IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. IO, OCTOBER 1992
99I
Estimation of the Spatio-Temporal Correlations of
Biological Electrical Sources from Their
Magnetic Fields
Warren E. Smith, Member, IEEE
Abstract-Quasi-static electromagnetic systems, such as those
found in biological systems, produce electric and magnetic fields
whose temporal and spatial correlations reflect the source correlations in a straightforward manner. These fields can be noninvasively measured, providing information about the coherence properties of the source, which may directly represent
ordered physiological processes of the organism. The description “biocoherence” will be adopted here to refer to the manifestation of the coherence in the magnetic measurements of
these sources due solely to physiological processes.
In this paper a general formulation linking the spatial and
temporal coherence of measurable magnetic fields with the corresponding spatial and temporal coherence of the inaccessible
current sources is derived in the quasi-static model. A method
for reconstructing the spatial and temporal coherence of the
source distribution is then presented. Such coherence maps
would be useful descriptors of physiological processes occurring over time and space, and would represent more information than an image of the current sources frozen in time, or
even a temporal sequence of such images.
INTRODUCTION
EN one structure of the nervous system is active
or suppressed at time t , , another structure’s activity
at another spatial location may be directly affected at some
later time t 2 . The cause-effect relationship may be mediated by a direct connection or by some intermediate
structures. A relatively low-level example of such a process is the cortical and motor rhythms taking place during
walking. These are periodic activities of all of the structures involved (e.g., cortex, thalamus, cerebellum, brainstem, muscles of the body), each structure performing its
own periodic function with varying amplitudes and
phases, contributing to the overall coordinated effect. Another possible example is the evoked response in the cortex to a sight, sound, or touch sensation. Many areas of
the cortex and subcortex may take part at different times
in response to the stimulus.
By examining the types of spatio-temporal correlations
that normally exist between structures during a specific
task, two things can be learned. First, any hidden corre-
w
Manuscript received July 13, 1990; revised August 27, 1991. This work
was supported by a Public Health Service Biomedical Research Support
Grant and the Joint Services Optics Program Grant #DAAL 03-88-K-0182.
The author is with the Institute of Optics, University of Rochester, Rochester, NY 14627.
IEEE Log Number 9202572.
lation (in space and time) between structures, whether by
direct or indirect linkage, might become evident. Second,
pathology of the structures may be detectable as perturbations from the norm of the correlations. Spatio-temporal correlation maps will contain much more information than an image of source activity at a fixed time, or
even a movie of the source activity over several time
frames. In such movies the human observer qualitatively
forms temporal correlations of the spatial processes, but
may miss subtle correlations widely separated in time. For
spatio-temporal processes such as nervous-system function the determination of the source correlations, and not
just source snapshots, is an important added dimension of
analysis.
How can such spatio-temporal processes be measured?
Systems that image anatomy in vivo, such as X-ray computed tomography (CT) and magnetic resonance imaging
(MRI), have spatial resolution in the millimeter range and
temporal resolution that is approaching tens of milliseconds, but they do not measure function. In nuclear
medicine [positron emission tomography (PET) and single photon emission computed tomography (SPECT)] it
is possible to measure physiological function with spatial
resolutions on the order of 5-10 mm with a temporal resolution on the order of many seconds or tens of minutes,
respectively , but this temporal resolution is much larger
than the time constants of the active processes. Electrical
potential measurements (electroencephalography, electrocardiography) [ 11 detect, with a temporal resolution on
the order of milliseconds, physiological processes that
produce global potential differences on the surface of the
body. The spatial resolution of the sources has been poor
(many centimeters), however, due to the smearing of the
electric fields by the conductivities of the tissues of the
body. More recently magnetic measurements have been
made of the electrical activity of the body [2]-[4] using
superconducting quantum interference devices (SQUID’S)
[ 5 ] . The production of magnetic fields by the electrical
processes within living tissue is called biomagnetism. The
magnetic fields can be measured with the excellent temporal resolution of the electrical-potential measurements,
but there is evidence that magnetic measurements may offer better spatial localization of the electrical sources than
potential measurements [6]. Thus, biomagnetism may be
0018-9294/92$03.00 0 1992 IEEE
998
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 10. OCTOBER 1992
able to provide us with a unique capability impossible with
other imaging modalities: a totally noninvasive data-acquisition technique that measures the spatio-temporal information of nervous-system function.
