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Quantitative analysis of dendritic branching pattern of granular cells from human dentate gyrus
EXPERIMENTAL
NEUROLOGY
Quantitative
Analysis
Granular ROBERT
Departments
52, 295-310 (1976)
Cells D.
of Dendritic
Branching
from
Dentate
LINDSAY
Human AND
ARNOLD
of Anatomy and Psychiatry and of California, Center for
University
B.
Pattern Gyrus
SCHEIBEL
the Brain the Health
of
Research Sciences,
l
Institute,
Los Angeles, California 90624 Received
February
9, 1976
The three-dimensional structure of Golgi-impregnated neurons was studied using modern data-collecting techniques. Branch length and branch angle distributions were examined and found to have a wide range of observed values. These types of distributions imply a stochastic design for the bifurcating structure. No apparent pattern was found in the branching configuration. Branch lengths were studied using both centrifugal and centripetal ordering. Results of this analysis indicate that branching probability is not uniform over the entire dendritic tree and may be dependent on the dendritic surface area.
INTRODUCTION The young neuron in the central nervous system of higher evolved animals has relatively short and thick dendritic processes which develop into a complex system of branches. The description of size, shape, and density of the dendritic arborization seemsto be characteristic for groups of neurons. The subjective characteristics derived from observational techniques have been used to classify neurons into types. However, closer examination of dendritic branching patterns reveals continuous structural variation within each neuron type. The classification of neurons in a precise and logical manner will require an understanding of the design principles for dendritic structure. These complex dendritic structures are studied objectively by using statistical analysis of quantitative data. Early attempts at using statistical 1 This study was supported by USPHS Grant NS 10567, National Institute of Neurological and Communicative Disorders and Stroke. We thank Barbara Lindsay for her assistance in the preparation of this paper. 235 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
296
LINDSAY
AND
SCHEIBEL
analysis to study dendritic structures were made by Sholl (9). Recent quantitative studies on the branch length and branch angle analysis of dendritic structures have been reported by Smit et al. (lo), Uylings and Smit (ll), Hollingworth and Berry (4), and Lindsay and Scheibel (6). Our study combines classical histological methods and modern datacollecting techniques to determine the three-dimensional structure of neurons, The branch lengths and branch angles are calculated from the threedimensional structural data. Frequently distributions and statistical parameters are determined for various categories of branch lengths and branch angles. Expe&ental
Methods
The histological preparations used in this study were rapid Golgiimpregnated thick serial sections of adult human dentate gyrus. Small blocks of tissue were fixed in Lillie’s neutral buffered formalin for 2 days (8). The blocks were placed in the rapid Golgi chromating solution for 3 days, #then in a silvering solution for 24 hr. They were dehydrated and embedded in Parlodion (3). The blocks were serially sliced on a sliding microtome at a thickness of 120 pm in a transverse orientation. The tissue slices were mounted on slides using Permount and cover glasses. Serial sections carefully prepared by the above method can be used to trace individual fibers from one slice to the next. This technique makes possible the complete reconstruction of the dendritic domain and as much of the axonal domain as is desired (5). For this quantitative study, a group of 20 granular cells was selected from the dentate gyrus of a 51-year-old female. All of these neurons were located in a block of tissue 1 mm thick that was cut transversely to the hippocampal formation one-third of the way distal from the anterior pole. The structures of the selected neurons were reconstructed using the Video Computer Microscope and the Anatomical Reconstruction Graphics Operating System developed by one of the authors (7). The data model used to represent the bifurcating structure of neurons is a stick or wire model. The three-dimensional coordinates in space were given for each inflection, branch, and end point. A code associated with the coordinate point and the sequential order of the points determine the connectivity of the points and hence the stick-like structure (Fig. 1). The neuron was viewed with a microscope coupled to a television camera. An x-y recording device attached to the television monitor and a potentiometer connected to the fine focus control of the microscope provided the computer with spatial coordinates of a point brought into best focus. The computer was programmed with an operating system that guided the operator through the measuring procedure and presented him
DENDRITIC
BRANCHING
IN
DENTATE
CYRUS
297
FIG. 1. A simplified sticklike figure representing the measured data. Each dot indicates a three-dimensional coordinate point. A code associated with each point is used to signify inflection, branch, or endpoint.
with a dynamic visual presentation of the data. When the structural data had been recorded in a permanent file on magnetic tape, the data could be analyzed using a standard higher-order computer language such as Fortran. Analyticul Methods Branch lengths and branch angles were easily calculated from the structural data using a Fortran program. Each dendrite was processed individually. The sequenceof points was scanned for branch and end codes. Fiber lengths between these points were calculated and stored in a two-dimensional array in such a manner as to preserve the topological structure of the dendrite. For each branch point there was an incoming branch and two outgoing branches. An angle was formed by each outgoing branch and the extended direction of the incoming branch. Thus a branch angle was associated with each outgoing branch of the structure. The angles were also stored in a two-dimensional array. The branch length and branch angle arrays were stored on magnetic tape for statistical analysis. A statistical analysis program was developed to read the branch length and branch angle arrays from magnetic tape and to select measuresto form a data set. The data set was considered as a distribution of some measure of the structure and the mean, variance, standard deviation, and standard deviation of the mean were calculated. The frequency distribution was also determined. An arbitrary theoretical function could be fitted to the experimental distribution, and the “goodness of fit,” the chi-square statistic, was calculated.
298
LINDSAY
AND
SCHEIBEL
Fro. 2. Photographs of Golgi-impregnated granular cells of human dentate gyrus. (a) Classical three-layer construction (X 25). (b) Somata of the granular cells in the granular layer and their dendrites projecting into the molecular layer (X 100). (c) Initial segments of dendrites protruding randomly from the surfaces of the somata (X 400). (d) Dendritic protrusions arching rapidly and projecting toward the molecular layer (X 400). (e) Development of one dendrite into a major structure compared to the other dendrites of a single cell (X 400). (f) Dendrites of granular celk densely covered with spines (X 400). (g) Dendritic terminal branches
DENDRITIC
BRANCHING
IN
TABLE Statistical
DENTATE
299
GYRUS
1
Parameters
for Major Ascending Dentate Cyrus Granular
Dendrites Cells”
of Human
n
F
UP
(r
x2
v
P
a
0
S-N N-N N-T TL S-T S-LN BA
20 107 147 20 147 147 2.54
22.6 61.3 144.7 1413.8 284.4 139.7 40.6
3.3 3.9 5.6 138.6 4.9 5.5 1.4
14.6 40.8 67.8 619.9 59.5 66.5 23.0
0.55 0.47 0.98 0.95 1.22 1.72
3 1.5 27 23 27 10
0.64 0.96 0.68 0.46 0.20 0.07
0.107 0.037 0.031 0.004 0.080 0.032 0.077
2.5 2.5 4.5 5.0 23.0 4.5 3.0
D See text
for symbol
definitions.
