278
IEEE TRANSACTIONS ON BIOMEDICAL
Theoretical Analysis of
ENGINEERING, VOL. BME-20, NO. 4, JULY 1973
Field
Potentials in
Anisotropic Ensembles of Neuronal Elements
CHARLES NICHOLSON
Abstract-Synchronous excitation of an ensemble of elements in the
nervous system generates field potentials in and around the ensemble.
Experimental analysis of these potentials can provide information
about neuronal function.
This paper derives a theoretical relationship between field potentials
and the volume source current density produced by transmembrane
currents of members of the neuronal ensemble, under typical experimental conditions. It is shown to be plausible that the potentials produced by a dense bounded ensemble of excited neuronal elements are
adequately represented by Poisson's equation. The ensemble is envisaged as embedded in an anisotropic conductive medium. When the
ensemble possesses some anatomical symmetry and is synchronously
excited, Poisson's equation may be solved to give the field potential as
a one-dimensional weighted integral of the volume source current
density. The weighting is a function of the conductivity tensor, the
neuronal population geometry, and the location of the observation
point relative to the source points.
When the neuronal elements are represented by a core conductor
model, the field potentials may be computed and are shown to agree
with experiments. The present theory should facilitate field potential
analysis by demonstrating effects of anisotropy and ensemble geometry
in a three-dimensional conductive medium.
[12] have not been wholly satisfactory when applied to
certain cerebellar problems [3] .
This paper attempts to put the genesis of field potentials on
a more rigorous theoretical basis and will deal specifically with
the relationship between the field potentials generated by an
ensemble of oriented and simultaneously activated neuronal
elements and their individual transmembrane currents. The
theory will take account of anisotropy in the conductivity of
the neuronal tissue and the size and shape of the neuronal
population. The effect of an interface in an anisotropic
medium will not be discussed here; the effect in an isotropic
medium has been dealt with elsewhere [6] .
Ensembles of oriented neuronal elements are often found
within the central nervous system of vertebrates; apart from
the cerebellum, familiar examples are the retina, olfactory
bulb, tectum, and cerebral cortex. In this paper the cerebellum
will be used for illustrative purposes, although the concepts
presented here are applicable to all the above-mentioned structures. Ensembles of elements can be simultaneously activated
by the use of electrical stimulation; consequently, the requirements of orientation and simultaneous activation are not restrictive, since in most experiments of interest they are
satisfied. These requirements do, however, enable the theory
to be developed in a relatively simple form.
INTRODUCTION
F IELD POTENTIALS may be defined as the bioelectric
potentials generated by an ensemble of neuronal elements
and recorded from the extracellular space. The neuronal
elements comprise the dendrites, somata, and axons of nerve
cells and in the present context the recording electrode is enTHEORY
visaged as a saline-filled micropipette having a tip diameter of
about 2 ,um. In practice, the relatively large tip of the recordThe elements of the theory to be developed here are as foling electrode produces localized destruction of the neural lows. The neuronal tissue can be divided into an extracellular
elements and so creates a minute conductive space about the and an intracellular space, the current flow within each of
tip, free from generators of large unitary potentials that might these regions being governed by different physical factors.
otherwise distort the field potentials [11 . This effect may also
Field potentials are generated by current flow in the extrabe enhanced by leakage of the concentrated saline solution cellular space, which is a three-dimensional conductive medium.
from the electrode tip.
The extracellular current arises from a system of sources and
When a population of neurons possesses some anatomical sinks resulting from the passage of current through the memsymmetry, field potentials can provide important information branes of the individual neuronal elements. This transmemabout the functional properties of the neuronal circuits. Re- brane current may be generated by synaptic activation and by
cent studies of field potentials have been particularly success- action potentials produced either by synaptic activity or by
ful in elucidating the neuronal properties and interactions in direct electrical stimulation. It will be shown that the extrathe cerebella of different vertebrates [2]-[6]. Nonetheless, cellular current flow can be adequately described by Poisson's
the interpretation of field potentials rests mainly on an em- partial differential equation applied to the extracellular space.
pirical understanding [2], [7]. As detailed elsewhere [61, The symmetry of the neuronal ensemble enables the solution
attempts at theoretical explanations of field potentials [8]- of the three-dimensional Poisson equation to be given as a
one-dimensional weighted integral of the current source denManuscript received January 18, 1972; revised July 24, 1972. This sity term. The weighting function in this integral contains all
work was supported in part by the U.S. Public Health Service under
Grant NS09916-01 from the National Institute of Neurological Dis- the information about the anisotropy of the extracellular
eases and Stroke.
The author is with the Division of Neurobiology, Department of medium and the size and shape of the neuronal population.
Physiology and Biophysics, University of Iowa, Iowa City, Iowa 52242. The field potential may be evaluated if the current source
279
NICHOLSON: FIELD POTENTIALS IN NEURONAL ELEMENTS
density term is known, and this may be derived either from a
model of the intracellular space of the neuronal elements or
from experimental data.
Equation for Potential in an Anisotropic Medium Containing
an Ensemble of Excited Neurons
It has been shown that when a current having a frequency
below 35 kHz flows in an extended neuronal tissue, it is distributed as though the medium were purely resistive [13] and,
at the frequencies normally encountered in nervous tissue, the
electric field may be regarded as quasi-static [14] -[16]. These
experimental facts imply that little extracellular current flows
through the cell membranes, since all membranes have a large
capacity, and hence that the division of the medium into extracellular and intracellular regions is justified. Thus for the
purposes of this paper, the extracellular space is taken to be
electrically equivalent to an extended homogeneous anisotropic
conductive medium, the properties of which remain constant
for the duration of plausible experimental measurements. This
representation gives useful results for field potential analysis,
but does not imply any statement about specific anatomical
channels for extracellular current and does not preclude a
more complex representation of extracellular space when
adequate data are available.
From the point of view of the extracellular medium, current
can be envisaged as disappearing at certain points (flows in
through a membrane) and appearing at other points (flows out
through a membrane) [17]. To a good approximation it has
been shown that, in the case where neuronal elements may be
represented as core conductor cables (i.e., where the length is
much greater than the diameter), the transmembrane current
flow depends only on the intracellular parameters [18], [19].