Magnetic fields produced by the currents found in the
brain and heart have been measured with the techniques
of magnetoencephalography and magnetocardiography,
respectively. One approach to source reconstruction has
been to fit only a single current dipole to the measured
magnetic fields. This is particularly useful in isolating
epileptogenic foci of the brain. More recently interest has
been developing in forming three-dimensional images of
large numbers of source current dipoles [7]-[14]. The
SQUID hardware has also been evolving to facilitate more
simultaneous magnetic measurements. Systems with up to
37 independent SQUID channels arranged in a hexagonal
lattice have been built recently by BTi and Siemens, reducing data-acquisition times for some experiments from
hours to minutes.
Many experiments have been performed to measure the
magnetic fields associated with evoked responses of the
brain (see [4] for recent papers). This procedure involves
stimulating the subject with a brief audio, visual, or somatic impulse, and observing the magnetic fields produced by the brain in response to the impulse. The idea
is to observe these fields at many different spatial locations near the head over several hundred milliseconds after
the impulse, with a temporal resolution on the order of
milliseconds. There are many variations on the types of
impulses that can be presented and whether or not the subject is asked to perform cognitive tasks based upon the
type of impulse. By observing the magnetic fields and estimating current sources from them an idea of the brain
structures that are participating in the evoked response can
be developed. Thus, the evoked response in biomagnetism is a very powerful tool for studying spatio-temporal
relationships of the brain, and it will form the model of
the kind of data from which the coherence maps discussed
below will be derived. A further advantage of the evoked
response is that it is repeatable: thus the time sequences
of many different evoked responses are in practice averaged to give a good signal-to-noise ratio (SNR) in the
measured magnetic fields.
Once the magnetic data has been measured over the
evoked response, it can be mathematically correlated with
itself both in space and time. Knowing the physical model
mapping the unknown current sources to the magnetic
measurements, the spatio-temporal correlation map of the
sources, referred to in this paper as the “biocoherence”
of the source, can then be estimated from the correlation
of the measurements. Such a map would show for any
source point at location r l the relative probability that any
other source point at r, is positively active, negatively active, or inactive at a time delay T in relation to the point
at r l , averaged over the evoked response. The biocoherence is a natural specialization of the well-known theory
of mutual coherence in optics [15], [16] to quasi-static
fields. The quasi-static approximation is a good approxi-
mation in biology, and implies three constraints that are
satisfied by the system [17]: capacitive effects are negligible (i.e., the system is purely resistive), inductive effects are negligible (i.e., the electric field is due solely to
the gradient of the electric potential), and propagation delays can be ignored.
With the biocoherence map it would be possible to automatically search for strong positive or negative correlations at all spatial separations and temporal delays in a
particular evoked response, and observe whether these
correlations are similar for normal subjects and whether
they show evidence of abnormal function in subjects exhibiting pathology. Correlations of structures at large delay times may indicate previously unrecognized causeand-effect relationships between the structures. As is
shown later, the biocoherence map at zero delay time can
be used as prior knowledge for a linear estimation reconstruction of the spatial current density at a given time.
An important direction in electroencephalography related to the theme of this paper has been to look at the
correlation of voltage signals from electrodes placed at
various parts of the scalp during different observer tasks
[ 181, [ 191. This paper extends some of these ideas as follows: 1) a source-to-data model, utilizing the physical
principles of the magnetic-data formation in the quasistatic case, is used to relate the correlations of the magnetic data directly to the source correlations, 2) a method
is presented to reconstruct the source correlations from
the data correlations, and 3) the estimated source-correlation matrix at zero delay time is used to form reconstructions of the current sources.
RELATIONOF THE MAGNETICFIELDTO THE
CURRENTDENSITY
The Biot-Savart law [20] relates the magnetic induction B(r, t ) (referred to subsequently as the magnetic field)
to the current density J(r, t ) :
where r is the position of the mgnetic field B ( r , t ) occurring at time t , r’ is the position of the current density
J ( r ’ , t ) occurring at time t , and po is the permeability of
free space. The position vector r is considered to be outside the source volume in this development, so that
B(r, t ) can be noninvasively measured. Bold quantities
will indicate vectors or tensors.
