701
J. Anat. (1977), 124, 3, pp. 701-715 With 2 figures Printed in Great Britain
Complexity of branching in dendritic trees: dependence on number of trees per cell and effects of branch loss during sectioning MYRA L. SAMUELS,* JAY E. MITTENTHAL,f GEORGE P. MCCABE, JR.* AND PAUL D. COLEMANt
*Department of Statistics, Purdue University,
tDepartment of Biological Sciences, Purdue University,
;Department of Anatomy, University of Rochester (Accepted 29 October 1976) INTRODUCTION
Quantitative characterization of the morphology of individual neurons provides precise indices for the influence of various factors on development of the nervous system. Recent studies have examined effects on dendritic branching of several factors extrinsic to the local neighbourhood of a developing neuron - characteristics of the animal's environment (Coleman & Riesen, 1968; Greenough, Volkmar & Juraska, 1973), aspects of its nutrition (Griffin & Woodward, 1975), and levels of hormones (Clendinnen & Eayrs, 1961; Zamenhof, Mosley & Schuller, 1966; Greenough & Carter, 1975). Effects on development of dendrites of factors intrinsic to the neuron and its local neighbourhood have been studied by use of neurological mutants (Sotelo, 1975; Williams, Ferrante & Caviness, 1975), of isogenic organisms (Macagno, LoPresti & Levinthal, 1973), and of irradiation (Altman, 1973). Such studies can reveal correlations between a treatment and a morphological index, or correlations among indices. Patterns of correlations, in turn, may suggest developmental mechanisms involved in generating neural organization. In our search for structural invariance in dendritic trees we examined the relationship between the complexity of individual dendritic trees and the number of trees per neuron. We have observed a noteworthy correlation between these two morphological indices: Neurons of a given type bearing fewer trees tend to have more complex trees.§ For some classes of neurons from the cerebral cortex of cat and rat this tendency maintains the average number of high-order branches per neuron (with centrifugal ordering) roughly constant in spite of twofold variation in number of trees per neuron. As an adjunct to the analysis underlying this conclusion, we have developed a statistical method designed to compensate for loss of parts of dendritic trees during sectioning. This report is organized as follows: First we give a description of the data base. In a preliminary analysis we examine the relation of number of branches per tree to number of trees per cell without correcting for the cutting of distal branches. We then discuss statistical approaches to the problem of compensating for cutting. Next we present the results obtained by applying these statistical methods to the data in order to estimate the average number of branches of each order per tree before § Throughout this paper 'tree' refers to the dendritic arborization of a single primary dendrite. 45
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cutting. Finally we discuss the biological implications of our findings and examine the general applicability of our methods of compensating for cutting. ANALYSIS OF DATA
Data base In this investigation four groups of neurons were studied: stellate cells from layer IV of visual cortex in dark-reared and control cats, and pyramidal cells from layer V of somatosensory cortex of rats, Berkeley strains SI and S3. Stellate cells of the spiny and non-spiny classes (LeVay, 1973) were not distinguished. Neurons were stained by a Golgi-Cox method described previously (Van der Loos, 1959). Sections were 100 ,um thick; cell body diameter was 20-30 /tm. Sections were cut in a standard plane which was perpendicular to the brain surface. Neurons of each specified type were selected as follows - microscope stage X, Y coordinates were written on a separate slip of paper for each cell that satisfied the following requirements: (1) cell body in centre (in Z axis) of section; (2) dendrites unobscured by glia, blood vessels or excessive numbers of processes of other cell bodies. The desired number of cells was selected by drawing randomly from this pool of acceptable neurons. For the selected neurons, projections of all dendrites of stellate cells and all basal dendrites of pyramidal cells were traced using a camera lucida. Transection of a branch (cutting) at a boundary of a section was noted on each drawing. Distal portions of cut branches were not traced in adjacent serial sections. The number of dendritic trees per cell, and the branching and cutting patterns in each tree were determined. Tables 1 and 2 summarize information about the data base.
Preliminary analysis As an initial step in examining the dependence of tree complexity on the number of trees per cell we chose, as a measure of the complexity of a tree, the number of branches it possesses. The total number of branches in the camera lucida drawing of each tree was counted; each branch-segment between bifurcation nodes was counted as a separate branch. (Of course, some of these branches were cut, so that any additional branches beyond the cut were missing.) Combining the counts for each cell we computed the average number of branches per tree for each cell. The cells were then divided into groups according to the number of dendritic trees on the cell; the mean and standard deviation of the average number of branches per tree was then calculated for each group. These are shown in Figure 1. The number of cells in each group can be determined from Table 2. The graphs of Figure 1 (except perhaps for the Pyramidal S1 data set) suggest that cells with a small number of trees tend to have more branches per tree than those with large numbers of trees. The effect of cutting, however, cannot be ignored. Of the trees in the data sets 45-65 % had one or more branches cut. If cutting is ignored a significant bias would probably be introduced. If cells with a large number of trees had trees bearing longer branches, which would be more likely to be cut, then one would expect that cutting alone could produce, as an artifact, the apparent relationships evident in Figure 1. On the other hand, ignoring differences in branch length among cells, one would expect that trees with more branches would probably have more branches excised through cutting. Hence if cells having a small number of trees tended to have more
Complexity of dendritic branching
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Table 1. Description of data Name of data set
Stellate Control Stellate Dark Pyramidal SI Pyramidal S3
Type of animal Cat Cat Rat Rat
No. of No. of No. of dendritic animals cells trees
Type of cell
(control) (dark reared) (Berkeley SI) (Berkeley S3)
Layer lV stellate Layer IV stellate Layer V pyramidal Layer V pyramidal
45 45 44 44
3 3 3 3
232 226 202 202
Table 2. Number of neurons bearing various numbers of trees (For pyramidal cells only basal trees are counted.) Number of trees per cell Data set
2
3
4
5
6
7
8
9
Stellate Control Stellate Dark Pyramidal SI Pyramidal S3
0 0 0 1
2 4 6 5
12 8 15 15
17 20 16 15
7 11 5 6
5 1 2 2
2 0 0 0
0 1 0 0
complex trees, then the true relationship between average number of branches per tree and number of trees per cell would actually be more pronounced than that indicated in Figure 1. Thus an investigation of the relation between tree complexity and number of trees per cell cannot proceed unless effects of cutting are considered. In the next section we describe statistical methods of compensating for the loss of branches through cutting. The following section summarizes results of applying these methods to the data.
Statistical methods of compensating for cut branches In order to take a statistical approach to the problem of cutting we divided the cells of each data set into three groups: cells bearing four or fewer trees, cells bearing five trees, and cells bearing six or more trees. With this grouping each group contained sufficient trees to allow estimation of the average number of branches per tree within the group by statistical methods. The number of trees in any group may be easily calculated from Table 2. For example, the number of trees on Stellate Control cells with < 4 trees per cell is 3(2) + 4(12) = 54. Thus we have clustered the trees into 12 groups: four data sets, each divided three ways according to the number of trees per cell. We consider each of these 12 groups to be a sample from a population, and we assume each population to be homogeneous in the sense that the probabilistic rules governing tree branching structure are the same for each tree in the population. (This assumption is only an approximation, since the number of trees per cell within a group may vary. The extent of heterogeneity of the groups' < 4 trees per cell' and' > 6 trees per cell' can be judged from Table 2.) For each of the 12 groups separately, then, we may abstract the statistical problem as follows: given data on a sample of trees (some of which have cut branches) from a homogeneous population, we wish to estimate the average number of branches per tree in the population. It is convenient to attack this problem in stages, considering dendritic branches of various orders separately. We order branches by a centrifugal 45-2
704 8
MYRA L. SAMUELS AND OTHERS 8 B x 7 6 I -
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Fig. 1. Mean ± 1 standard deviation of average number of branches per dendritic tree (BIT) as a function of the number of trees per cell (TIC). Where standard deviation is not shown, the group contained only one or two cells. Computed from raw data, without compensating for cut branches. (A) Stellate Control, (B) Stellate Dark, (C) Pyramidal Si, (D) Pyramidal S3. See Tables 1 and 2 and accompanying text for descriptions of these data sets.
method: * The primary dendrite is the first-order branch, the two (if any) arising from it are second-order branches, and so on. We denote by Nk the average number of kth-order branches per tree (before cutting). Thus N1 = 1 always. ik denotes the estimate of Nk obtained from data by statistical inference. Models for estimating Nk. We first discuss the estimation of N2. In order to estimate N2 it is necessary to make some assumptions regarding the probability that a given first-order branch is cut. Since there is evidence that terminal branches are longer than bifurcating branches (see below), and since longer branches have a greater chance of being cut, we assume that the cutting probability depends upon whether the given branch bifurcates or is terminal; thus, we let t, = probability that a first-order terminal branch is cut, b, = probability that a first-order bifurcating branch is cut. For estimating N2, we have available three pieces of data: xi = number of bifurcating uncut first-order branches, Yi = number of terminal uncut first-order branches, z, = number of cut first-order branches. * Centripetal methods of ordering (summarized by Uylings, Smit, & Veltman, 1975) were not used because cutting removes distal branches, preventing correct centripetal ordering. Cutting does not affect the centrifugal ordering used here. Furthermore, centrifugal ordering reflects distance from the cell body, and the probability of cutting presumably depends on distance from the cell body. Thus, the assumption which we make in our models that the probability of cutting depends on order is a natural one when centrifugal ordering is used. -
-
Complexity of dendritic branching
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Let T = x1 +y + z1 = the total number of trees in the sample. The estimate N2 of the parameter N2 will be of the form
-
2(x1+pzl)
T where p is the fraction of the cut first-order branches presumed to have bifurcated. If a first-order terminal branch is as likely to be cut as a first-order bifurcating branch (t1 = b1), then a reasonable value forp would bep = x1/(xl+yj). On the other hand, p would be less than this value if bifurcating branches were less likely to be cut (b1 < t1), andp would be more than this value if b1 > t1. The extreme cases are b1 = 0 (in which case p = 0) and t1 = 0 (in which case p = 1). It is intuitively obvious that the data (xl, Yi, zj) are equally compatible with either of these extreme cases, as well as with any intermediate case. The above discussion can be summarized as follows: Let A1 = t1/bl. If a value of A1 is assumed, N2 can be estimated from the data. There is, however, no way of estimating A1 itself from the data. The considerations in the estimation of N3, N4, etc. are similar but more complex. Samuels (1977) and McCabe & Samuels (1977) have considered several possible models for describing the stochastic structure of the data. We will now briefly describe how some of these models are used in the estimation of N3, N4, etc. The simplest model is called by Samuels (1977) the Binomial Branching and Binomial Cutting Model (BB and BC). This model assumes that the several branches of a given order (a) bifurcate independently of each other and (b) are cut or not cut independently of each other. These are called the Binomial Branching (BB) assumption and the Binomial Cutting (BC) assumption. The parameters of the model are:
flk= probability that a kth order branch bifurcates,
tk = probability that a kth order terminal branch is cut, bk = probability that a kth order bifurcating branch is cut.
