementwise local shape functions to represent the local 
supporting functions. 
d) The use of a compressed data structure to obviate the need to allocate a 
very large memory space so that large size problems can be solved on 
desktop computer workstations. 
e) Use of a C-language sparse matrix solver incorporating a temporary 
working array in the process of Cholesky Decomposition to speed up the 
process of computing the numerical solution. 
Advantages of this approach are: 1) it handles the Dirac Delta Function 
.cuberoot..multidot.J smoothly by transforming equation (1) into a 
variational form when J represents a dipole-like kind of source; 2) it 
allows modeling of the complicated geometries of different tissues within 
the head by generating finite elements using contours on pairs of adjacent 
MR images; and 3) it produces a sparse matrix in which many entries are 
zeros and thus is convenient and fast to compute on a desktop computer 
workstation. For example, a brain model with 10,087 FEM nodes can be 
solved in about 100 minutes on a SUN Sparc-1 workstation (12 MIPS) with 8 
MB of memory (FIG. 5). 
III. Spatial Enhancement Techniques to Reduce Blur Distortion 
Electrical currents generated by sources in the brain are volume conducted 
through brain, cerebrospinal fluid, skull and scalp to the recording 
electrodes. The principal source of distortion in this process is produced 
by the skull, which has low conductance relative to the other tissues. 
Because of this spatial low-pass distortion, the potential distribution of 
a localized cortical source is spread over a considerable area of scalp 
and appears blurred or out of focus. For example, using a 4-shell 
spherical head model, we estimated the "point spread" for a radial dipole 
in the cortex to be about 2.5 cm. There are several methods to reduce this 
distortion which involve a tradeoff between the amount of information 
about head shape and tissue properties required versus the accuracy of the 
distortion correction. 
A. Laplacian Derivation 
The simplest method is the spatial Laplacian operator or Laplacian 
Derivation (LD). It involves computing the second derivative in space of 
the potential field at each electrode. This converts the potential into a 
quantity proportional to the current entering and exiting the scalp at 
each electrode site, and eliminates the effect of the reference electrode 
used during recording. An approximation to the Laplacian Derivation, 
introduced by Hjorth (1975, 1980) assumes that electrodes are equidistant 
and at right angles to each other. Although this approximation is fairly 
good for some electrode positions such as midline central (Cz), it is less 
accurate for others such as midtemporal (T5). We have developed two 
methods which produce more accurate estimates of the Laplacian using the 
actual measured electrode positions. 
1. Planar Projection and Least Squares Solution 
The first method is based on projecting the measured electrode positions 
onto a two-dimensional surface and estimating the surface Laplacian with a 
least-squares procedure. 
The objective of the LD calculation is an accurate estimate of 
##EQU1## 
Assume we wish to compute the LD at position q.sub.o using N surround 
electrodes in an arbitrary configuration. Let the voltage at position 
q.sub.i be designated by V.sub.i. Put the origin of a cartesian coordinate 
system at q.sub.o with the z axis pointing perpendicular to the plane 
nearest the electrodes. Let (q.sub.ix,q.sub.iy) represent the x and y 
components of the electrode position within this plane. 
Then E(x,y)=-.gradient.V(x,y) is the electric field in the plane or 
equivalently: 
##EQU2## 
Where C is any curve in the plane connecting q.sub.o to q.sub.i. If we let 
E.sub.x and E.sub.y designate the x and y components of E then the 
two-dimensional laplacian of the voltage is given by: 
##EQU3## 
The field E can be expanded in a two-dimensional Taylor series (Rudin, 
1976; Apostol, 1969) and expressed in a single matrix equation: 
EQU QF=V (7) 
In this equation the Q matrix is composed of the x and y coordinates 
electrode positions and remains fixed during the entire recording. The 
voltage vector V is measured at each time point and vector F composed of 
the electric field components and their derivatives is unknown. 