If the current sources are in a conducting medium, the
total current density J ( r ’ , t ) will contain two parts:
+
J(r‘, t ) = Jp(r’,t ) u(r’, t ) E ( r f ,t)
(2)
where J p ( r r ,t ) is the primary source current, u ( r r , t ) is
the conductivity of the current-containing medium (assumed to be linear and isotropic), and E ( r ‘ , t ) is the electric field induced in the conducting medium by the primary currents. The quantity oE is often referred to as the
return current. This current is necessary to avoid build up
of charges due to the primary current.
SMITH: ESTIMATION OF SPATIO-TEMPORAL CORRELATIONS OF BIOLOGICAL ELECTRICAL SOURCES
If the conducting medium is made of a series of N homogeneous conducting regions ok, each region separated
by the N - 1 surfaces Sk, then (1) can be written as
[22]:
N- I
B(r, t ) =
999
tion will be referred to as biocoherence. The biocoherence
tensor is first defined for the magnetic measurements, and
then it is linked to the biocoherence tensor of the source.
The Cartesian components of the biocoherence tensor
rBrln(7)
of the measured magnetic field can be defined as
the following ensemble average:
+ 7) -
[rBrln(7)l,/j’
(Ba(r1,
t
+ 7 ) >)
- Bp(r2, 0 - (Bp(r2, 0 ) ) )
where
Bp(r,t ) =
r - rr
Jp(rr,t ) x ) r - r r I 3d 3 r ’ ,
4~ source
-
V ( r ‘ , t ) is the electric potential on the surface of interest
= -VV(r’, t ) ) ,and Ak(r’)is the outer normal to
the kth surface separating the regions of homogeneous
conductivity ak from a k + I .
(E(rr,t )
BIOCOHERENCE
The relationship between the source currents and the
magnetic measurements reviewed above constitute the
forward problem. Usually the inverse problem is to be
solved, which involves estimating the source currents
from the magnetic fields. The biomagnetic inverse problem is ill-posed because there are essentially an infinite
number of current sources that produce no external magnetic fields. Because of this nonuniqueness of the current
distribution it is necessary to impose prior knowledge on
the reconstruction. Methods for estimating the source currents at a fixed time from the measured magnetic fields at
(5)
where the tildes indicate that the magnetic fields B(r, t )
are random processes, a distinction that has not been made
until now. The brackets indicate an ensemble averge over
the time t of the biological response. Specifically, where
a discrete time sampling of integer length T is assumed
for a one-dimensional random scalar f ( x , t ) , the temporal
ensemble average is defined as
1
c
T-7
(”%, t + 7 ) f ( Y y , 0 ) = T - T k = I f(x,
k
+M
Y , 4.
(6)
Greek subscripts will refer to the x, y, and z components
represents the
of vectors or tensors. The tensor rErln(7)
correlation of the vector magnetic field at position rl with
the vector magnetic field at position r2 at a delayed time
7 . This correlation can be obtained computationally after
all of the magnetic data has been collected at all measurement positions over the time of the biological response.
This is the tensor form of the mutual coherence function
that was first defined by Wolf for optical fields [15]. It
will be assumed that all random processes are zero mean
in the following development to simplify the expressions.
Equation (5) then becomes
(7)
the corresponding time and the types of prior knowledge
used in these estimates have been reported extensively in
the literature [7]-[ 141.
Instead of forming images of the sources at fixed times
the quantitative spatio-temporal coherence, or correlation, between the sources that arises solely from their biological behavior can be defined. A direct approach is to
determine the spatio-temporal coherence of the magnetic
measurements, and from this estimate the coherence of
the source, without the need to form intermediate images
of the source. That is the approach taken here. The coherence of the sources due solely to their biological func-
where (1) has been used, and the primary and return currents have been lumped together into J to simplify the
development. The labels s ’ and s r r both represent the
source volume. Expressions similar to that above can be
defined to describe the coherence of various spatial derivatives of the magnetic field as well.