Two theoretical functions were used in this study. The first was the gamma density function (2). A continuous random variable x with range O
=
(~T,2)t~-(z-!m2u2
,
_
~
where all parameters have already been defined above. When the statistical calculations had been completed and a theoretical function had been determined, the results were printed out by the computer. The computer also plotted as a graph the experimental distribution in the form of a histogram and the theoretical density function normalized to the experimental data. projecting development protruding
into the white matter above the molecular layer with loss of spines and of frequent nodules (X 400). (h) Unusual granular cell with two axons from the lower portion of its soma (X 400).
300
LINDSAY
ArjD
SCHEtBtt
TABLE Statistical
Parameters for Minor Dentate Gyrus
S-N N-N N-T TL S-T S-LN BA
20 40 80 25 86 86 122
10.4 65.4 133.4 640.7 271.3 134.3 43.7
a See text
for symbol
7.2 8.5 7.9 65.6 6.3 7.2 2.1
2 Ascending Dendrites Granular Cellsa
32.1 53.5 70.3 328.1 58.1 66.9 23.1
0.39 0.89 0.95 0.97 0.92 2.09 1.29
9 11 22 15 20 23 9
of Human
0.94 0.55 0.53 0.48 0.56 0.01 0.24
0.069 0.023 0.027 0.006 0.800 0.030 0.082
5.0 1.5 3.5 4.0 22.0 4.0 3.5
definitions.
RESULTS Descriptive 0 bservations. The dentate gyrus consists of three layers : a molecular layer, a granular layer, and a polymorphic layer. Layers of the dentate gyrus are arranged in a “U”-shaped configuration in which the open portion is directed toward ‘the fimbria in transverse sections. The granular layer, made up of closely arranged spherical or oval bodied neurons, gives rise to axons which pass through the polymorphic layer to terminate upon dendrites of pyramidal cells in the hippocampus. Almost all dendrites of granular cells enter the molecular layer. Cells of the polymorphic layer are of several types, including modified pyramidal cells and so-called basket cells. The dentate gyrus does not give rise to fibers passing beyond the hippocampal formation. Figure 2a shows the classical three-layer construction of the dentate gyrus. In the center of the picture is a light band, the molecular layer. Just below the molecular layer in the dorsal direction is the granular layer and below this the polymorphic layer. At the top of the picture are the apical dendrites of the hippocampal pyramids.
a FIG.
3. Diagram
b of link
ordering.
(a)
Centrifugal;
(b)
Centripetal.
DENDRITIC
"1
m n
BRANCHING
IN
DENTATE
BRANCH
ORDER
1
BRANCH
ORDER
2
CYRUS
301
m 1
LENGTH
z
CUICRONSl
ERflNCH
ORDER
3
BRFINCH
ORDER
4
m
lb n
BRFtNCH
ORDER
S
m
FIG. 4. Distribution of major dendritic branch lengths using centrifugal ordering. With greater magnification, Fig. 2b shows the somata of the granular cells in the granular layer and their dendrites projecting into the molecular layer. The initial segments of the dendrites protrude randomly from the
302
LINDSAY
AND
BRANCH
SCHEIBEE
ORDER
1
x = Lo g E : E w
LENGTH
I 7
1
:
BRANCH
cNICRONSI
ORDER
LENGTH
CHICRONSI
LENGTH
CllICRONSl
LENGTH
CUICRONSI
BRANCH
ORDER
2
5
1 tn n
LENGTH
FIG. 5. Distribution
of major dendritic
CHICRONS,
branch lengths using centripetal
ordering.
surfaces of the somata as seen in Fig. 2c. However, most of the dendritic protrusions, regardless of their initial direction, arch rapidly and project toward the molecular layer, as seen in Fig. 2d. There seemsto be a major
DENDRITIC
BRANCHING
IN
TABLE Statistical
Order 1 2 3 4 5 6 ” See text
Parameters
P
ofi
Q
22.6 47.8 68.7 65.7 79.0 41.0
3.3 5.2 8.1 6.3 13.1 1.0
14.6 30.9 51.7 28.8 34.7 1.4
for symbol
303
GYRUS
3
for Major Ascending Dendrites lar Cells Using Centrifugal
n 20 36 41 21 7 2
DENTATE
of Human Ordering”
Dentatr
x2
y
P
0.55 0.39 0.72 0.75 0.00 -
3 8 12 8 8 -
0.64 0.93 0.74 0.65 1.oo -
___-
Gyrus
ff 0.107 0.050 0.026 0.079 0.066
Granu-
6 2.5 2.5 2.0 5.0 5.0
defmitions.
polarizing force present during the development of these dendrites. Virtually all branches of the dendrites terminate in the distal portion of the molecular layer. The resulting shape of the dendritic domain is that of an inverted cone. Usually one of the dendrites protruding from the surface of the soma nearest the molecular layer, as seen in Fig. Ze, develops into a structure larger than the other dendrites. This major dendrite has a larger initial diameter and larger total fiber length than the other dendrites protruding from the same soma. As seen in Fig. 2f, the dendrites of the granular cells are densely covered with spines. Frequently the dendrites terminate as a nodule with a protruding hair at the tip. As seen in Fig. 2g, some of the dendritic terminal branches project into the white matter above the molecular layer where the! lose their spines and develop frequent nodules. Occasionally we find a dendrite that projects obliquely into the polymorphic layer. TABLE Statistical
1 2 3 4 5 6 n See text
Parameters
147 5.5 34 20 12 5 for symbol
4
for Major Ascending Dendrites lar Cells Using Centripetal
144.7 78.7 45.6 36.6 23.7 21.4 definitions.