Multiplying the above by the transpose of Q gives 
EQU Q.sup.T QF=Q.sup.T V (8) 
If Q premultiplied by its transpose Q.sup.T is nonsingular then we can form 
the least squares estimate of F as 
EQU F=[Q.sup.T Q].sup.-1 Q.sup.T V (9) 
If we let C be the sum of the third and fifth rows of [Q.sup.T Q].sup.-1 
Q.sup.T then from the above equation we can estimate the laplacian LD as: 
##EQU4## 
This method accounts for both variations in distance and variations in 
angle between electrodes in a computationally efficient manner, since once 
the coefficients are computed based on the positions of the electrodes, 
application of the method only requires a single matrix multiplication for 
each point of a time series. Although this procedure produces a dramatic 
improvement in topographic detail, some problems remain because of the 
projection onto a two-dimensional plane, and the assumption that the 
current gradient is uniform over the region encompassed by the surrounding 
electrodes used to estimate the Laplacian of an electrode. 
2. Three Dimensional Spline Method 
A more accurate method of estimating the Laplacian Derivation does not rely 
on the two-dimensional projection. 
We compute the Laplacian of a distribution function U(x,y,z) (defined on a 
convex surface S) at its tangent plane attached to S at point 
(x.sub.0,y.sub.0,z.sub.0) where the surface S is defined by: 
EQU x=f(.xi.,.eta.); y=g(.mu.,.eta.); z=h(.xi., .eta.) 
We want to compute: 
##EQU5## 
such that 
EQU x.sub.0 =f(.xi..sub.0,.eta..sub.0); y.sub.0 =g(.xi..sub.0,.eta..sub.0); 
z.sub.0 =h(.xi..sub.0,.eta..sub.0) 
(.xi..sub.0,.eta..sub.0) is a point in .xi..eta.-plane. For convenience, we 
denote .differential.x/.differential..xi. by f.sub.86, etc. By the 
definition of the Laplacian here, f, g and h have to be the transformation 
functions for points on the .xi..eta.-plane to points on the surface S 
defined in the current xyz-Cartesian coordinates. General forms of f, g 
and h for all points on .xi..eta.-plane are difficult to find. However for 
every point (x.sub.0, y.sub.0, z.sub.0) on S, explicit forms of f, g and h 
can be found for translation and rotation transformations of points from 
.xi..eta.-plane to the tangent plane of S at (x.sub.0, y.sub.0, z.sub.0) 
in xyz-Cartesian coordinates. The explicit form of the potential 
distribution function U(x,y,z) on a scalp surface S is not available in 
EEG recordings. However a set of scalp-recorded potential V.sub.i (i=1,2, 
. . . n) is available on a set of recording channels on S, (x.sub.i, 
y.sub.i, z.sub.i), i=1,2, . . . , n. Therefore the potential distribution 
can be computed by interpolation from the scalp-recorded potentials. First 
a set of basis functions .phi..sub.i (x,y,z) must be computed such that 
##EQU6## 
Since EEGs can be described by Maxwell's Equation, the open solution of 
that equation can be used for the basis functions. Accordingly, we define: 
##EQU7## 
Here (x.sub.i0, y.sub.i0, z.sub.i0) is the center of the best fitting 
sphere of radius 1 to the point (x.sub.i, y.sub.i, z.sub.i) on the surface 
S. Algorithmically we choose this center point 1 unit from the i-th 
channel position (x.sub.i, y.sub.i, z.sub.i) in the opposite direction of 
the normal orientation of surface S at (x.sub.i, y.sub.i, z.sub.i). The 
basis function .phi..sub.i is a local support function and is smooth 
enough in the neighborhood of (x.sub.i, y.sub.i, z.sub.i). For a given 
channel list (x.sub.i, y.sub.i, z.sub.i) and its corresponding scalp 
potential data set V.sub.i, (i=1,2, . . . , n), we 
1) construct a polygon montage spanned by these channels, that defines the 
surface S; 
2) estimate the normal directions for every channel position by averaging 
the normal orientations of polygons surrounding this channel position; 
3) construct a set of basis functions .phi..sub.i (x,y,z) and solve the 
weights a.sub.i through systems of equation in (13) by the Gaussian 
Elimination Method; 
4) for every point on S, the desired surface tangent laplacian can be 
obtained by equations (11) and (12). 
Equation (12) suggests that the desired surface tangent laplacian of 
U(x,y,z) can be computed by taking the surface tangent Laplacians of 
.phi..sub.i (x,y,z) for every point (x,y,z) on S, and then linearly 
combining them with weights a.sub.i. 
