1 Institute of Neuroinformatics, University of Zürich and ETH Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland and 2 Henry Wellcome Building for Neuroecology, University of Newcastle upon Tyne NE2 4HH, UK
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Abstract |
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Key Words: cell types dendrogram 3D reconstruction fractal analysis GaltonWatson branching process HortonStrahler method
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Introduction |
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Nearly absent from the literature are metrical or topological comparisons of the axonal arborizations of different cortical neurons within the same cortical area. While it is clear that axons of different neural types have different laminar preferences and different lateral extents, what we do not know is whether these differences are simply variations on a theme, like different breeds of horses or dogs, or whether they reflect fundamentally different species, each with its own rule of pattern formation. By analogy, simply measuring the dimensions of axons will not reveal whether they have structural properties in common. In this paper we compare structures of the axons of different neurons drawn from a single cortical area, with the aim of establishing the similarities and differences in the structure of different axons, beyond the obvious differences of their patterns of laminar innervation and horizontal spread. We therefore include in our analyses metrical parameters, e.g. the collateral length, and topological parameters, which are sensitive only to the number and structural arrangement of the collaterals within the tree, e.g. the number of segments between origin and tip. Analyses of parameters that rely on the 3D shape of the axonal tree, e.g. the branch angles and branch locations in the neuropil, are beyond the scope of the present work.
Establishing whether there are relationships between the different types will contribute to current debates of diversity versus stereotypy of cortical neurons. These issues are important, because they address implicitly a deep problem: how do cortical neuronal circuits assemble themselves using a relatively small amount of genetic instruction? Here we discovered that axons of different neurons share more topological and metrical properties than are immediately apparent from their traditional 2D gestalts.
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Material and Methods |
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The axons examined in this study were obtained from anaesthetized adult cats that had been prepared for in vivo intracellular recording (for details, see Martin and Whitteridge, 1984; Douglas et al., 1991
). The same set of axons was also used in another study (Anderson et al., 2002
). All experiments were carried out by Kevan Martin and colleagues under the authorization of animal research licenses granted by the Home Office of the UK and the Cantonal Veterinary Authority of Zürich. We first recorded from each cell extracellularly and mapped the receptive field orientation preference, size, type, binocularity and direction preference by hand. The mapping was repeated intracellularly and horseradish peroxidase (HRP) was then ionophoresed into the cell. Thalamic afferents were classified as X- or Y-type using a battery of tests (Friedlander and Stanford, 1984
; Martin and Whitteridge, 1984
). After appropriate survival times, the brains were fixed, sectioned at 80 µm with a Vibratome and processed to reveal the HRP and then osmicated and embedded in resin to eliminate differential shrinkage. The reconstructions were done in 3D so that the correct lengths of axons could be accurately measured. Our estimate of the shrinkage of the tissue is 11%. This is far lower than is usual in material prepared for light microscopy with the more common method of air-drying, which on our estimate results in a shrinkage of 80% in the thickness of the section and less severe, but variable amounts in the X- and Y-dimensions. The lengths given in the dendrograms are the Euclidean lengths estimated from the 3D coordinates of the axon and are corrected for the shrinkage.
Cell Reconstructions
Neurons were reconstructed in three dimensions at x400 magnification with the aid of a light microscope (Leitz Dialux 22) with drawing tube attached to an in-house 3D reconstruction system (TRAKA). TRAKA was written in PASCAL by Rodney Douglas and Danie Botha. The reconstructions were characterized by a list of data points consisting of a code describing the digitized structure (axon, bouton or dendrite) and its three spatial co-ordinates and thickness (where relevant). The somata of the neurons each gave rise to only one axon. The axonal arborizations were complex and often extended through many histological sections. The pieces in each section were merged to form a single tree. Occasionally labeled collaterals could not be connected: these were ignored in the analysis. The measurement error of the digitized structures was estimated by measuring four boutons ten times. The standard deviation was smaller than 0.6 µm in all three dimensions. The data were rotated in order to bring all reconstructed cells into the same coordinate system.
Metrical and Topological Analysis
Axonal trees are complex structures and it is impossible to provide a single compact and coherent description of all their attributes. Here we focus on collateral length, collateral frequency and the arrangement of these collaterals within the tree. This information is contained within the 3D tree, but it is much more conveniently represented by the simpler 2D binary tree, called the dendrogram. (In the present context the word axogram would seem more appropriate, but we defer to the generic technical term derived from Greek for tree). A dendrogram is simply a flattened version of the three dimensional axonal tree, in which the polygons representing the axon collaterals are stretched out and arranged so that the collaterals do not intersect (Figs 13). The dendrogram is the appropriate representation to use here, because it is independent of the actual three-dimensional embedding of the axonal tree in the neuropil. In the process of making the dendrogram, the relationships between the different collaterals were carefully maintained. If two collaterals in the axonal tree branched from the same mother collateral, then the same two collaterals also branch from the same mother collateral in the dendrogram.
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Magnitude, Depth, Height and Exterior Path Length
An axonal tree, or its representation as a dendrogram (Figs 2 and 3), is a binary tree consisting of three types of collaterals. The root collateral connects the origin of the tree (i.e. the cell body) with the first branch point, the end collaterals are the collaterals that have no children and thus terminate the branching, and the inner collaterals are all collaterals which are not end-collaterals (i.e. the root collateral is also an inner collateral).