It is useful at this point to write down the expression
for the cross product of two vectors in terms of the LeviCivita symbol E [23] so that (8) can be developed further:
[ A X Bl,
=
cP r
eolPyAaBy
(9)
where €123 = €231 = €312 = 1, € 1 3 2 = €213 = €321 = - 1,
with all other permutations = 0. Thus (8) can be written
as
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 39, NO. 10. OCTOBER 1992
loo0
where
[rjr,,,(7)lYp
E ( l y ( r f ,t + 7)l , , ( r f f ,4 )
(13)
the exact counterpart of the magnetic coherence function
of (7). Equation (12) links the coherence of the sources
with the coherence of their mgnetic fields. Note that
rJr,r.
( 7 ) includes the biocoherence of both the primary and
return currents. Because the return currents depend linearly upon the primary currents, the biocoherence of the
primary currents alone can be extracted.
The scalar and vector biocoherences can be directly derived from the tensor biocoherence. This is demonstrated
for the magnetic fields:
where “ ” and “ X ” represent the vector dot and cross
products, respectively.
DISCRETEFORMULATION
To implement the calculation of the magnetic biocoherence from actual measurements, and to develop estimates of the source biocoherence, it is necessary to discretize the entire problem. In doing so, each Cartesian
component of the continuous magnetic data can be approximated by an expansion over a finite set of orthonormal basis functions &(r),i = 1 * * * M , defined in the
measurement space (see Appendix A of [14]):
M
Ba(r, t )
=
C B,i(t) d)i(r)?
(16)
1
with a similar expansion for each Cartesian component of
the continuous source distribution in terms of a finite set
N , deof orthonormal basis functions $ i ( r ) ,i = 1
fined in the source space:
-
t)
=
6..
rJ =
IajCt)
Ic.j(r).
(17)
Note that these orthonormal basis sets may not be complete, and the expansions represented in (16) and (1 7) will
in general provide us with a least squares approximation
to the quantity of interest. It is also necessary to point out
that the 4i (r)basis set is not related to the sensitivity functions (i.e., lead fields) of the mgnetic detectors. The
quantities to be associated with the detector lead fields
’j
+i(r) 4jjcr) d3r9
(6ii = 1 when i = j , it is zero otherwise) the Bai(t)are
given by
they are describing are random processes.
Linking this discussion with experiment, the +i(r) could
represent the spatial sensitivity of a SQUID coil at location i to the magnetic field at the SQUID coil, so that
Bai( t )would be the discrete temporal signal from that particular coil for the a t h component of the field. Again, the
~ $ ~ are
( r )not the lead fields of the system. The lead fields
link the sensitivity of a detector to the source elements,
and are not generally orthonormal. In contrast, the &(r)
basis functions map the continuous magnetic field in a
particular region near a SQUID coil to a discrete detector
output from that coil. The 4i(r)
are independent of the
source model: they are sampling functions. The
are
orthogonal to each other as long as the SQUID coils do
not overlap.
To connect the expansion coefficients of the magnetic
fields to the expansion coefficients of the sources, it is first
necessary to write down the continuous linear expression
relating one vector field B(r) to another vector field J ( r ) :
BJr)
N
la(r9
will be discussed below, when the full matrix formalism
is introduced.
The time-dependent expansion coefficient Bai(1) is
found as follows. Multiplying the left and right sides of
(16) with 4j(r),integrating both sides of the equation over
the domain of 4j ( r ) ,and using the orthonormality condition
=
5
P
s
s’
Qap(r,
r ’ )J p ( r ’ ) d 3 r f
(20)
where Qap(r,r f) is the ath, Pth component of the 3 x 3
shift-variant tensor Q(r,r ’ ) relating the two fields, and s ’
is the region over which Ja(r’)is non-zero. Comparing
the form of (20) with (3), Q(r,r ‘ ) can be written as the
sum of two terms: the first representing the simple vector
product associated with the primary current, and the second representing the conduction-dependent retum current, which also depends linearly upon the primary cur-
SMITH: ESTIMATION OF SPATIO-TEMPORAL CORRELATIONS OF BIOLOGICAL ELECTRICAL SOURCES
rent. Analytic expressions for this second term will in
general be complicated, even for simple conducting
boundaries. Because of this complexity, empirical mapping of n(r, r f) would be more practical and realistic than
its analytic determination. This is discussed briefly below.
