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The Sholl analysis of neuronal cell images: Semi-log or log–log method?
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Journal of Theoretical Biology 245 (2007) 130–140 www.elsevier.com/locate/yjtbi
The Sholl analysis of neuronal cell images: Semi-log or log–log method? Nebojsˇ a T. Milosˇ evic´, Dusˇ an Ristanovic´ Department of Biophysics, School of Medicine, University of Belgrade, Visegradska 26/2, 11000 Belgrade, Serbia Received 19 June 2006; received in revised form 19 August 2006; accepted 20 September 2006 Available online 24 September 2006
Abstract Although the Sholl analysis is a quantitative method for morphometric neuronal studies and its application provides many benefits to neurobiology since it is obvious, common and meaningful, there are many unresolved theoretical issues that need to be addressed. Nevertheless, it can be practiced without much background or sophistication. The two different methods of the Sholl analysis—log–log and semi-log—have been applied previously without a clear basis as to what to use. To make an adequate choice of the method, one should try and accept that one which gives a better result. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be found in basic mathematics. In order to compare the results of applications of the semi-log and log–log methods to the same neuron, we introduce the concept of determination ratio as the ratio of the coefficient of determination for the semi-log method and that for the log–log method. If the semi-log method is better as related to the log–log method, the value of this parameter is larger than 1. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method is more frequently utilizable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log–log method. We also compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns and found that between the Sholl and fractal analysis exist various and important analogies. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fractal analysis; Inverse power laws; Morphometric parameters; Sholl analysis; Topological trees
1. Introduction One of the major goals in neurobiology is morphologic analysis of neuronal dendritic and axonal structure. The most standard approach is applying the Sholl analysis. The pioneering work of Sholl (1953) formed the basis of modern quantitative techniques in morphometric characterization of Golgi-impregnated and intracellulary stained neurons. The Sholl analysis is a method for quantitative study of radial distribution of some properties of neuronal dendritic arborization pattern around the cell’s perikaryon. This consecutive-circles (cumulative intersection) analysis specifies dendritic geometry, ramification richness, and dendritic branching patterns (Caserta et al., 1995). The Sholl analysis has long been widely applied and used for quantitative morphologic studies of impregnated and Corresponding author. Tel.: +381 11 3615775; fax: +381 11 3615767.
E-mail addresses: [email protected] (N.T. Milosˇ evic´), [email protected] (D. Ristanovic´). 0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.09.022
stained cells, and achieved widespread application in all fields of neuroscience (Lowndes et al., 1990; Neale et al., 1993; Caserta et al., 1995; Duan et al., 2003; Jelinek et al., 2005). It provides information on the number of branches relative to distance from the cell’s perikaryon. This analysis consists of (i) construction of concentric and equidistantly organized circles, which are centred in the perikaryon, (ii) counting the numbers of intersections of dendrites with the circles of increasing radii, (iii) defining suitable variables using the circle radius and the number of intersections of dendrites with circles of corresponding radii (such as, the Schoenen ramification index, number of intersections per circle area, maximum number of intersections, etc.), and (iv) choosing appropriate mathematical techniques for data processing and presenting (e.g., the histograms or plots in co-ordinate systems). Three main modifications of the Sholl analysis are in use: (i) the plot of the number (frequency) of dendritic intersections with the circles which can be calculated per circle area (Sholl, 1953) or per circle length (Neale et al., 1993), versus the circle
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radius (the linear Sholl method), (ii) the plot of the log of the number of intersections per circle area or circle length, versus the circle radius (the semi-log Sholl method), and (iii) the plot of the log of the number of intersections per circle area or circle length, versus log of the circle radius (the log–log Sholl method). Semi-log and log–log methods give different results when used to analyse the same structure (Caserta et al., 1995). When plotting the values either as the semi-log or as log–log, Sholl (1953) demonstrated that at least in one of these cases the data points approximately lie on a straight line. He used that analysis, which provides better results and calculated the slope k of the regression straight line (the Sholl regression coefficient) to characterize the morphology of the neuron that he studied. The slope of fitted straight line is a measure of the decay rate of the number of branches with distance from the perikaryon. The aim of the analysis was to present the radial distribution of dendritic branches with the distance from the perikaryon and extract the main morphologic characteristics of the neuron. The semi-log and log–log Sholl methods have been applied previously without a clear basis as to what to use. When performing the Sholl methods, it is good to note that different Sholl’s methods work best for different neuron types. Even Richardson (1961), who first suggested the log–log method when presenting straight-line plots of various coastline lengths in which the apparent length (the log of the total length) is graphed versus the log of the measuring units, does not provide details of why this choice was made. Therefore, to make an adequate choice of the method, one should try and accept that one which gives the best result. Sholl applied the double logarithmic method to apical dendrites and semi-log method to basal dendrites of pyramidal neurons from the visual and motor cortices of the cat because only these methods, respectively, produced approximately linear results for these types of neurons (Sholl, 1953). Although the Sholl analysis represents a quantitative method for morphometric neuronal studies, it is not established exactly. Lima and Coimbra (1986) have argued that the two different methods of the Sholl analysis— log–log and semi-log—have been applied previously without clear basis as to which to choose. Nevertheless, it can be practiced without much background or sophistication. It has provided many benefits to neurobiology because it is obvious, common and meaningful. That is probably one of the reasons why it is still currently in use (Duan et al., 2003; Cook and Wellman, 2004; Vega et al., 2004; Jelinek et al., 2005; Kheirandish et al., 2005; Martinez-Tellez et al., 2005; Nelson et al., 2005). In the present study, we analyse the results of application of both the log–log and semi-log method to the same neurons. We ask if one method is better that the other, or, whether these two methods can be equally treated and used in different dendroarchitectonics of arborization of various neurons. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method
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is more frequently usable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log–log method. We also examine the relationship between these two methods of the Sholl analysis and compare the results with some main concepts of the fractal analysis. 2. Methods and results It has been generally agreed that the Sholl analysis is not established exactly. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be met in basic mathematics. 2.1. Surface density of intersections Let y be the function specified by N , (1) r2 p where N is a constant and r is the radius of a circle. The expression in denominator of Eq. (1) is the area S of the circle of radius r. Then y(r) is an inverse function of the circle area. If we take the logarithms to the basis 10 of both sides of Eq. (1), the following relationship is obtained: yðrÞ ¼
log y ¼ 2 log r þ K,
(2)
where K ¼ log Nlog p. Thus, the inverse power (hyperbolic) function, given by Eq. (1), is revealed as a straightline plot when the values of y are plotted on log–log axes against the values of r. If, for example, N ¼ 6 and r takes the values 1, 2, 3, y, 10 (in any units) and if ordered pairs of coordinates (r, N), calculated from Eq. (1), are graphed on log–log axes, the line fitting these points is a straight line with the coefficient of correlation RLL ¼ 1.000 (Fig. 1C). If all the values of the variables satisfy such an equation exactly, one can say that the variables are perfectly correlated or that there is a perfect correlation between them. The drawing in Fig. 1A simulates a hypothetical neuron with six equally sized primary dendrites in 2D. According to the Sholl methodology, over this neuronal ‘‘perikaryon’’, using the radii in arbitrary units, we construct a series of equidistantly arranged concentric circles which are centred in the perikaryon, and measure the number of intersections of dendrites with these circles. In a plane on which a rectangular co-ordinate system has been constructed we plot the number of intersections N as a function of radius r for such dendritic (topographic) tree (Fig. 1A) and the corresponding scatter diagram present in Fig. 1B (open circles). This scatter diagram indicates that a horizontal straight line would perfectly fit this set of points. If we determine the surface density of intersections y as the number of intersections N of dendritic branches with a circle per its area (S) and plot on log–log axes this variable (y ¼ N/S) versus radius (r) of the circle, we obtain just the
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same graph as shown in Fig. 1C. We also include the plot of the log of N/S versus the radius r in this figure as Fig. 1B (the semi-log Sholl method) log
N ¼ k r þ m, S
(3)
where k is the Sholl regression coefficient. We find that the correlation coefficient RSL of a regression straight line to these data is less than RLL and equals 0.952o1 since it is evidently impossible to achieve here a perfect linearity. 2.2. Image preparation The drawings of neurons to be considered here were converted into digitized images using a scanner with a resolution of 600 dpi. All transformations were carried out on a PC computer using the public domain Image J software (www.rcb.info.nih.gov/ij) developed at the US National Institute of Health. All scanned images were imported into this software. Each dendrite was filled with pixels. Since the drawings were analysed as ‘‘skeleton’’ tracings, the software performed a skeletonization of the image to a stick figure. Such image of the neuron (with correctly inscribed scale bar) was scaled to fit A4 size of the paper (i.e., 15 21 cm) and then printed. In order to study the way in which the number of neuronal branches varies with the distance from the cell body, we used a series of concentric circles (after Sholl) with a common centre in the cell’s perikaryon as coordinates of reference, and traced the network of such circles over a single neuronal drawing, with radii increasing at regular steps of 10 mm in all cases. Adopting such a choice of the steps we avoided a possible effect of changes in network density on the values of the correlation coefficient for a single neuron. Analysing our data we noticed that magnifying the steps between the circles brought about smaller values of the correlation coefficients. 2.3. Determination ratio
Fig. 1. Application of Sholl methods to an idealized neuronal tree. (A) Topological tree consisting of six equally-sized dendrites, with a system of concentric circles superimposed over the drawing. (B) The graphs of linear (open circles) and semi-log Sholl analyses (filled circles). The first series of data points illustrates the relationship between the number of intersections N with the circles and the corresponding radius r in arbitrary units. The second graph is a regression straight line fitting the data points (filled circles) on semi-log axes. (C) The graph of data points calculated using the log–log Sholl method. RSL and RLL are the correlation coefficients for samples of 10 pairs of r, N-values calculated for application to the semi-log and log–log Sholl methods, respectively.