The magnitude (0) of a tree is defined as the number of end collaterals it contains, and is related to the total number of collaterals (
1) in the tree by
1 = 2
0 1. (It follows that the number of inner collaterals equals the number of end collaterals minus 1.)
The depth of a particular end collateral is the number of collaterals (including the root collateral and the end collateral itself) along the shortest path between the root and the particular termination (Fig. 1A). The height is the maximum depth of all the end collaterals. The exterior path length is the sum of the depths of all the end collaterals.
It is easy to see that both the height and the exterior path length are not independent of the magnitude of the tree. A large exterior path length can be achieved if there are a large number of end collaterals (i.e. the magnitude of the tree is large) or if the end collaterals have a large depth. With increasing magnitude, the height will tend to increase too. Thus, direct comparison of the height or exterior path length is of little significance for trees with very different magnitudes. We therefore plotted these two parameters as a function of magnitude.
Tree Asymmetry Index
Pelt et al. (1992) proposed a topological index that measures the asymmetry of a binary tree. The definition is based on the partition asymmetry index
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HortonStrahler Analysis
For a more detailed analysis of different parts of the dendrogram, we ordered the collaterals using the method proposed by Strahler (1952). He first introduced it in studies of the topology of river networks, where ordering begins naturally at the sources, i.e. at the smallest drainage streams that have no tributaries themselves. This centripetal ordering convention of the HortonStrahler (HS) numbering scheme may seem counterintuitive for the description of axonal organization or growth, where the natural inclination is to begin numbering at the origin. However, as many studies show, key features about the branching behavior of river networks, dendritic trees and other natural occurring binary trees can often be characterized by only two ratios (the bifurcation and length ratio) when applying this ordering system. Capturing essential features of axonal trees in a few key numbers greatly facilitates comparisons between the axons of different types of neurons.
In this method, each collateral in a binary tree is given an order in the following way (Fig. 1C). The end collaterals have all order 1. If the two children of an inner collateral have order k, then the inner collateral is assigned order k + 1. If one child has order k and the other an order smaller than k, then the inner collateral is assigned to the larger order k. Each path formed by consecutive collaterals of the same order k is called an HS segment of order k. Note again that segments can include many collaterals.
The segment with the maximum order I contains the root collateral. This number is called Strahler number. A tree has to be pruned I 1 times in order to remove all the branches so that only the stem of the tree remains (i.e. the segment of order I in the original tree). Pruning a tree is the operation that forms a new tree by cutting all end collaterals. The second order segments of the old tree become the end collaterals of the new tree and the numbering system is adjusted in the new tree (Fig. 1C,D).
A subtree of order k consists of a segment of order k together with all the branches that emerge directly or indirectly from this segment (Fig. 1C). The Strahler number of this subtree is k.
Let Nk be the number of segments of order k in the tree, and Lk the average length of these segments. N1 is the number of end collaterals, and of course NI = 1. All other Nk are between these two values. The bifurcation ratio of order k [1, ... ,I 1} is defined as Nk/Nk+1. This ratio measures the relative change of the number of segments as one moves from a higher order to a lower order. The length ratio of order k
[1, ... ,I 1} is defined as Lk+1/Lk. This ratio measures the relative change of segment length from a higher order to a lower order.
We call a binary tree topological self-similar if all bifurcation ratios are similar. For a topological self-similar tree with bifurcation ratio b, the logarithm of the segment numbers Nk, plotted versus HS segment order k, forms a straight line with slope log(b). This is an immediate consequence of the basic equation From this equation it also follows that
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Fractal Dimension
We used the box-counting procedure (Mandelbrot, 1983) to assign each axonal tree, embedded in the three dimensional space, a fractal dimension. A similar procedure was used to determine the fractal dimension of dendritic trees (Caserta et al., 1995
). The three-dimensional space is covered with cubes of side length lk and the number of boxes Mk which intersect with the axonal branches of the tree are counted (Fig. 8A). We determined Mk for the values lk = 20 x 2k/2 µm, k = 0,1, ... ,12, i.e. lk is between 20 and 1280 µm. For ideal fractal objects the points of the curve fk = (log(lk), log(Mk)) form a straight line and its negative slope defines the fractal dimension of the object. For natural occurring fractal objects the curve fk forms a straight line only for a limited range. In order to find this region for an axonal tree, we first fitted a straight line (regression line) through four consecutive points fu, f1+u, f2+u, f3+u (u = 0,1, ... ,9) of the curve and determined the slope S(u) of this line (Fig. 8B,C, inset). If the curve fk is a straight line, all local slopes S(u) would have the same value. We therefore looked for four consecutive local slopes S(u) that had the least variance, and defined the mean of these local slopes as the fractal dimension of the axonal tree.