5.6 6.2 4.0 4.9 5.0 6.4
67.8 46.0 23.2 21.9 17.4 14.3
0.98 0.62 0.45 0.29 0.31 -
of Human Ordering*
27 14 7 5 2 -
Dentate
0.68 0.85 0.87 0.92 0.74 -
Gyrus
0.031 0.037 0.085 0.076 0.078 0.104
Granu-
4.5 3.0 4.0 3.0 2.0 2.0
304
LINDSAY
AND
TABLE Statistical
Order 1 2 3 4 Q See text
Parameters
SCHEIBEL
5
for Minor Ascending Dendrites lar Cells Using Centrifugal
of Human Ordering”
Dentate
Gyrus
Granu-
n
P
gfi
(r
x2
IJ
P
Q
B
21 23 12 4
77.3 62.0 69.5 80.3
9.7 11.4 15.9 28.8
44.4 54.7 55.0 57.7
0.53 1.54 0.99 -
9 8 3
0.85 0.14 0.40 -
0.039 0.021 0.023 0.024
3.0 1.5 1.5 2.0
for symbol
-
definitions.
The axons of the granular cells protrude from the soma or a proximal portion of a dendrite and turn obliquely to enter the pathway to the hippocampal pyramids. These axons have many collateral branches which form a plexus in the polymorphic layer. The most unusual observations we have made in these preparations are granular cells with two axons protruding from the soma. Figure 2h shows one of these unusual cells. Statistical Analysis. Two groups of branch-length and branch-angle data were studied. They were the major ascending and the minor ascending dendrites of the granular cells. For each group the following structural measures were statistically analyzed : soma-to-first-node fiber lengths (S-N) ; node-to-node fiber lengths (N-N) ; node-to-terminus fiber lengths (N-T) ; total fiber length of each dendrite (TL) ; soma-to-terminus fiber lengths (S-T) ; soma-to-last-node fiber lengths (S-LN) ; and branch angles in degrees (BA). All lengths are in micrometers. For each set of parameters the following statistics were calculated: the number of measures (B) ; the mean value (a) ; the standard deviation of the mean (up) ; the standard deviation (u) ; the (Y constant of the gamma density distribution (a) ; the /3 constant of the gamma density function (/3) ; the reduced chi-square statistic from the distribution and theoretical TABLE Statistical
Order 1 2 3 4 5 tl See text
Parameters
6
for Minor Ascending Dendrites lar Cells Using Centripetal
of Human Ordering”
Dentate
Gyrus
f-2
n
P
,Jlc
0
x2
y
P
86 32 1.5 10 3
137.0 88.0 48.1 52.8 38.0
7.9 10.0 9.5 10.9 8.7
72.8 56.5 36.7 34.4 15.1
0.87 0.54 3.00 0.72 0.00
24 12 4 4 4
0.65 0.89 0.02 0.58 1.00
for symbol
definitions.
0.026 0.028 0.036 0.044 0.166
Granu-
B 3.5 2.5 1.5 2.5 6.5
DENDRITIC
BRANCHING
IN
DENTATE
305
GYRUS
3mI 0
is
Se
js
1
iee
vi5
de HERN
I
17s SUM
,
200
21s
I
256
24s
I
388
315
t
356
I
37s
I
tee
CMICRONSI
FIG. 6. Graph of major dendritic branch length means using centrifugal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
function (x2) ; the number of degrees of freedom used in chi-square calculation (v) ; and the probability of observing a value of chi-square greater than x2 for a random sample of n observations with I/ degrees of freedom (PI.
MEAN
SUtl
CMICRONSI
FIG. 7. Graph of major dendritic branch length means using centripetal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
306
LINDSAY
AND
SCHEIBEL
FIG. 8. Graph of minor dendritic branch length means using centrifugal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
The results of these statistical calculations are given for the two groups of data in Tables 1 and 2. In addition to the statistical calculations, the experimental frequency distribution histograms and theoretical density functions were plotted for each set of structural measures of the two data groups.
I D I d
2'5
58
ds
de
11s
Al flERN
l?J.s zbs zis 2ss SUM CflICRONSl
2?53/8
315
361335
4.a
FIG. 9. Graph of minor dendritic branch length means using centripetal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
DENDRITIC
BRANCHING
IN
DENTATE
GYRUS
307
The orders of the internodal branching lengths, called links, were determined. Since the branching patterns in general are not symmetric, the ordering of the links can be defined in several ways (1, 12). Two different schemes were selected to assign branch orders to the links in the asymThe first starts with the initial metrical bifurcating dendritic structures. projection from the soma and continues sequentially numbering the links outwards toward the termini. This scheme will be referred to as centrifugal ordering (Fig. 3a). The second orders the links from the termini inward towards the soma. Since the pattern is asymmertic, inner links may be assigned different orders derived from different termini. The selection is made unique by assigning the largest number as the order of the link (Fig. 3b). The second scheme is referred to as centripetal ordering. Using these two ordering schemes, the two groups of data were subdivided by branch order and statistically analyzed. The frequency distributions for the major ascending dendrites are presented in Figs. 4 and 5. The results of the statistical analysis is presented in Tables 3-6. Another method of presenting the internodal branch length data is given in Figs. 6-9. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graphs. The small cross is the value of the mean, the inner bracket is the standard error of the mean, and the outer bracket is the standard deviation of the distribution. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values. This graphic presentation gives a sense of the change of the branching probability as we move from soma to termini for centrifugal ordering and from termini toward soma for centripetal ordering. DISCUSSION In examining the branch length and branch angle distributions, the most obvious observation is that each distribution has a wide continuous range of observed values. The values of the lengths range over several orders of magnitude and the values of the angles range over 120”. The distribution data do not suggest any preferred discrete values for the structural measures under consideration. When the branch lengths and branch angles of individual dendrites are examined in the topological sequence as a tree-like structure, no discernible patterns are observed to indicate a highly organized structure. Some earlier work by one of the authors (5, 6) using these measures to examine the bifurcating structure of cortical neurons also failed to find any rigid structural design. Some similar observations carried out using visual measurements through the microscope were made by Sholl (9). He also found nothing to suggest any rigid structural design in the dendritic branching pattern.