From Fig. 1C it is obvious that the log–log plot for an artificial neuronal cell shown in Fig. 1A is theoretically perfect since the correlation coefficient RLL equals unity. The plot for the semi-log method gives worse result (experimental points scattered about the fitted straight line—Fig. 1B filled points). In order to compare the results of applications of the two methods to the same neuron, we define the determination ratio D as the ratio of the coefficient of determination for the semi-log method (RLS2) and that for the log–log method (RLL2) D¼
R2LS . R2LL
(4)
If the log–log method is better then the semi-log one, the value of the parameter D is less than 1; on the contrary, this value will be larger than 1. In Fig. 1, where an ideal case is
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presented, this value is D ¼ 0.906o1 showing that the log–log method gives better result. Similar finding can be obtained from the theoretical examples (topological trees) shown in Fig. 2A and B (left). They are the models of the two-dimensional branching patterns reduced to skeletons of branch points and segments in the form of rooted trees. Contrary to the scatter diagram in Fig. 1B (open circles) where all the points lie on a horizontal straight line (not shown), in Fig. 2A (right) the values for N increase regularly, say, in an arithmetic progression (the first circle cuts the dendrites six times, the next circle curs the dendrites seven times etc.). This can be seen from the scatter in Fig. 2A. In that case, the value of D is 0.878. In Fig. 2B a little more complex theoretical model of a nondecreasing sequence is presented. The value of D ¼ 0.861. To our experience, such theoretically depicted neurons can be met in reality very occasionally. From these theoretical examples it is possible to conclude that the log–log method might be successfully applied to neurons whose dendrites are rather long, poorly ramified, and all of them along with their branches terminate on a circle of the largest radius. The next step to make topographic models of neurons with more complex dendritic trees are shown in Fig. 3A and C (left): some dendrites reach the circles of the largest
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radii, but some of them, being a consequence of dendritic ramifications, terminate earlier. The model in Fig. 3C is a bit more complex than that in Fig. 3A. But these models are more realistic as compared to those in Fig. 2: in most of the identified types of neurons the scatter diagrams are irregular, but mainly convex-upward. Indeed, the consequent distributions of the numbers of intersections N against the circle radii r are shown on right sides of these models. Fig. 3B represents an alpha neuron from the cat retinal ganglion (left) and the corresponding frequency distribution of data points (right). In cells of alpha-type, filled with Lucifer Yellow, the dendritic fields are relatively large and basically monostratified within the inner plexiform layer. Several stout primary dendrites emerge radially from the soma. They branch successfully without significant crossing and evenly fill the dendritic field. This example corresponds to the model shown in Fig. 3A (right) since both have similar distributions (convexupward) of data points (compare Figs. 1A and 1B, right). Fig. 3D, related to the model in Fig. 3C, shows a typical Purkinje neuron in the mouse cerebellum (left) stained with rapid Golgi method. Stained Purkinje cells found in the primary fissure were selected and drawn with a camera lucida. The Purkinje neuron in Fig. 3D has a more profuse branching pattern than the cell in Fig. 3B; therefore, the
Fig. 2. Frequency distribution of number of intersections N versus the radius r of concentric circles (right) for the two idealized topological trees (left). The trees and frequency distributions of the data where the number of intersections increase as an arithmetic progression (A) and as a no decreasing series of data (B).
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Fig. 3. Topological trees (A and C) and related neurons (B and D) (left), as well as the corresponding frequency distributions of the number of intersections N with the circles superimposed over the neuronal trees (right). Neuron in (B) represents an Alpha neuron from the cat retinal ganglion (from Jelinek and Spence, 1997—with permission), and the neuron in (D) is a typical Purkinje cell in the mouse cerebellum (From Takeda et al., 1992—with permission). D is the determination ratio.
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correlation coefficient RLL for the Purkinje cell is a bit smaller than that for the alpha neuron from the cat retinal ganglion cell showing that the scattering in the case of the Purkinje cell is a little more pronounced. In all cases shown in Fig. 3, the values of the determination ratio are larger than 1 which means that in these (real) cases the semi-log method gives better results and the D is larger than 1 (see insets in Figs. 3B and D, right).
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2.4. Pyramidal cells From the preceding considerations one can get an impression that the semi-log Sholl method is practically the only one which should be of use in practice. But despite the fact that the log–log Sholl method has a theoretical significance, this can also be efficient in practical explorations.
Fig. 4. Frequency distributions of numbers of intersections N against the radii r for some characteristic pyramidal cells. The scatter diagram of number of intersections N against the radius r of related circles of an isolated pyramidal neuron from layer V (from Cohen and Sherman, 1988, Fig. 19.2) (A), the scatter plots for a typical pyramidal neurons from layer V in the rat motor cortex (B and D), with a thick apical dendrites terminating in layer I (From Gao and Zhang, 2004 (A and C), and the scatter diagrams obtained when using the main apical bifurcation points in the same neurons, as centres of concentric circles (C and E).