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We used a GaltonWatson branching process with parameters pst, pel and pbr (Jagers, 1975) to generate randomly branching axonal trees. The parameters describe the probability of stopping growth, elongating, or branching. In order to generate a tree, we started with one segment of arbitrary length
l = 1 µm and extended this segment by either adding a segment of length
l or by forming a branch point and adding two children of length
l. If the segment has elongated or branched, one or two segments are added, which we call end-tips. Each of these end-tips is elongated, branched or stopped with the same probabilities pel, pbr and pst. This process produces newly formed end-tips (excluding the end-tips that stopped growing). A binary tree grows by repeating this procedure to the newly formed end-tips (Fig. 9A). Whether one end-tip elongates, branches or stops is assumed to be statistically independent from the elongation of other end-tips. A newly formed end-tip has exactly three possibilities for its fate (to stop growing, branch or elongate), therefore
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Results |
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Calculations of average axon lengths in neocortex have been based mainly on estimates of the percentage of neuropil occupied by axon-like profiles (Foh et al., 1973; Braitenberg and Schüz, 1991
). Here we could measure the total lengths (i.e. the summed length of the axon collaterals in the tree) of individual axons directly. These ranged between 11.7 and 125.5 mm (average length 41.3 ± 20.9 mm, mean ± SD; Fig. 4A). The axonal arborization of smooth cells were restricted to a much smaller volume than spiny cells and thalamic afferents and this was reflected in their average total axonal length (31.8 ± 11.7 mm, range 11.752.1 mm), which was smaller than that of spiny cells (47.4 ± 23.8 mm, range 18.7125.5 mm) and thalamic afferents (35.5 ± 11.0 mm, range 27.048.0 mm). Only the basket cells in layer 2/3 (36.0 ± 2.7 mm, range 32.937.9 mm; n = 5) had a total axonal length comparable to spiny cells. The biggest axonal arbors in terms of total length were made by the pyramidal cells in layers 2 and 3 (3/5) and one layer 5 pyramidal cell, which had total axon lengths in excess of 60 mm.
The number of collaterals of the axonal trees varied widely, from 103 to 1403 (Fig. 4B). Although their average total axon length was greater, spiny cells and thalamic afferents tended to have fewer axon collaterals than the smooth cells. Basket cells in layer 2/3 had about twice as many collaterals as the maximum number of collaterals seen for spiny neurons and thalamic axons. The average number of collaterals of the smooth cells was 803 ± 329 (range 3251403 ), that of the spiny cells and thalamic afferents 306 ± 140 (range 103641) and the collateral lengths were correspondingly different for these two groups (Fig. 4C). The smooth cells had median lengths for their collaterals that varied between 23 and 50 µm (30 ± 7 µm). The spiny cells had median lengths ranging from 64 to 167 µm (104 ± 22 µm). The thalamic afferents had intermediate collateral lengths, with the median ranging from 39 to 50 µm (43 ± 6 µm). The longest collaterals were found on a layer 6 pyramidal cell whose axon was restricted to layer 6 (see Fig. 4C) and a star pyramidal cell of layer 4 with diffuse branching pattern (p4). The star pyramid cell had the lowest number of collaterals of any cell.
Length Distribution of Collaterals
The distribution of collateral lengths over the entire axonal tree was roughly similar for neurons within the group of spiny neurons, smooth neurons and thalamic afferents. By inspection, the distributions are approximately exponential, but have a relative lack of very short (<20 µm) collaterals. Such short collaterals would easily be seen in the light microscope, so their unexpectedly low occurrence is not an artifact of reconstruction. Beyond this lack of very short collaterals, smooth neurons and thalamic afferents had a higher proportion of short (<
50 µm) collaterals than spiny neurons, reflecting the results we found for the median collateral lengths (Fig. 4DF).
Tree Magnitude and Length of End-collaterals
The end collaterals had median lengths that were generally larger than the lengths of the inner collaterals (absolute difference 28 ± 40 µm, range of differences between 19 and 203 µm). An extreme example was the star pyramidal cell of layer 4 (p4), where the median length of the end collaterals was 200 µm longer than that of the inner collaterals. However, for many neurons (24/39) the absolute difference was <20 µm and the two distributions were often (23/39) not significantly different (significance level 0.01). The number of end collaterals is referred to as the magnitude of the tree. The total number of collaterals in the tree is twice the magnitude minus 1 (see Materials and Methods). The magnitude and the mean length of the end collaterals will be used in the topological and HS analysis of the axonal trees and are shown in Figures 5 and 6.
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In order to make an objective comparison of the axonal branching pattern of the 39 neurons, we applied a number of well-established methods to define the topology of the axonal trees. However, while these methods are frequently used to describe natural structures (e.g. rivers and root systems), they are rare in applications to neurons and thus require some introduction. We apply them here, because these topological methods offer compact descriptions based on quantitative measures of some key aspects of the branching patterns of axons that are independent of the embedding in the three dimensional space. Thus, topological analysis provides simple and direct comparisons between the different axon types that goes beyond the obvious differences of axonal innervation patterns, such as the differential selection of cortical layers or the formation of axonal patches. Constancies of organizational form hidden from normal metrical analyses may be revealed by topological analyses.
Topological Depth, Height, Exterior Path Length and Magnitude
The depth of a particular end collateral is a convenient measure of the number of collaterals (including the root collateral and the end collateral itself) along the shortest path between the root and the given end collateral (see Fig. 1A). The maximum depth of the end collaterals considered over the whole arbor is called the height. The average height of all axons was 21 ± 6 (range 1237). Although the overlap was large, the spiny neurons had a smaller height (18 ± 4, range 1224) than the smooth neurons (25 ± 7, range 1737) or thalamic afferents (25 ± 3, range 2227).
A measure of the degree of collateral branching is given by the sum of the depths of all the end collaterals: this sum is referred to as the exterior path length. The average exterior path length of the whole population of cells was 2999 ± 2420 (range 42010467). It was smallest for the spiny neurons, 1624 ± 936 (range 4204141), followed by the LGN axons with 3426 ± 486 (range 28693766). The largest exterior path length was formed by the smooth neurons, with 5333 ± 2706 (range 186210467).