Consequently, it is realistic to regard the neuronal membranes
as providing a system of sinks and sources of current with respect to the extracellular medium. In the cerebellum, for example, the neuronal elements are densely packed [Fig. l(a)
and (c)] and consequently when the ensemble is simultaneously activated, the sources and sinks of current are numerous and dense and may be regarded as being continuously
distributed within the finite volume defined by the ensemble
of active cells. This will now be discussed in more detail.
Consider a closed surface S, within the neuronal tissue, containing a volume V [Fig. 1(d)]. Let the volume contain m
core conductors each having a surface Mi within S, and let
these cores intersect the surface at n disks Nj (assume no cores
lie wholly within S, but that some may terminate there; then
2m > n > m). Let the surface S, less the disks Nj, be S', i.e.,
S'= S - Lln I Nj. Let the current flowing out across the surface S' be denoted by the vector J and the current flowing out
across Mi be Jm (the transmembrane current is defined here to
be outward with respect to the inside of the core conductor).
Then the region enclosed by S' + Zs= Mi is closed and contains no sources or sinks; hence, from continuity of the
current,
fJ-ds- E
S'
=1
i
Jm*ds=O
(1)
Recording
Electrod e
-Local
+
Electrode
2!i~-
0,
1a)
(c)
(bI
-Z
/
L
i,th Core Conductor
-
Synaptic
/activation
Population of
synapticolly
octivated
cells
I
2b
(e)
Cd)
Fig. 1. Neuronal ensembles and their representations. (a) Simplified
diagram of the region of the cerebellum; Purkinje cell (Pc) dendrites
lie in planes at right angles to the parallel fibers (pf), which are tshaped axonal extensions of granule cell (gr. c) axons (ax). A local
electrode can stimulate a superficial beam of parallel fibers and
field potentials resulting from the activation of pf-Pc synapses are
measured by the micropipette (recording electrode) and amplifier.
(b) Accurate drawing of a single Purkinje cell (alligator) showing extensive dendritic ramification. (c) Superimposition of nine Purkinje
cells to show the density of dendrites, which approximate a continuous distribution of core conductors. (d) Derivation of the
Poisson equation. The arbitrary region is bounded by surface S containing volume V, through which core conductors with surface Mi intersecting S in disk Nj are passing. Current J crosses S and Jm crosses
Mi. (e) Representation of the neuronal population for the derivation
of the weighting function. The synaptically excited population is confined to a rectangular volume 2a 2b c. A synaptic input causes a
distributed current sink in the x-y plane at z = 0, while the rest of the
volume becomes a distributed current source, due to vertical orientation of the core conductors (dendrites).
-
-
where ds is an element of surface having a direction parallel to
the outward normal at the center of the element.
Consider an arbitrary vector field P in the volume V enclosed
by S; then the Gauss' divergence theorem implies that
fP.ds+ fP-ds=fP-ds=f V Pd3x
S
j=l NjS
where d3x is a volume element of V.
Define P to be J on S' and zero elsewhere; then from (1),
VVPd3x= fJm ds.
~~~~i=li
V
(2)
Define a volume source current density Im in V equal to the
"smoothed out" transmembrane current flux such that
f Imd3x=
V
~~~i=i Mi
Jm ds.
(3)
Then, since V is arbitrary, using (2) and (3),
v -P=Im.
However, on the boundary of S, P = J, so that, external to V,
the field due to the current emanating from V will be identical
for the vector fields P and J. Inside V, the smoothing process (3) and the fact that P exists in regions of V where J does
not will mean that P and J are not identical; however, V may
be made very small, so that P and J are identical over almost
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, JULY 1973
280
all the tissue. Hence, the equation for the current density
within the neuronal tissue may be written
V - J=Im
where Im is defined by (3).
Since the medium is purely ohmic, the relation
J = uE
Then in the t space, this defines a volume U(z, ,B, y) outside of
which I(t) = 0; hence, using Green's theorem,
(4)
(tdd 3 t
(h')r)=
= 4 JutI-t
(7)
This is the potential at an observation point t', produced by a
source distribution I(Q), and is a standard result for the solution of Poisson's equation.
holds, where a is the conductivity tensor and E is the electric
field. The quasi-static assumption implies that the electric
field is solenoidal; hence, it may be represented as the negative Simplification ofIntegral for a Simultaneously Activated
gradient of a scalar potential 0; thus E = - Vq. Then (4) may Population of Oriented Neuronal Elements
The triple integral (7) may be reduced to a single integral by
be written
making use of anatomical symmetry. Fig. l(a) and (e) illusV * OV = -Im.
(5) trates the anatomical basis for this in a typical cerebellar exIn general, a will be a tensor with nine components. In periment. A beam of parallel fibers is excited by direct
several neuronal tissues the local geometry conforns to a electrical stimulation and the wave of excitation activates the
natural rectangular Cartesian coordinate system. In the cere- superficial dendritic synapses of a population of Purkinje cells.
bellar cortex, for example, the Purkinje cells are oriented The activity of the parallel fibers may be ignored here, but the
perpendicular to the pial surface with their dendrites forming behavior of the Purkinje cells, which form a population of
a system of parallel planes, while the parallel fiber system then oriented core conductors, is of importance in the study of
runs perpendicular to these planes [see Fig. l(a)] [20] -[231. cerebellar function. It will be assumed that the parallel fibers
The presence of a natural system of Cartesian axes implies are parallel to the Cartesian x axis and the dendritic trees of
that a will only have three principal components aii (i. = x, the Purkinje cells lie parallel to the y-z plane, with the main
y, and z). In the case of more complex tissues, a will be a soma-dendritic axis parallel to the vertical z axis. The form of
symmetric tensor [24]; thus there is always a rotation of the stimulation described above synchronizes activation of the
coordinate system to the principal axes, which reduces the superficial Purkinje cell dendritic synapses, creating an inward
tensor to its three main components [25]. It is assumed that synaptic current at that level together with an outward transthe components of the tensor are constant; then, designating membrane current at lower regions, govemed by the passive
membrane properties of the cell; in some species there may
the components by ox, ay, and ao, (5) becomes
also be active processes [3], [6], [27]. Within the excited
neuronal
population the current entering or leaving the extra=
z).
(6) cellular medium
OIX ax2
is a function of only a single spatial coordinate
az
Y
the
since
current
is determined by intracellular cable properz,
The conductivity coefficients may be eliminated by a coties
and
not
by
extracellular
conditions. Hence,
ordinate transformation defined by
aX2
'
J
a2 +aa -Im (x,y,
j
t = x/V , n =y/lV ,
z
=zl/V
whence
V20 -I(tN¢
I(t) =M
Then integral (7) becomes
(h')4=7-
¢
I(¢) dD 33
0
in the t space, where
Q,
This is Poisson's equation [25], [26] for the potential due
dqd~
to a continuous source distribution.