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Fig. 4A shows the scatter diagram of an isolated large pyramidal neuron used from ganglionic (internal pyramidal) layer V of the cerebral cortex, as obtained by Golgiimpregnated method (from Cohen and Sherman, 1988, Fig. 19.2). Pyramidal cells are the most frequent type of neocortical neurons. Such type of cells has a single, very large and poorly ramified apical dendrite often reaching layer I and even extending to the pial surface. The vertical extent of this dendrite is markedly greater than that of dendrites in horizontal or tangential directions. Basal dendrites of pyramidal cells commonly descend toward the white matter. The dendrites are usually covered with spiny protrusions that are thought to be postsynaptic specializations. This dendritic branching pattern is analysed using Sholl’s methodology of regularly spaced concentric circles centred on the perikaryon. It has, as a consequence, rather large rectilinear horizontal part of the plot (Fig. 4A, the tail). It should be noticed that such a horizontal plot is also seen in Fig. 1B (open circles). In connection with it, Sholl (1953) has expressed that the apical dendrites do not terminate as fast as other dendrite branches; therefore, the log–log plot is better. Each branch, as it grows longer, has an increasing tendency either to form a new pair of branches or to terminate. Branch points from which more than two branches arise occur with a very low frequency (van Pelt, 1997). For such sparsely-branched dendrite and from the corresponding scatter diagram it is expectable that the determination ratio be less than 1 (indeed, D ¼ 0.927, see Fig. 4A), showing that in this particular case the log–log method of the Sholl analysis gives better result. Similar situation can be observed from Fig. 4B (from Gao and Zheng, 2004, Fig. 4A). A typical corticospinal (pyramidal) neuron in deep layer Vb of the rat motor cortex has been analysed. The large soma had a thick apical dendrite that tapered as it ascended toward the pial surface and bifurcated twice to produce three parallel apical dendritic trunks that terminated as small horizontal tufts in layer I. Reconstruction of the neuron was performed using a camera lucida drawing tube. In Fig. 4B the dendritic profile of a typical pyramidal neuron from layer V in the rat motor cortex is illustrated as a scatter diagram. In the corresponding scatter diagram of Fig. 4B similar nearly horizontal row of points (or, better still, nearly monotonically increasing row, as in Fig. 2A) can also be observed. Thanks to such a shape of the scatter diagram, being the consequence of three parallel and sparse dendritic trunks, the determination ratio is again less that 1. But, when using the main apical bifurcation point as a centre of concentric circles (similar example has been shown by Sholl (1953), as it is shown in Fig. 4C, the D is larger than 1. This is due to the fact that the three large apical trunks terminate as small horizontal tufts in lamina I. An aspiny corticothalamic pyramidal neuron located in the superficial part of layer Va is also analysed for the selection of the Sholl method (from Gao and Zheng, 2004, Fig. 4C). These cells are generally small or medium-sized
with thin apical dendritic trunks that gave rise to several oblique dendritic branches. In Fig. 4D the scatter diagram for that neuron is presented. The scatter diagram constructed for a whole cell shows that the log–log method gives even better results than that in Fig. 4B since D ¼ 0.858. Corresponding Sholl’s analysis of the pair of the apical dendrite after the main apical bifurcation point gives now D ¼ 0.907 because thin apical dendritic trunks give rise to several oblique dendritic branches (the dendritic branches are now rectilinear, not tufted). It is well known that dendrites exhibit enormously diverse forms. When using the logarithmic Sholl analyses and looking at the morphologies of so many different neuronal types, one can be in dilemma which of the two Sholl’s methods (semi-log or log–log) is better. The log–log method is probably better if the dendritic arbor satisfies the following conditions: (i) the dendrites of the cell should be sparsely ramified and terminated (along with their branches) on a circle of the largest radius (see Figs. 1 and 2); the dendrites may not be straight-lined and equallysized, and (ii) one or a few dendrites should be much longer than the rest ones irrespective of their number, enabling long horizontal series of data points (the ‘‘tail’’) in their scatter diagrams (see Fig. 4). 2.5. Statistics Assumptions made in the use of the correlation analysis are well known. It is not unusual, however, to encounter samples from populations in which some of these assumptions are violated. Therefore, the researcher must resort to other methods one of which is the change of the scale of measurement by means of a corresponding transformation. In order to apply the test of significance involving a normal distribution, R.A. Fisher used the logarithmic transformation to transform the coefficient of correlation R into the quantity Z (Fischer’s Z-statistic) which has an approximately normal distribution. Similar transformation has been used by Sholl (1953) to convert his original data (the number of intersections N of dendrites with the concentric circles and radius r of these circles) to their more workable logarithms. In fact, he noticed that the number of intersections per unit area falls off exponentially with the distance. He tested this crucial finding by plotting the logarithms of these numbers against the distance from the perikaryon, and the result was clearly linear. Such a method (the semi-log Sholl method) brought about an important parameter (the Sholl regression coefficient). Generally speaking, there could appear larger variations in the values transformed than their original values. The wrong result for the coefficient of correlation could be a consequence. For that reason, we calculated the coefficients of correlation R0 using untransformed (actual) bivariate data and tested our correlation data with Student’s (onetailed) t-test for differences between RLL and R0 as well as between RSL and R0 from the plots in Figs. 3 and 4. The results showed that all the values of R0 are significantly
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smaller that RLL as well as than RSL at the level of at least 0.05. For example, for the Purkinje cell shown in Fig. 3D, RSL ¼ 0.953, RLL ¼ 0.830 but R0 ¼ 0.191. For all our samples depicted in Figs. 1–4 having between 10 and 14 pairs of N, r-values and for all corresponding correlation coefficients RSL and RLL we calculated the values of Student’s t-statistic and found that all the correlation coefficients calculated can be considered significant on the basis of the 0.001 level of significance. 3. Comparison of the Sholl analysis to the fractal analysis The cumulative intersection (Sholl) analysis has long been used for quantitative morphometric studies of the neuronal dendritic arborization. Recently, in addition to the Sholl analysis, researchers use the fractal analysis to quantify the properties of dendritic patterns. Some authors compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns (Caserta et al., 1995; Duan et al., 2003; Uylings and van Pelt, 2002; Elston and Zietsch, 2005; Jelinek et al., 2005). Caserta et al. (1995) have stated that the Sholl analysis is similar to fractal analysis. In this respect, it is reasonable to presume that underlying basis of both methods relies on the simplicity and regularity of mathematical objects, and to complexity and irregularity of natural forms. In fractal geometry, two types of fractals are being used: mathematical (geometric) and statistical (natural) fractals. Mathematical fractals are geometric constructions characterized by never-ending cascades of similar structural details repeating themselves at progressively smaller scales (Mandelbrot, 1982; Bassingthwaighte et al., 1994), while statistical fractals are more restricted than are mathematical ones. The common measure of complexity of forms found in Euclidean geometry and nature is the fractal dimension (D). The larger the value of the fractal dimension of an object, the more complex it will be. To clarify the meaning of this important parameter, let us consider a border of a geometric or natural object in plane. Suppose that a scale (e.g. a ruler) of length r is used for the measurement of a border length. The measurement is performed in such a way that the ruler is moved along the border N times until the border is completely traversed. The obtained approximate length L of the border is then given as L ¼ Nr. It is known that so obtained length of the border depends inversely on the ruler length. This inverse proportionality relationship is known as an inverse power law, the general form of which is LðrÞ ¼
B rD1
,
(5)
where D41 is the fractal dimension of the object (border) and B is a constant. This law is the principal relationship in fractal geometry. The condition given by Eq. (5) is necessary for an object to be fractal (Mandelbrot, 1982; Bassingthwaighte et al., 1994; Ristanovic et al., 2002). We used here the classical ‘‘coastline method’’ of Richardson
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(Smith et al., 1996) but all fractal analysis methods rely on determining the relationship between the length of an image with changes in the scale of the measure (Jelinek and Ferna´ndez, 1998). Some objects in geometry are mathematical fractals (for review see Mandelbrot, 2004). The function given by Eq. (5) can be transformed to a straight-line form log L ¼ ðD 1Þ log r þ log B,
(6)