Axonal trees with similar magnitude (the number of end collaterals) had similar height and exterior path length (Fig. 5A,B). The dependence on the magnitude 0 (or equivalently on the number of end collaterals) of the height and the exterior path length can be described reasonably well by a power function of the form
Based on the regression lines in Figure 5A,B, we estimated
and ß. For the height we got
= 3.3 and ß = 0.34, for the exterior path length
= 3.80 and ß = 1.21. The dashed lines in Figure 5A,B indicate the maximum and minimum height and the maximum and minimum exterior path length for trees of a given magnitude.
Figure 5C provides a comparison of the degree of topological asymmetry of the axonal trees. Intuitively, the tree asymmetry index is an average measure of the size difference between the two subtrees that emerge from the branch points of the tree. For example, a herringbone tree (Fig. 1B) of high magnitude is maximally asymmetric (i.e. has an index of 1), because for any branch point one emerging subtree is very small (consisting of one end collateral), while the other emerging subtree is very big (having many end collaterals). In contrast, a dichotomous tree (Fig. 1B) is maximally symmetric (i.e. has an index of 0), because for any branch point the two emerging subtrees have an equal amount of end collaterals. Despite their differences in size and complexity, the asymmetry index is very similar for all the axonal trees (0.53 ± 0.05, range 0.420.65, Fig. 5C). Typically, one subtree emerging from a branch point of an axonal tree has about three times as many end collaterals as the other subtree.
HortonStrahler Analysis
The analysis of the collateral length showed that end-collaterals are typically shorter than the inner collaterals. This indicates that axon collaterals form a heterogeneous population that divided into classes of different metrical and topological properties. The topological and metrical measures applied so far are global in the sense that they do not distinguish collaterals of different groups. We therefore use here the HortonStrahler method to order collaterals of an axonal tree into a hierarchy and to compare the different levels. Although collateral takes a slightly different meaning in this context (and is called segment), in the analysis we basically compare the change in segment number and segment length between the different levels, and also compare these changes between the different axonal trees.
The Strahler number I is a measure of the degree of branching of a tree. It is derived by determining how many times the tree has to be pruned in order to cut off all branches. Each pruning removes all the end collaterals. The second order collaterals of the old tree become the end collaterals of the new tree and the numbering system is adjusted in the new tree (Fig. 1C,D). The more complex the branching, the higher the Strahler number. For the axonal trees of smooth cells, the Strahler number ranged between 6 and 7 with one exception, a basket cell in layer 4, where I = 5. Spiny cells and thalamic afferents had Strahler numbers between 5 and 6. (It should be noted that the thalamic afferent root was taken as the entry point to the cortical grey matter and not the origin of the axon in the thalamus, where additional branching might occur.) With one exception, the spiny cells of type p2/3 had a Strahler number of 6, and, again with one exception, layer 6 pyramidal cells had a Strahler number of 5.
In the HortonStrahler (HS) numbering scheme, the end collaterals are the first order segments of the axonal tree for which the number (N1) and average length (L1) are shown in Figure 6A,B. The number of first order segments (the end collaterals) is linearly related to the total number of collaterals. This number varied widely, from 54 for the star pyramid of layer 4, to 702 for a layer 3 basket cell. As might be expected, the average length of end collaterals dominates the distribution and thus is close to the median length of all collaterals (Fig. 4C).
Bifurcation Ratio
A basic property of binary trees is that the number of segments of a given order increases as the order number decreases. The topological measure of the relative increase of segments from a higher to a lower order is termed the bifurcation ratio. More exactly, the bifurcation ratio of order k is the ratio Nk/Nk+1 of the number of segments of order k and k + 1. Bifurcation ratios are always >2, but otherwise unlimited (in theory). For many natural occurring trees [including dendrites of rat neocortex (MacDonald, 1983)] the bifurcation ratios are rather independent of the order and have a value between 2 and 5.
The first and second order bifurcation ratios of the reconstructed axonal trees are very similar (Fig. 6C). The first order bifurcation ratio (N1/N2, circles) ranges between 2.8 and 4.5 (mean 3.3 ± 0.3), the second order ratio (N2/N3, squares) between 2.9 and 4.5 (3.4 ± 0.4). For the neurons with dissimilar bifurcation ratios, no clear pattern could be observed concerning the size relation. For example, the bifurcation ratio of order 1 was bigger than the bifurcation ratio of order 2 in the case of the double bouquet cell, but smaller for one of the spiny stellate cells which had its major axonal arborization within layer 4.
The bifurcation ratios of higher orders are more varied, but the vast majority of the ratios (92%) still lie between 2 and 4 (Fig. 6D). The mean of the pooled ratios of order >2 is 2.9 ± 0.9 (range 2.07.0). The mean of the pooled ratios of all orders is 3.1 ± 0.8 (range 2.07.0).
We also estimated the common bifurcation ratio b, by plotting for each tree the number of segments of order k + 1 against the number of segments of order k (Fig. 7B). The points lie on a straight line through the origin, which confirms that our entire sample of axons (both spiny and smooth) have similar bifurcation ratios (b = 3.32, r = 0.99). This result suggests that all the axons are topologically self-similar. Indeed, Figure 7A demonstrates that, despite the variances in higher order bifurcation ratios mentioned above, the overall relationship between log(Nk) and order k for individual axons is linear. Furthermore, the HS index predicted by log(N1)/log(b) + 1 is in good agreement with the observed values (Fig. 7C).