308
LINDSAY
AND
SCHEIBEL
The soma-to-node and node-to-node fiber length distributions have a most interesting shape. The theoretical fit is almost that of the decreasing exponential function. For this case, the function is to be interpreted as meaning that for each unit length along the dendritic fiber there is equal probability of finding a branching node. This probability per unit length is equal to the inverse of the mean of the internodal length distribution. The node-to-terminus fiber length distributions have a much larger range of values than the node-to-node fiber length distributions. When the distribution is fitted with the gamma density function, the fi parameters are also larger as seen in Tables 1 and 2, which indicates a more bell-like shape. We will not imply that the shape of this distribution is the result of a probability density function which is the linear sum of p independent random variables (in this case, lengths), each having a negative exponential density with a mean of l/cy. Instead, we suggest the distribution is the composite of a number of different random processes and/or that there is a strong variable dependence in the process of terminating branches. Perhaps the process that determines the terminus is dependent on the soma-to-terminus fiber length. Such a variable might be related to metabolic support of the fiber from the soma and may be length-dependent. When the soma-to-terminus fiber length distribution is examined a strong tendency for a central value is indeed found. The soma-to-last-node fiber length distribution has been fitted with a low order (small p) gamma density function. This function is in accordance with our hypothesis that the node-to-node fiber length probability density function is almost unconditional. The distribution of the total fiber lengths of each dendrite does not have a strong central tendency. When this distribution is examined, a wide range of values is found. This wide range is not the result of a population of neurons being included in one distribution. The dendrites of each neuron contribute to the full range of the distribution. The range for each of the branch angle distributions extends from 0 to 120”. When examining the distribution of a stochastic parameter, one often finds it instructive to consider the hypothetical case in which the probability density is unconditional. For this case we consider a distribution in which all directions in space are equally probable. For this hypothetical case we find that the frequency distribution for the cosine of the branch angles is a flat distribution, i.e., all values of the cosine of the angle are equally probable. The branch angle cosine distributions show the highest frequency for small branch angles and rapidly decreasing frequency as the angle approaches 90”. To further investigate the branching probability, the data have been subdivided by the order of the branch. Figure 6 and Table 3 show the results of the centrifugal ordering for the major dendrites. Notice that the mean
DENDRITIC
BRANCHING
IN
DENTATE
GYRUS
309
value increases for the first three branch orders and then remains nearly constant for the higher branch orders. As mentioned previously, the Ftzajor ascending dendrites protrude from the soma with a large diameter and taper down to a more constant diameter. One possible explanation for these results is that the probability of making a node is proportional to the unit surface area of the dendritic membrane and not the unit length of fiber. It was also mentioned previously that the minor ascending dendrites protrude from the soma with a smaller diameter with much less tapering. If our hypothesis is correct, then a similar graph for the minor ascending dendrites should show a more constant mean value for all branch orders. Indeed the graph in Fig. 8 is in accord with this prediction. The constant mean value for the minor ascending dendrites is approximately equal to the higher order mean values of the major ascending dendrites. The centrifugal ordering is useful in examining branching probability near the soma, but has limited use near the distal portion of the dendritic tree. Hence the nearly constant value for the higher order links is misleading. Centripetal ordering is more useful in examining branching probability in the distal portion of the dendritic tree. The results of this analysis as seen in Figs. 7 and 9 indicate a continued decrease in branching probability in the distal portion of the dendritic tree. The analytical results agree with our observations that dendritic processes continue to taper to their termini. Previously, it was mentioned that the theoretical fit of the node-to-node fiber length distribution is &most a decreasing exponential function. Experimentally, it was observed that there are less short segments than expected (Figs. 4 and 5). This decrease from the theoretical expectation value can be accounted for with the hypothesis that when a node develops, there is a local inhibitory surround where the probability of forming another node is decreased. Another explanation for this result is that the processes of forming branching nodes and of the linear growth of the dendrites are dependent on time. If the two dynamic processes were not uniform over the entire structure and over the entire developmental period, then distortion of the expected internodal distribution would result, even though our basic surface area hypothesis is correct. REFERENCES 1. BERRY, M., T. HOLLINGWORTH, E. M. ANDERSON, and R. M. FLINN. 1975. Application of network analysis to the study of the branching patterns of dendritic fields, pp. 217-245. 1~ “Advances in Neurology,” Vol. 12. G. W. Kreutzberg [Ed.]. Raven Press, New York. 2. CLARKE, A. B., and R. L. DISNEY. 1970. “Probability and Random Processes for Engineers and Scientists,” John Wiley & Sons, New York. 3. DAVENPORT, H. A. 1960. “Histological and Histochemical Techniques,” Saunders, Philadelphia.
310
LINDSAY
AND
SCHEIBEL
4. HOLLINGWORTH, T., and M. BERRY. 1975. Network analysis of dendritic fields of pyramidal cells in neocortex and Purkinje cells in the cerebellum of the rat. Phil. Trans. Royal Sot. Lond. Series B. 270. of the Cerebral Cortex,” Dissertation, 5. LINDSAY, R. D. 1971. “Connectivity Physics Department, Syracuse University, Syracuse, New York. 6. LINDSAY, R. D., and A. B. SCHEIBEL. 1974. Quantitative analysis of the dendritic branching pattern of small pyramidal cells from adult rat somesthetic and visual cortex. Exfi. Neural. 45, 424-434. 7. LINDSAY, R. D. Video Computer Microscope and A.R.G.O.S. In “Computer Analysis of Neuronal Structures,” R. D. Lindsay [Ed.]. Plenum Press, New York, (in press.) 8. MCMANUS, J. F., and R. W. MOWRY. 1960. “Staining Methods Histological and Histochemical,” Paul B. Hoeber, New York. 9. SHOLL, D. A. 1967. “The Organization of the Cerebral Cortex,” Hafner Publishing, New York. 10. SMIT, G. J., H. B. M. UYLINGS, and L. VELDMAAT-WANSINK. 1972. The branching pattern in dendrites of cortical neurons. Acta Morphot. Neerl. Stand. 9: 253-274. 11. UYLINGS, H. B. M., and G. J. SMIT. 1975. Three-dimensional branching structure of pyramid cell dendrites. Brain Res. 86. 12. UYLINGS, H. B. M., G. J. SMIT, and W. A. M. VELTMAN. 1975. Ordering methods in quantitative analysis of branching structures of dendritic trees, pp. 247-254. In “Advances in Neurology,” Vol. 12. G. W. Kreutzberg [Ed.], Raven Press, New York.