corresponding to Eq. (2). According to Eq. (6), the law given by Eq. (5) is revealed as a straight line plot when the results of the measurements are plotted on log–log axes against the values of the ruler lengths at which they are measured. If the r is made progressively smaller and L(r) always satisfies the inverse power-law (Eq. (5)), the object is said to be a mathematical fractal. But if this value (r) enables that the inverse power-law holds only for the r from a finite interval [r1, r2], the object represents a statistical fractal, i.e. a fractal over a finite interval of values r. When performing such an analysis to real objects, say neuronal trees, no perfect linear regions over such log–log plots can be noticed since dendrites exhibit enormously diverse forms (Milosˇ evic´ et al., 2005). But, if the corresponding correlation coefficient R for a sample of n pairs of r, L-values is slightly less than 1, say more than 0.995, we have shown that under these circumstances the neurons with sparse arborization could be considered fractals over three decades of the scale (Milosˇ evic´ and Ristanovic´, 2006). We have observed that between the fractal and Sholl analyses exist various and important analogies: (i) The data obtained by an application of the fractal analysis to mathematical fractals as well as of the Sholl analysis to idealized neurons (for example, to that shown in Fig. 1A) are revealed as straight-line plots when the results of measurements are plotted on log–log axes against the values of the ruler lengths in the first case, or the circle radii in the second case. (ii) If the data are obtained from real neurons considered as statistical fractals, a restricted fractality can be exposed. Similarly, a restriction is observed when the Sholl analysis is performed to real neurons: the data points plotted on log–log axes are not placed on fitting straight lines. Under such circumstances, acceptable results could be obtained when the semi-log method is used. (iii) The fractal dimension D corresponds to some extent to the determination ratio D since both parameters measure complexity of an object, but each of them measures different aspects of the complexity: while the fractal dimension, obtained using length-related techniques, measures the amount of details of dendritic tree and provides a measure of how completely the branches of a neuron fill its dendritic field (Ferna´ndez and Jelinek, 2001), the determination ratio measures intensity of the data points scattering about the fitted straight line, obtained using the semi-log method, in relation to such a scattering of the data points for the same neuron, found using the log–log method. The less the RLL, the larger the D will be (Eq. (4)). Using our neurons, there is a direct correlation between these parameters but it is not statistically
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significant. (iv) It is possible to have two quite different branching structures which have the same fractal dimension (Murray, 1995). Actually, some biological structures that look different and are structurally different may have the same value of the fractal dimension (Ferna´ndez et al., 1999). On the other hand, Uylings and van Pelt (2002) have argued that the Sholl analysis is not a sensitive method, since it can assign the same series of intersection numbers to a set of concentric circles for different trees. 4. Discussion 4.1. The logarithmic Sholl methods There are many qualitative and quantitative methods to analyse the morphologic properties of neuronal dendritic arborization patterns. The most standard quantitative method is the Sholl analysis. This analysis basically describes the radial distribution of some morphologic characteristics of neuronal trees around the cell’s perikaryon. The three main methods of the Sholl analysis are in use: the linear, semi-log and log–log method. One can distinguish between the data offered by the linear Sholl method and these given by both logarithmic Sholl methods, but the semi-log and log–log methods have been applied without a clear basis as to which to use under different circumstances. In the present study we attempted, using basic mathematics, to promote an approximate and general method how to cope with this problem. The choice between these two logarithmic methods depends of the coefficient of correlation values calculated. The semi-log Sholl method can be successfully used in studying how the number of neuronal branches varies with the distance from the perikaryon. The law given by Eq. (3) is revealed as a straight-line plot. The slope k of the regression line, given by this equation (Sholl’s regression coefficient), can be determined as a measure of the rate of decay in log (N/S) being associated with a unit change in r. Therefore, the larger the regression coefficient k, the greater is the gradient of changes in the function log (N/S). Shortly, the coefficient k measures the change in dendritic density with the distance r from the cell’s perikaryon. Similar property gives the log–log Sholl method but interpretation of such changes is not so obvious. This is an important advantage of these two methods over other quantitative methods. Sholl (1953) has pointed out significant differences between the slope k of the regression line for the visual stellate cells and the slopes of the lines for pyramidal and stellate cells of the motor area of the cortex in the cat. We have notices that this coefficient can successfully discriminate neuronal populations among different laminae of the cat spinal cord (unpublished material). 4.2. The linear Sholl method When using the linear Sholl method for data presentation (number of intersections versus radius—see Fig. 3,
right scatters and Fig. 4), the authors usually choose groups of neurons of the same type, (identified cells) and of similar dendritic organization. The network of concentric circles, traced with radii increasing at regular steps, is superimposed on the drawing of a neuron from a given type and the intersections with the circles of increasing radii (r) are counted (N). The same procedure is applied to the next neuron of the same type etc. Finally, the means of the numbers of intersections for all cells are calculated for each value of the radius on the network and the data obtained are presented as a frequency histogram or a graph on the r, N-plane of a Descartes coordinate system (Sholl, 1953; Schoenen, 1982; Neale et al., 1993; Galhardo and Lima, 1999; Duan et al., 2003; Jelinek et al., 2005). Using such figure, for an arborisation pattern of a corresponding type of a neuron one can estimate the following parameters: (i) the critical value (rc) of the radius r which defines the place of a possible circle intersecting maximum number of dendrites, (ii) the maximum of the function N(r) (maximum of the diagram), denoted as Nm, which measures the maximum density of dendritic arbor specified by the distance (rc) from the cell body, where this maximum density could be expected (iii) the Schoenen ramification index (Schoenen, 1982) calculating the quotient of the maximum number of dendritic intersections and the number of primary dendrites of the cell (the branches arising directly from the cell’s perikaryon). It is seen that the logarithmic Sholl methods provide different data (e.g. the Sholl regression coefficient, coefficient of determination, etc.) as compared to the linear Sholl one. In view of that, it is to be noted that these methods are complementary: in order to achieve the entire results using the Sholl analysis it is necessary to perform both the linear and the logarithmic Sholl method. From the results of applying the linear Sholl analysis to neuronal dendritic trees, the following conclusions can generally be drawn: (i) Use of linear Sholl analysis proves that similar plots of the relationship between the mean number of intersections and the radius of the circle can be found for every type of cells; (ii) Most of the authors apply the Sholl method to pyramidal cells (Elston, 2001; Cook and Wellman, 2004; Vega et al., 2004; Kheirandish et al., 2005; Martinez-Tellez et al., 2005) and reported differences in their Sholl coefficients depending on the experimental set-up (Vega et al. 2004; Kheirandish et al., 2005; MartinezTellez et al., 2005; Nelson et al., 2005); (iii) The main field of interest is the central nervous system, such as, the frontal cortex (Kheirandish et al., 2005) and prefrontal cortex (Cook and Wellman, 2004; Vega et al., 2004; MartinezTellez et al., 2005). 4.3. Applicability of the log–log Sholl method to neurons Some islet neurons in substantia gelatinosa of the cat spinal cord could be analysed using the log–log method (Gobel, 1978). Analysing one of them we have showed that RSL ¼ 0.950 and RLL ¼ 0.973, so that the determination
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ratio is D ¼ 0.953. Islet cells are found in small clusters in layer II. Their dendrites and axons are largely confined in that layer. One of the main characteristics of dendritic branching patterns is the extent to which the dendrites fill the spatial domain of their arborization. The dendrites of dendritic arbor lie on a continuum of values. At one extreme, a dendrite connects the cell body to a single remote target. An olfactory sensory cell is used to illustrate this being an example of a selective arborization (Fiala and Harris, 2001). Such a neuron can be successfully analysed using the log–log method. Also, using the same method the Lugaro and bipolar cells of the cortex, having two dendrites emerging from opposite poles of the cell’s perikaryon and few branches, can be effectively treated. Retinal horizontal cell is also an example where dendrites of approximately equal lengths radiate from the cell’s perikaryon in all directions within a thin domain. Dendrites of pallidal and reticular neurons ramify from a central soma in a thick disk-shaped domain (Fiala and Harris, 2001). All of these cells could attain better results when analysing using the log–log method. In fact, all of them have sparse, equally sized and poorly ramified dendritic trees. By inspecting the drawings of the cells on the cat retina (Wa¨ssle et al., 1987) we noticed that most of them satisfy the proposed conditions for cells to be successfully analysed with the log–log method. They have straight-lined, equally sized and very long dendrites (Wa¨ssle et al., 1987; Fig. 6). On the other hand, some of these cells have symmetrically distributed and densely packed dendrites of nearly equal sizes so that all of them can be surrounded by a circle of a large radius (Wa¨ssle et al., 1987; Fig. 7). A theoretical reason why to insist on an importance to analyse dendritic trees using the log–log Sholl method could be the abovementioned analogy between the Sholl and fractal analysis. In the special case shown in Fig. 1 the prefactor of log r in Eq. (2) is 2, but that in Eq. (6), which follows from an inverse power law (Eq. (5)) of the fractal theory, is (D1) where D is the fractal dimension. It is interesting to emphasize that the corresponding prefactors of equations related to Eq. (2), calculated for the mentioned neurons in the present study (Fig. 3B and D, as well as Fig. 4) are around 2 (more precisely, between 1.38 and 3.13). In this respect, we also analysed a Gamma retinal cell taken from Jelinek and Spence (1997) with the Sholl analysis and found that RLL ¼ 0.960 and RSL ¼ 0.987 so that D ¼ 1.058. When the determination ratio is approximately equal to 1, both methods can be successfully used. The reason for that is a very long dendrite of that neuron (Jelinek and Spence, 1997; Fig. 2). It should be noticed that D ¼ 1.266 (Fig. 3B) for the alpha neuron, thus much larger than that for the gamma neuron considered. 4.4. The diameter exponent Mandelbrot (2004) has argued that trees cannot be self similar but they involve a parameter to be called diameter
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exponent d. When in special cases the tree is self similar, the d coincides with the fractal dimension D of the branch tips. Otherwise, D and d are separate characteristics so that doD. If d is a diameter of the trunk before bifurcation and d1 and d2 these after bifurcation (dichotomous branching is the main mode of dendritic bifurcation), these quantities satisfy the relation d d ¼ d d1 þ d d2 .