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The length ratio measures the relative increase of the length of the collaterals from a lower to a higher order. The length ratio of order k is the ratio Lk+1/Lk of the average length of segments of order k + 1 and k. Length ratios can take any value >0. For naturally occurring tree structures [including dendrites of rat Purkinje cells (MacDonald, 1983)] the length ratio is relatively independent of the segment order and typically has values between 1 and 2.
For the reconstructed axonal trees, the average length ratios of order 1 (L2/L1, Fig. 6E, circles) is 1.7 ± 0.6 (range 0.84.2). For the second order length ratio (L3/L2, Fig. 6E, squares) the average is quite similar, 1.5 ± 0.7 (range 0.54.1). Most ratios are >1. For the first order ratios this means that the end collaterals (the segments of order 1) are typically smaller than the second order segments. As with the bifurcation ratios, no clear pattern was observed concerning the size relationship of the first and second order length ratios [i. e. Fig. 6E, db2/3, ss4(L4)].
As with the bifurcation ratio, the higher order length ratios are more varied (1.2 ± 1.1, range 0.035.9, Fig. 6F). The average of the pooled length ratios of all orders is 1.4 ± 1.0 (range 0.05.9). The average length of segments of order k + 1 and the average length of segments of order k, pooled for all neurons and all orders, did not correlate (r = 0.39). Thus, in contrast to the branching ratio (Fig. 7B), there is no global length ratio that applies to all axons and all orders.
Fractal Analysis
Whereas HS methods measure the topological complexity of a tree in terms of the number of times a tree can be pruned, fractal methods measure its dimensional complexity. The higher the fractal dimension of the tree, the bushier it appears (Fernández and Jelinek, 2001). To the extent that axons are fractal at all, they have the fractal property of self-similarity. Natural objects are not ideal mathematical objects, and so they are not expected to exhibit exact fractal self-similarity. Nevertheless, they may show some degree of self-similarity over a limited range of scales (in our case, spatial scale). A large axonal tree with constant HS bifurcation and length ratios should display some scale invariance (Tarboton et al., 1988
). Although our axons have quite strong fluctuations of higher order branching and length ratios, we expected to find evidence of this invariance by fractal analysis.
We used the box-counting algorithm to determine the fractal dimension of axonal trees embedded in their three dimensional space (Fig. 8A,B). The fractal dimension of axons of all cells was rather similar, with a slight tendency of smooth cells to be higher (Fig. 8B,C,D). Spiny cells and thalamic afferents have an average fractal dimension of 1.5 ± 0.1 (range 1.21.7), for smooth cells it is 1.7 ± 0.2 (range 1.41.9). Some neurons show a fairly straight line in the box-counting approach (Fig. 8B,C), indicative of fractal self-similarity. However, we also found axons for which the curve l M(l) hardly contained a straight segment (
7/39), indicating a lack of fractal self-similarity.
Growth Model
Apart from some specific lengthening of some high-order axonal segments, the metrical and topological aspects of the axonal data can be described by a simple growth model. The model we used is a GaltonWatson branching process, which is one of the simplest and best understood (Jagers, 1975). A tree is grown (Fig. 9A) by repeatedly elongating its end-tips by 1 µm (with probability pel) or by branching the end-tips into two new branches of length 1 µm (with probability pbr), causing two new end-tips. It is also possible to stop the growth of an end-tip (with probability pst).
Estimation of the Parameters pst, pel and pbr for the Reconstructed Trees
To test whether the GaltonWatson branching process is a reasonable model of the terminal branching of axonal trees, we estimated pst, pel and pbr for the two major subdivisions of the axonal types those originating from spiny cells and thalamic afferents, and those of smooth cells. Note that because of relation (1) it is enough to estimate pel and pbr. We modeled only the growth of the terminal arbors (i.e. subtrees of order 2 or 3), which constitute most of the axon. We did not model the main axonal trunks because they are rather few in number and varying considerably in length (see Figs 2 and 3), thus compromising any statistical comparison with the simple GaltonWatson branching process. We described each of the two axonal types (spiny and smooth) by the histogram of branch length (), the first bifurcation and length ratio, and for the subtrees of order 3 also the second order bifurcation and length ratio (Fig. 9B,C).
In order to estimate pel, we fitted for each population the function to the histogram of branch length (equation 4). Reasonable fits were obtained for pel = 0.9927 (spiny population) and pel = 0.9780 (smooth population). The best fit was found by minimizing the least square error using the LevenbergMarquardt algorithm. In order to estimate pbr we picked values in the interval
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Discussion |
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Simple measures, such as the lengths of axons and the order, number, and length of branches are needed, since there are so few direct measurements of filled cortical axons in the literature. Here the 2D dendrogram is invaluable as a means of organizing and representing the axonal tree. In the dendrogram, the axonal branches and their relative lengths, and the tree-like structure of the axons are readily apparent, quantitative measurements are easily made, and the topology of the axonal trees can be characterized.