NEUROLOGY
Quantitative
Analysis
Granular ROBERT
Departments
52, 295-310 (1976)
Cells D.
of Dendritic
Branching
from
Dentate
LINDSAY
Human AND
ARNOLD
of Anatomy and Psychiatry and of California, Center for
University
B.
Pattern Gyrus
SCHEIBEL
the Brain the Health
of
Research Sciences,
l
Institute,
Los Angeles, California 90624 Received
February
9, 1976
The three-dimensional structure of Golgi-impregnated neurons was studied using modern data-collecting techniques. Branch length and branch angle distributions were examined and found to have a wide range of observed values. These types of distributions imply a stochastic design for the bifurcating structure. No apparent pattern was found in the branching configuration. Branch lengths were studied using both centrifugal and centripetal ordering. Results of this analysis indicate that branching probability is not uniform over the entire dendritic tree and may be dependent on the dendritic surface area.
INTRODUCTION The young neuron in the central nervous system of higher evolved animals has relatively short and thick dendritic processes which develop into a complex system of branches. The description of size, shape, and density of the dendritic arborization seemsto be characteristic for groups of neurons. The subjective characteristics derived from observational techniques have been used to classify neurons into types. However, closer examination of dendritic branching patterns reveals continuous structural variation within each neuron type. The classification of neurons in a precise and logical manner will require an understanding of the design principles for dendritic structure. These complex dendritic structures are studied objectively by using statistical analysis of quantitative data. Early attempts at using statistical 1 This study was supported by USPHS Grant NS 10567, National Institute of Neurological and Communicative Disorders and Stroke. We thank Barbara Lindsay for her assistance in the preparation of this paper. 235 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
296
LINDSAY
AND
SCHEIBEL
analysis to study dendritic structures were made by Sholl (9). Recent quantitative studies on the branch length and branch angle analysis of dendritic structures have been reported by Smit et al. (lo), Uylings and Smit (ll), Hollingworth and Berry (4), and Lindsay and Scheibel (6). Our study combines classical histological methods and modern datacollecting techniques to determine the three-dimensional structure of neurons, The branch lengths and branch angles are calculated from the threedimensional structural data. Frequently distributions and statistical parameters are determined for various categories of branch lengths and branch angles. Expe&ental
Methods
The histological preparations used in this study were rapid Golgiimpregnated thick serial sections of adult human dentate gyrus. Small blocks of tissue were fixed in Lillie’s neutral buffered formalin for 2 days (8). The blocks were placed in the rapid Golgi chromating solution for 3 days, #then in a silvering solution for 24 hr. They were dehydrated and embedded in Parlodion (3). The blocks were serially sliced on a sliding microtome at a thickness of 120 pm in a transverse orientation. The tissue slices were mounted on slides using Permount and cover glasses. Serial sections carefully prepared by the above method can be used to trace individual fibers from one slice to the next. This technique makes possible the complete reconstruction of the dendritic domain and as much of the axonal domain as is desired (5). For this quantitative study, a group of 20 granular cells was selected from the dentate gyrus of a 51-year-old female. All of these neurons were located in a block of tissue 1 mm thick that was cut transversely to the hippocampal formation one-third of the way distal from the anterior pole. The structures of the selected neurons were reconstructed using the Video Computer Microscope and the Anatomical Reconstruction Graphics Operating System developed by one of the authors (7). The data model used to represent the bifurcating structure of neurons is a stick or wire model. The three-dimensional coordinates in space were given for each inflection, branch, and end point. A code associated with the coordinate point and the sequential order of the points determine the connectivity of the points and hence the stick-like structure (Fig. 1). The neuron was viewed with a microscope coupled to a television camera. An x-y recording device attached to the television monitor and a potentiometer connected to the fine focus control of the microscope provided the computer with spatial coordinates of a point brought into best focus. The computer was programmed with an operating system that guided the operator through the measuring procedure and presented him
DENDRITIC
BRANCHING
IN
DENTATE
CYRUS
297
FIG. 1. A simplified sticklike figure representing the measured data. Each dot indicates a three-dimensional coordinate point. A code associated with each point is used to signify inflection, branch, or endpoint.
with a dynamic visual presentation of the data. When the structural data had been recorded in a permanent file on magnetic tape, the data could be analyzed using a standard higher-order computer language such as Fortran. Analyticul Methods Branch lengths and branch angles were easily calculated from the structural data using a Fortran program. Each dendrite was processed individually. The sequenceof points was scanned for branch and end codes. Fiber lengths between these points were calculated and stored in a two-dimensional array in such a manner as to preserve the topological structure of the dendrite. For each branch point there was an incoming branch and two outgoing branches. An angle was formed by each outgoing branch and the extended direction of the incoming branch. Thus a branch angle was associated with each outgoing branch of the structure. The angles were also stored in a two-dimensional array. The branch length and branch angle arrays were stored on magnetic tape for statistical analysis. A statistical analysis program was developed to read the branch length and branch angle arrays from magnetic tape and to select measuresto form a data set. The data set was considered as a distribution of some measure of the structure and the mean, variance, standard deviation, and standard deviation of the mean were calculated. The frequency distribution was also determined. An arbitrary theoretical function could be fitted to the experimental distribution, and the “goodness of fit,” the chi-square statistic, was calculated.
298
LINDSAY
AND
SCHEIBEL
Fro. 2. Photographs of Golgi-impregnated granular cells of human dentate gyrus. (a) Classical three-layer construction (X 25). (b) Somata of the granular cells in the granular layer and their dendrites projecting into the molecular layer (X 100). (c) Initial segments of dendrites protruding randomly from the surfaces of the somata (X 400). (d) Dendritic protrusions arching rapidly and projecting toward the molecular layer (X 400). (e) Development of one dendrite into a major structure compared to the other dendrites of a single cell (X 400). (f) Dendrites of granular celk densely covered with spines (X 400). (g) Dendritic terminal branches
DENDRITIC
BRANCHING
IN
TABLE Statistical
DENTATE
299
GYRUS
1
Parameters
for Major Ascending Dentate Cyrus Granular
Dendrites Cells”
of Human
n
F
UP
(r
x2
v
P
a
0
S-N N-N N-T TL S-T S-LN BA
20 107 147 20 147 147 2.54
22.6 61.3 144.7 1413.8 284.4 139.7 40.6
3.3 3.9 5.6 138.6 4.9 5.5 1.4
14.6 40.8 67.8 619.9 59.5 66.5 23.0
0.55 0.47 0.98 0.95 1.22 1.72
3 1.5 27 23 27 10
0.64 0.96 0.68 0.46 0.20 0.07
0.107 0.037 0.031 0.004 0.080 0.032 0.077
2.5 2.5 4.5 5.0 23.0 4.5 3.0
D See text
for symbol
definitions.