Using the BB and BC model, estimates of the parameters /3k, tk and bk are easily obtained from the data, so long as the ratios Ak = tk/bk are specified. The parameters Ak cannot be estimated from the data under the BB and BC model. Smit, Uylings & Veldmaat-Wansink (1972) used the BB and BC model (implicitly) to obtain estimates Of /3k, tk and bk for their data, using a more or less arbitrary value of Ak = 2 for each k. The BB and BC model can be criticized on two grounds: (a) the Binomial Branching assumption may or may not be true; and (b) the Binomial Cutting assumption is assuredly false, since the several branches of a given order are cut, if at all, by the same plane, namely the surface of the section containing the neuron. In Samuels (1977) a more general model, called Modified Binomial Cutting (MBC) is developed for k = 2 (i.e. for cutting and bifurcation of second-order branches, and thus for estimation of N3). In the MBC model, (a) bifurcation of sister branches may be stochastically dependent; and (b) the cutting probabilities t2 and b2 are allowed to depend upon the position of the tree with respect to the cutting plane, but only in such a way that the ratio t2/b2 = A2 iS constant. Under the MBC model, the parameter A2 may or may not be estimable, depending on the data set. For example, if a data set fits the BB and BC model closely, then it contains virtually no information about A2; on the other hand, a data set with strong correlation between bifurcation of sister branches may allow close estimation of A2. This unusual statistical situation is discussed in detail by Samuels (1977). The data sets used in the present study contain
706
MYRA L. SAMUELS AND OTHERS some information about A2, but the information is poor; in fact, the only statistically significant finding about A2 was that the group 'Stellate Control > 6 trees per cell' was not compatible with A2 values in the range 0-3-1 8. In view of the paucity of information about A2, it was decided to treat A2 as an arbitrary parameter in the present work. The MBC model can be extended to cover higher orders of branching, i.e. k = 3, .... Because of the complexity of the computations and because of the relatively small numbers of higher-order branches in our data, this extension has not been carried out. Rather, we have used, for our estimation of N4 and N5, the Binomial Branching (BB) model. The BB model takes the BB assumption (a) of the BB and BC model and combines. it with an assumption about spatial constancy of A,k = tklbk, analogous to assumption (b) of the MBC model. The compatibility of the data with the BB assumption was examined for the case k = 2 by testing the BB model against the MBC model (which lacks the BB assumption) for each of the 12 groups of trees. The Binomial Branching hypothesis was not rejected (at the 5 % significance level) for any group for any value of A2 > 1. Furthermore, numerical investigation indicated that the numerical values of ki were nearly the same under the BB and the MBC models. It is felt that these results justify use of the BB model for k 3 and k = 4 in the present study. To summarize: in the present study N3 is estimated using the MBC model; N4 and N5 are estimated using the more restrictive BB model; for estimation of N2 all the models are equivalent. For the reader who wishes to estimate the Nk for his own data, we have provided in the Appendix the necessary formulae for estimating the branching parameters /k and Nk under the BB model, and also the cutting probabilities tk and bk under the BB and BC model. Details of the models, and also formulae for estimating second-order branching and cutting parameters under the MBC model, are given by Samuels (1977). Whichever model is used, in the present study the Ak are treated as arbitrary parameters; that is, the Ak are pre-specified, rather than estimated from the data. The estimation of each Nk requires specificationof A,k for all lower orders; thus estimation of N2requires specificationofAL, estimation of N3 requires specification ofALandA2, etc. ChoiceofAk. To get an idea of what may be 'reasonable' values for Ak, we turn to the literature. There is considerable evidence that terminal branches of a given order tend to be longer than bifurcating branches of the same order. This phenomenon was observed, for example, by Smit et al. (1972) for basal dendrites of pyramidal cells from the striate cortex of adult rabbits, and by Peters & Bademan (1963) for stellate cells from the sensory-motor cortex of newborn and adult guinea-pigs. While these studies contain unknown biases because of cutting, it appears highly unlikely that the phenomenon is an artifact of cutting. Lindsay & Scheibel (1974), studying pyramidal cells from somesthetic and visual cortex of adult albino rats, made length measurements on dendritic trees reconstructed from serial sections, so that no cutting occurred. They also found that terminal branches were longer than bifurcating branches. Their data is not broken down by order of branching, but this probably has little effect, since Smit et al. (1972) and Peters & Bademan (1963) found little dependence of branch length upon order of branching. The above-mentioned authors found that terminal branches of a given order were, on the average, from twice as long to more than four times as long as bifurcating branches of the same order. Since longer branches have a greater chance of being cut, it appears very likely that, for each k, tk > bk, so that Ak > 1.
4,
Complexity of dendritic branching
707
In applying our statistical methods to the data, we have actually let each of the Ak vary from 0 (meaning tk = 0) to oo (meaning bk = 0). In the following section we illustrate the results of this analysis with selected values of the Ak, and in addition we indicate the range of Ak values within which the results are similar to those illustrated.
Application of statistical method to data In this section we examine the dependence of tree complexity upon the number of trees per cell, denoted henceforth by TIC. We first consider the estimates N3 of the average number of third-order branches per tree. These estimates were generated by the MBC method, letting A1 and A2 take various values. The graphs of Fig. 2 show the estimates N3 obtained when A1 and A2 have equal values, ranging from 0 to 4; the asymptotic value of N3 for A1 = A2 = Xo is also shown. For each data set, N3 is shown as a function of the common value of A1 and A2, plotted as the abscissa, and as a function of TIC by separate curves. The most striking feature of Figure 2 is that N3 is a decreasing function of TIC, no matter what value of A1 and A2 is chosen. (The only exception to this is the Pyramidal Strain SI data set in the range A1 = A2 < 0-8 which are, as we have noted above, probably unrealistic values of A1 and A2.) It is possible that the value of A1 = A2 varies as a function of TIC. However, examination of Figure 2 shows that, unless such variation is extremely large, 3 still decreases as a function of TIC. For example, in the Stellate Control data set, if A1 = A2 > 4 for TIC < 4, and A1 = A2 = 1 for TIC = 5, then the 93 pattern between these two groups is reversed. However, such a large variation of the A1, A2 values seems implausible. It is also possible that A1 and A2 are not equal. We have examined the dependence of N3 on TIC for various combinations of A1 and A2 in the range 0-8 to co. The pattern is in all cases the same as that indicated by Figure 2: The more trees a cell possesses, the fewer the third-order branches borne, on the average, by each tree. The standard errors of the estimates N3 are of the order 0-1554X25 when A1 = A2 =1 (see McCabe & Samuels, 1977). However, since A1 and A2 are not known, and since 3 depends on A1 and A2, these standard errors are of questionable relevance. In any event, the strong statistical significance of the trend in R3 as a function of TIC can be established by a non-parametric argument, without reference to standard errors. This argument is as follows: There are 3! = 6 possible patterns of variation of 3 with T/C-monotone increasing, monotone decreasing, and four others. In each of our four data sets the same (monotone decreasing) pattern is observed. If the six possible patterns are equally likely, then the probability of observing the same pattern in four data sets is 6(0)4, which is < 0'005. Thus the null hypothesis of no trend can be rejected at the 0-5 % significance level, in favour of the alternative hypothesis of a monotone decreasing trend. We now turn to other orders of branching. Table 3 shows the estimates R2, N3, N4 and R. as a function of TIC for the four data sets. (The data contained no branches of order higher than 5.) These estimates were generated for A1 = A2 = A3 = A4 = 2. No consistent pattern of dependence of N2 on TIC is evident in Table 3. If such a pattern exists it is apparently too weak to be detected on the basis of samples as small as ours (the standard errors of R2 are of the order of 0-05-0-10). On the other hand, Table 3 shows that the downward trend of N3 as a function of TIC is generally repeated by R4 and N,. The only exceptions are N4 for the Pyramidal S3 data set, and N3 for the Pyramidal SI data set, where the trend is not maintained.