If (17) is substituted into (20), and this result then substituted into (18), the following expression relating the
source and magnetic-field coefficients emerges:
~ a i ( t )=
3
N
B
j
c
j’ j’
detector
source
+i(r)Qafi(rT r ’ )
$ , ( r f )d 3 r rd3r Jfij(t).
(21)
By introducing the compound indices m = a + 3(i - 1)
and n = p 3 ( j - l), the above expression can be written in a more compact form as
+
3N
gm(t) =
(22)
where g m ( t ) = Bai(t),L(t)= JBj(t),
and
Am =
=
4 + 3 ( i - 1 ) fi+3(j-1)
1
+;(r)QaP(r,
r f ) $ j ( r ’d)3 r fd3r.
detector
source
(23)
The elements of the 3M X 1 vector g(t) and the 3N X 1
vector f (t) represent a particular ordering of the expansion coefficients of the magnetic fields and the source
fields, respectively, which is defined by the compound indexes m and n. For the case of the data vector this ordering takes the form:
g%>= [B,,(t), BYl(t),Bzl(~)7
Bx2(t),By2(~),
Bz2(t),
*
-
is general enough to include the effects of the conducting
boundaries, assuming that the conduction current is a linear function of the primary current. The A matrix can be
found empirically from phantom studies by mapping out
the magnetic fields produced by a small current dipole situated at sequentially different locations inside a conducting volume. The conducting volume need not be homogeneous: the effect on the return currents of any realizable
conductivity distribution can be built into the A matrix.
The formalism is quite powerful in this regard; the difficult part is actually finding the elements of A for a particular problem.
At this point the connection between the discrete coherence and the continuous coherence can be made. The
covariance matrix (which could also be called the coherence matrix in the present context) of the noisy, discretized measurements can be defined in the following way:
Kg(T)
AmnSn(t)
B,,@), B,,(t),
(24)
with a similar expressin forfT(t). The “T” superscript
indicates the transpose operation. The g ( t ) vector is an
ordered list of the SQUID measurements.
Equation (22) can be written even more succinctly as
g(t) = Af(t)
+ ii(t)
(25)
where A is a 3M X 3N array with elements Amn.The zeromean noise vector Z ( t ) , ordered exactly as the data vector
g ( t ) , has been introduced to represent additive noise processes unavoidably present in the detection operation.
This noise will be correlated and nonstationary in general.
Physically, each row i of the A matrix represents the
lead field for the corresponding ith detector. The lead field
is the sensitivity of a given detector element to the entire
source distribution in the presence of conducting boundaries. Each columnj of the A matrix represents the “pointspread function” (PSF) due to a given source element$(t)
in the presence of conducting boundaries. The PSF (also
referred to as a Green’s function) is the response of all of
the detectors due to a single source element. The A matrix
1001
= (g(t
+” 7
(26)
where it is assumed as before that all random quantities
are zero mean, and again the brackets indicate an ensemble average over time. The matrix K,(7) is a 3M X 3M
matrix. Using (1 8) and (26) and the definitions below (22),
the relationship between the m, nth element of the discrete
covariance matrix of (26) and the a, 0th element of the
continuous biocoherence tensor r B r r ’ (7)is given by
[ K g ( ~ ) I m n= ( g m ( t
+ 7)Sn(t))
(27)
where the indexes m, n are related to CY, 0 and i, j as
discussed under (21). Note that noise correlations are implicitly contained within r B r r ’ ( T ) , because noise will be
present in the magnetic data.
As a specific example of (27) assume that the basis
functions represent shifted delta functions:
4;(r) = 6(r - r ; ) ,
4 j ( r ’ )= 6(rf - 5 )
(28)
so that (27) becomes:
rKg (7)Jmn = IrBr,q (7)14-
(29)
In a real experiment the covariance matrix of (26) will be
formed directly from the discrete data. Equation (27) justifies our interpretation of this matrix as the discrete representation of the tensor I’Brrr (7)for a given choice of
measurement functions +;(r).