N/Gt _ 2 + (711 772
Poisson's equation may be reduced to an integral by the use
of Green's theorem [26]. Assume that the extracellular Evaluation of the Two-Dimensional iq-t Integral
medium is -very large so that the surface integral, implicit in
Define the new variables as
Green's theorem, is neglibly small, and let the population of
active cells be confined to a rectangular volume defined by
p=_ q=q=7t 7?? -=
a
> x > -a, b > y > - b, c > z > O
and let
A2 =(x' a)/
a.
so that
-
B2 = (y' - b)/\
bc
:-b/
/ > ba/0-.
> rl0.by-
C2 =(z'--c)/O
Then integral (7) becomes
+ Na)/
A
=A(x'
1
B1 = (y' + b)/
Cl=z'/y.
+
It
t
281
NICHOLSON: FIELD POTENTIALS IN NEURONAL ELEMENTS
JBfBi
2
Al
dqdp
Considering now the integral
L=ln(q2 +r2)dq
B~~~~~~~2
Since 1/Ir2 +
+ p2 may have a singularity whenever any
coordinate passes through zero, it follows that if the domain integration by parts and a trigonometric substitution of the
of integration includes this point it is convenient to split the form q = r tan 0 leads to the expression
range of integration into two parts; e.g., if A1 > 0 > A2, the
L = q ln (q2 + r2) + 2r * arctan (q/r) - 2q.
integral is split into the two ranges A1 > p > 0 and A2 >
p > 0, and similarly for case B1 > 0 > B2. Thus three cases Noting that
arise: 1) A 1 > 0 > A2, B1> 0 > B2 ; 2) A 1 > 0 > A2, B1 > arctan {-(A2 +r2 +A VA2 + q2 +r2)/rq}- arctanq/r
B2 > 0 (the case A 1 > A2 > O, B1 > 0 > B2 is symmetrical
=arctan{rN/A2+q2+r2/Aq}
since 1/r2 + q2 + p2 is a symmetrical function, and need not
be considered separately); and 3) AI >A2 > 0,Bj >B2 > 0. and writing
Ki1 = VA? +I B2I + r2, then substituting for B1 and
Case 1) will be considered in detail; the other results may be B2 and collecting
terms, the integral H(r) is finally derived as
derived by similar methods. Let
H(r) =[M(A I )+ M(A2 ) + L] B2
F A=
dp
-
FJA
2
up2+q2 +r2'
+(Al KKll) (A2 +K21)If
-B1ln {(Ai
B2+r2
Then [28, appendix E, integral 192] yields
F = [ln (p + vp2+q2 + r2
Hence, splitting the range of integration, as described above,
and settingA2 = IA2 I,
F=ln(A + A +q2 +r2)+In(A2 + A +q2 +r2)
f=
F(q) dq= f
B2
dq +
F(q)
0
F(q) dq.
rK
(A
+ N/A2 + q2
+r2) dq.
+
A2+q2+r2
Case 2) is Al > A2 > 0; BI > 0 > B2 . In this case it is not
modify the range of the p integral to incorporate
the origin, and it may be shown that
/A2 +qq2 +r2) Jf
w
H(r) = A In {(B
dw
and
W=w2 -2Aw-r2
Then from [28, appendix E; integrals 260, 241, and 2481,
A In
(2VBW- + 2w
-A2
K
1) (B2
(A 1+
+B1,ln +A'
K,1)
(A2 +K21 )
-
r arcsin
2A)
-
(Aw
r2
+ const
whence, resubstituting for W and w and converting the arcsin
to arctan,
M(A) = q In (A + N/A 2 +q2 + r2) q + A ln (q + V/A2 + q2 +r2)
+r- arctan {-(A2 +r2 +A V/A2 +q2+r2)/rq}
-
+
-
+
K12)l
{(Bi +K21)(B2 +K22
n
+ r larctan
const.
(8)
necessary to
where
+
2- 27rr.
A2B2
ln
-
arctan.r2
+ arctantrI
A2B1
A1B2
rK22
+ arctan A
M(A)=qln(A
fJ_ dw = VhW
+Al lIn nt (B1 +Kll)(B2
Al +r2 +K12)}
+
+ rfarctan
{AIB,
Integrating by parts,
w=A+
K12)
1) (B2 +K22)l
{(1 K2
+A2 In { (B1+
A2 +r2
Consider the integral
M(A)
+
-ln(q2 +r2).
Now let
FB2
(A22I
+K22)I
(Al
{(A1B2+r
+B2
In
+B2ln
arctan
rK,
1
A1B,
rK22
B .
A2B2
+
+ B2 lnI
arctan
(Al1+K12)
(A2
rKl 2
A1B2
+K2
-
arctan
rK2I
A2B,
(9)
For the case A 1 > 0 > A2,B1 >B2 > 0, it is only necessary
to interchange A1 with B1 and A2 with B2 in the above
equation.
Case 3)isA1 >A2> 0;B1 >B2 >0. In this case the origin
does not enter into either range of integration and thus
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, JULY 1973
282
B2 In
H(r) = B In (A-K1
+
(A2
K21)
ctan
A2B2
- arctan
25%
16%
7%
4%
44%
33%
21%
13%
8%
4%
A2B1
1,1,16
.KB2I
(10)
A,B2f
Field Potentials as a Weighted Integral of Volume Source
Current Density
Equations (8)-(10) are explicit functions of r = z' - z /and implicit functions of x' and y' through their dependence
on Ai and Bj; thus from (7), the potentials within an infinite
medium may be expressed in the x space in the form
(c
(x' y z ) = fu
-
G (x', y',z', z)Im(z) dz
or
Jc
(x') =
14%
2-
A1B1
0
30%
0
(B2 + K22)
.arctan
- arctan
2.0
1.0
63%
100%
2'
(B1 +K21)
(B2 +K12)
0.5
_wmm~
(A2 + K22)
(B1 +K11)
+r
0.0
A'K2
f
G(x',z)Im(z)dz
(11)
0
where
G (x', z) =
I
o
H(r)
and H(r) is one of the expressions (8)-(10). This is a weighted
integral of the volume source current density, where G is the
weighting function that describes the "importance" which the
recording electrode assigns to sources and sinks of current in
its vicinity. Since G is a function of z' z, (I1) may also be
regarded as a convolution integral.