Similarity of Branching Rules
Our topological (Fig. 5) and HS analysis (Fig. 6) show that all the axons occupy a relatively restricted region of the possible parameter space. Within this sub-space we could not detect a consistent and distinct signature for the different cell-types. For example, the axonal tree height and exterior path length tended to be smaller for the smooth neurons than the spiny neurons, suggesting that a topological distinction between these two classes might be possible. However, both these numbers are strongly correlated with the magnitude of the tree (Fig. 5A,B) because, for trees of a given magnitude, the height and exterior path length must fall within a range whose lower and upper limits themselves increase with magnitude (Fig. 5A,B, lower stippled lines). Of course, within this range, the values could still be arbitrary. But they are not. Instead, for all axons examined, the dependence on the magnitude can be described by just two functions; one for the height and the other for the exterior path length (Fig. 5A,B). This surprising regularity suggests that the axonal branching pattern of all neurons, spiny or smooth, is related and follow similar branching (growth) rules. This hypothesis is further strengthened by the analysis of the HS bifurcation and length ratios.
HortonStrahler Analysis
The HS method has been applied successfully to both physical and biological tree-like structures (for example, river networks, blood vessels, and trees). The HS ordering was useful for our application, because it expresses the hierarchy between segments based on patterns of branching, and therefore allows us to investigate axonal structure on a finer level. In the original HS application of river systems, a higher order segment provides a branch that drains equivalent lower order (more peripheral) tributaries. In the case of the axons, the higher order segment can be interpreted as a source branch that gives rise to (possibly many) hierarchically equivalent lower order branching patterns. The highest order segment is the trunk whose origin is at the soma.
The HS bifurcation ratio measures the relative change in the number of hierarchically equivalent segments when moving distally from one segment order to the next. The length ratio measures the relative change in average segment length when moving distally from one segment order to the next lower, and so indicates the degree of extension of the tree. The HS method has proved useful, because the bifurcation ratios and length ratios of natural tree-like structures turned out to be relatively insensitive to segment order, i.e. natural trees can often be described by a constant length ratio and a constant bifurcation ratio, which indicates topological self-similarity.
We found that the bifurcation ratios of all orders and all investigated neurons can be reasonably well described with one global bifurcation ratio (Fig. 7A,B). Thus each axonal tree is, to a first approximation, topological self-similar. This offers a great simplification in that the number of segments of similar order is fully determined by only three variables: the Strahler number, the global bifurcation ratio, and the segment order (see Materials and Methods). In fact, the number of segments of similar order is only dependent on the Strahler number and the segment order, because the bifurcation ratio applies globally to all axons. In particular, the Strahler number is the only determinant of the number of end-collaterals (end-collaterals have always segment order 1), and therefore of the total number of axonal branches that an axonal tree can form (Fig. 7C).
The situation for the length ratio was different in that we did not discover a global length ratio that applied to all axons and all segment orders. Nevertheless, the first and second order length ratios are also rather similar for the different cells (Fig. 6E), fluctuating around a mean value 1.6. The global bifurcation ratio was 3.3. These values are within the range for many naturally occurring tree-like structures (MacDonald, 1983).
These observations suggest that the topology of axons may be dominated by fundamental principles of construction [for example, simple growth rules that optimize the distribution of metabolic materials through a space-filling branching tree (Changizi, 2001)], rather than by the need to make connections to very specific targets.
Of course, it is also clear that neurons do have a strategy of making some intra-areal long range connections, and so one might expect greater variances in the ratios for higher order segments. Indeed, higher-order (>2) bifurcation and length ratios of axons do vary more than those of the lower orders (Fig. 6CF). Their increased variance is partly explained on numerical grounds. The data necessarily contain a smaller number of high order segments than low ones. For example, orders 57 never have >7 segments for the individual axonal trees. The increased variance in the length ratio is also partly due to the inhomogeneous extension of higher order segments. It is often one or a few of these high order segments that extend much further than other segments of similar order. Usually, these long segments are formed by the vertical or horizontal running axons of spiny neurons or thalamic afferents that innervate different layers or functional columns. Basket cells can also form such segments.
If these few segments all have the same order, the result will be a large length ratio [i.e. Fig. 6F, p2/3:C-E, ss4(L4):B, lgnY]. However, these obviously extending segments are often of different orders. The admixture of very local smaller segments results in a smaller length ratio [i.e. Fig. 6F, p6(L4)]. In this sense, the large variation of length ratios of higher orders indicates that there is at least some regional specificity in the axonal tree. That is, a few collaterals will break the default axonal growth rule in the interests of extending the arbor to a more distant cortical region. However, there are clearly many ways how generation of branches can be introduced in a tree, and the question to what extent the methods of Horton and Strahler captures the true biological development of axonal trees needs further analysis.
Tree Complexity
The Strahler number (i.e. the number of times a tree can be pruned) gives an indication of the complexity of axonal branching. The number can vary up or down with increasing magnitude. For example, for the herringbone tree the Strahler number is a constant 2, which is the lowest possible Strahler number of a binary tree. For the dichotomous tree the Strahler number equals the height of the tree and no other binary tree of smaller magnitude can form higher Strahler numbers. Smooth cells, which tend to have higher magnitudes (i.e. more end collaterals) than spiny cells, also tended to have a Strahler number that was 1 or 2 greater than the typical number of 56 for most spiny cells and thalamic afferents (Fig. 7C). The layer 6 pyramidal cells had the least complex branching and thus the lowest Strahler number (5).
Another more frequently applied measure of the complexity of axonal shape is the fractal dimension, which showed a weak tendency to be higher for smooth cells (Fig. 8D). Again, this suggests that the axons of smooth cells are slightly more complex than spiny neurons.