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MYRA L. SAMUELS AND OTHERS 3
3
2
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N3
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N3
1%
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c
3 -
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Fig. 2. Estimated number of third-order branches per tree (N3) as a function of the common value of A1 and A2 and of the number of trees per cell. ---- -, < 4 trees per cell; 5 trees per cell; - --, 6 trees per cell. (A) Stellate Control, (B) Stellate Dark, (C) Pyramidal SI, (D) Pyramidal S3. See text for definition of A1 and A2. ,
In view of the relatively large standard errors of the estimates S4 and R., we do not regard these exceptions as significant. We have examined the dependence of S4 and R, on TIC for other choices of the Ak. For Ak in the range 1-4, with all Ak's equal, the pattern is essentially the same as that seen in Table 3, namely that the number of fourth- and fifth-order branches per tree, like the number of third-order branches per tree, is smaller for cells with more trees. Having established that N3, N4, and N5 are decreasing functions of TIC, we turn to a more refined question: Is the amount of decrease in these Nk sufficient to compensate for the increase in TIC? That is, are the numbers of third-, fourth-, and fifthorder branches per cell approximately constant? To answer this question, we define Wk to be the average (over all cells in a population) number of kth order branches per cell. We estimate Wkfor each of our 12 data groups by Wk = RAk(TIC), where T the number of trees in the group and C = the number of cells in the group. Table 4 displays these estimates for the two stellate data sets. The quantities shown are W1 (= TIC), W2, W3, W4, J5 and (W2+W3+IVt4+W,); these estimates have been computed with all Ak = 2. In considering Table 4, the reader should bear in mind that if there were no dependence of bifurcation pattern on the number of trees per =
Complexity of dendritic branching
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Table 3. 54 as a function of TIC (computed with all Ak = 2) Stellate Control
Stellate Dark Pyramidal Sil
TIC
N2
< 4 5 > 6 6 4 5
200 1-93 1-74 1-90 1-95 1-77 1-79 1-86
6
169 200 1-83 1-87
>
Pyramidal S3
< 4
5 > 6
2-11 1-72 1-09 1.91 1-39 1-02 1 38 1-26 1-21 1-78 1-15 1-06
N4
N5
090 0 74 047 0 95 0 55 0-39 0-61 0 45 043 0-52 044 045
0-25 0-24 004 0-22 0-14 000 0 07 0 04 007 0-15 0-14 0-06
Table 4. Ik as a function of TIC for Stellates (computed with all A k = 2) TIC Stellate Control
< 4
Stellate Dark
5 > 6 < 4 5 > 6
W, = TIC W2 39 7-7 50 9-6 6-6 11-5 3-7 70 9-8 5-0 6-3 11-2
W3
W4
W5
12 Wk
8-1 8-6 7-3 70 7-0 6-5
3-5 3-7
1.0 1-2 03 0-8 07 0-0
20-3 23-2 22-2 18-3 20-2 20-1
3-1 3-5 2-8 2-5
cell, then the values of k would be directly proportional to TIC (within each data set). For this reason, variations in the Jtk should be considered on a percentage basis. The $k of Table 4 shows a striking pattern. As W, = TIC increases, W2 also increases, although in a somewhat smaller ratio. On the other hand, in many cases W3, H'4, and W5 tend to decrease as TIC increases. This decrease is not very great for P3,, which is remarkably constant compared to J- and 0T2; the decrease is relatively larger for W4J and TV5. The total effect of these downward trends in T-3, W4 and W5 is remarkable: they virtually cancel the upward trend in JV2. This is shown by the last column of Table 4: for each data set the quantity (02+ W. + + J+5) hardly changes as TIC increases. Thus, for example, in the Stellate Control data set, although the number of trees per cell varies from 3-9 to 6-6, an increase of 70 %, the quantity 2 k varies by only about 14 % (and not monotonically). The variation in 5 kk is even less, about 10 %, for the Stellate Dark data set. It would appear, then, that for the Stellate data sets, the decreases in 53, N4, and R5 actually overcompensate for the increase in TIC, so that a cell bearing many trees actually produces, on the average, fewer third-, fourth- and fifth-order branches than a cell bearing few trees. In this way, the higher-order trees balance the second-order trees, with the result that, for the stellate cells, the average total number of second- and higher-order branches per cell is approximately constant, even though these branches may be borne on few or many trees. Table 5 shows the Wk for the Pyramidal data sets (also computed with all Ak = 2). The last column of Table 5 shows (O3 + W4+ TV5) rather than (2+ T3+ -4 + TV5). The reason for this difference is that the pattern of the Tkk for the Pyramidal cells is not
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MYRA L. SAMUELS AND OTHERS
Table 5.
J-k as a function of TIC for Pyramidals (computed with all Ak
Pyramidal
S,
Pyramidal S,
= 2) Wk
TIC
W1= TIC
W2
W3
W4
W5
3
S4 5 > 6 < 4 5 > 6
3-7 50 6-3 3-7 50 6-2
6.7 93 10-6 7.3 9.1 11.7
5.1 6-3
2.3 2.2 2-7 1.9 2-2 2-8
03 0-2 04 05 07 03
77 8-8 10-1 90 8-7 9-8
7-6 65 5*8 6-6
the same as that shown by the Stellates. As W1 and JV2 increase, there is no tendency for iW3, W4, and W5 to decrease; in the Pyramidal S3 data set, they are relatively constant as a function of TIC, while in the Pyramidal Si data set, they increase, although at a smaller rate than TIC. The result is that for the Pyramidal S3 data set, the quantity (Jk3+ 4+ 's) is relatively constant, varying by only 13 % (and not monotonically) while TIC increases by 68 %. In the Pyramidal SI data set, the variation in 15 It is much larger, about 39 %, but still smaller than the variation in TIC. The pattern of constancy of the kk is thus less clear for the Pyramidal cells than for the Stellate cells. There is a tendency for the average total number of third- and higher-order branches per Pyramidal cell to be maintained constant regardless of the number of trees on a cell. This tendency is present in both Pyramidal data sets, but for some reason it is less clearly evident in the Pyramidal S1 data set. We have examined tables analogous to Tables 4 and 5 for other values of the Ak, varying from 1 to oo. In all cases the patterns were essentially the same as those seen in Tables 4 and 5. It is worth noting that, while the patterns of dependence of such quantities as 2 Wk, etc., upon TIC are maintained for a wide range of values of the Ak, the absolute value of these measures of tree complexity may depend strongly upon Ak. The values of S3, for example, may vary by as much as 40 % as A1 and A2 range from 1 to 4. (This may be seen in the graphs of Fig. 2.) This emphasizes the importance of compensation for cutting, and at the same time reveals its limits: groups of data can only be compared if the differences among them are large compared to the range of uncertainty (often itself large) resulting from ignorance of the values of the Ak. DISCUSSION
We have investigated the relation between the number of dendritic trees (primary dendrites) per neuron and the complexity of those trees in samples from four populations of neurons from mammalian cerebral cortex. We defined a more complex tree as a tree bearing a larger number of high-order branches. To characterize complexity we estimated the average number of branches of each order per tree, using statistical methods to compensate for loss of portions of trees during cutting of the section containing the neuron. For all four groups of neurons the estimated mean number of branches of orders 3, 4 and 5 per tree decreased as the number of trees per cell (TIC) increased, with minor exceptions. That is, we have found that the more trees a neuron bears, the less complex those trees tend to be. Consequently, the number of high-order branches per neuron varies less than TIC. Indeed, in three of the four populations the number of high-order branches is nearly independent of TIC; in the fourth, cells with larger TIC tend to have more high-order branches.
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Number oJ branches The inverse relation between tree complexity and TIC indicates that, for purposes of morphometric study of dendrites, trees from a sample of neurons are not necessarily all equivalent. Trees of a sample should not be pooled unless it is verified that all neurons of the sample are equivalent with respect to the properties of interest. In particular, our results show that topological properties such as bifurcation probabilities can depend on TIC. Consequently, pooling of cells bearing different numbers of trees may not always be justifiable in an analysis of topological properties. If metric properties (branch lengths, interbranch angles) are to be studied, suitable compensation must be made for correlation between metric and topological properties. The observation that terminal branches tend to be longer than bifurcating branches is an example of such metric-topological correlation. It is possible that the broad dispersions which have been observed in distributions of branch length (Smit et al. 1972; Lindsay & Scheibel, 1974) would be substantially reduced if neurons were grouped according to values of TIC. It might also be found that clustering of unusual metric properties in certain neurons (e.g. Smit et al. 1972, Table 5) can be interpreted with knowledge of the topological characteristics of these neurons. Our conclusion, that the average number of high-order branches per cell is approximately independent of TIC, suggests that developmental mechanisms may regulate the development of the total dendritic arborization of a neuron, rather than generating each tree on a primary dendrite independently. The occurrence of regional clustering of afferent synapses on the arborization suggests an adaptive utility for developmental control of the branching patterns of the whole neuron. Anatomical and physiological evidence (summarized by Shepherd, 1974) shows that in many cell types each class of afferents to a given type of neuron synapses on a characteristic region of the neuron's surface. In particular, some afferents may synapse preferentially on proximal dendrites, and others on distal dendrites (e.g. LeVay, 1973). Rall (1964) has shown theoretically that the proximodistal position of synaptic input on a dendrite may strongly affect the influence of that input on initiation of impulses in the post-synaptic neuron. If some afferents synapse preferentially on distal dendrites, a developmental mechanism regulating the total surface area of distal dendrites available for these synapses would make input on those afferents elicit more nearly the same output from any neuron in the population, regardless of fluctuations in details of dendritic branching patterns among neurons. (Regulation of post-synaptic surface area of distal dendrites would be manifest in near-constancy of the number of high-order branches if the mean area per high-order branch were nearly independent of TIC). By contrast, if all neurons of the population had trees of the same complexity, regardless of fluctuations in TIC, then afferent input would have a greater postsynaptic effect on neurons with larger TIC. The distribution of input-output relations over the population would then show a larger dispersion, which might result in less satisfactory performance of function. Because of the statistical approach which the presence of cut branches necessitated, we have been able to describe only the average behaviour of neurons - that is, the average number of high-order branches per neuron is more or less independent of TIC. The extent to which the number of high-order branches actually varies among individual neurons can only be determined from knowledge of the complete arborization pattern of individual neurons. Complete reconstruction of neurons by Mannen (1966) yielded results compatible with the hypothesis that the dendritic surface area of
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individual neurons is regulated. Mannen characterized the morphology of neurons from the nucleus gigantocellularis of the bulbar and pontine reticular formation. He traced every dendrite through all sections in which it extended. He observed (loc. cit. p. 159) that "cells which belong to the same type in certain region of the brain may have nearly the same volume and the same area, no matter how irregular their shape may be". Compensation for cutting We have examined the feasibility of compensating statistically for the effects of cutting of dendritic branches. Such statistical compensation requires specification of the relative likelihood of cutting terminal and bifurcating branches. Data such as we have examined, from neurons with cut trees, provide virtually no basis for estimating this relative likelihood. Consequently, estimates of such quantities as the average number of third-order branches per tree are subject to a range of uncertainty which may amount to a considerable fraction of the estimate itself. We found that those comparisons we wished to make, namely among groups of neurons bearing different numbers of trees, were nevertheless conclusive because the differences among groups were large compared with the range of uncertainty. Our statistical method for compensating for cut branches (detailed in the Appendix and by Samuels, 1977) might be useful in analysing future experiments in which relatively large intergroup differences are found. However, if the intergroup differences are small enough to be confounded by cutting, some other approach, such as complete reconstruction of neurons from serial sections, must be used if valid conclusions are to be drawn. SUMMARY
We have investigated whether the complexity of dendritic trees is correlated with the number of primary dendrites per neuron (trees per cell). In estimating the average number of branches of centrifugal orders 1-5 per tree we used statistical methods to compensate for loss of parts of trees during sectioning. Limitations of these methods are discussed. Neurons from four populations, stained by the Golgi-Cox method, were examined: stellate cells from layer IV, area 17 of visual cortex, in normal and dark-reared cats; and pyramidal cells from layer V, somatosensory cortex, in two strains of rats. In all four groups of neurons the average number of branches of higher orders (3, 4, 5) per tree tended to be smaller in neurons bearing more trees. Thus all trees from a population of neurons should not be assumed to be equivalent. The decrease in high-order branches per tree tended to offset the increase in number of trees per cell. In three of the four groups these opposed tendencies maintained the average number of high-order branches per neuron nearly independent of the number of trees per cell. Natural selection may have favoured near-constancy in the number of high-order branches to reduce dispersion among neurons of one type in functional input-output relations.