-
THE NATURE OF THE WEIGHTING FUNCTION AND ITS
RELATIONSHIP TO FIELD POTENTIALS
The weighting function incorporates information about the
anisotropy of the extracellular medium and the size and shape
of the neuronal population. In evaluating the weighting function it is convenient to introduce spatial variables X, Y, and Z
in terms of the length constant (see Appendix I) of the
neuronal elements, i.e., X = x/X, Y = y/X, and Z = z/X. It has
been shown [6] that in the cerebellum, the depth of the
Purkinje cell dendrites is about 2 X and a typical population of
excited neurons is roughly 1 X in diameter. Consequently, the
weighting function will be evaluated for i Z' - Z < 2 and a
typical neuronal population will be taken as having a square
cross section in the X-Y plane defined by a = b = 0.5, except
when the size is specifically varied.
The weighting functions corresponding to four locations
along the X axis from X' = 0 to X' = 2 are shown in Fig. 2.
The observation point corresponds to the peak of the function.
The upper row of figures shows the weighting function for
an isotropic medium. At the center of the population the
function peaks sharply, while at the edge there is less rapid
fall off of the curve. The magnitude of the peak of the func-
16,1,1
0
4
gV
1,16,1
44%V
25%
.
0:
Fig. 2. Weighting functions G for different anisotropies and distances
from the center of the neuronal ensemble. The functions are all
plotted on the same scale for Z < 2 A, with the four different
anisotropy values indicated by the triads (ax, ay, and uz) to the left
of the figure. Horizontally, the four sets of functions correspond to
weighting functions, parallel to the Z axis, with origins 0 X, 0.5 X,
1.0 X, and 2.0 X along the X axis. Thus these functions represent progressively more lateral recording positions with respect to the axis of
symmetry of the neuronal ensemble. The ensemble has a square
cross section in the X-Y plane of side 1 X (shading indicates extent of
ensemble). The percentages indicate the peak values as a function of
the central isotropic value (x = 0.0 X, a = 1, 1, 1).
tion is diminished (expressed as percentage of the peak value
in the center for the isotropic case) with distance from the
center. Within the population there is a discontinuity in the
curvature, at the observation point, which is not present outside the population; this is in accordance with the presence of
sources inside the population and their absence outside.
The second row of curves corresponds to a 16-fold increase
in the a, component of conductivity relative to the other
components. This results in a less rapid fall off of the function
with distance from the observation point and a diminished
peak amplitude. On the other hand, if the a. component of
conductivity undergoes a 16-fold increase with respect to
the other components (third row down, Fig. 2), the weighting
function becomes more localized with respect to the isotropic
A 16-fold increase in the ay component leaves the
case.
central weighting function the same as in the last case, but
from the center the function becomes less localized.
Two effects are apparent. Firstly, all weighting functions
are most sharply localized along the axis of symmetry of the
neuronal population and then flatten out with increasing
laterality. Secondly, anisotropy can modify the shape of the
weighting function, particularly anisotropy in the direction of
the axis of symmetry of the population, i.e., in the a,
away
component.
When the cross section of the population is ten times smaller
(Fig. 3, A = 0.05) than normal (A = 0.5), the weighting function is more sharply localized, while for a tenfold increase in
size (A = 5.0) the curve falls off much less rapidly. When the
283
NICHOLSON: FIELD POTENTIALS IN NEURONAL ELEMENTS
A=B-0.5
A=B,0.05 X
V
0.0
0.4
=--------=
08
----
------
---
1. 2
1.6
A=B=50.0
A=B=5.0
b~
~~ ~
~
~
~
~
=
2.0
~~~~~i
(a)
Fig. 3. Weighting functions G for different sizes of the excited neuronal
ensemble. The ensemble has a square cross section in the X-Y plane
of side 2A; the value of A is indicated above each graph. In each case,
the weighting function has been normalized so that the peak values
all have the same magnitude, thus emphasising differences in shape of
weighting functions. In the three sets of curves, i corresponds to
isotropic case a = 1, 1, 1, a corresponds to a = 1, 1, 16, and b corresponds to a = 16, 1, 1. In order to normalize the curves, a has been
multiplied by 4.0 relative to i and b has been multiplied by 2.3. For
different square ensembles, the peaks are proportional to the dimension of the cross section [see (8)]. Only half the weighting function
is shown (c.f., Fig. 2), since the function is symmetrical.
population is 100 times the normal size (A - 50.0), the weighting functions approach a straight line parallel to the z axis and
differences induced by anisotropy are very small. The magnitude of the peak value of the weighting function is linearly related to the population size [see (8)-(1 1)].
By making a and b unequal in (8)-(l 1), the effect of an
arbitrary rectangular population can be calculated. It may be
noted that there is a duality between certain anisotropy problems and certain population shape calculations. The parameters that actually determine the weighting function, in relation
to variables in the X-Y plane, are A1, A2, B1, and B2 [see
(8)-(l 1)]. These parameters are functions of X', Y', a, b, ax,
and ay and any combinations of these variables that result in
the same values of A and B will produce the same weighting
function. Precisely, suppose that the two sets of values X',
Y', a, b, ax, and ay and X', Y',a, b, ax, and ay result inthe
same values of A1, A2, B1, and B2; then the following relations must hold between X' and X':
,
X'\f=X'V=
and
aV= Va
.
Similarly for Y' and Y',
Y'Vj=
Y'V%j
and
b
y
=b fy.
Relationship Between Transmembrane Potentials, Currents,
and Field Potentials
In order to compute the field potential, the volume source
current density term Im(z) must be known. This implies
either the selection of a model to represent the neuronal elements or independent experimental measurements. A model
(b)
{c)
Fig. 4. Comparison of the transmembrane potential and current for a
core conductor and extracellular field potential resulting from an ensemble of such conductors. (a) V: Transmembrane potential (potential difference across membrane for single core in infinite medium).
Potential measured at distances indicated from the closed end of the
core. (b) Im: Transmembrane current. The inward current is total
inward currenit; the outward current is the current density per unit
length [see (14)]. (c) 1: Field potentials are measured with respect to
remote indifferent electrode (effectively at infinity). Calibrations result from numerical values given in Appendix I. Isotropic medium
and neuronal ensemble of square cross section, side 1 X, used for
field computation.
is chosen here and the neuronal element represented as a uniform, unbranched, vertically oriented core conductor terminated by a sealed end and extending infinitely in the other
direction. An ensemble of such core conductors approximates
a population of Purkinje cell dendrites in the cerebellum [6].