Two theoretical functions were used in this study. The first was the gamma density function (2). A continuous random variable x with range O
=
(~T,2)t~-(z-!m2u2
,
_
~
where all parameters have already been defined above. When the statistical calculations had been completed and a theoretical function had been determined, the results were printed out by the computer. The computer also plotted as a graph the experimental distribution in the form of a histogram and the theoretical density function normalized to the experimental data. projecting development protruding
into the white matter above the molecular layer with loss of spines and of frequent nodules (X 400). (h) Unusual granular cell with two axons from the lower portion of its soma (X 400).
300
LINDSAY
ArjD
SCHEtBtt
TABLE Statistical
Parameters for Minor Dentate Gyrus
S-N N-N N-T TL S-T S-LN BA
20 40 80 25 86 86 122
10.4 65.4 133.4 640.7 271.3 134.3 43.7
a See text
for symbol
7.2 8.5 7.9 65.6 6.3 7.2 2.1
2 Ascending Dendrites Granular Cellsa
32.1 53.5 70.3 328.1 58.1 66.9 23.1
0.39 0.89 0.95 0.97 0.92 2.09 1.29
9 11 22 15 20 23 9
of Human
0.94 0.55 0.53 0.48 0.56 0.01 0.24
0.069 0.023 0.027 0.006 0.800 0.030 0.082
5.0 1.5 3.5 4.0 22.0 4.0 3.5
definitions.
RESULTS Descriptive 0 bservations. The dentate gyrus consists of three layers : a molecular layer, a granular layer, and a polymorphic layer. Layers of the dentate gyrus are arranged in a “U”-shaped configuration in which the open portion is directed toward ‘the fimbria in transverse sections. The granular layer, made up of closely arranged spherical or oval bodied neurons, gives rise to axons which pass through the polymorphic layer to terminate upon dendrites of pyramidal cells in the hippocampus. Almost all dendrites of granular cells enter the molecular layer. Cells of the polymorphic layer are of several types, including modified pyramidal cells and so-called basket cells. The dentate gyrus does not give rise to fibers passing beyond the hippocampal formation. Figure 2a shows the classical three-layer construction of the dentate gyrus. In the center of the picture is a light band, the molecular layer. Just below the molecular layer in the dorsal direction is the granular layer and below this the polymorphic layer. At the top of the picture are the apical dendrites of the hippocampal pyramids.
a FIG.
3. Diagram
b of link
ordering.
(a)
Centrifugal;
(b)
Centripetal.
DENDRITIC
"1
m n
BRANCHING
IN
DENTATE
BRANCH
ORDER
1
BRANCH
ORDER
2
CYRUS
301
m 1
LENGTH
z
CUICRONSl
ERflNCH
ORDER
3
BRFINCH
ORDER
4
m
lb n
BRFtNCH
ORDER
S
m
FIG. 4. Distribution of major dendritic branch lengths using centrifugal ordering. With greater magnification, Fig. 2b shows the somata of the granular cells in the granular layer and their dendrites projecting into the molecular layer. The initial segments of the dendrites protrude randomly from the
302
LINDSAY
AND
BRANCH
SCHEIBEE
ORDER
1
x = Lo g E : E w
LENGTH
I 7
1
:
BRANCH
cNICRONSI
ORDER
LENGTH
CHICRONSI
LENGTH
CllICRONSl
LENGTH
CUICRONSI
BRANCH
ORDER
2
5
1 tn n
LENGTH
FIG. 5. Distribution
of major dendritic
CHICRONS,
branch lengths using centripetal
ordering.
surfaces of the somata as seen in Fig. 2c. However, most of the dendritic protrusions, regardless of their initial direction, arch rapidly and project toward the molecular layer, as seen in Fig. 2d. There seemsto be a major
DENDRITIC
BRANCHING
IN
TABLE Statistical
Order 1 2 3 4 5 6 ” See text
Parameters
P
ofi
Q
22.6 47.8 68.7 65.7 79.0 41.0
3.3 5.2 8.1 6.3 13.1 1.0
14.6 30.9 51.7 28.8 34.7 1.4
for symbol
303
GYRUS
3
for Major Ascending Dendrites lar Cells Using Centrifugal
n 20 36 41 21 7 2
DENTATE
of Human Ordering”
Dentatr
x2
y
P
0.55 0.39 0.72 0.75 0.00 -
3 8 12 8 8 -
0.64 0.93 0.74 0.65 1.oo -
___-
Gyrus
ff 0.107 0.050 0.026 0.079 0.066
Granu-
6 2.5 2.5 2.0 5.0 5.0
defmitions.
polarizing force present during the development of these dendrites. Virtually all branches of the dendrites terminate in the distal portion of the molecular layer. The resulting shape of the dendritic domain is that of an inverted cone. Usually one of the dendrites protruding from the surface of the soma nearest the molecular layer, as seen in Fig. Ze, develops into a structure larger than the other dendrites. This major dendrite has a larger initial diameter and larger total fiber length than the other dendrites protruding from the same soma. As seen in Fig. 2f, the dendrites of the granular cells are densely covered with spines. Frequently the dendrites terminate as a nodule with a protruding hair at the tip. As seen in Fig. 2g, some of the dendritic terminal branches project into the white matter above the molecular layer where the! lose their spines and develop frequent nodules. Occasionally we find a dendrite that projects obliquely into the polymorphic layer. TABLE Statistical
1 2 3 4 5 6 n See text
Parameters
147 5.5 34 20 12 5 for symbol
4
for Major Ascending Dendrites lar Cells Using Centripetal
144.7 78.7 45.6 36.6 23.7 21.4 definitions.