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REFERENCES
ALTMAN, J. (1973). Experimental reorganization of the cerebellar cortex. 1V. Parallel fiber reorientation following regeneration of the external germinal layer. Journal of Comparative Neurology 149, 181-192. CLENDINNEN, B. G. & EAYRS, J. T. (1961). The anatomical and physiological effects of prenatally administered somatotropin on cerebral development in rats. Journal of Endocrinology 2, 183-193. COLEMAN, P. D. & RIESEN, A. H. (1968). Environmental effects on cortical dendritic fields. I. Rearing in the dark. Journal of Anatomy 102, 363-374. GREENOUGH, W. T. & CARTER, C. S. (1975). Androgen-dependent sexual dimorphism in dendritic branching pattern (clustering) of preoptic area neurons in the hamster. Neuroscience Abstracts 1, 789. GREENOUGH, W. T., VOLKMAR, F. R. & JURASKA, J. M. (1973). Effects of rearing complexity on dendritic branching in frontolateral and temporal cortex of the rat. Experimental Neurology 41, 371-379. GRIFFIN, W. S. T. & WOODWARD, D. J. (1975). Dendritic alterations of developing Purkinje cells during neonatal malnutrition in rat. Neuroscience Abstracts 1, 756. LEVAY, S. (1973). Synaptic patterns in the visual cortex of the cat and monkey. Electron microscopy of Golgi preparations. Journal of Comparative Neurology 150, 53-86. LINDSAY, R. D. & SCHEIBEL, A. B. (1974). Quantitative analysis of the dendritic branching pattern of small pyramidal cells from adult rat somesthetic and visual cortex. Experimental Neurology 45, 424-435. Loos, H. VAN DER. (1959). Dendrodendritische Verbindungen in deschors grote Hersenen. N. V. de Technische Uitgeverij. Haarlem: H. Stam. MACAGNO, E. R., LOPRESTI, V. & LEVINTHAL, C. (1973). Structure and development of neuronal connections in isogenic organisms: Variations and similarities in the optic system of Daphnia magna. Proceedings of the National Academy of Science 70, 57-61. MANNEN, H. (1966). Contribution to the morphological study of dendritic arborization in the brain stem. Progress in Brain Research 21A, 131-162. MCCABE, G. P. & SAMUELS, M. L. (1977). Random censoring and dendritic trees. Biometrics (in the Press). PETERS, H. G. & BADEMAN, H. (1963). The form and growth of stellate cells in the cortex of the guinea pig. Journal of Anatomy 97, 111-117. RALL, W. (1964). Theoretical significance of dendritic trees for neuronal input-output relations. In Neural Theory and Modelling (ed. R. F. Reiss), pp. 73-97. Stanford: Stanford University Press. SAMUELS, M. L. (1977). Statistical methods for estimating the parameters of cut dendritic trees. (In preparation.) SHEPHERD, G. M. (1974). The Synaptic Organization of the Brain. New York: Oxford University Press. SMIT, G. J., UYLINGS, H. B. M. & VELDMAAT-WANSINK, L. (1972). The branching pattern in dendrites of cortical neurons. Acta morphologica neerlando-scandinavica 9, 253-274. SOTELO, C. (1975). Dendritic abnormalities of Purkinje cells in the cerebellum of neurological mutant mice (weaver and staggerer). In Advances in Neurology, 12: Physiology and Pathology of Dendrites (ed. G. W. Kreutzberg), pp. 335-351. New York: Raven Press. UYLINGS, H. B. M., SMrr, G. J. & VELTMAN, W. A. M. (1975). Ordering methods in quantitative analysis of branching structures of dendritic trees. In Advances in Neurology, 12: Physiology and Pathology of Dendrites (ed. G. W. Kreutzberg), pp. 247-254. New York: Raven Press. WILLIAMS, R. S., FERRANTE, R. J. & CAVINESS, V. S. JR. (1975). Neocortical organization in human cerebral malformations: A Golgi study. Neuroscience Abstracts 1, 776. ZAMENHOF, S., MOSLEY, J. & SCHULLER, E. (1966). Stimulation of the proliferation of cortical neurons by prenatal treatment with growth hormone. Science 152, 1396-1397.
APPENDIX
We present here the equations for calculating estimates of the branching parameters flk and Nk under the BB model, and estimates of the cutting probabilities tk and bk under the BB and BC model. We shall use the following notation for the data. For each order k of branching, let xk = number of uncut bifurcating kth order branches, Yk = number of uncut terminal kth order branches, and Zk = number of cut kth order branches.
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First, we present an intuitive derivation of the estimates of the parameters ,8k, tk and bk under the BB and BC model. (This derivation is equivalent to that used by Smit et al. (1972), except that Smit et al. set Ak = 2, while our formulation leaves Ak arbitrary.) Of all the kth order branches which existed before cutting, a total of (Xk + YK + ZJ) are visible in the data; i.e. they were not excised by cuts on branches of previous orders. Of these (Xk + Yk + Z) branches, we expect under the BB and BC model fractions hJ(1 - bk) to bifurcate and not be cut, (1 - /1k) (1 - tk) to terminate and not be cut, and 1/ bk+ (l -/hv) tk to b- cut. Equating the first two of these to the corresponding fractions observed in the data gives bk),
(1)
X}, + Yk + Zk(1)
(
Xk+Yk+Zk
h 8tc
2
Setting tkc = Ak bk and eliminating bk from (1) and (2) yields /3. (1 -Ak) (Xk+yk+ Zk) + /k[Ak(2xk+yk+ zk) - (Xk+ Zk)] -Ak Xk = 0(3) Once A, is specified, equation (3) has a unique* solution for I8 between 0 and 1; this solution is the estimate Ik. The cutting probabilities are then estimated by
bk
(4)
lA
=
IN(Xk+YVk+ Zk)
and tk
=
(5)
Ak bk
In the extreme case Ak = oo, equations (3)-(5) are replaced by xXk = b= 0
(4')
and tk
(3')
Zk Yk +
Zk-(5 )
Under the BB model, the cutting probabilities tk and bk are allowed to depend upon the position of the tree with respect to the cutting plane, in such a way that Ak, = tk/bk is constant. Under the BB model, then, the intuitive derivation given above does not work. Nevertheless, as shown by Samuels (1977), when the method of maximum likelihood is applied to the BB model, the resulting equation for the estimated branching probability /3,k is exactly the same as that presented above, equation (3) (or (3')). That is, once Ak is specified, the estimate 1k is the same under the BB as under the BB and BC model. * We assume throughout that unique for certain values of Ak-
Xk and Yk are both non-zero; if either is zero, the solution may not be
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Estimates of the Nk under the BB or the BB and BC model are obtained from the /]k as:
N1= 1
R=41= = N4 N4 = 8/J1l/?3(6 id1424 A
A
(6)
2kflAfl2 ...fik-1. Equations (3) (or (3')) and (6) were used for calculating all estimates Rk discussed in the text, except for R3. The equations for estimating S under the MBC model (in which the assumption of binomial branching is dropped) are more complex, and are presented by Samuels (1977). Nk=
J. Anat. (1977), 124, 3, pp. 701-715 With 2 figures Printed in Great Britain
Complexity of branching in dendritic trees: dependence on number of trees per cell and effects of branch loss during sectioning MYRA L. SAMUELS,* JAY E. MITTENTHAL,f GEORGE P. MCCABE, JR.* AND PAUL D. COLEMANt
*Department of Statistics, Purdue University,
tDepartment of Biological Sciences, Purdue University,
;Department of Anatomy, University of Rochester (Accepted 29 October 1976) INTRODUCTION
Quantitative characterization of the morphology of individual neurons provides precise indices for the influence of various factors on development of the nervous system. Recent studies have examined effects on dendritic branching of several factors extrinsic to the local neighbourhood of a developing neuron - characteristics of the animal's environment (Coleman & Riesen, 1968; Greenough, Volkmar & Juraska, 1973), aspects of its nutrition (Griffin & Woodward, 1975), and levels of hormones (Clendinnen & Eayrs, 1961; Zamenhof, Mosley & Schuller, 1966; Greenough & Carter, 1975). Effects on development of dendrites of factors intrinsic to the neuron and its local neighbourhood have been studied by use of neurological mutants (Sotelo, 1975; Williams, Ferrante & Caviness, 1975), of isogenic organisms (Macagno, LoPresti & Levinthal, 1973), and of irradiation (Altman, 1973). Such studies can reveal correlations between a treatment and a morphological index, or correlations among indices. Patterns of correlations, in turn, may suggest developmental mechanisms involved in generating neural organization. In our search for structural invariance in dendritic trees we examined the relationship between the complexity of individual dendritic trees and the number of trees per neuron. We have observed a noteworthy correlation between these two morphological indices: Neurons of a given type bearing fewer trees tend to have more complex trees.§ For some classes of neurons from the cerebral cortex of cat and rat this tendency maintains the average number of high-order branches per neuron (with centrifugal ordering) roughly constant in spite of twofold variation in number of trees per neuron. As an adjunct to the analysis underlying this conclusion, we have developed a statistical method designed to compensate for loss of parts of dendritic trees during sectioning. This report is organized as follows: First we give a description of the data base. In a preliminary analysis we examine the relation of number of branches per tree to number of trees per cell without correcting for the cutting of distal branches. We then discuss statistical approaches to the problem of compensating for cutting. Next we present the results obtained by applying these statistical methods to the data in order to estimate the average number of branches of each order per tree before § Throughout this paper 'tree' refers to the dendritic arborization of a single primary dendrite. 45
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cutting. Finally we discuss the biological implications of our findings and examine the general applicability of our methods of compensating for cutting. ANALYSIS OF DATA