Growth Model
Based on topological and metrical properties of adult trees alone, it is difficult to deduce the precise growing rule that generates the tree. Nevertheless, growing rules which are not consistent with the observed parameters can be excluded. For example, the measured fractal dimension of retinal ganglion dendrites is about 1.68 (Caserta et al., 1995), which is similar to the fractal dimension of a tree structure grown by diffusion limited aggregation (DLA cluster) in two dimensions (Tolman and Meakin, 1989
). This has led to the suggestion that dendrites are created by this process (Caserta et al., 1990
). For cortical axons, this growth process can be ruled out. The fractal dimension of the reconstructed axonal trees is always <2 (Fig. 8D), while three dimensional DLA clusters have a fractal dimension of
2.5 (Tolman and Meakin, 1989
).
GaltonWatson branching processes are the oldest and best understood branching processes. They appear in many variants and are applied in many sciences (Jagers, 1975), including the modeling of dendritic trees (Kliemann, 1987
). We used it here in its simplest form to simulate axonal branching. Its key properties are constant elongation, branching and stopping probabilities, and the assumption that these events are statistically independent for the end-tips of the growing tree. Despite its simplicity, the model produces trees with bifurcation ratios, length ratios and branch length distributions that are similar to the ones observed for the spiny and smooth population of the reconstructed neurons (Fig. 9B,C).
The model fails to reproduce finer details of the axonal trees, such as their lack of very small branch lengths (Fig. 9B,C). This indicates that for a better fit, the assumption of constant probabilities should be relaxed. In addition, the model simulates only the distal subtrees of the axonal trees (i.e. subtrees which have a root of order 2 or 3) and fails to predict bifurcation and length ratios of higher order (see Fig. 9B,C). The model's prediction of the second order bifurcation ratio of the subtrees of order 3 fitted worst. The reason for this failure is that the simple model does not contain a mechanism for atypical elongation of a few high-order branches (specific branching), as described above.
While many studies model dendritic trees, for references see Van Pelt et al. (2001), there is only one study that attempted to model the local branching of axonal trees (Nelken, 1992
). The model used in the study of Nelken (1992)
is also a GaltonWatson branching process, but slightly more complicated than our approach because it involves the distinction of several types of axonal branches. Although the model results were compared with data from axons of the somatosensory cortex of the mouse, the comparison was qualitative and no details about the fitted model parameters were given.
Conclusion
At face value, the structure of the axons of various types of cortical neurons seems to be distinctly different from one another. However, the topological structure of axons is not easily detected by simple observation. Therefore, we have applied a battery of analyses that are sensitive to the topological characteristics of the reconstructed axonal trees.
Surprisingly, our results revealed no dramatic differences in the fundamental organization of the axonal arbors of neurons as distinct as basket cells and pyramidal cells, except for their scales of collateral length. Instead, we found a marked topological resemblance between the trees of all axons, suggesting that they all grow according to the same basic rules. We have demonstrated that even a simple three-parameter axonal growth model can generate trees with characteristics that are nearly indistinguishable from those of intra-areal cortical axons.
Obviously there are many other branching processes that might also produce the observed length ratios, bifurcation ratios and branch length distributions of the distal subtrees of the measured axons. Much more analysis is needed to specify more completely the branching process of axonal trees. In particular, a simple mechanism for the atypical collateral extension that seems to support clustering of the tree branches in 3D space, would be a useful next step.
Although we now can describe the generic structure of axons, the particular instantiation of an axon in 3D space is another matter. Presumably, the 3D space instantiation will take the topological rules as growth constraints, and configure themselves to satisfy connection constraints, which have not been addressed here.
That even our simple model reproduces the observed values raises the possibility that only a minimal set of specific rules is actually used when the axon of any type of cortical neuron grows and branches to form connections with other neurons. Small parametric adjustments in the rules could be sufficient to explain the different macroscopic structures of axons that are (superficially) characteristic of different neuronal types. In this context it is interesting to note that a simple generation rule was also found for the placement of boutons along the branches of cortical axons (Braitenberg and Schüz, 1991; Anderson et al., 2002
). These simplifications have interesting implications for models of the development of complex neuronal networks such as cortex.
Address correspondence to Tom Binzegger, School of Biology, Henry Wellcome Building for Neuroecology, University of Newcastle upon Tyne NE2 4HH, UK. Email: tom.binzegger{at}ncl.ac.uk.
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Bart AG, Kozhanov VM, Chmykhova NM, Botchkina NA, Ternovaya LA, Clamann GP (2000) Topological and statistical analysis of 3-D reconstructions of axon collaterals. Neurophysiology 32:249259.[CrossRef][ISI]
Braitenberg V, Schüz A (1991) Anatomy of the cortex. Berlin: Springer.