(7)
If, for example, two branches of a trunk, put together, are equal in thickness to the trunk before branching, it is easy to point out that d 2. For the river width, when one river empts into the other, the d is also equal to 2. For lung’s bronchial tree the d 3, for arteries d 2.7, etc. Our inspection of photographs of neurons presented by Peichl (1989) and Wa¨ssle et al. (1987) clearly show that the diameters of dendrites, measured from the cell body to their tips, slowly diminish so that it is expectable that d should be larger than 2. There is little published information on use of the bifurcation diameter in analysing neuronal branching patterns. Therefore, we cannot offer at present any definite conclusion dealing with this problem since we tested with the Sholl analysis only skeletonized neurons. References Bassingthwaighte, J.B., Liebovitch, L.S., West, B.J., 1994. Fractal Physiology. Oxford University Press, New York, Oxford. Caserta, F., Eldred, W.D., Ferna´ndez, E., Hausman, R.E., Stanford, L.R., Bulderev, S.V., et al., 1995. Determination of fractal dimension of physiologically characterized neurons in two and three dimensions. J. Neurosci. Meth. 56, 133–144. Cohen, D.H., Sherman, S.M., 1988. The nervous system. In: Berne, R.M., Levy, M.N. (Eds.), Physiology, second ed. The C.V. Mosby Co., St. Louis, Washington DC, Toronto, pp. 69–311. Cook, S.C., Wellman, C.L., 2004. Chronic stress alters dendritic morphology in rat medial prefrontal cortex. J. Neurobiol., 236–248. Duan, H., Wearne, S.L., Rocher, A.B., Macedo, A., Morrison, J.H., Hof, P.R., 2003. Age-related dendritic and spine changes in corticocortically projecting neurons in macaque monkeys. Cereb. Cortex 13, 950–961. Elston, G.N., 2001. Interlaminar differences in the pyramidal cell phenotype in cortical areas 7m and STP (the superior temporal polysensory area) of the macaque monkey. Exp. Brain Res. 138, 141–152. Elston, G.N., Zietsch, B., 2005. Fractal analysis as a tool for studying specialization in neuronal structure: the study of the evolution of the primate cerebral cortex and human intellect. Adv. Complex Syst. 8, 217–227. Ferna´ndez, E., Bolea, J.A., Ortega, G., Louis, E., 1999. Are neurons multifractals? J. Neurosci. Meth. 89, 151–157. Ferna´ndez, E., Jelinek, F., 2001. Use of fractal theory in neuroscience: mathods, advantages and potintialproblems. Methods 24, 309–321. Fiala, J.C., Harris, K.M., 2001. Dendrite structure. In: Stuart, G., Spruston, N., Ha¨usser, M. (Eds.), Dendrites. Oxford University Press, Oxford, New York, pp. 1–34. Galhardo, V., Lima, D., 1999. Structural characteristics of marginal (lamina I) spinal cord neurons in the cat: a Golgi study. J. Comp. Neurol. 414, 315–333. Gao, W.J., Zheng, Z.H., 2004. Target-specific differences in somatodendritic morphology of Layer V pyramidal neurons in rat motor cortex. J. Comp. Neurol. 476, 174–175.
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Gobel, S., 1978. Golgi studies of the neurons in layer II of the dorsal horn of the medulla (trigeminal nucleus caudalis). J. Comp. Neurol. 180, 395–413. Jelinek, H.F., Elston, G.N., Zietsch, B., 2005. Fractal analysis: pitfalls and revelations in neuroscience. In: Losa, G.A., Merlini, D., Nonnenmacher, T.F., Weibel, E.R. (Eds.), Fractals in Biology and Medicine IV. Birkha¨user Verlag, Basel, Boston, Berlin, pp. 85–94. Jelinek, H.F., Ferna´ndez, E., 1998. Neurons and fractals: how reliable and useful are calculations of fractal dimensions? J. Neurosci. Meth. 81, 9–18. Jelinek, H.F., Spence, I., 1997. Categorization of physiologiucally and morphologically characterized non-a/non-b cat retinal ganglion cells using fractal geometry. Fractals 5, 673–684. Kheirandish, L., Gozal, D., Pequignot, J.M., Pequignot, J., Row, B.W., 2005. Intermittent hypoxia during development induces long-term alterations in spatial working memory, monoamines, and dendritic branching in rat frontal cortex. Pediatr. Res. 58, 594–549. Lima, D., Coimbra, A., 1986. A Golgi study of the neuronal population of the marginal zone (lamina I) of the rat spinal cord. J. Comp. Neurol. 244, 53–71. Lowndes, M., Stanford, D., Stewart, M.G., 1990. A system for the reconstruction and analysis of dendritic fields. J. Neurosci. Meth. 31, 235–245. Mandelbrot, B.B., 1982. The Fractal Geometry of Nature. WN Freeman, New York. Mandelbrot, B.B., 2004. The Fractal Geometry of Nature, 20th ed. NW Freeman and Co., New York. Martinez-Tellez, R., Gomez-Villalobos, Med.J., Flores, G., 2005. Alteration in dendritic morphology of cortical neurons in rats with diabetes mellitus induced by streptozotocin. Brain Res. 1048, 108–115. Milosˇ evic´, N.T., Ristanovic´, D., 2006. Fractality of dendritic arborization of spinal cord neurons. Neurosci. Lett. 396, 172–176. Milosˇ evic´, N.T., Ristanovic´, D., Stankovic´, J.B., 2005. Fractal analysis of the laminar organization of spinal cord neurons. J. Neurosci. Meth. 146, 198–204.
Murray, J.D., 1995. Use and abuse of fractal theory in neuroscience. J. Comp. Neurol. 361, 369–371. Neale, E.A., Bowers, L.M., Smith Jr., T.G., 1993. Early dendrite development in spinal cord cell cultures: a quantitative study. J. Neurosci. Res. 34, 54–66. Nelson, A.J., Juraska, J.M., Musch, T.I., Iwamoto, G.A., 2005. Neuroplastic adaptations to exercise: neuronal remodeling in cardiorespiratory and locomotor areas. J. Appl. Physiol. 99, 2312–2322. Peichl, L., 1989. Alpha and delta ganglion cells in the rat retina. J. Comp. Neurol. 286, 120–139. Richardson, L.F., 1961. The problem contiguity, an appendix to Statistics of Deadly Quarrels. Gen. Sys. 6, 139–187. Ristanovic´, D., Nedeljkov, V., Stefanovic´, B.D., Milosˇ evic´, N.T., Grgurevic´, M., Sˇtulic´, V., 2002. Fractal and nonfractal analysis of cell images: comparison and application to neuronal dendritic arborization. Biol. Cybern. 87, 278–288. Schoenen, J., 1982. The dendritic organization of the human spinal cord: the dorsal horn. Neuroscience 7, 2057–2087. Sholl, D.A., 1953. Dendritic organization in the neurons of the visual and motor cortices of the cat. J. Anat. 87, 387–406. Smith Jr., T.G., Lange, G.D., Marks, W.B., 1996. Fractal metods and results in cellular morphology: dimensions, lacunarity and multifractals. J. Neurosci. Meth. 69, 123–136. Takeda, T., Ishikawa, A., Ohtomo, K., Kobayashi, Y., Matsuoka, T., 1992. Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development. Neurosci. Res. 13, 19–31. Uylings, H.B.M., van Pelt, J., 2002. Measures for quantifying dendritic arborizations. Network: Comput. Neural Syst. 13, 397–414. van Pelt, J., 1997. Effect of pruning on dendritic tree topology. J. Theor. Biol. 186, 17–31. Vega, E., Gomez-Villalobos, MdeJ., Flores, G., 2004. Alteration in dendritic morphology of pyramidal neurons from the prefrontal cortex of rats with renovascular hypertension. Brain Res. 1021, 112–118. Wa¨ssle, H., Voigt, T., Patel, B., 1987. Morphological and immunosythochemical identification of indoleamine-accumulating neurons in the cat retina. J. Neurosci. 7, 1574–1585.
Journal of Theoretical Biology 245 (2007) 130–140 www.elsevier.com/locate/yjtbi
The Sholl analysis of neuronal cell images: Semi-log or log–log method? Nebojsˇ a T. Milosˇ evic´, Dusˇ an Ristanovic´ Department of Biophysics, School of Medicine, University of Belgrade, Visegradska 26/2, 11000 Belgrade, Serbia Received 19 June 2006; received in revised form 19 August 2006; accepted 20 September 2006 Available online 24 September 2006
Abstract Although the Sholl analysis is a quantitative method for morphometric neuronal studies and its application provides many benefits to neurobiology since it is obvious, common and meaningful, there are many unresolved theoretical issues that need to be addressed. Nevertheless, it can be practiced without much background or sophistication. The two different methods of the Sholl analysis—log–log and semi-log—have been applied previously without a clear basis as to what to use. To make an adequate choice of the method, one should try and accept that one which gives a better result. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be found in basic mathematics. In order to compare the results of applications of the semi-log and log–log methods to the same neuron, we introduce the concept of determination ratio as the ratio of the coefficient of determination for the semi-log method and that for the log–log method. If the semi-log method is better as related to the log–log method, the value of this parameter is larger than 1. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method is more frequently utilizable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log–log method. We also compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns and found that between the Sholl and fractal analysis exist various and important analogies. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fractal analysis; Inverse power laws; Morphometric parameters; Sholl analysis; Topological trees
1. Introduction One of the major goals in neurobiology is morphologic analysis of neuronal dendritic and axonal structure. The most standard approach is applying the Sholl analysis. The pioneering work of Sholl (1953) formed the basis of modern quantitative techniques in morphometric characterization of Golgi-impregnated and intracellulary stained neurons. The Sholl analysis is a method for quantitative study of radial distribution of some properties of neuronal dendritic arborization pattern around the cell’s perikaryon. This consecutive-circles (cumulative intersection) analysis specifies dendritic geometry, ramification richness, and dendritic branching patterns (Caserta et al., 1995). The Sholl analysis has long been widely applied and used for quantitative morphologic studies of impregnated and Corresponding author. Tel.: +381 11 3615775; fax: +381 11 3615767.