The flow of transmembrane current may be initiated by the
activation of the peripheral synapses of the Purkinje cells following a synchronized volley of action potentials in the superficial parallel fibers [Fig. 1(a)] . The formulas and parameters
used are detailed in Appendix I. Using (13), the transmembrane
potentials of a core conductor were calculated at a sequence
of locations from the sealed end, and are shown in Fig. 4(a).
The waveforms are characteristic of electrotonic potentials in
a leaky cable, following an inward current at the point Z = 0.
The potentials are positive-going monophasic and decrement
rapidly with an increase in the latency of the peak. with depth.
The transmembrane current densities associated with the
above potentials [(12) and (14)] are shown in Fig. 4(b). The
inward (synaptic) current (i2) is shown superimposed on the
outward current at the point Z = 0. Note that the inward
current is total inward current, while outward current is current density per unit length in this figure. The transmembrane
currents have sharper waveforms than the potentials, corresponding to the fact that they are the second spatial derivative
of the transmembrane potential. The current is outward across
the membrane at all points except the synapse; thus there is a
current sink at the synapse with respect to the extracellular
mediumn and a distributed source elsewhere.
Fig. 4(c) shows the field potentials computed from (12) and
(14) and the weighting function given by (8) and (11) for observation points along the central axis of a square population
of vertically oriented core conductors. The main features of
these waveforms are 1) the superficial negativity; 2) the reversal of the negative wave to a positive one with depth; 3) the
earliest part of the negative wave reverses first; and 4) the peak
positivity attains a distinct maximum, as a function of depth.
284
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, JULY 1973
5msec
1 25
jio mV
|0.3mV
-04
<
i. 2
1.2
__/
1.6
2.0
X=OX
X=1
X=2)<
Fig. 5. Field potentials at different distances from the axis of symmetry of the excited neuronal ensemble. The ensemble has a square
cross section of side 1 X and the medium is isotropic. Potentials are
calculated on the axis of the ensemble (X = 0 X) and at positions outside the ensemble on the X axis (X = 1 X and X = 2 X). Field potentials external to the ensemble are scaled so that surface negativity has
approximately the same magnitude as the central value (scaling indicated by calibration). See Appendix I for numerical values used in
the computation. The depth at which the field potential is calculated
(Z coordinate) is indicated on left.
0. 0
-------- -----
jIomv
5msec
12m
2.5 mV
--
0.A
0.8
1.2
-11;w-.....
ponents. Fig. 6 shows the field potentials resulting from independent 16-fold increases in the a,, and a, conductivity
components, with respect to the isotropic case (Figs. 4 and. 5).
Increasing the ax component produces a faster reversal of the
field with depth and decrease in overall magnitude, especially
in the positivity. Increasing the a, component diminishes the
amplitude of the negativity with respect to the isotropic case,
but the positivity is less affected; thus the negativity and positivity have similar magnitudes.
The above examples enable the effect of population size on
the field potential to be predicted. When the population is
small the weighting function is small and sharply localized; the
superficial negativity will therefore be much larger than the
positivity. As the population size increases, the weighting
function flattens out and the size of the positivity relative to
the negativity increases.
DISCUSSION
The basic concepts of the genesis of field potentials, expressed in this paper, are that in neuronal tissue containing a
densely packed population of excited neuronal elements, the
field potentials are generated by current flowing in a threedimensional resistive medium formed by the extracellular
medium. The potentials may be described by Poisson's partial
differential equation with the current source density term for
this equation supplied by the transmembrane current of the
neuronal elements. When there is sufficient anatomical symmetry the solution of Poisson's equation can be reduced to a
one-dimensional integral:
)=
o=16,1,1
Fig. 6. Effect of anisotropy
X
c
G(X', Z)Im(Z) dZ
o=1,1,16
on field potentials. Neuronal ensemble of
section with side 1.0 X. The relative anisotropy indicated
below the figure ("1" corresponds to a a value of 0.004 mhos/cm).
Right-hand set of potentials is scaled to give approximately the same
superficial negativity as the left-hand set. The scaling is indicated by
calibrations. See Appendix I for numerical values used in the
computation.
where Im(Z) is the volume source current density term and G
is a weighting function containing structural information
about the neuronal tissue. In practice, G takes account of
anisotropy in the conductivity and size and shape of the
neuronal ensemble. G is a function of the distance between
observation point Z' and source point Z, with a maximum
All these features resemble those found in experimentally in- value when this distance is zero. This implies that the weightduced field potentials following local stimulation of the cere- ing function is a measure of the "importance" that a microbellum [2], [4], [6]. On the other hand, it is clear that the electrode attaches to the current density at varying distances
field potentials are dissimilar in waveshape from both the from its tip (Fig. 7). The shape of the weighting function thus
determines the relative contribution of sources and sinks of
transmembrane potential and the current density.
current at different distances from the tip.
Variations in Field Potentials as a Function of Observation
In one experimental study [13], the three components of
Point in an Isotropic Medium and as a Function of
the conductivity tensor were measured for the frog cerebelConductivity in an Anisotropic Medium
lum. The experimental values (ax = 0.014 mhos/cm, ay =
The field potentials corresponding to recordings parallel to 0.002 mhos/cm, and a, = 0.005 mhos/cm) have been inserted
the Z axis at the center of the neuronal ensemble, and 1 X and in (8), and the resulting weighting function together with the
2 X along the X axis, are shown in Fig. 5. The neuronal popu- isotropic function are shown in Fig. 8(a). It is seen that for
lation was assumed to be a column (1.0-X square cross section) the above data, anisotropic and isotropic weighting functions
and the extracellular medium was assumed isotropic. These are virtually indistinguishable in shape; thus the field potenresults show that the superficial negativity wanes with distance tials would be almost the same. This result is for observation
from the center, while the positivity is accentuated. The over- points along the axis of symmetry; off this axis computation
all magnitude of the field potentials drops rapidly outside the shows that the similarity of the anisotropic and isotropic
neuronal ensemble.
cases is less marked, but still close enough to make it difficult
The modifications of the field potentials due to anisotropy to distinguish between the two sets of resulting field potencan be illustrated by varying the a, and a_ conductivity comtials. Thus in this case at least, the assumption of isotropicity,
square cross
NICHOLSON: FIELD POTENTIALS IN NEURONAL ELEMENTS
285
cylindrical cross section of radius B, embedded in an isotropic
medium of conductance a. The expression in the latter
case was
G(Z,Z) =(1/2a) (Z('Z)2 +B2 - IZ' Z ).