5.6 6.2 4.0 4.9 5.0 6.4
67.8 46.0 23.2 21.9 17.4 14.3
0.98 0.62 0.45 0.29 0.31 -
of Human Ordering*
27 14 7 5 2 -
Dentate
0.68 0.85 0.87 0.92 0.74 -
Gyrus
0.031 0.037 0.085 0.076 0.078 0.104
Granu-
4.5 3.0 4.0 3.0 2.0 2.0
304
LINDSAY
AND
TABLE Statistical
Order 1 2 3 4 Q See text
Parameters
SCHEIBEL
5
for Minor Ascending Dendrites lar Cells Using Centrifugal
of Human Ordering”
Dentate
Gyrus
Granu-
n
P
gfi
(r
x2
IJ
P
Q
B
21 23 12 4
77.3 62.0 69.5 80.3
9.7 11.4 15.9 28.8
44.4 54.7 55.0 57.7
0.53 1.54 0.99 -
9 8 3
0.85 0.14 0.40 -
0.039 0.021 0.023 0.024
3.0 1.5 1.5 2.0
for symbol
-
definitions.
The axons of the granular cells protrude from the soma or a proximal portion of a dendrite and turn obliquely to enter the pathway to the hippocampal pyramids. These axons have many collateral branches which form a plexus in the polymorphic layer. The most unusual observations we have made in these preparations are granular cells with two axons protruding from the soma. Figure 2h shows one of these unusual cells. Statistical Analysis. Two groups of branch-length and branch-angle data were studied. They were the major ascending and the minor ascending dendrites of the granular cells. For each group the following structural measures were statistically analyzed : soma-to-first-node fiber lengths (S-N) ; node-to-node fiber lengths (N-N) ; node-to-terminus fiber lengths (N-T) ; total fiber length of each dendrite (TL) ; soma-to-terminus fiber lengths (S-T) ; soma-to-last-node fiber lengths (S-LN) ; and branch angles in degrees (BA). All lengths are in micrometers. For each set of parameters the following statistics were calculated: the number of measures (B) ; the mean value (a) ; the standard deviation of the mean (up) ; the standard deviation (u) ; the (Y constant of the gamma density distribution (a) ; the /3 constant of the gamma density function (/3) ; the reduced chi-square statistic from the distribution and theoretical TABLE Statistical
Order 1 2 3 4 5 tl See text
Parameters
6
for Minor Ascending Dendrites lar Cells Using Centripetal
of Human Ordering”
Dentate
Gyrus
f-2
n
P
,Jlc
0
x2
y
P
86 32 1.5 10 3
137.0 88.0 48.1 52.8 38.0
7.9 10.0 9.5 10.9 8.7
72.8 56.5 36.7 34.4 15.1
0.87 0.54 3.00 0.72 0.00
24 12 4 4 4
0.65 0.89 0.02 0.58 1.00
for symbol
definitions.
0.026 0.028 0.036 0.044 0.166
Granu-
B 3.5 2.5 1.5 2.5 6.5
DENDRITIC
BRANCHING
IN
DENTATE
305
GYRUS
3mI 0
is
Se
js
1
iee
vi5
de HERN
I
17s SUM
,
200
21s
I
256
24s
I
388
315
t
356
I
37s
I
tee
CMICRONSI
FIG. 6. Graph of major dendritic branch length means using centrifugal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
function (x2) ; the number of degrees of freedom used in chi-square calculation (v) ; and the probability of observing a value of chi-square greater than x2 for a random sample of n observations with I/ degrees of freedom (PI.
MEAN
SUtl
CMICRONSI
FIG. 7. Graph of major dendritic branch length means using centripetal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
306
LINDSAY
AND
SCHEIBEL
FIG. 8. Graph of minor dendritic branch length means using centrifugal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
The results of these statistical calculations are given for the two groups of data in Tables 1 and 2. In addition to the statistical calculations, the experimental frequency distribution histograms and theoretical density functions were plotted for each set of structural measures of the two data groups.
I D I d
2'5
58
ds
de
11s
Al flERN
l?J.s zbs zis 2ss SUM CflICRONSl
2?53/8
315
361335
4.a
FIG. 9. Graph of minor dendritic branch length means using centripetal ordering. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graph. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values.
DENDRITIC
BRANCHING
IN
DENTATE
GYRUS
307
The orders of the internodal branching lengths, called links, were determined. Since the branching patterns in general are not symmetric, the ordering of the links can be defined in several ways (1, 12). Two different schemes were selected to assign branch orders to the links in the asymThe first starts with the initial metrical bifurcating dendritic structures. projection from the soma and continues sequentially numbering the links outwards toward the termini. This scheme will be referred to as centrifugal ordering (Fig. 3a). The second orders the links from the termini inward towards the soma. Since the pattern is asymmertic, inner links may be assigned different orders derived from different termini. The selection is made unique by assigning the largest number as the order of the link (Fig. 3b). The second scheme is referred to as centripetal ordering. Using these two ordering schemes, the two groups of data were subdivided by branch order and statistically analyzed. The frequency distributions for the major ascending dendrites are presented in Figs. 4 and 5. The results of the statistical analysis is presented in Tables 3-6. Another method of presenting the internodal branch length data is given in Figs. 6-9. The mean value of the internodal lengths for each branch order is plotted along the ordinate of the graphs. The small cross is the value of the mean, the inner bracket is the standard error of the mean, and the outer bracket is the standard deviation of the distribution. The abscissa value is the branch order mean value plus the sum of all preceding branch order mean values. This graphic presentation gives a sense of the change of the branching probability as we move from soma to termini for centrifugal ordering and from termini toward soma for centripetal ordering. DISCUSSION In examining the branch length and branch angle distributions, the most obvious observation is that each distribution has a wide continuous range of observed values. The values of the lengths range over several orders of magnitude and the values of the angles range over 120”. The distribution data do not suggest any preferred discrete values for the structural measures under consideration. When the branch lengths and branch angles of individual dendrites are examined in the topological sequence as a tree-like structure, no discernible patterns are observed to indicate a highly organized structure. Some earlier work by one of the authors (5, 6) using these measures to examine the bifurcating structure of cortical neurons also failed to find any rigid structural design. Some similar observations carried out using visual measurements through the microscope were made by Sholl (9). He also found nothing to suggest any rigid structural design in the dendritic branching pattern.