Caserta F, Stanley HE, Eldred WD, Daccord G, Hausman RE, Nittmann J (1990) Physical mechanism underlying neurite outgrowth: a quantitative analysis of neuronal shape. Phys Rev Letters 64:9598.[CrossRef][ISI][Medline]
Caserta F, Eldred WD, Fernández E, Hausman RE, Stanford LR, Bulderev SV, Schwarzer S, Stanley HE (1995) Determination of fractal dimension of physiologically characterized neurons in two and three dimensions. J Neurosci Meth 56:133144.[CrossRef][ISI][Medline]
Changizi MA (2001) Principles underlying mammalian neocortical scaling. Biol Cybern 84:207215.[CrossRef][ISI][Medline]
Douglas RJ, Martin KAC, Whitteridge D (1991) An intracellular analysis of the visual responses of neurons in cat visual cortex. J Physiol Lond 440:659696.[Abstract]
Fernández E, Jelinek HF (2001) Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods 24:309321.[CrossRef][ISI][Medline]
Foh E, Haug H, König M, Rast A (1973) Quantitative Bestimmung zum feineren Aufbau der Sehrinde der Katze, zugleich ein methodischer Beitrag zur Messung des Neuropils. Microsc Acta 75:148168.[Medline]
Friedlander MJ, Stanford LR (1984) Effects of monocular deprivation on the distribution of cell types in the LGNd: a sampling study with fine-tipped micropipettes. Exp Brain Res 53:451461.[ISI][Medline]
Gupta A, Wang Y, Markram H (2000) Organization principles for a diversity of GABAergic interneurons and synapses in the neocortex. Science 287:273278.
Innocenti GM, Lehmann P, Houzel JC (1994) Computational structure of visual callosal axons. Eur J Neurosci 6:918935.[ISI][Medline]
Jagers P (1975) Branching processes with biological applications. London: John Wiley.
Jones EG (1975) Varieties and distribution of non-pyramidal cells in the somatic sensory cortex of the squirrel monkey. J Comp Neurol 160:205268.[ISI][Medline]
Kisvárday ZF, Martin KAC, Whitteridge D, Somogyi P (1985) Synaptic connections of intracellularly filled clutch neurons: a type of small basket neuron in the visual cortex of the cat. J Comp Neurol 241:111137.[ISI][Medline]
Kisvárday ZF, Martin KAC, Freund TF, Maglóczky ZS, Whitteridge D, Somogyi P (1986) Synaptic targets of HRP-filled layer III pyramidal cells in the cat striate cortex. Exp Brain Res 64:541552.[ISI][Medline]
Kliemann W (1987) A stochastic dynamical model for the characterization of the geometrical structure of dendritic processes. Bull Math Biol 49:135152.[ISI][Medline]
Koulakov AA, Chklovskii DB (2001) Orientation preference patterns in mammalian visual cortex: a wire length minimization approach. Neuron 29:519527.[ISI][Medline]
Lorente de Nó R (1949) Cerebral cortex: architecture, intracortical connections, motor projections. In: Physiology of the nervous system (Fulton J, ed.), pp. 288313. New York: Oxford University Press.
Lund JS, Boothe RG (1975) Interlaminar connections and pyramidal neuron organisation in the visual cortex, area 17, of the macaque monkey. J Comp Neurol 159:305334.[ISI]
Lund JS, Henry GH, MacQueen CL, Harvey AR (1979) Anatomical organization of the primary visual cortex (area 17) of the cat. A comparison with area 17 of the macaque monkey. J Comp Neurol 184:599618.[ISI][Medline]
MacDonald N (1983) Trees and networks in biological models. New York: John Wiley.
Mandelbrot BB (1983) The fractal geometry of nature. New York: Freeman.
Martin KAC, Whitteridge D (1984) Form, function and intracortical projections of neurones in the striate visual cortex of the cat. J Physiol Lond 353:463504.[Abstract]
Mitchison G, Crick F (1982) Long axons within striate cortex: their distribution, orientation and patterns of connections. Proc Natl Acad Sci USA 79:36613665.[Abstract]
Nelken I (1992) A probabilistic approach to the analysis of propagation delays in large cortical axonal trees. In: Information processing in the cortex: experiments and theory (Aertsen A, Braitenberg V, eds), pp. 2949. Berlin: Springer.
Pelt JV, Uylings HB, Verwer RW, Pentney RJ, Woldenberg MJ (1992) Tree asymmetry a sensitive and practical measure for binary topological trees. Bull Math Biol 54:759784.[ISI][Medline]
Ramón y Cajal S (1908) Histologie du système nerveux. Madrid: CSIC (reprinted 1972).
Rockland KS (1995) Morphology of individual axons projecting from area V2 to MT in the macaque. J Comp Neurol 355:1526.[CrossRef][ISI][Medline]
Strahler AN (1952) Hypsometric (area-altitude) analysis of erosional topology. Bull Geol Soc Am 63:11171142.[ISI]
Szentagothai J (1975) The module-concept in cerebral cortex architecture. Brain Res 95:475496.[CrossRef][ISI][Medline]
Tarboton DG, Bras RL, Rodriguez-Iturbe I (1988) The fractal nature of river networks. Water Resour Res 24:13171322.[ISI]
Tettoni L, Gheorghita-Baechler F, Bressoud R, Welker E, Innocenti GM (1998) Constant and variable aspects of axonal phenotype in cerebral cortex. Cereb Cortex 8:543552.[Abstract]
Tolman S, Meakin P (1989) Off-lattice and hypercube-lattice models for diffusion-limited aggregation in dimensionalities 28. Phys Rev A 40:428437.[CrossRef][ISI][Medline]
Triller A, Korn H (1986) Variability of axonal arborizations hides simple rules of construction: a topological study from HRP intracellular injections. J Comp Neurol 253:500513.[CrossRef][ISI][Medline]
Uylings HBM, Van Pelt J (2002) Measures for quantifying dendritic arborizations. Network 13:397414.[ISI][Medline]
Van Pelt J, van Ooyen A, Uylings HBM (2001) The need for integrating neuronal morphology databases and computational environments in exploring neuronal structure and function. Anat Embryol 204:255265.[CrossRef][ISI][Medline]