E-mail addresses: [email protected] (N.T. Milosˇ evic´), [email protected] (D. Ristanovic´). 0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.09.022
stained cells, and achieved widespread application in all fields of neuroscience (Lowndes et al., 1990; Neale et al., 1993; Caserta et al., 1995; Duan et al., 2003; Jelinek et al., 2005). It provides information on the number of branches relative to distance from the cell’s perikaryon. This analysis consists of (i) construction of concentric and equidistantly organized circles, which are centred in the perikaryon, (ii) counting the numbers of intersections of dendrites with the circles of increasing radii, (iii) defining suitable variables using the circle radius and the number of intersections of dendrites with circles of corresponding radii (such as, the Schoenen ramification index, number of intersections per circle area, maximum number of intersections, etc.), and (iv) choosing appropriate mathematical techniques for data processing and presenting (e.g., the histograms or plots in co-ordinate systems). Three main modifications of the Sholl analysis are in use: (i) the plot of the number (frequency) of dendritic intersections with the circles which can be calculated per circle area (Sholl, 1953) or per circle length (Neale et al., 1993), versus the circle
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radius (the linear Sholl method), (ii) the plot of the log of the number of intersections per circle area or circle length, versus the circle radius (the semi-log Sholl method), and (iii) the plot of the log of the number of intersections per circle area or circle length, versus log of the circle radius (the log–log Sholl method). Semi-log and log–log methods give different results when used to analyse the same structure (Caserta et al., 1995). When plotting the values either as the semi-log or as log–log, Sholl (1953) demonstrated that at least in one of these cases the data points approximately lie on a straight line. He used that analysis, which provides better results and calculated the slope k of the regression straight line (the Sholl regression coefficient) to characterize the morphology of the neuron that he studied. The slope of fitted straight line is a measure of the decay rate of the number of branches with distance from the perikaryon. The aim of the analysis was to present the radial distribution of dendritic branches with the distance from the perikaryon and extract the main morphologic characteristics of the neuron. The semi-log and log–log Sholl methods have been applied previously without a clear basis as to what to use. When performing the Sholl methods, it is good to note that different Sholl’s methods work best for different neuron types. Even Richardson (1961), who first suggested the log–log method when presenting straight-line plots of various coastline lengths in which the apparent length (the log of the total length) is graphed versus the log of the measuring units, does not provide details of why this choice was made. Therefore, to make an adequate choice of the method, one should try and accept that one which gives the best result. Sholl applied the double logarithmic method to apical dendrites and semi-log method to basal dendrites of pyramidal neurons from the visual and motor cortices of the cat because only these methods, respectively, produced approximately linear results for these types of neurons (Sholl, 1953). Although the Sholl analysis represents a quantitative method for morphometric neuronal studies, it is not established exactly. Lima and Coimbra (1986) have argued that the two different methods of the Sholl analysis— log–log and semi-log—have been applied previously without clear basis as to which to choose. Nevertheless, it can be practiced without much background or sophistication. It has provided many benefits to neurobiology because it is obvious, common and meaningful. That is probably one of the reasons why it is still currently in use (Duan et al., 2003; Cook and Wellman, 2004; Vega et al., 2004; Jelinek et al., 2005; Kheirandish et al., 2005; Martinez-Tellez et al., 2005; Nelson et al., 2005). In the present study, we analyse the results of application of both the log–log and semi-log method to the same neurons. We ask if one method is better that the other, or, whether these two methods can be equally treated and used in different dendroarchitectonics of arborization of various neurons. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method
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is more frequently usable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log–log method. We also examine the relationship between these two methods of the Sholl analysis and compare the results with some main concepts of the fractal analysis. 2. Methods and results It has been generally agreed that the Sholl analysis is not established exactly. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be met in basic mathematics. 2.1. Surface density of intersections Let y be the function specified by N , (1) r2 p where N is a constant and r is the radius of a circle. The expression in denominator of Eq. (1) is the area S of the circle of radius r. Then y(r) is an inverse function of the circle area. If we take the logarithms to the basis 10 of both sides of Eq. (1), the following relationship is obtained: yðrÞ ¼
log y ¼ 2 log r þ K,
(2)
where K ¼ log Nlog p. Thus, the inverse power (hyperbolic) function, given by Eq. (1), is revealed as a straightline plot when the values of y are plotted on log–log axes against the values of r. If, for example, N ¼ 6 and r takes the values 1, 2, 3, y, 10 (in any units) and if ordered pairs of coordinates (r, N), calculated from Eq. (1), are graphed on log–log axes, the line fitting these points is a straight line with the coefficient of correlation RLL ¼ 1.000 (Fig. 1C). If all the values of the variables satisfy such an equation exactly, one can say that the variables are perfectly correlated or that there is a perfect correlation between them. The drawing in Fig. 1A simulates a hypothetical neuron with six equally sized primary dendrites in 2D. According to the Sholl methodology, over this neuronal ‘‘perikaryon’’, using the radii in arbitrary units, we construct a series of equidistantly arranged concentric circles which are centred in the perikaryon, and measure the number of intersections of dendrites with these circles. In a plane on which a rectangular co-ordinate system has been constructed we plot the number of intersections N as a function of radius r for such dendritic (topographic) tree (Fig. 1A) and the corresponding scatter diagram present in Fig. 1B (open circles). This scatter diagram indicates that a horizontal straight line would perfectly fit this set of points. If we determine the surface density of intersections y as the number of intersections N of dendritic branches with a circle per its area (S) and plot on log–log axes this variable (y ¼ N/S) versus radius (r) of the circle, we obtain just the
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same graph as shown in Fig. 1C. We also include the plot of the log of N/S versus the radius r in this figure as Fig. 1B (the semi-log Sholl method) log
N ¼ k r þ m, S
(3)
where k is the Sholl regression coefficient. We find that the correlation coefficient RSL of a regression straight line to these data is less than RLL and equals 0.952o1 since it is evidently impossible to achieve here a perfect linearity. 2.2. Image preparation The drawings of neurons to be considered here were converted into digitized images using a scanner with a resolution of 600 dpi. All transformations were carried out on a PC computer using the public domain Image J software (www.rcb.info.nih.gov/ij) developed at the US National Institute of Health. All scanned images were imported into this software. Each dendrite was filled with pixels. Since the drawings were analysed as ‘‘skeleton’’ tracings, the software performed a skeletonization of the image to a stick figure. Such image of the neuron (with correctly inscribed scale bar) was scaled to fit A4 size of the paper (i.e., 15 21 cm) and then printed. In order to study the way in which the number of neuronal branches varies with the distance from the cell body, we used a series of concentric circles (after Sholl) with a common centre in the cell’s perikaryon as coordinates of reference, and traced the network of such circles over a single neuronal drawing, with radii increasing at regular steps of 10 mm in all cases. Adopting such a choice of the steps we avoided a possible effect of changes in network density on the values of the correlation coefficient for a single neuron. Analysing our data we noticed that magnifying the steps between the circles brought about smaller values of the correlation coefficients. 2.3. Determination ratio
Fig. 1. Application of Sholl methods to an idealized neuronal tree. (A) Topological tree consisting of six equally-sized dendrites, with a system of concentric circles superimposed over the drawing. (B) The graphs of linear (open circles) and semi-log Sholl analyses (filled circles). The first series of data points illustrates the relationship between the number of intersections N with the circles and the corresponding radius r in arbitrary units. The second graph is a regression straight line fitting the data points (filled circles) on semi-log axes. (C) The graph of data points calculated using the log–log Sholl method. RSL and RLL are the correlation coefficients for samples of 10 pairs of r, N-values calculated for application to the semi-log and log–log Sholl methods, respectively.