3. Finite Element Method 
A better improvement in distortion correction is possible by using a Finite 
Element Method to represent the true geometry of cortex, cerebrospinal 
fluid, skull and scalp, and estimates of the conductivities of these 
tissues, to model the potential activity described by Maxwell's equations 
in the scalp and skull layers. This allows estimation of the potentials 
which would actually be recorded on the surface of the brain. This can be 
done without introducing an arbitrary model of the actual number and 
location of sources because, regardless of where they are generated in the 
brain, potentials recorded at the scalp must arise from volume conduction 
from the cortical surface through the skull and scalp. 
We use Maxwell's equation and our Finite Element Method to relate the 
scalp-recorded potentials with the cortical potentials without introducing 
an explicit source function. For this application, the region of interest 
is limited to the scalp volume and skull volume. The difference between 
the inner surface of the skull and the outer convexity of the cortical 
surface is assumed to be small enough to be neglected. We also assume that 
the potential activity in the region of interest is described by Maxwell's 
equations, that the boundary condition on the air-scalp surface is stated 
by equation (2), i.e., current cannot flow out of the scalp, and that the 
boundary condition on the cortical surface is stated by the following 
equation which describes the cortical surface potentials: 
EQU u=G(x,y,z) on S.sub.2. (15) 
We also assume that there is no generating source within the scalp or skull 
which means the function J in equation (1) is zero. Although of course 
there is no general solution to the "inverse problem" because of the 
assumed boundary conditions there is a unique solution of equation (1), 
which may be obtained as follows. Applying the Finite Element Method to 
equation (1) with boundary conditions in (2) and (15), the following 
matrix vector relation is obtained: 
EQU Au=f. (16) 
Here u is a vector of dimension n, which is a numerical approximation to 
the analytical potential distribution function u in equation (1). The 
value of n corresponds to the total number of vertices on all the finite 
elements in the region of interest. Since we are assuming that no 
generating sources are present in the skull and scalp, J=0 and therefore 
f=0. We then decompose u into three sets, the potentials which correspond 
to the electrodes on the scalp, the potentials which correspond to the 
cortical surface, and the potentials in the rest of the region, and denote 
them by u.sub.1, u.sub.3 and u.sub.2 respectively, If we decompose the 
matrix A correspondingly, then equation (16) becomes: 
##EQU8## 
Solving u.sub.1 with respect to u.sub.3, we get 
EQU (A.sub.11 -A.sub.12 A.sub.22.sup.-1 A.sub.21)u.sub.1 =(A.sub.12 
A.sub.22.sup.-1 A.sub.23 -A.sub.13)u.sub.3 (18) 
and solving u.sub.3 with respect to u.sub.1, we get 
EQU (A.sub.33 -A.sub.32 A.sub.22.sup.-1 A.sub.23)u.sub.3 =(A.sub.32 
A.sub.22.sup.-1 A.sub.21 -A.sub.31)u.sub.1. (19) 
For a given G.sub.k, u.sub.3.sup.k is known. Then u.sub.1.sup.k can be 
computed via equation (18) and the residual is calculated from 
u.sub.i.sup.k and U(x,y,z) which correspond to the predicted and measured 
potentials, respectively. 
Equation (19) puts forward an alternative to the above approach. For each 
u.sub.1, we can compute the cortical potential u.sub.3. If there are a 
large number of recorded potential observations, such as hundreds of 
sampled time points, the following process will reduce the calculation 
time dramatically. Suppose e.sub.i is a vector of the same dimension as 
u.sub.1 with all its components as zero except 1 at i-th position. Then 
##EQU9## 
If we solve the u.sub.3 in equation (19) with u.sub.1 replaced by e.sub.i 
and denote the resulting solution vector by p.sub.i, then for an arbitrary 
u.sub.i =(s.sub.1,s.sub.2, . . . ,s.sub.n.sbsb.1).sup.t, the corresponding 
solution is: 
##EQU10## 
which is a simple linear combination of n.sub.1 vectors and therefore much 
faster to compute. 
4. Surface Integral Method 
An alternative to performing deblurring (or dipole source localization) 
through using numerical solutions to the differential form of Maxwell's 
equations is to use numerical solutions to the integral form of Maxwell's 
equations. 