The cylindrical weighting function is compared with that derived in the present paper for the axis of a square population
in Fig. 8(b). The peak magnitude of the cylinder function is
simply 0.5 B/u, while from (8) the peak of the square function
(with side 2B) is
([2 ln (I +V)] fir) (B/a) = 0.561 B/a.
Thus from Fig. 8(b) and a comparison of peak values, it is
seen that the cylinder and square functions are very similar.
Fig. 7. Weighting function concept. Typical Purkinje cell outlined at This demonstrates that the exact shape of the population
left; this cell is one member of a bounded neuronal ensemble, which
receives synchronous synaptic input near superficial dendritic tips. cross section is not very significant, for a typical neuronal
This input causes transmembrane currents, which flow through ensemble, in determining the weighting function. This result,
the extracellular anisotropic resistive medium producing the field potentials. The potential is recorded by an electrode at the recording together with the data shown in Fig. 8(a), also shows that the
point. The geometry of the neuronal population and anisotropy of simple model consisting of a cylindrical population in an
the medium determine the "weight" which electrode attaches to
current sources and sinks at different distances from the tip (weight- isotropic medium is an adequate approximation for many
ing function). Integrated product of weighting function and trans- cerebellar problems.
membrane current gives the field potential. Current and potential
Several previous theoretical analyses [161, [18], [29], [30]
profiles are calculated for time 1.0 T. The weighting function, current
density, and field potential are in arbitrary units.
have dealt with the extracellular potential arising from a single
cell or cable model in a volume conductor. The case of an extended bounded ensemble of neuronal elements considered
here has not received so much attention. Previous theoretical
explorations of the latter problem [8]-[12] have been reviewed and are discussed elsewhere [6].
The effects of conductive anisotropy on field potentials are
usually ignored, in part because of the difficulty of measuring
three conductivity components in a small region of the brain
and in part because of a lack of a simple method of appreciating the effects of anisotropy. The weighting function con(a)
cept put forward here provides a relatively easy method of
displaying the possible influence of anisotropy and may encourage more extensive experimental determination of conductivity components. The present paper demonstrates that
anisotropy can have a substantial effect on field potentials;
the effect of dissimilar components of anisotropy is not intuitively obvious, however, as is evident from the example of
frog cerebellar data given above.
The field potentials derived in the present paper show good
in their general features with those recorded exagreement
(b)
perimentally [2], [4] -[6], [22], [23]. They also throw light
Fig. 8. Validity of theory. (a) Comparison of weighting function a for on some points of practical significance. First it is clear that
experimentally measured anisotropy in frog cerebellar cortex (a =
0.014 mhos/cm; ay = 0.002 mhos/cm; uz = 0.005 mhos/cm; [1]). for an ensemble of neurons, synaptically activated in their
Isotropic weighting function i is scaled to give the same peak value. superficial dendrites, the ratio of the ensuing superficial nega(b) Comparison of weighting function for cylindrical neuronal population of radius 0.5 (curve b) and square populations just circum- tive and deeper positive field potentials is significantly affected
scribing cylinder (curve a) and inscribing cylinder (curve c). Weight- by the geometry of the neuronal ensemble and the anisotropy
ing functions are scaled to give the same peak values.
of the medium. Thus variation in this ratio is a rather complicated function of change in synaptic activation. A second
although false, would lead to the computation of quite accu- point is that the deep positivity will increase relative to the
rate field potentials.
superficial negativity as the observation point becomes more
The present paper has derived the weighting function for re- lateral with respect to the axis of symmetry of the neuronal
cording positions both on and off the axis of symmetry of a ensemble (Fig. 5). A third point is that, despite the relative inpopulation of cells of arbitrary rectangular cross section em- crease of the deep positivity, a superficial negativity can still
bedded within an anisotropic medium. An earlier publica- be recorded outside the ensemble of synaptically activated
tion [6] derived the weighting function for recording positions cells (Fig. 5). Thus the presence of a negativity is not an inon the axis of a population of neuronal elements having a fallible sign of a current sink. This effect has also been noted
X
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, JULY 1973
286
.L
Synoptic
|
I'
|Extracellular
4-
Dendritic
cable
medium
(13)
Introcellular
m
Membrone
applied at a depth A below the closed end of a cable, then the
transmembrane potential difference V at the point Z is
V -eX(Ri + Re) e-T { e-(A -Zz+K)214Te-(A+Z+K)214T}
Im
-
activation
medium
X
R
1 1
The transmembrane current per unit volume may then be
written as [6]
RCm
E
R
Re
Fig. 9. Dendric core conductor and its electrical analog. The core conductor consists of a membrane having resistive Rm and capacitative
Cm elements and a source of EMF E, which separate the longitudinal
resistive core Ri from the resistive sheath of the extracellular medium
Re associated with the core conductor for the purpose of calculating
transmembrane quantities.
for a single neuron in a volume conductor [29]. All the above
effects are reflected in the shape of the weighting function
and are a direct consequence of the behavior of an electric
current in a three-dimensional resistive medium containing
extended current sources and sinks. It is thus important to
recognize these effects before trying to make inferences about
underlying neuronal mechanisms.
It is frequently possible to utilize anatomical information to
assess which factors are likely to dominate a given recording
situation; in this case, the theory presented here may be useful
in interpreting the physiological process occurring within the
neuronal elements. Prior use of the theory may also be helpful
in designing experiments, in that it can be used to establish
whether field potential measurements are likely to yield useful
information in a given situation.
APPENDIX I
NUMERICAL VALUES
The application of core conductor theory to Purkinje cell
dendrites and the solution of the cable equations is detailed in
[6]. This Appendix lists the formulas derived in [6] and
describes the numerical values used in the present calculations.