Data base In this investigation four groups of neurons were studied: stellate cells from layer IV of visual cortex in dark-reared and control cats, and pyramidal cells from layer V of somatosensory cortex of rats, Berkeley strains SI and S3. Stellate cells of the spiny and non-spiny classes (LeVay, 1973) were not distinguished. Neurons were stained by a Golgi-Cox method described previously (Van der Loos, 1959). Sections were 100 ,um thick; cell body diameter was 20-30 /tm. Sections were cut in a standard plane which was perpendicular to the brain surface. Neurons of each specified type were selected as follows - microscope stage X, Y coordinates were written on a separate slip of paper for each cell that satisfied the following requirements: (1) cell body in centre (in Z axis) of section; (2) dendrites unobscured by glia, blood vessels or excessive numbers of processes of other cell bodies. The desired number of cells was selected by drawing randomly from this pool of acceptable neurons. For the selected neurons, projections of all dendrites of stellate cells and all basal dendrites of pyramidal cells were traced using a camera lucida. Transection of a branch (cutting) at a boundary of a section was noted on each drawing. Distal portions of cut branches were not traced in adjacent serial sections. The number of dendritic trees per cell, and the branching and cutting patterns in each tree were determined. Tables 1 and 2 summarize information about the data base.
Preliminary analysis As an initial step in examining the dependence of tree complexity on the number of trees per cell we chose, as a measure of the complexity of a tree, the number of branches it possesses. The total number of branches in the camera lucida drawing of each tree was counted; each branch-segment between bifurcation nodes was counted as a separate branch. (Of course, some of these branches were cut, so that any additional branches beyond the cut were missing.) Combining the counts for each cell we computed the average number of branches per tree for each cell. The cells were then divided into groups according to the number of dendritic trees on the cell; the mean and standard deviation of the average number of branches per tree was then calculated for each group. These are shown in Figure 1. The number of cells in each group can be determined from Table 2. The graphs of Figure 1 (except perhaps for the Pyramidal S1 data set) suggest that cells with a small number of trees tend to have more branches per tree than those with large numbers of trees. The effect of cutting, however, cannot be ignored. Of the trees in the data sets 45-65 % had one or more branches cut. If cutting is ignored a significant bias would probably be introduced. If cells with a large number of trees had trees bearing longer branches, which would be more likely to be cut, then one would expect that cutting alone could produce, as an artifact, the apparent relationships evident in Figure 1. On the other hand, ignoring differences in branch length among cells, one would expect that trees with more branches would probably have more branches excised through cutting. Hence if cells having a small number of trees tended to have more
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Table 1. Description of data Name of data set
Stellate Control Stellate Dark Pyramidal SI Pyramidal S3
Type of animal Cat Cat Rat Rat
No. of No. of No. of dendritic animals cells trees
Type of cell
(control) (dark reared) (Berkeley SI) (Berkeley S3)
Layer lV stellate Layer IV stellate Layer V pyramidal Layer V pyramidal
45 45 44 44
3 3 3 3
232 226 202 202
Table 2. Number of neurons bearing various numbers of trees (For pyramidal cells only basal trees are counted.) Number of trees per cell Data set
2
3
4
5
6
7
8
9
Stellate Control Stellate Dark Pyramidal SI Pyramidal S3
0 0 0 1
2 4 6 5
12 8 15 15
17 20 16 15
7 11 5 6
5 1 2 2
2 0 0 0
0 1 0 0
complex trees, then the true relationship between average number of branches per tree and number of trees per cell would actually be more pronounced than that indicated in Figure 1. Thus an investigation of the relation between tree complexity and number of trees per cell cannot proceed unless effects of cutting are considered. In the next section we describe statistical methods of compensating for the loss of branches through cutting. The following section summarizes results of applying these methods to the data.
Statistical methods of compensating for cut branches In order to take a statistical approach to the problem of cutting we divided the cells of each data set into three groups: cells bearing four or fewer trees, cells bearing five trees, and cells bearing six or more trees. With this grouping each group contained sufficient trees to allow estimation of the average number of branches per tree within the group by statistical methods. The number of trees in any group may be easily calculated from Table 2. For example, the number of trees on Stellate Control cells with < 4 trees per cell is 3(2) + 4(12) = 54. Thus we have clustered the trees into 12 groups: four data sets, each divided three ways according to the number of trees per cell. We consider each of these 12 groups to be a sample from a population, and we assume each population to be homogeneous in the sense that the probabilistic rules governing tree branching structure are the same for each tree in the population. (This assumption is only an approximation, since the number of trees per cell within a group may vary. The extent of heterogeneity of the groups' < 4 trees per cell' and' > 6 trees per cell' can be judged from Table 2.) For each of the 12 groups separately, then, we may abstract the statistical problem as follows: given data on a sample of trees (some of which have cut branches) from a homogeneous population, we wish to estimate the average number of branches per tree in the population. It is convenient to attack this problem in stages, considering dendritic branches of various orders separately. We order branches by a centrifugal 45-2
704 8
MYRA L. SAMUELS AND OTHERS 8 B x 7 6 I -
A
7
6
I
I xI
5
B/T 4
5
B/T 4 _
x
XI
3
3
2
2-
x
x
I
0
1
2
3 4
T/C 8
t B/T
5
6
7
8 9
1
0
D
8
7
7
6
6
I
3-
X
BIT
I
6
7
8 9
7
8 9
I
4
3 2
2
_ 0
xl X
-~~
5
x1
4
5
3 4
T/C -o
C
X
2
--.
1
2
3
4
5
T/C
->-
6
7
8 9
0
I
I
I
1
2
3
4
5
6
TIC o.
Fig. 1. Mean ± 1 standard deviation of average number of branches per dendritic tree (BIT) as a function of the number of trees per cell (TIC). Where standard deviation is not shown, the group contained only one or two cells. Computed from raw data, without compensating for cut branches. (A) Stellate Control, (B) Stellate Dark, (C) Pyramidal Si, (D) Pyramidal S3. See Tables 1 and 2 and accompanying text for descriptions of these data sets.
method: * The primary dendrite is the first-order branch, the two (if any) arising from it are second-order branches, and so on. We denote by Nk the average number of kth-order branches per tree (before cutting). Thus N1 = 1 always. ik denotes the estimate of Nk obtained from data by statistical inference. Models for estimating Nk. We first discuss the estimation of N2. In order to estimate N2 it is necessary to make some assumptions regarding the probability that a given first-order branch is cut. Since there is evidence that terminal branches are longer than bifurcating branches (see below), and since longer branches have a greater chance of being cut, we assume that the cutting probability depends upon whether the given branch bifurcates or is terminal; thus, we let t, = probability that a first-order terminal branch is cut, b, = probability that a first-order bifurcating branch is cut. For estimating N2, we have available three pieces of data: xi = number of bifurcating uncut first-order branches, Yi = number of terminal uncut first-order branches, z, = number of cut first-order branches. * Centripetal methods of ordering (summarized by Uylings, Smit, & Veltman, 1975) were not used because cutting removes distal branches, preventing correct centripetal ordering. Cutting does not affect the centrifugal ordering used here. Furthermore, centrifugal ordering reflects distance from the cell body, and the probability of cutting presumably depends on distance from the cell body. Thus, the assumption which we make in our models that the probability of cutting depends on order is a natural one when centrifugal ordering is used. -
-
Complexity of dendritic branching
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Let T = x1 +y + z1 = the total number of trees in the sample. The estimate N2 of the parameter N2 will be of the form
-
2(x1+pzl)
T where p is the fraction of the cut first-order branches presumed to have bifurcated. If a first-order terminal branch is as likely to be cut as a first-order bifurcating branch (t1 = b1), then a reasonable value forp would bep = x1/(xl+yj). On the other hand, p would be less than this value if bifurcating branches were less likely to be cut (b1 < t1), andp would be more than this value if b1 > t1. The extreme cases are b1 = 0 (in which case p = 0) and t1 = 0 (in which case p = 1). It is intuitively obvious that the data (xl, Yi, zj) are equally compatible with either of these extreme cases, as well as with any intermediate case. The above discussion can be summarized as follows: Let A1 = t1/bl. If a value of A1 is assumed, N2 can be estimated from the data. There is, however, no way of estimating A1 itself from the data. The considerations in the estimation of N3, N4, etc. are similar but more complex. Samuels (1977) and McCabe & Samuels (1977) have considered several possible models for describing the stochastic structure of the data. We will now briefly describe how some of these models are used in the estimation of N3, N4, etc. The simplest model is called by Samuels (1977) the Binomial Branching and Binomial Cutting Model (BB and BC). This model assumes that the several branches of a given order (a) bifurcate independently of each other and (b) are cut or not cut independently of each other. These are called the Binomial Branching (BB) assumption and the Binomial Cutting (BC) assumption. The parameters of the model are:
flk= probability that a kth order branch bifurcates,
tk = probability that a kth order terminal branch is cut, bk = probability that a kth order bifurcating branch is cut.