The following equation is used for the source localization problem: 
##EQU11## 
where .sigma. denotes conductivity; x.sub.0 is a point on the outer 
surface of the scalp, u is potential; R.sub.j is the vector from a source 
to x.sub.0 and R.sub.j is its magnitude; J is the moment of a dipole 
source; 1 is the number of interfaces, S.sub.i, between regions differing 
in conductivity; n.sub.i is the unit vector from the in region to the out 
region; and R is the vector from x.sub.0 to the point of integration with 
R denoting the unit vector and R denoting the magnitude. A numerical 
solution of this equation involves obtaining a discrete approximation to 
the integral. Boundary element methods can be used to do this. An 
alternative is to use efficient integration methods that have been 
developed for the sphere and methods of computational geometry for 
projecting a non-spherical surface on a sphere. 
One advantage of using the integral method for source localization is that 
no nodes are close to the sources, a locus where potential fields vary a 
great deal. This is a problem for the finite element method. 
The following equation is used for the deblurring problem: 
##EQU12## 
where the notation is that for Equation (20). After regularization, the 
numerical integration techniques used for source localization are applied. 
An advantage of this formulation when applied to deblurring is that it 
accounts for the geometry of the whole conductor whereas the finite 
element technique only accounts for the portion of the conductor that 
includes the electrodes on the scalp and the tissue immediately below. 
5. Estimating Scalp and Skull Thickness for Models 
In the finite element or surface integral methods, the local geometry of 
scalp and skull are each explicitly represented. If an MRI or CT brain 
scan is not available, an alternative method of measuring tissue thickness 
can be used based on low-frequency ultrasound technology. 
For a low-resolution representation of skull thickness, the "end echo" in 
standard echoencephalography is used to estimate the thickness of the 
skull. This requires placing the ultrasonic probe at a position on the 
opposite side of the head from the place of desired measurement, normal to 
the scalp at the probe, and normal to the skull at the place of skull 
thickness measurement on the opposite side of the head from the probe. 
Skull thickness at each of the major cranial bones is measured in the 
following locations: 
______________________________________ 
cranial bone 
position probe position 
______________________________________ 
left temporal 
T3 near T4, directly above the ear 
right temporal 
T4 near T3, directly above the ear 
frontal (left) 
aF1-aF3 roughly 1 cm below O2 
frontal (right) 
aF2-aF4 roughly 1 cm below O1 
left occipital 
O1 .2-.5 cm right of FPZ 
right occipital 
O2 .2-.5 cm left of FPZ 
left parietal 
aP3 1 cm below M2, with upward tilt 
right parietal 
aP4 1 cm below M1, with upward tilt 
______________________________________ 
The end-echo consists primarily of the echo due to reflection from 
soft-tissue to inner skull interface and the echo due to outer skull to 
scalp interface (followed by a very small scalp-to-air echo). The delay 
prior to the end-echo varies, according to scalp location and 
inter-subject variation, roughly from 90 to 120 microseconds. The delay 
between the echos from either surface of the skull may vary from 1.7 to 5 
microseconds. The second echo is attenuated with respect to the first, and 
requires a gain increase of 2.5 dB per MHz (typical) to be comparable in 
size to the first. Pulse widths of 0.25 to 1 microseconds are used with 
burst frequencies of 2 to 10 MHz, adjusted to produce the best 
discrimination between the two echos. 
As necessary, passive receptors are used to scan the area of the intended 
skull thickness measurement to determine the accuracy of aim of the 
transmitted beam. When necessary, an average is taken over a small cluster 
of locations (3 to 6) near the listed probe site, in order to average over 
the variations in thickness for a given bone near the target area. In 
those instances when the aimed beam cannot be focused well on the parietal 
bones, the average thickness of these bones is estimated from normalized 
data on the relationship between parietal bone thickness and the average 
thickness of the other bones. 
For finer spatial resolution in measurement of skull thickness, a near side 
detector can be used. It has two to four transmitters adjacent to the 
transmitter/receiver probe, spaced far enough away so that the critical 
angle (27 degrees) prevents energy from the adjacent transmitters from 
entering the skull at an angle that could be reflected back to the probe. 