The dendrites of a Purkinje cell are represented by a population of one-dimensional cylindrical core conductors having the
form and electrical analog depicted in Fig. 9. It is customary
to introduce the time constant T = CmRm and the length constant X = VRm/(Ri + Re), where Cm is the membrane capacity per unit length of core conductor, Rm is the membrane
resistance across a unit length of core conductor membrane,
Ri is the internal resistance per unit length of the core conductor, and Re is the external resistance per unit length of the
region of extracellular space per unit core conductor. Then
time and distance may be expressed in dimensionless units:
T = t/r and Z = z/X. The synaptic input current may be approximated by the expression
Iin (T) =
-e
eT eK24T
(12)
where e is a constant having the dimension of current and K is
a factor, measured in units of X, which determines the shape
of the synaptic input function. If this synaptic input is then
Im(Z,T)
-v{2 (R
R)aZ2
+ IT
6(Z - A)}
I [(A - Z I +K)2
2X/t2 L
2T
vee T
+
I
[(A +Z+K)
e-(IA-z+K)2/4T
e-(A +Z+K)2/4T
_
(14)
-K5 (Z - A) e-K2/4T}
where 6 (Z - A) is the Dirac delta function and v represents
the number of core conductors passing through a unit volume.
The field potential will then be given by (11) for each instant
of time.
In order to assign numerical values to the results, the various
parameter values must be inserted. The values chosen for the
parameters used in the calculations were rm = 6750 2 * cm2;
cm = 1.0 ,F/cm2; ri = 500 Q cm; re = 250 Q cm; p = I ,m;
and p' = S gim. The value of re is not critical for the calculation of intracellular parameters, so that an isotropic value
suffices. For the purpose of intracellular core conductor
calculations it is assumed that a cylinder of extracellular
medium, of radius p', is associated with each core conductor
and Rm, Ri, Re, and Cm are derived from cylinder geometry
(Appendix II).
The above values result in a time constant = 6.75 ms, a length
constant X = 260 ,im, and a packing factor v = 106 core conductors/cm2. The remaining parameters were chosen to be
e = 0.05 nA; K = 1.6 X, and, in most calculators, A = 0,
a = b = 0.5 X, and a, = ay = a, = 0.004 mhos/cm. These
values give the potential and current magnitudes illustrated in
Figs. 4-6. The values of the chosen parameters and resulting
magnitudes of dependent variables are of a similar order of
magnitude to those found experimentally in more accessible
neurobiological preparations [31] [33]. Since, however, none
of the crucial parameters have been measured for Purkinje
cells, the values used in the computations must not be assumed
accurate; they merely demonstrate the consistency of the
theoretical equations.
It should be noted that the cable model used here is a very
simplified model of a dendritic tree, and branching and tapering are probably not adequately represented [6], [29]. Since,
however, the weighted integral smooths out local features of
the transmembrane current, the results are not critically dependent on the type of intracellular model used, and the cable
model is adequate for the present discussion of extracellular
phenomena.
r
-
287
NICHOLSON: FIELD POTENTIALS IN NEURONAL ELEMENTS
A
A1
A2
a
B
B1
B2
b
C
cc
cO
Cl
C2
APPENDIX IL
NOMENCLATURE
Synaptic input location on cc.
=(x' + a)lNax.
=(x' - #Na/x-.
Integration limit for x.
Radius of cylindrical neuronal population.
=(y' + b)l-\l
Ie(y'aib)ltr
=
(z' c)/VaN
-
1=1
=
t
w
Time.
Volume of neuronal population in t space.
Transmembrane potential difference.
Variable of integration.
Variable of integration.
X
=x/X.
W
x
x
x'
a
'e
6
77
Rectangular Cartesian coordinate.
= z/X.
Rectangular Cartesian coordinate.
Integration limit for t.
Integration limit for 7.
Integration limit for t.
Dirac delta function.
Input factor for synaptic current.
=
ZI-VT.
= x/-VX.
Source point.
Observation
point.
t,
of
Radius
cc.
p
Radius of extracellular medium associated with cc.
p
a, a11 Conductivity tensor for extracellular medium.
Field potential.
Time constant of cc = CmRm.
T
,t
REFERENCES
[1] J. I. Hubbard, R. Llina's, and D. M. J. Quastel, Electrophysiologi[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
t/-.
Rectangular Cartesian coordinate.
Source point.
Observation point.
=Y/X.
Variable of integration.
Length constant of cc = N/Rm/(Ri + Re).
Number of cc per unit area, normal to x-y plane.
= z'/az.
T
V
z
Integration limit for y.
Integration limit for Z.
Core conductor.
= 2irp cm.
.
cm
Membrane capacity.
ds
Surface element.
d3x Volume element.
Volume element.
E
Electric field.
F
Integral.
G
Weighting function.
H
Integral.
I, Im Volume source current density.
Iin Synaptic input current.
J
Current density in extracellular medium.
Jm
Transmembrane current density.
K
Shape factor for synaptic input function.
L
Integral.
Mi
Surface of ith cc within S.
m
Number of cc within S.
Nj
Surface of intersection of ith cc with S.
n
Number of Nj.
p
Arbitrary vector field.
p
q
= re/1T (p'2 - p2).
Re
Ri
=ri/rp2.Rm = rm/2-rp.
r
Extracellular longitudinal resistance for cc.
re
ri
Intracellular longitudinal resistance for cc.
rm
Membrane resistance for cc.
S
Arbitrary surface described in neuronal tissue.
m
=S- E Nj.
S'
U
y
y
z
[14]
[15]
[16]
[17]
cal Analysis of Synaptic Transmission. Baltimore, Md.: Williams
& Wilkins, 1969.
J. C. Eccles, R. Llinas, and K. Sasaki, "Parallel fibre stimulation
and responses induced thereby in the Purkinje cells of the cerebellum," Exp. Brain Res., vol. 1, pp. 17-39, 1966.
R. Llinas, C. Nicholson, J. A. Freeman, and D. E. Hillman, "Dendritic spikes and their inhibition in alligator Purkinje cells,"
Science, vol. 160, pp. 1132-1135, 1968.
R. Llina's, J. R. Bloedel, and D. E. Hillman, "Functional characterization of the neuronal circuitry of the frog cerebellar cortex,"
J. Neurophysiol., vol. 32, pp. 847-870, 1969.
T. Shimono, D. T. Kennedy, and S. T. Kitai, "Field potential
analysis of the inhibitory pattern in a reptilian cerebellar cortex
(Lacerta viridis)," Brain Res., vol. 22, pp. 386-391, 1970.
C. Nicholson and R. Llinas, "Field potentials in the alligator
cerebellum and theory of their relationship to Purkinje cell dendritic spikes," J. Neurophysiol., vol. 34, pp. 509-531, 1971.