Using (25) and (26) it is straightforward to show that,
for noise independent of the object (without restricting
ourselves to stationary noise or stationary object ensembles), the covariance matrix of the noisy data can be written as:
~ ~ (=7 A) K ~ ( ~ +
) K,(T)
A~
(30)
where the covariance matrix of the source is defined as
Kf(7)= ( f ( t
+~)f(t)~>,
(31)
1002
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 10, OCTOBER 1992
a 3N
is
X
3N matrix, and the covariance matrix of the noise
&(7)
=
(ii(t
+ 7)ii(QT)
(32)
a 3M X 3M matrix. Equation (30) is the desired result of
this section, namely a link between the source and data
coherences, and it is the discrete equivalent of (12) with
the explicit addition of noise. The A matrix contains the
physical modeling of both the primary current and returncurrent contributions to the magnetic fields. The source
covariance matrix Kf(7)represents the coherence of the
pirmary currents alone, because all of the return-current
dependencies are lumped into the A matrix. This is possible because the return currents are linearly dependent
upon the primary currents given our assumption of a linear conductor.
The relationship between the source covariance matrix
Kf(7)and the biocoherence tensor of the sources [see (13)]
is analogous to the expression in ( 2 7 ) , with s and s’ both
representing the source volume:
[Kf(~)~mn
( fm(t
p:. The SVD representation of the Q
pseudoinverse A+ of A is
X
P Moore-Penrose
R
A+
e
1
c-viyT
i
Pi
(36)
where the p i are nonzero for i = 1
* R . In practice, the
pseudoinverse of (36) must be regularized to account for
noise amplification and limitations in numerical precision
for small singular values p i . One way this regularization
can be implemented is by replacing the coefficient l / p i
with p i / @ ? + E ) , and take the limit as E goes to zero. A
value of E can usually be found to optimize the performance of the pseudoinverse. A method for modifying
these coefficients to include statistical prior knowledge
about the object and noise has been demonstrated [ 2 5 ] ,
[261*
Using ( 3 0 ) , the MP estimate of the matrix Kf(7)ATis
Taking the transpose of both sides,
+ 7)JI,(l))
again operating from the left with A + , the M P solution
for Kf(7)is obtained:
(33)
K;.(T)= A+(K,(7) - Kn(7))AfT
(39)
where the indicates that the quantity is an estimate. The
ESTIMATION
OF THE SOURCE
COHERENCE
FROM THE
Kg(7)
covariance is derived directly from the magnetic data
MAGNETIC
COHERENCE
[cf. ( 2 6 ) ] . The Kn(7) covariance is derived directly from
The estimation of the source coherence Kf(7),like the noise measurements in the absence of signal. The absence
estimation of the source itself, is an ill-posed inverse of signal could mean that the patient is not present, or that
problem in biomagnetism. This ill-posedness is due to the the patient is present, but is not being stimulated to form
nature of the Biot-Savart operator relating the sources to evoked responses. The A matrix can be found either from
the fields. It is thus necessary to impose prior knowledge mathematical modeling, or as stated earlier from phantom
on the quantity being estimated to regularize the solution. studies by mapping out the magnetic fields produced by a
The approach to estimating K’(T)here will be to find current dipole situated at sequentially different locations
the minimum-norm, least squares solution, also referred inside a conducting volume. The important point is that
to as the Moore-Penrose (MP) pseudoinverse [ 2 4 ] . This the expressions on the right-haFd side of the equation can
choice is made primarily because of its simplicity, and be found explicitly, allowing Kf(7)to be estimated. Note
because of the lack of knowledge of the properties of Kf(7) that 447) may not represent the complete tensor rBr,,J7)
in general. The singular value decomposition (SVD) [24] because not all of the Cartesian components of the magof the A matrix will be used to implement this pseudoin- netic field are measured independently at each *measureverse.
ment position with standard SQUID techniques. Kf(7)may
If a P X Q matrix A is of rank R , it can be expanded still represent the full source tensor rJrjr+),
however, if
in terms of R unit-rank P X Q outer-product matrices the A matrix is constycted generally enough.
formed by the vectors yi and vi:
The source matrix KAT) is in essence forming a highly
parametrized “signature” for a given patient doing a speR
cific neural task. This matrix represents a seven-dimenA =C
piyiq‘
(34) sional data set for a three-dimensional source. The phys1
ical significance of the source matrix can be described as
where
follows. Given any three-dimensional source location of
ATAvi = p?vi and AA’y, = p:yi.