Using the BB and BC model, estimates of the parameters /3k, tk and bk are easily obtained from the data, so long as the ratios Ak = tk/bk are specified. The parameters Ak cannot be estimated from the data under the BB and BC model. Smit, Uylings & Veldmaat-Wansink (1972) used the BB and BC model (implicitly) to obtain estimates Of /3k, tk and bk for their data, using a more or less arbitrary value of Ak = 2 for each k. The BB and BC model can be criticized on two grounds: (a) the Binomial Branching assumption may or may not be true; and (b) the Binomial Cutting assumption is assuredly false, since the several branches of a given order are cut, if at all, by the same plane, namely the surface of the section containing the neuron. In Samuels (1977) a more general model, called Modified Binomial Cutting (MBC) is developed for k = 2 (i.e. for cutting and bifurcation of second-order branches, and thus for estimation of N3). In the MBC model, (a) bifurcation of sister branches may be stochastically dependent; and (b) the cutting probabilities t2 and b2 are allowed to depend upon the position of the tree with respect to the cutting plane, but only in such a way that the ratio t2/b2 = A2 iS constant. Under the MBC model, the parameter A2 may or may not be estimable, depending on the data set. For example, if a data set fits the BB and BC model closely, then it contains virtually no information about A2; on the other hand, a data set with strong correlation between bifurcation of sister branches may allow close estimation of A2. This unusual statistical situation is discussed in detail by Samuels (1977). The data sets used in the present study contain
706
MYRA L. SAMUELS AND OTHERS some information about A2, but the information is poor; in fact, the only statistically significant finding about A2 was that the group 'Stellate Control > 6 trees per cell' was not compatible with A2 values in the range 0-3-1 8. In view of the paucity of information about A2, it was decided to treat A2 as an arbitrary parameter in the present work. The MBC model can be extended to cover higher orders of branching, i.e. k = 3, .... Because of the complexity of the computations and because of the relatively small numbers of higher-order branches in our data, this extension has not been carried out. Rather, we have used, for our estimation of N4 and N5, the Binomial Branching (BB) model. The BB model takes the BB assumption (a) of the BB and BC model and combines. it with an assumption about spatial constancy of A,k = tklbk, analogous to assumption (b) of the MBC model. The compatibility of the data with the BB assumption was examined for the case k = 2 by testing the BB model against the MBC model (which lacks the BB assumption) for each of the 12 groups of trees. The Binomial Branching hypothesis was not rejected (at the 5 % significance level) for any group for any value of A2 > 1. Furthermore, numerical investigation indicated that the numerical values of ki were nearly the same under the BB and the MBC models. It is felt that these results justify use of the BB model for k 3 and k = 4 in the present study. To summarize: in the present study N3 is estimated using the MBC model; N4 and N5 are estimated using the more restrictive BB model; for estimation of N2 all the models are equivalent. For the reader who wishes to estimate the Nk for his own data, we have provided in the Appendix the necessary formulae for estimating the branching parameters /k and Nk under the BB model, and also the cutting probabilities tk and bk under the BB and BC model. Details of the models, and also formulae for estimating second-order branching and cutting parameters under the MBC model, are given by Samuels (1977). Whichever model is used, in the present study the Ak are treated as arbitrary parameters; that is, the Ak are pre-specified, rather than estimated from the data. The estimation of each Nk requires specificationof A,k for all lower orders; thus estimation of N2requires specificationofAL, estimation of N3 requires specification ofALandA2, etc. ChoiceofAk. To get an idea of what may be 'reasonable' values for Ak, we turn to the literature. There is considerable evidence that terminal branches of a given order tend to be longer than bifurcating branches of the same order. This phenomenon was observed, for example, by Smit et al. (1972) for basal dendrites of pyramidal cells from the striate cortex of adult rabbits, and by Peters & Bademan (1963) for stellate cells from the sensory-motor cortex of newborn and adult guinea-pigs. While these studies contain unknown biases because of cutting, it appears highly unlikely that the phenomenon is an artifact of cutting. Lindsay & Scheibel (1974), studying pyramidal cells from somesthetic and visual cortex of adult albino rats, made length measurements on dendritic trees reconstructed from serial sections, so that no cutting occurred. They also found that terminal branches were longer than bifurcating branches. Their data is not broken down by order of branching, but this probably has little effect, since Smit et al. (1972) and Peters & Bademan (1963) found little dependence of branch length upon order of branching. The above-mentioned authors found that terminal branches of a given order were, on the average, from twice as long to more than four times as long as bifurcating branches of the same order. Since longer branches have a greater chance of being cut, it appears very likely that, for each k, tk > bk, so that Ak > 1.
4,
Complexity of dendritic branching
707
In applying our statistical methods to the data, we have actually let each of the Ak vary from 0 (meaning tk = 0) to oo (meaning bk = 0). In the following section we illustrate the results of this analysis with selected values of the Ak, and in addition we indicate the range of Ak values within which the results are similar to those illustrated.
Application of statistical method to data In this section we examine the dependence of tree complexity upon the number of trees per cell, denoted henceforth by TIC. We first consider the estimates N3 of the average number of third-order branches per tree. These estimates were generated by the MBC method, letting A1 and A2 take various values. The graphs of Fig. 2 show the estimates N3 obtained when A1 and A2 have equal values, ranging from 0 to 4; the asymptotic value of N3 for A1 = A2 = Xo is also shown. For each data set, N3 is shown as a function of the common value of A1 and A2, plotted as the abscissa, and as a function of TIC by separate curves. The most striking feature of Figure 2 is that N3 is a decreasing function of TIC, no matter what value of A1 and A2 is chosen. (The only exception to this is the Pyramidal Strain SI data set in the range A1 = A2 < 0-8 which are, as we have noted above, probably unrealistic values of A1 and A2.) It is possible that the value of A1 = A2 varies as a function of TIC. However, examination of Figure 2 shows that, unless such variation is extremely large, 3 still decreases as a function of TIC. For example, in the Stellate Control data set, if A1 = A2 > 4 for TIC < 4, and A1 = A2 = 1 for TIC = 5, then the 93 pattern between these two groups is reversed. However, such a large variation of the A1, A2 values seems implausible. It is also possible that A1 and A2 are not equal. We have examined the dependence of N3 on TIC for various combinations of A1 and A2 in the range 0-8 to co. The pattern is in all cases the same as that indicated by Figure 2: The more trees a cell possesses, the fewer the third-order branches borne, on the average, by each tree. The standard errors of the estimates N3 are of the order 0-1554X25 when A1 = A2 =1 (see McCabe & Samuels, 1977). However, since A1 and A2 are not known, and since 3 depends on A1 and A2, these standard errors are of questionable relevance. In any event, the strong statistical significance of the trend in R3 as a function of TIC can be established by a non-parametric argument, without reference to standard errors. This argument is as follows: There are 3! = 6 possible patterns of variation of 3 with T/C-monotone increasing, monotone decreasing, and four others. In each of our four data sets the same (monotone decreasing) pattern is observed. If the six possible patterns are equally likely, then the probability of observing the same pattern in four data sets is 6(0)4, which is < 0'005. Thus the null hypothesis of no trend can be rejected at the 0-5 % significance level, in favour of the alternative hypothesis of a monotone decreasing trend. We now turn to other orders of branching. Table 3 shows the estimates R2, N3, N4 and R. as a function of TIC for the four data sets. (The data contained no branches of order higher than 5.) These estimates were generated for A1 = A2 = A3 = A4 = 2. No consistent pattern of dependence of N2 on TIC is evident in Table 3. If such a pattern exists it is apparently too weak to be detected on the basis of samples as small as ours (the standard errors of R2 are of the order of 0-05-0-10). On the other hand, Table 3 shows that the downward trend of N3 as a function of TIC is generally repeated by R4 and N,. The only exceptions are N4 for the Pyramidal S3 data set, and N3 for the Pyramidal SI data set, where the trend is not maintained.