The phase and time delay between the adjacent transmitters and the probe is 
adjusted to minimize the reverberation at the probe. Thus, at the probe 
site the adjacent transmitters are contributing energy solely through 
reflections from the outer skull surface, and in a way to minimize 
reverberation at the probe. Only energy transmitted by the probe itself 
can enter the skull and be reflected back to the probe. 
The thickness of the near skull can then be measured either by subtracting 
independent measures of the reflection time from the outer skull surface 
(adjacent transmitters inactive) and reflection time of the echo from the 
inner skull surface (reverberation minimization active), or by 
interferometry. The interferometry option sweeps the sound frequency 
(readjusting reverberation minimization for each frequency), and 
determines the change in frequency required for one oscillation in the 
variation in reflected energy due to interference of the inner and outer 
reflections from the skull. The measured thickness of the skull is one 
half the speed of sound in the skull divided by the change in frequency 
required to move through one "interference fringe". 
6. Estimating Conductivities 
Up to now, estimates of conductivity for the scalp, skull, CSF and brain 
have been from published sources (Geddes & Baker, 1967; Hosek, 1970). More 
accurate values are needed to realize the full benefit of the FEM head 
model. This can be done by solving for the ratio of skull-to-scalp and 
skull-to-brain conductivities using the scalp potential distribution 
produced by a single compact population of neurons, represented as an 
equivalent current dipole, whose location is approximately known. In one 
procedure, a steepest-descent, non-linear, least-squares fit between the 
measured potentials and the forward solution of an equivalent current 
dipole in a three-sphere model is performed (Fender, 1987). The main 
unknown parameters are the ratios of conductivities of the tissues, as 
well as the dipole strength (which is not of interest here). The 
availability of reasonable initial guesses, as well as upper and lower 
bounds, from existing experimental data result in realistic solutions for 
the unknown parameters. A highly localized source suitable for this 
purpose is the somatosensory evoked potential peak at 20 msec produced by 
mild stimulation of the contralaterial median nerve at the wrist. This 
potential, negative posterior to the central sulcus and positive anterior 
to it, is generated by a single compact population of neurons oriented 
tangentially in the hand area of somatosensory area 3b and located about 
2.5-3.0 cm below the surface of the scalp. Published values of the 
amplitude of the cortical potential and the ratio of cortical to scalp 
potentials are also available (Dinner et al., 1983). The global 
conductivity parameter is then adjusted for each of the finite elements in 
the skull using local skull thickness measures determined from the low 
frequency ultrasound, MRI or other means. 
An alternative method for estimating tissue conductivity is based on 
measuring the reverse electromotive force produced by eddy currents 
induced in the tissue by an applied external field generated by a coil 
placed against the head of each patient. The magnitude of this emf depends 
on the conductivity of the tissue and the frequency of the magnetic field, 
i.e. higher conductivity means more eddy current is induced which gives a 
larger reverse emf. Very low frequencies induce eddy currents in all 
tissues whereas high frequencies induce currents only in the tissues 
closest to the coil. Scanning a spectrum of frequencies yields data from 
which the conductivities of each tissue type can be computed. 
IV. Source Localization 
We have also used FEM to: 1) localize single equivalent dipole sources of 
recorded scalp potentials; 2) obtain the potential distribution within the 
whole head given the localized source. We choose a specific function J and 
solve the corresponding potential distribution u using FEM such that the 
difference between u on S.sub.1 and U(x,y,z) defined on S.sub.1 is 
minimized. The following source localization procedure (SLP) outlines an 
iterative process used to find J and u. For a chosen J.sub.0 and K=0, 1, 
2, . . . , repeat the following: 1) find the numerical solution u.sub.k of 
equation (1) on .OMEGA. coupled with the boundary condition in equation 
(2) using FEM; 2) measure the difference between u.sub.k and U(x,y,z) on 
S.sub.1 and choose J.sub.k+1 ; 3) compute the distance between J.sub.k 
-J.sub.k+1. If the distance is less than some tolerance, stop the loop. We 
applied the SLP to the case that J represents a single dipole. We used the 
Simplex algorithm (Nelder and Mead, 1965) for computing J.sub.k+1. 
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