From Fig. 1C it is obvious that the log–log plot for an artificial neuronal cell shown in Fig. 1A is theoretically perfect since the correlation coefficient RLL equals unity. The plot for the semi-log method gives worse result (experimental points scattered about the fitted straight line—Fig. 1B filled points). In order to compare the results of applications of the two methods to the same neuron, we define the determination ratio D as the ratio of the coefficient of determination for the semi-log method (RLS2) and that for the log–log method (RLL2) D¼
R2LS . R2LL
(4)
If the log–log method is better then the semi-log one, the value of the parameter D is less than 1; on the contrary, this value will be larger than 1. In Fig. 1, where an ideal case is
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presented, this value is D ¼ 0.906o1 showing that the log–log method gives better result. Similar finding can be obtained from the theoretical examples (topological trees) shown in Fig. 2A and B (left). They are the models of the two-dimensional branching patterns reduced to skeletons of branch points and segments in the form of rooted trees. Contrary to the scatter diagram in Fig. 1B (open circles) where all the points lie on a horizontal straight line (not shown), in Fig. 2A (right) the values for N increase regularly, say, in an arithmetic progression (the first circle cuts the dendrites six times, the next circle curs the dendrites seven times etc.). This can be seen from the scatter in Fig. 2A. In that case, the value of D is 0.878. In Fig. 2B a little more complex theoretical model of a nondecreasing sequence is presented. The value of D ¼ 0.861. To our experience, such theoretically depicted neurons can be met in reality very occasionally. From these theoretical examples it is possible to conclude that the log–log method might be successfully applied to neurons whose dendrites are rather long, poorly ramified, and all of them along with their branches terminate on a circle of the largest radius. The next step to make topographic models of neurons with more complex dendritic trees are shown in Fig. 3A and C (left): some dendrites reach the circles of the largest
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radii, but some of them, being a consequence of dendritic ramifications, terminate earlier. The model in Fig. 3C is a bit more complex than that in Fig. 3A. But these models are more realistic as compared to those in Fig. 2: in most of the identified types of neurons the scatter diagrams are irregular, but mainly convex-upward. Indeed, the consequent distributions of the numbers of intersections N against the circle radii r are shown on right sides of these models. Fig. 3B represents an alpha neuron from the cat retinal ganglion (left) and the corresponding frequency distribution of data points (right). In cells of alpha-type, filled with Lucifer Yellow, the dendritic fields are relatively large and basically monostratified within the inner plexiform layer. Several stout primary dendrites emerge radially from the soma. They branch successfully without significant crossing and evenly fill the dendritic field. This example corresponds to the model shown in Fig. 3A (right) since both have similar distributions (convexupward) of data points (compare Figs. 1A and 1B, right). Fig. 3D, related to the model in Fig. 3C, shows a typical Purkinje neuron in the mouse cerebellum (left) stained with rapid Golgi method. Stained Purkinje cells found in the primary fissure were selected and drawn with a camera lucida. The Purkinje neuron in Fig. 3D has a more profuse branching pattern than the cell in Fig. 3B; therefore, the
Fig. 2. Frequency distribution of number of intersections N versus the radius r of concentric circles (right) for the two idealized topological trees (left). The trees and frequency distributions of the data where the number of intersections increase as an arithmetic progression (A) and as a no decreasing series of data (B).
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Fig. 3. Topological trees (A and C) and related neurons (B and D) (left), as well as the corresponding frequency distributions of the number of intersections N with the circles superimposed over the neuronal trees (right). Neuron in (B) represents an Alpha neuron from the cat retinal ganglion (from Jelinek and Spence, 1997—with permission), and the neuron in (D) is a typical Purkinje cell in the mouse cerebellum (From Takeda et al., 1992—with permission). D is the determination ratio.
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correlation coefficient RLL for the Purkinje cell is a bit smaller than that for the alpha neuron from the cat retinal ganglion cell showing that the scattering in the case of the Purkinje cell is a little more pronounced. In all cases shown in Fig. 3, the values of the determination ratio are larger than 1 which means that in these (real) cases the semi-log method gives better results and the D is larger than 1 (see insets in Figs. 3B and D, right).
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2.4. Pyramidal cells From the preceding considerations one can get an impression that the semi-log Sholl method is practically the only one which should be of use in practice. But despite the fact that the log–log Sholl method has a theoretical significance, this can also be efficient in practical explorations.
Fig. 4. Frequency distributions of numbers of intersections N against the radii r for some characteristic pyramidal cells. The scatter diagram of number of intersections N against the radius r of related circles of an isolated pyramidal neuron from layer V (from Cohen and Sherman, 1988, Fig. 19.2) (A), the scatter plots for a typical pyramidal neurons from layer V in the rat motor cortex (B and D), with a thick apical dendrites terminating in layer I (From Gao and Zhang, 2004 (A and C), and the scatter diagrams obtained when using the main apical bifurcation points in the same neurons, as centres of concentric circles (C and E).
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Fig. 4A shows the scatter diagram of an isolated large pyramidal neuron used from ganglionic (internal pyramidal) layer V of the cerebral cortex, as obtained by Golgiimpregnated method (from Cohen and Sherman, 1988, Fig. 19.2). Pyramidal cells are the most frequent type of neocortical neurons. Such type of cells has a single, very large and poorly ramified apical dendrite often reaching layer I and even extending to the pial surface. The vertical extent of this dendrite is markedly greater than that of dendrites in horizontal or tangential directions. Basal dendrites of pyramidal cells commonly descend toward the white matter. The dendrites are usually covered with spiny protrusions that are thought to be postsynaptic specializations. This dendritic branching pattern is analysed using Sholl’s methodology of regularly spaced concentric circles centred on the perikaryon. It has, as a consequence, rather large rectilinear horizontal part of the plot (Fig. 4A, the tail). It should be noticed that such a horizontal plot is also seen in Fig. 1B (open circles). In connection with it, Sholl (1953) has expressed that the apical dendrites do not terminate as fast as other dendrite branches; therefore, the log–log plot is better. Each branch, as it grows longer, has an increasing tendency either to form a new pair of branches or to terminate. Branch points from which more than two branches arise occur with a very low frequency (van Pelt, 1997). For such sparsely-branched dendrite and from the corresponding scatter diagram it is expectable that the determination ratio be less than 1 (indeed, D ¼ 0.927, see Fig. 4A), showing that in this particular case the log–log method of the Sholl analysis gives better result. Similar situation can be observed from Fig. 4B (from Gao and Zheng, 2004, Fig. 4A). A typical corticospinal (pyramidal) neuron in deep layer Vb of the rat motor cortex has been analysed. The large soma had a thick apical dendrite that tapered as it ascended toward the pial surface and bifurcated twice to produce three parallel apical dendritic trunks that terminated as small horizontal tufts in layer I. Reconstruction of the neuron was performed using a camera lucida drawing tube. In Fig. 4B the dendritic profile of a typical pyramidal neuron from layer V in the rat motor cortex is illustrated as a scatter diagram. In the corresponding scatter diagram of Fig. 4B similar nearly horizontal row of points (or, better still, nearly monotonically increasing row, as in Fig. 2A) can also be observed. Thanks to such a shape of the scatter diagram, being the consequence of three parallel and sparse dendritic trunks, the determination ratio is again less that 1. But, when using the main apical bifurcation point as a centre of concentric circles (similar example has been shown by Sholl (1953), as it is shown in Fig. 4C, the D is larger than 1. This is due to the fact that the three large apical trunks terminate as small horizontal tufts in lamina I. An aspiny corticothalamic pyramidal neuron located in the superficial part of layer Va is also analysed for the selection of the Sholl method (from Gao and Zheng, 2004, Fig. 4C). These cells are generally small or medium-sized