708
MYRA L. SAMUELS AND OTHERS 3
3
2
2
N3
B
N3
1%
1
1
1
2
1
0
3'1
-
Al and A2
c
3 -
oo
4
1
0
r'F
2
3 2 A1 and A2
4
00
3 2 A1 and A2 -
4 '
00
D
2
N
R3
N3 1
1
I
0
I
1
Al
I
2 3 and A2
I
#,
4 ' o00
0
1
Fig. 2. Estimated number of third-order branches per tree (N3) as a function of the common value of A1 and A2 and of the number of trees per cell. ---- -, < 4 trees per cell; 5 trees per cell; - --, 6 trees per cell. (A) Stellate Control, (B) Stellate Dark, (C) Pyramidal SI, (D) Pyramidal S3. See text for definition of A1 and A2. ,
In view of the relatively large standard errors of the estimates S4 and R., we do not regard these exceptions as significant. We have examined the dependence of S4 and R, on TIC for other choices of the Ak. For Ak in the range 1-4, with all Ak's equal, the pattern is essentially the same as that seen in Table 3, namely that the number of fourth- and fifth-order branches per tree, like the number of third-order branches per tree, is smaller for cells with more trees. Having established that N3, N4, and N5 are decreasing functions of TIC, we turn to a more refined question: Is the amount of decrease in these Nk sufficient to compensate for the increase in TIC? That is, are the numbers of third-, fourth-, and fifthorder branches per cell approximately constant? To answer this question, we define Wk to be the average (over all cells in a population) number of kth order branches per cell. We estimate Wkfor each of our 12 data groups by Wk = RAk(TIC), where T the number of trees in the group and C = the number of cells in the group. Table 4 displays these estimates for the two stellate data sets. The quantities shown are W1 (= TIC), W2, W3, W4, J5 and (W2+W3+IVt4+W,); these estimates have been computed with all Ak = 2. In considering Table 4, the reader should bear in mind that if there were no dependence of bifurcation pattern on the number of trees per =
Complexity of dendritic branching
709
Table 3. 54 as a function of TIC (computed with all Ak = 2) Stellate Control
Stellate Dark Pyramidal Sil
TIC
N2
< 4 5 > 6 6 4 5
200 1-93 1-74 1-90 1-95 1-77 1-79 1-86
6
169 200 1-83 1-87
>
Pyramidal S3
< 4
5 > 6
2-11 1-72 1-09 1.91 1-39 1-02 1 38 1-26 1-21 1-78 1-15 1-06
N4
N5
090 0 74 047 0 95 0 55 0-39 0-61 0 45 043 0-52 044 045
0-25 0-24 004 0-22 0-14 000 0 07 0 04 007 0-15 0-14 0-06
Table 4. Ik as a function of TIC for Stellates (computed with all A k = 2) TIC Stellate Control
< 4
Stellate Dark
5 > 6 < 4 5 > 6
W, = TIC W2 39 7-7 50 9-6 6-6 11-5 3-7 70 9-8 5-0 6-3 11-2
W3
W4
W5
12 Wk
8-1 8-6 7-3 70 7-0 6-5
3-5 3-7
1.0 1-2 03 0-8 07 0-0
20-3 23-2 22-2 18-3 20-2 20-1
3-1 3-5 2-8 2-5
cell, then the values of k would be directly proportional to TIC (within each data set). For this reason, variations in the Jtk should be considered on a percentage basis. The $k of Table 4 shows a striking pattern. As W, = TIC increases, W2 also increases, although in a somewhat smaller ratio. On the other hand, in many cases W3, H'4, and W5 tend to decrease as TIC increases. This decrease is not very great for P3,, which is remarkably constant compared to J- and 0T2; the decrease is relatively larger for W4J and TV5. The total effect of these downward trends in T-3, W4 and W5 is remarkable: they virtually cancel the upward trend in JV2. This is shown by the last column of Table 4: for each data set the quantity (02+ W. + + J+5) hardly changes as TIC increases. Thus, for example, in the Stellate Control data set, although the number of trees per cell varies from 3-9 to 6-6, an increase of 70 %, the quantity 2 k varies by only about 14 % (and not monotonically). The variation in 5 kk is even less, about 10 %, for the Stellate Dark data set. It would appear, then, that for the Stellate data sets, the decreases in 53, N4, and R5 actually overcompensate for the increase in TIC, so that a cell bearing many trees actually produces, on the average, fewer third-, fourth- and fifth-order branches than a cell bearing few trees. In this way, the higher-order trees balance the second-order trees, with the result that, for the stellate cells, the average total number of second- and higher-order branches per cell is approximately constant, even though these branches may be borne on few or many trees. Table 5 shows the Wk for the Pyramidal data sets (also computed with all Ak = 2). The last column of Table 5 shows (O3 + W4+ TV5) rather than (2+ T3+ -4 + TV5). The reason for this difference is that the pattern of the Tkk for the Pyramidal cells is not
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MYRA L. SAMUELS AND OTHERS
Table 5.
J-k as a function of TIC for Pyramidals (computed with all Ak
Pyramidal
S,
Pyramidal S,
= 2) Wk
TIC
W1= TIC
W2
W3
W4
W5
3
S4 5 > 6 < 4 5 > 6
3-7 50 6-3 3-7 50 6-2
6.7 93 10-6 7.3 9.1 11.7
5.1 6-3
2.3 2.2 2-7 1.9 2-2 2-8
03 0-2 04 05 07 03
77 8-8 10-1 90 8-7 9-8
7-6 65 5*8 6-6
the same as that shown by the Stellates. As W1 and JV2 increase, there is no tendency for iW3, W4, and W5 to decrease; in the Pyramidal S3 data set, they are relatively constant as a function of TIC, while in the Pyramidal Si data set, they increase, although at a smaller rate than TIC. The result is that for the Pyramidal S3 data set, the quantity (Jk3+ 4+ 's) is relatively constant, varying by only 13 % (and not monotonically) while TIC increases by 68 %. In the Pyramidal SI data set, the variation in 15 It is much larger, about 39 %, but still smaller than the variation in TIC. The pattern of constancy of the kk is thus less clear for the Pyramidal cells than for the Stellate cells. There is a tendency for the average total number of third- and higher-order branches per Pyramidal cell to be maintained constant regardless of the number of trees on a cell. This tendency is present in both Pyramidal data sets, but for some reason it is less clearly evident in the Pyramidal S1 data set. We have examined tables analogous to Tables 4 and 5 for other values of the Ak, varying from 1 to oo. In all cases the patterns were essentially the same as those seen in Tables 4 and 5. It is worth noting that, while the patterns of dependence of such quantities as 2 Wk, etc., upon TIC are maintained for a wide range of values of the Ak, the absolute value of these measures of tree complexity may depend strongly upon Ak. The values of S3, for example, may vary by as much as 40 % as A1 and A2 range from 1 to 4. (This may be seen in the graphs of Fig. 2.) This emphasizes the importance of compensation for cutting, and at the same time reveals its limits: groups of data can only be compared if the differences among them are large compared to the range of uncertainty (often itself large) resulting from ignorance of the values of the Ak. DISCUSSION
We have investigated the relation between the number of dendritic trees (primary dendrites) per neuron and the complexity of those trees in samples from four populations of neurons from mammalian cerebral cortex. We defined a more complex tree as a tree bearing a larger number of high-order branches. To characterize complexity we estimated the average number of branches of each order per tree, using statistical methods to compensate for loss of portions of trees during cutting of the section containing the neuron. For all four groups of neurons the estimated mean number of branches of orders 3, 4 and 5 per tree decreased as the number of trees per cell (TIC) increased, with minor exceptions. That is, we have found that the more trees a neuron bears, the less complex those trees tend to be. Consequently, the number of high-order branches per neuron varies less than TIC. Indeed, in three of the four populations the number of high-order branches is nearly independent of TIC; in the fourth, cells with larger TIC tend to have more high-order branches.