308
LINDSAY
AND
SCHEIBEL
The soma-to-node and node-to-node fiber length distributions have a most interesting shape. The theoretical fit is almost that of the decreasing exponential function. For this case, the function is to be interpreted as meaning that for each unit length along the dendritic fiber there is equal probability of finding a branching node. This probability per unit length is equal to the inverse of the mean of the internodal length distribution. The node-to-terminus fiber length distributions have a much larger range of values than the node-to-node fiber length distributions. When the distribution is fitted with the gamma density function, the fi parameters are also larger as seen in Tables 1 and 2, which indicates a more bell-like shape. We will not imply that the shape of this distribution is the result of a probability density function which is the linear sum of p independent random variables (in this case, lengths), each having a negative exponential density with a mean of l/cy. Instead, we suggest the distribution is the composite of a number of different random processes and/or that there is a strong variable dependence in the process of terminating branches. Perhaps the process that determines the terminus is dependent on the soma-to-terminus fiber length. Such a variable might be related to metabolic support of the fiber from the soma and may be length-dependent. When the soma-to-terminus fiber length distribution is examined a strong tendency for a central value is indeed found. The soma-to-last-node fiber length distribution has been fitted with a low order (small p) gamma density function. This function is in accordance with our hypothesis that the node-to-node fiber length probability density function is almost unconditional. The distribution of the total fiber lengths of each dendrite does not have a strong central tendency. When this distribution is examined, a wide range of values is found. This wide range is not the result of a population of neurons being included in one distribution. The dendrites of each neuron contribute to the full range of the distribution. The range for each of the branch angle distributions extends from 0 to 120”. When examining the distribution of a stochastic parameter, one often finds it instructive to consider the hypothetical case in which the probability density is unconditional. For this case we consider a distribution in which all directions in space are equally probable. For this hypothetical case we find that the frequency distribution for the cosine of the branch angles is a flat distribution, i.e., all values of the cosine of the angle are equally probable. The branch angle cosine distributions show the highest frequency for small branch angles and rapidly decreasing frequency as the angle approaches 90”. To further investigate the branching probability, the data have been subdivided by the order of the branch. Figure 6 and Table 3 show the results of the centrifugal ordering for the major dendrites. Notice that the mean
DENDRITIC
BRANCHING
IN
DENTATE
GYRUS
309
value increases for the first three branch orders and then remains nearly constant for the higher branch orders. As mentioned previously, the Ftzajor ascending dendrites protrude from the soma with a large diameter and taper down to a more constant diameter. One possible explanation for these results is that the probability of making a node is proportional to the unit surface area of the dendritic membrane and not the unit length of fiber. It was also mentioned previously that the minor ascending dendrites protrude from the soma with a smaller diameter with much less tapering. If our hypothesis is correct, then a similar graph for the minor ascending dendrites should show a more constant mean value for all branch orders. Indeed the graph in Fig. 8 is in accord with this prediction. The constant mean value for the minor ascending dendrites is approximately equal to the higher order mean values of the major ascending dendrites. The centrifugal ordering is useful in examining branching probability near the soma, but has limited use near the distal portion of the dendritic tree. Hence the nearly constant value for the higher order links is misleading. Centripetal ordering is more useful in examining branching probability in the distal portion of the dendritic tree. The results of this analysis as seen in Figs. 7 and 9 indicate a continued decrease in branching probability in the distal portion of the dendritic tree. The analytical results agree with our observations that dendritic processes continue to taper to their termini. Previously, it was mentioned that the theoretical fit of the node-to-node fiber length distribution is &most a decreasing exponential function. Experimentally, it was observed that there are less short segments than expected (Figs. 4 and 5). This decrease from the theoretical expectation value can be accounted for with the hypothesis that when a node develops, there is a local inhibitory surround where the probability of forming another node is decreased. Another explanation for this result is that the processes of forming branching nodes and of the linear growth of the dendrites are dependent on time. If the two dynamic processes were not uniform over the entire structure and over the entire developmental period, then distortion of the expected internodal distribution would result, even though our basic surface area hypothesis is correct. REFERENCES 1. BERRY, M., T. HOLLINGWORTH, E. M. ANDERSON, and R. M. FLINN. 1975. Application of network analysis to the study of the branching patterns of dendritic fields, pp. 217-245. 1~ “Advances in Neurology,” Vol. 12. G. W. Kreutzberg [Ed.]. Raven Press, New York. 2. CLARKE, A. B., and R. L. DISNEY. 1970. “Probability and Random Processes for Engineers and Scientists,” John Wiley & Sons, New York. 3. DAVENPORT, H. A. 1960. “Histological and Histochemical Techniques,” Saunders, Philadelphia.
310
LINDSAY
AND
SCHEIBEL
4. HOLLINGWORTH, T., and M. BERRY. 1975. Network analysis of dendritic fields of pyramidal cells in neocortex and Purkinje cells in the cerebellum of the rat. Phil. Trans. Royal Sot. Lond. Series B. 270. of the Cerebral Cortex,” Dissertation, 5. LINDSAY, R. D. 1971. “Connectivity Physics Department, Syracuse University, Syracuse, New York. 6. LINDSAY, R. D., and A. B. SCHEIBEL. 1974. Quantitative analysis of the dendritic branching pattern of small pyramidal cells from adult rat somesthetic and visual cortex. Exfi. Neural. 45, 424-434. 7. LINDSAY, R. D. Video Computer Microscope and A.R.G.O.S. In “Computer Analysis of Neuronal Structures,” R. D. Lindsay [Ed.]. Plenum Press, New York, (in press.) 8. MCMANUS, J. F., and R. W. MOWRY. 1960. “Staining Methods Histological and Histochemical,” Paul B. Hoeber, New York. 9. SHOLL, D. A. 1967. “The Organization of the Cerebral Cortex,” Hafner Publishing, New York. 10. SMIT, G. J., H. B. M. UYLINGS, and L. VELDMAAT-WANSINK. 1972. The branching pattern in dendrites of cortical neurons. Acta Morphot. Neerl. Stand. 9: 253-274. 11. UYLINGS, H. B. M., and G. J. SMIT. 1975. Three-dimensional branching structure of pyramid cell dendrites. Brain Res. 86. 12. UYLINGS, H. B. M., G. J. SMIT, and W. A. M. VELTMAN. 1975. Ordering methods in quantitative analysis of branching structures of dendritic trees, pp. 247-254. In “Advances in Neurology,” Vol. 12. G. W. Kreutzberg [Ed.], Raven Press, New York.