R. Lorente de No, "Action potential of the motoneurones of the
hypoglossus nucleus," J. Cell Comp. Physiol., vol. 29, pp. 207288, 1947.
W. Rall and G. M. Shepherd, "Theoretical reconstruction of field
potentials and dendrodendritic synaptic interactions in olfactory
bulb," J. Neurophysiol., vol. 31, pp. 884-915, 1968.
W. H. Calvin and D. Hellerstein, "Dendritic spikes vs. cable
properties," Science, vol. 163, pp. 96-97, 1969.
R. S. Zucker, "Field potentials generated by dendritic spikes and
synaptic potentials," Science, vol. 165, pp. 409-413, 1969.
W. H. Calvin, "Dendritic spikes revisited," Science, vol. 166,
pp. 637-638, 1967.
D. Hellerstein, "Cable theory and gross potential analysis,"
Science, vol. 166, pp. 638-639, 1969.
J. A. Freeman and J. Stone, "A technique for current density
analysis of field potentials and its application to the frog cerebellum," in Neurobiology of Cerebellar Evolution and Development,
R. Llinas, Ed. Chicago, Ill.: American Medical Association,
1969, pp. 421-430.
R. Plonsey and D. B. Heppner, "Considerations of quasistationarity in electrophysiological systems," Bull. Math. Biophys., vol. 29, pp. 657-664, 1967.
R. Plonsey, Bioelectric Phenomena. New York: McGraw-Hill,
1969.
P. Rosenfalck, "Intra- and extracellular potential fields of active
nerve and muscle fibers," Acta Physiol. Scand., suppl. 321, pp.
1-168, 1969.
W. Pitts, "Investigations on synaptic transmission," in Proc. 9th
Conf Cybernetics. New York: Josiah Macy, Jr., Foundation,
pp. 159-166, 1953.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, JULY 1973
288
[181 J. Clark and R. Plonsey, "A mathematical evaluation of the core
conductor model," Biophys. J., vol. 6, pp. 95-112, 1966.
[19] W. Rall, "Distributions of potential in cylindrical coordinates
and time constants for a membrane cylinder," Biophys. J., vol. 9,
pp. 1509-1541, 1969.
[20] S. Ramon y Cajal, Histologie du Systeme Nerveux de I'Homme et
des Vertebres. Paris, France: Maloine, 1909-1911.
[211 D. A. Fox, D. E. Hillman, K. A. Siegesmund, and D. R. Dutta,
"The primate cerebellar cortex: A Golgi and electron microscopical study," in Progress in Brain Research, C. A. Fox and
R. Snider, Eds., vol. 25. New York: Elsevier, 1967, pp. 174-225.
[22] J. C. Eccles, M. Ito, and J. Szentagothai, The Cerebellum as a
Neuronal Machine. New York: Springer-Verlag, 1967.
[231 R. Llinas and D. E. Hillman, "Physiological and morphological
organization of the cerebellar circuits in various vertebrates," in
Neurobiology of Cerebellar Evolution and Development,
R. Llinas, Ed. Chicago, Ill.: American Medical Association,
1969, pp. 43-73.
[24] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous
Media. Reading, Mass.: Addison-Wesley, 1960.
[25] P. M. Morse and H. Feshbach, Methods of Theoretical Physics.
New York: McGraw-Hill, 1963.
[26] J. D. Jackson, Classical Electrodynamics. New York: Wiley,
1962.
[27] R. Llina's and C. Nicholson, "Electrophysiological properties of
dendrites and somata in alligator Purkinje cells," J. Neurophysiol., vol. 34, pp. 532-551, 1971.
[28] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, 1968.
[291 W. Rall, "Electrophysiology of a dendritic neuron model," Biophys. J., vol. 2, pp. 145-167, 1962.
[30] R. Lorente de No, "Analysis of the distribution of the action
currents of nerve in volume conductors," in A Study of Nerve
Physiology, Part 2, vol. 132. Rockefeller Institute, New York,
N.Y.: 1947, pp. 384-477.
[31] B. Katz, Nerve, Muscle and Synapse. New York: McGraw-Hill,
1966.
[32] S. Jacobson and D. A. Pollen, "Electrotonic spread of dendritic
potentials in feline pyramidal cells," Science, vol. 161, pp. 13511353, 1968.
[33] P. G. Nelson and H. D. Lux, "Some electrical measurements of
motoneuron parameters," Biophys. J., vol. 10, pp. 55-73, 1970.
Short Communications.
Computer Graphics in Three Dimensions for Perspective
Reconstruction of Brain Ultrastructure
T. JOE WILLEY, ROBERT L. SCHULTZ, AND ALLAN H. GOTT
Abstract-A technique for computer reassembly and plotting of
morphological serial sections in three dimensions is described. The
computer reconstruction is obtained at a savings in cost and time compared to previous model-processing or isometric drawing methods.
Manuscript received March 16, 1972; revised June 21, 1972. This
work was supported in part by the National Institutes of Health under
General Research Support Grants FR-053203 and MH-18752-01.
T. J. Willey is with the Department of Physiology, Pharmacology,
and Biophysics, School of Medicine, Loma Linda University, Loma
Linda, Calif. 92354.
R. L. Schultz is with the Department of Anatomy, School of Medicine, Loma Linda University, Loma Linda, Calif. 92354.
A. H. Gott is with the Department of Applied Computing Technology, The Aerospace Corporation, San Bernardino, Calif. 92408.
INTRODUCTION
Despite the fact that classical silver methods provide visualization
of central nervous aborizations, the light microscopic techniques are
inherently handicapped in providing certain kinds of detailed morphological information. There is no choice but to resort to the electron microscope. However, sectioning brain tissue for electron microscopy reduces complex geometrical forms to two-dimensional images
that are flat and difficult, if not impossible, to spatially identify [1].
Therefore, three-dimensional analysis must be undertaken for many
problems in brain ultrastructure [3], [71-[10]. Unfortunately, to
obtain three dimensions in ultrastructure is commonly a tour de force,
and therefore not often undertaken by electron microscopists. In
general, the following steps are necessary to obtain three-dimensional
reconstructions. Thin serial sections are collected on grids, stained,
and the structures photographed on film after examination with the
electron microscope. Each photomicrograph is then traced on vellum for structural alignment by "best-fit" procedures. Selected sections or the entire collection of serial sections are retraced on a dis-