(35) interest and a desired delay time 7 , the three-dimensional
spatial map showing the correlation of all other sources
The Q X 1 vectors vi,i = 1 * * Q, and the P X 1 vectors with the source of interest at that delay time averaged over
yi,i=l
P , are eigenfunctions of A*A and AAT, the evoked response is obtained from the biocoherence
respectively, with eigenvalues (called singular values) matrix. This map can give information about which struc-
-
0
.
.
SMITH: ESTIMATION OF SPATIO-TEMPORAL CORRELATIONS OF BIOLOGICAL ELECTRICAL SOURCES
tures are linked directly or indirectly, on average, to the
source point of interest, and whether this linkage is excitatory or inhibitory. Particularly strong positive or negative correlations can be automatically found from the
matrix, with their associated delay times, and used as reference points for a given evoked response by a given subject. Comparison of biocoherences between normal subjects performing the same tasks would allow common
features to be determined. Significant departures from
these features might indicate abnormal function.
Once the estimate of the covariance matrix of the
sources has been obtained, it can be used, at zero delay
time 7, to produce an estimate of the source distribution
at any time t for which the data has been collected:
This expression is often referred to as a generalized Wiener filter, and has been demonstrated as a way in which
non-stationary sources imaged with shift-variant operators (of which the Biot-Savart law in the presence of conductors is an example) can be estimated [ 141, [25]. This
filter minimizes the mean square error between the source
and the reconstructed source averaged over the source and
noise ensembles. Because of the direct estimation of the
source covariant: from the magnetic covariance in (39),
reconstructions f ( t ) will be obtained that automatically
satisfy the measured magnetic coherences.
SUMMARY
The goal of this paper was to present a quantitative,
formally based framework for unifying the spatial and
temporal aspects of biological current sources. The relation between electric-current sources and the magnetic
fields that they produce was reviewed. Upon this framework the biocoherence tensor was defined, based upon the
mutual coherence function of optics. The term biocoherence was introduced to represent the quasi-static form of
the mutual coherence, suitable for the case of biological
current sources. A practical discretization procedure was
applied to the continuous problem. In the framework of
the discretized form, the Moore-Penrose estimate of the
source biocoherence was formed from the magnetic biocoherence, which can be directly determined from the
magnetic measurements. This source biocoherence may
contain clinical and cognitive information regarding the
performance of the nervous system over an evoked response. It was then demonstrated how the source biocoherence can itself be used as prior knowledge in the optimum linear estimator to form a reconstruction of current
sources at fixed times during the evoked response.
An effort is underway for practical applications of the
theoretical framework presented in this paper. There was
no discussion here of any prior knowledge relating to the
structure of the source and field coherences. Further investigation will be made of possible constraints that the
coherence matrices may be subject to given the effect of
1003
the conducting medium on the sources’ magnetic-field
vectors (e.g., the external magnetic fields due to radial
current components within a spherical conductor vanish
[22]) and the fact that the divergence of the total current
density must be zero. An example of a biocoherence tensor derived from actual SQUID data is unfortunately lacking here; this demonstration will be pursued with multichannel SQUID data when available.
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Warren E. Smith (M’88) received the Ph.D. degree in optical sciences from the University of Arizona, Tucson, in 1985.
He then visited Germany as a Humboldt Fellow
and joined the faculty of the Institute of Optics at
the University of Rochester in 1988 His research
interests span a wide range of topics in medical
imaging He has published in the areas of biomagnetism, automated chromosome analysis,
quantitative endoscopy, magnetic resonance imaging, and nuclear medicine. He is also active in
the application of fluorescent confocal microscopy to the study of biological and nonbiological objects.