Complexity of dendritic branching
711
Number oJ branches The inverse relation between tree complexity and TIC indicates that, for purposes of morphometric study of dendrites, trees from a sample of neurons are not necessarily all equivalent. Trees of a sample should not be pooled unless it is verified that all neurons of the sample are equivalent with respect to the properties of interest. In particular, our results show that topological properties such as bifurcation probabilities can depend on TIC. Consequently, pooling of cells bearing different numbers of trees may not always be justifiable in an analysis of topological properties. If metric properties (branch lengths, interbranch angles) are to be studied, suitable compensation must be made for correlation between metric and topological properties. The observation that terminal branches tend to be longer than bifurcating branches is an example of such metric-topological correlation. It is possible that the broad dispersions which have been observed in distributions of branch length (Smit et al. 1972; Lindsay & Scheibel, 1974) would be substantially reduced if neurons were grouped according to values of TIC. It might also be found that clustering of unusual metric properties in certain neurons (e.g. Smit et al. 1972, Table 5) can be interpreted with knowledge of the topological characteristics of these neurons. Our conclusion, that the average number of high-order branches per cell is approximately independent of TIC, suggests that developmental mechanisms may regulate the development of the total dendritic arborization of a neuron, rather than generating each tree on a primary dendrite independently. The occurrence of regional clustering of afferent synapses on the arborization suggests an adaptive utility for developmental control of the branching patterns of the whole neuron. Anatomical and physiological evidence (summarized by Shepherd, 1974) shows that in many cell types each class of afferents to a given type of neuron synapses on a characteristic region of the neuron's surface. In particular, some afferents may synapse preferentially on proximal dendrites, and others on distal dendrites (e.g. LeVay, 1973). Rall (1964) has shown theoretically that the proximodistal position of synaptic input on a dendrite may strongly affect the influence of that input on initiation of impulses in the post-synaptic neuron. If some afferents synapse preferentially on distal dendrites, a developmental mechanism regulating the total surface area of distal dendrites available for these synapses would make input on those afferents elicit more nearly the same output from any neuron in the population, regardless of fluctuations in details of dendritic branching patterns among neurons. (Regulation of post-synaptic surface area of distal dendrites would be manifest in near-constancy of the number of high-order branches if the mean area per high-order branch were nearly independent of TIC). By contrast, if all neurons of the population had trees of the same complexity, regardless of fluctuations in TIC, then afferent input would have a greater postsynaptic effect on neurons with larger TIC. The distribution of input-output relations over the population would then show a larger dispersion, which might result in less satisfactory performance of function. Because of the statistical approach which the presence of cut branches necessitated, we have been able to describe only the average behaviour of neurons - that is, the average number of high-order branches per neuron is more or less independent of TIC. The extent to which the number of high-order branches actually varies among individual neurons can only be determined from knowledge of the complete arborization pattern of individual neurons. Complete reconstruction of neurons by Mannen (1966) yielded results compatible with the hypothesis that the dendritic surface area of
712
MYRA L. SAMUELS AND OTHERS
individual neurons is regulated. Mannen characterized the morphology of neurons from the nucleus gigantocellularis of the bulbar and pontine reticular formation. He traced every dendrite through all sections in which it extended. He observed (loc. cit. p. 159) that "cells which belong to the same type in certain region of the brain may have nearly the same volume and the same area, no matter how irregular their shape may be". Compensation for cutting We have examined the feasibility of compensating statistically for the effects of cutting of dendritic branches. Such statistical compensation requires specification of the relative likelihood of cutting terminal and bifurcating branches. Data such as we have examined, from neurons with cut trees, provide virtually no basis for estimating this relative likelihood. Consequently, estimates of such quantities as the average number of third-order branches per tree are subject to a range of uncertainty which may amount to a considerable fraction of the estimate itself. We found that those comparisons we wished to make, namely among groups of neurons bearing different numbers of trees, were nevertheless conclusive because the differences among groups were large compared with the range of uncertainty. Our statistical method for compensating for cut branches (detailed in the Appendix and by Samuels, 1977) might be useful in analysing future experiments in which relatively large intergroup differences are found. However, if the intergroup differences are small enough to be confounded by cutting, some other approach, such as complete reconstruction of neurons from serial sections, must be used if valid conclusions are to be drawn. SUMMARY
We have investigated whether the complexity of dendritic trees is correlated with the number of primary dendrites per neuron (trees per cell). In estimating the average number of branches of centrifugal orders 1-5 per tree we used statistical methods to compensate for loss of parts of trees during sectioning. Limitations of these methods are discussed. Neurons from four populations, stained by the Golgi-Cox method, were examined: stellate cells from layer IV, area 17 of visual cortex, in normal and dark-reared cats; and pyramidal cells from layer V, somatosensory cortex, in two strains of rats. In all four groups of neurons the average number of branches of higher orders (3, 4, 5) per tree tended to be smaller in neurons bearing more trees. Thus all trees from a population of neurons should not be assumed to be equivalent. The decrease in high-order branches per tree tended to offset the increase in number of trees per cell. In three of the four groups these opposed tendencies maintained the average number of high-order branches per neuron nearly independent of the number of trees per cell. Natural selection may have favoured near-constancy in the number of high-order branches to reduce dispersion among neurons of one type in functional input-output relations.
Complexity of dendritic branching
713
REFERENCES
ALTMAN, J. (1973). Experimental reorganization of the cerebellar cortex. 1V. Parallel fiber reorientation following regeneration of the external germinal layer. Journal of Comparative Neurology 149, 181-192. CLENDINNEN, B. G. & EAYRS, J. T. (1961). The anatomical and physiological effects of prenatally administered somatotropin on cerebral development in rats. Journal of Endocrinology 2, 183-193. COLEMAN, P. D. & RIESEN, A. H. (1968). Environmental effects on cortical dendritic fields. I. Rearing in the dark. Journal of Anatomy 102, 363-374. GREENOUGH, W. T. & CARTER, C. S. (1975). Androgen-dependent sexual dimorphism in dendritic branching pattern (clustering) of preoptic area neurons in the hamster. Neuroscience Abstracts 1, 789. GREENOUGH, W. T., VOLKMAR, F. R. & JURASKA, J. M. (1973). Effects of rearing complexity on dendritic branching in frontolateral and temporal cortex of the rat. Experimental Neurology 41, 371-379. GRIFFIN, W. S. T. & WOODWARD, D. J. (1975). Dendritic alterations of developing Purkinje cells during neonatal malnutrition in rat. Neuroscience Abstracts 1, 756. LEVAY, S. (1973). Synaptic patterns in the visual cortex of the cat and monkey. Electron microscopy of Golgi preparations. Journal of Comparative Neurology 150, 53-86. LINDSAY, R. D. & SCHEIBEL, A. B. (1974). Quantitative analysis of the dendritic branching pattern of small pyramidal cells from adult rat somesthetic and visual cortex. Experimental Neurology 45, 424-435. Loos, H. VAN DER. (1959). Dendrodendritische Verbindungen in deschors grote Hersenen. N. V. de Technische Uitgeverij. Haarlem: H. Stam. MACAGNO, E. R., LOPRESTI, V. & LEVINTHAL, C. (1973). Structure and development of neuronal connections in isogenic organisms: Variations and similarities in the optic system of Daphnia magna. Proceedings of the National Academy of Science 70, 57-61. MANNEN, H. (1966). Contribution to the morphological study of dendritic arborization in the brain stem. Progress in Brain Research 21A, 131-162. MCCABE, G. P. & SAMUELS, M. L. (1977). Random censoring and dendritic trees. Biometrics (in the Press). PETERS, H. G. & BADEMAN, H. (1963). The form and growth of stellate cells in the cortex of the guinea pig. Journal of Anatomy 97, 111-117. RALL, W. (1964). Theoretical significance of dendritic trees for neuronal input-output relations. In Neural Theory and Modelling (ed. R. F. Reiss), pp. 73-97. Stanford: Stanford University Press. SAMUELS, M. L. (1977). Statistical methods for estimating the parameters of cut dendritic trees. (In preparation.) SHEPHERD, G. M. (1974). The Synaptic Organization of the Brain. New York: Oxford University Press. SMIT, G. J., UYLINGS, H. B. M. & VELDMAAT-WANSINK, L. (1972). The branching pattern in dendrites of cortical neurons. Acta morphologica neerlando-scandinavica 9, 253-274. SOTELO, C. (1975). Dendritic abnormalities of Purkinje cells in the cerebellum of neurological mutant mice (weaver and staggerer). In Advances in Neurology, 12: Physiology and Pathology of Dendrites (ed. G. W. Kreutzberg), pp. 335-351. New York: Raven Press. UYLINGS, H. B. M., SMrr, G. J. & VELTMAN, W. A. M. (1975). Ordering methods in quantitative analysis of branching structures of dendritic trees. In Advances in Neurology, 12: Physiology and Pathology of Dendrites (ed. G. W. Kreutzberg), pp. 247-254. New York: Raven Press. WILLIAMS, R. S., FERRANTE, R. J. & CAVINESS, V. S. JR. (1975). Neocortical organization in human cerebral malformations: A Golgi study. Neuroscience Abstracts 1, 776. ZAMENHOF, S., MOSLEY, J. & SCHULLER, E. (1966). Stimulation of the proliferation of cortical neurons by prenatal treatment with growth hormone. Science 152, 1396-1397.
APPENDIX
We present here the equations for calculating estimates of the branching parameters flk and Nk under the BB model, and estimates of the cutting probabilities tk and bk under the BB and BC model. We shall use the following notation for the data. For each order k of branching, let xk = number of uncut bifurcating kth order branches, Yk = number of uncut terminal kth order branches, and Zk = number of cut kth order branches.
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MYRA L. SAMUELS AND OTHERS
First, we present an intuitive derivation of the estimates of the parameters ,8k, tk and bk under the BB and BC model. (This derivation is equivalent to that used by Smit et al. (1972), except that Smit et al. set Ak = 2, while our formulation leaves Ak arbitrary.) Of all the kth order branches which existed before cutting, a total of (Xk + YK + ZJ) are visible in the data; i.e. they were not excised by cuts on branches of previous orders. Of these (Xk + Yk + Z) branches, we expect under the BB and BC model fractions hJ(1 - bk) to bifurcate and not be cut, (1 - /1k) (1 - tk) to terminate and not be cut, and 1/ bk+ (l -/hv) tk to b- cut. Equating the first two of these to the corresponding fractions observed in the data gives bk),
(1)
X}, + Yk + Zk(1)
(
Xk+Yk+Zk
h 8tc
2
Setting tkc = Ak bk and eliminating bk from (1) and (2) yields /3. (1 -Ak) (Xk+yk+ Zk) + /k[Ak(2xk+yk+ zk) - (Xk+ Zk)] -Ak Xk = 0(3) Once A, is specified, equation (3) has a unique* solution for I8 between 0 and 1; this solution is the estimate Ik. The cutting probabilities are then estimated by
bk
(4)
lA
=
IN(Xk+YVk+ Zk)
and tk
=
(5)
Ak bk
In the extreme case Ak = oo, equations (3)-(5) are replaced by xXk = b= 0
(4')
and tk
(3')
Zk Yk +
Zk-(5 )
Under the BB model, the cutting probabilities tk and bk are allowed to depend upon the position of the tree with respect to the cutting plane, in such a way that Ak, = tk/bk is constant. Under the BB model, then, the intuitive derivation given above does not work. Nevertheless, as shown by Samuels (1977), when the method of maximum likelihood is applied to the BB model, the resulting equation for the estimated branching probability /3,k is exactly the same as that presented above, equation (3) (or (3')). That is, once Ak is specified, the estimate 1k is the same under the BB as under the BB and BC model. * We assume throughout that unique for certain values of Ak-
Xk and Yk are both non-zero; if either is zero, the solution may not be
Complexity of dendritic branching
715
Estimates of the Nk under the BB or the BB and BC model are obtained from the /]k as:
N1= 1
R=41= = N4 N4 = 8/J1l/?3(6 id1424 A
A
(6)
2kflAfl2 ...fik-1. Equations (3) (or (3')) and (6) were used for calculating all estimates Rk discussed in the text, except for R3. The equations for estimating S under the MBC model (in which the assumption of binomial branching is dropped) are more complex, and are presented by Samuels (1977). Nk=