with thin apical dendritic trunks that gave rise to several oblique dendritic branches. In Fig. 4D the scatter diagram for that neuron is presented. The scatter diagram constructed for a whole cell shows that the log–log method gives even better results than that in Fig. 4B since D ¼ 0.858. Corresponding Sholl’s analysis of the pair of the apical dendrite after the main apical bifurcation point gives now D ¼ 0.907 because thin apical dendritic trunks give rise to several oblique dendritic branches (the dendritic branches are now rectilinear, not tufted). It is well known that dendrites exhibit enormously diverse forms. When using the logarithmic Sholl analyses and looking at the morphologies of so many different neuronal types, one can be in dilemma which of the two Sholl’s methods (semi-log or log–log) is better. The log–log method is probably better if the dendritic arbor satisfies the following conditions: (i) the dendrites of the cell should be sparsely ramified and terminated (along with their branches) on a circle of the largest radius (see Figs. 1 and 2); the dendrites may not be straight-lined and equallysized, and (ii) one or a few dendrites should be much longer than the rest ones irrespective of their number, enabling long horizontal series of data points (the ‘‘tail’’) in their scatter diagrams (see Fig. 4). 2.5. Statistics Assumptions made in the use of the correlation analysis are well known. It is not unusual, however, to encounter samples from populations in which some of these assumptions are violated. Therefore, the researcher must resort to other methods one of which is the change of the scale of measurement by means of a corresponding transformation. In order to apply the test of significance involving a normal distribution, R.A. Fisher used the logarithmic transformation to transform the coefficient of correlation R into the quantity Z (Fischer’s Z-statistic) which has an approximately normal distribution. Similar transformation has been used by Sholl (1953) to convert his original data (the number of intersections N of dendrites with the concentric circles and radius r of these circles) to their more workable logarithms. In fact, he noticed that the number of intersections per unit area falls off exponentially with the distance. He tested this crucial finding by plotting the logarithms of these numbers against the distance from the perikaryon, and the result was clearly linear. Such a method (the semi-log Sholl method) brought about an important parameter (the Sholl regression coefficient). Generally speaking, there could appear larger variations in the values transformed than their original values. The wrong result for the coefficient of correlation could be a consequence. For that reason, we calculated the coefficients of correlation R0 using untransformed (actual) bivariate data and tested our correlation data with Student’s (onetailed) t-test for differences between RLL and R0 as well as between RSL and R0 from the plots in Figs. 3 and 4. The results showed that all the values of R0 are significantly
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smaller that RLL as well as than RSL at the level of at least 0.05. For example, for the Purkinje cell shown in Fig. 3D, RSL ¼ 0.953, RLL ¼ 0.830 but R0 ¼ 0.191. For all our samples depicted in Figs. 1–4 having between 10 and 14 pairs of N, r-values and for all corresponding correlation coefficients RSL and RLL we calculated the values of Student’s t-statistic and found that all the correlation coefficients calculated can be considered significant on the basis of the 0.001 level of significance. 3. Comparison of the Sholl analysis to the fractal analysis The cumulative intersection (Sholl) analysis has long been used for quantitative morphometric studies of the neuronal dendritic arborization. Recently, in addition to the Sholl analysis, researchers use the fractal analysis to quantify the properties of dendritic patterns. Some authors compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns (Caserta et al., 1995; Duan et al., 2003; Uylings and van Pelt, 2002; Elston and Zietsch, 2005; Jelinek et al., 2005). Caserta et al. (1995) have stated that the Sholl analysis is similar to fractal analysis. In this respect, it is reasonable to presume that underlying basis of both methods relies on the simplicity and regularity of mathematical objects, and to complexity and irregularity of natural forms. In fractal geometry, two types of fractals are being used: mathematical (geometric) and statistical (natural) fractals. Mathematical fractals are geometric constructions characterized by never-ending cascades of similar structural details repeating themselves at progressively smaller scales (Mandelbrot, 1982; Bassingthwaighte et al., 1994), while statistical fractals are more restricted than are mathematical ones. The common measure of complexity of forms found in Euclidean geometry and nature is the fractal dimension (D). The larger the value of the fractal dimension of an object, the more complex it will be. To clarify the meaning of this important parameter, let us consider a border of a geometric or natural object in plane. Suppose that a scale (e.g. a ruler) of length r is used for the measurement of a border length. The measurement is performed in such a way that the ruler is moved along the border N times until the border is completely traversed. The obtained approximate length L of the border is then given as L ¼ Nr. It is known that so obtained length of the border depends inversely on the ruler length. This inverse proportionality relationship is known as an inverse power law, the general form of which is LðrÞ ¼
B rD1
,
(5)
where D41 is the fractal dimension of the object (border) and B is a constant. This law is the principal relationship in fractal geometry. The condition given by Eq. (5) is necessary for an object to be fractal (Mandelbrot, 1982; Bassingthwaighte et al., 1994; Ristanovic et al., 2002). We used here the classical ‘‘coastline method’’ of Richardson
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(Smith et al., 1996) but all fractal analysis methods rely on determining the relationship between the length of an image with changes in the scale of the measure (Jelinek and Ferna´ndez, 1998). Some objects in geometry are mathematical fractals (for review see Mandelbrot, 2004). The function given by Eq. (5) can be transformed to a straight-line form log L ¼ ðD 1Þ log r þ log B,
(6)
corresponding to Eq. (2). According to Eq. (6), the law given by Eq. (5) is revealed as a straight line plot when the results of the measurements are plotted on log–log axes against the values of the ruler lengths at which they are measured. If the r is made progressively smaller and L(r) always satisfies the inverse power-law (Eq. (5)), the object is said to be a mathematical fractal. But if this value (r) enables that the inverse power-law holds only for the r from a finite interval [r1, r2], the object represents a statistical fractal, i.e. a fractal over a finite interval of values r. When performing such an analysis to real objects, say neuronal trees, no perfect linear regions over such log–log plots can be noticed since dendrites exhibit enormously diverse forms (Milosˇ evic´ et al., 2005). But, if the corresponding correlation coefficient R for a sample of n pairs of r, L-values is slightly less than 1, say more than 0.995, we have shown that under these circumstances the neurons with sparse arborization could be considered fractals over three decades of the scale (Milosˇ evic´ and Ristanovic´, 2006). We have observed that between the fractal and Sholl analyses exist various and important analogies: (i) The data obtained by an application of the fractal analysis to mathematical fractals as well as of the Sholl analysis to idealized neurons (for example, to that shown in Fig. 1A) are revealed as straight-line plots when the results of measurements are plotted on log–log axes against the values of the ruler lengths in the first case, or the circle radii in the second case. (ii) If the data are obtained from real neurons considered as statistical fractals, a restricted fractality can be exposed. Similarly, a restriction is observed when the Sholl analysis is performed to real neurons: the data points plotted on log–log axes are not placed on fitting straight lines. Under such circumstances, acceptable results could be obtained when the semi-log method is used. (iii) The fractal dimension D corresponds to some extent to the determination ratio D since both parameters measure complexity of an object, but each of them measures different aspects of the complexity: while the fractal dimension, obtained using length-related techniques, measures the amount of details of dendritic tree and provides a measure of how completely the branches of a neuron fill its dendritic field (Ferna´ndez and Jelinek, 2001), the determination ratio measures intensity of the data points scattering about the fitted straight line, obtained using the semi-log method, in relation to such a scattering of the data points for the same neuron, found using the log–log method. The less the RLL, the larger the D will be (Eq. (4)). Using our neurons, there is a direct correlation between these parameters but it is not statistically
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significant. (iv) It is possible to have two quite different branching structures which have the same fractal dimension (Murray, 1995). Actually, some biological structures that look different and are structurally different may have the same value of the fractal dimension (Ferna´ndez et al., 1999). On the other hand, Uylings and van Pelt (2002) have argued that the Sholl analysis is not a sensitive method, since it can assign the same series of intersection numbers to a set of concentric circles for different trees. 4. Discussion 4.1. The logarithmic Sholl methods There are many qualitative and quantitative methods to analyse the morphologic properties of neuronal dendritic arborization patterns. The most standard quantitative method is the Sholl analysis. This analysis basically describes the radial distribution of some morphologic characteristics of neuronal trees around the cell’s perikaryon. The three main methods of the Sholl analysis are in use: the linear, semi-log and log–log method. One can distinguish between the data offered by the linear Sholl method and these given by both logarithmic Sholl methods, but the semi-log and log–log methods have been applied without a clear basis as to which to use under different circumstances. In the present study we attempted, using basic mathematics, to promote an approximate and general method how to cope with this problem. The choice between these two logarithmic methods depends of the coefficient of correlation values calculated. The semi-log Sholl method can be successfully used in studying how the number of neuronal branches varies with the distance from the perikaryon. The law given by Eq. (3) is revealed as a straight-line plot. The slope k of the regression line, given by this equation (Sholl’s regression coefficient), can be determined as a measure of the rate of decay in log (N/S) being associated with a unit change in r. Therefore, the larger the regression coefficient k, the greater is the gradient of changes in the function log (N/S). Shortly, the coefficient k measures the change in dendritic density with the distance r from the cell’s perikaryon. Similar property gives the log–log Sholl method but interpretation of such changes is not so obvious. This is an important advantage of these two methods over other quantitative methods. Sholl (1953) has pointed out significant differences between the slope k of the regression line for the visual stellate cells and the slopes of the lines for pyramidal and stellate cells of the motor area of the cortex in the cat. We have notices that this coefficient can successfully discriminate neuronal populations among different laminae of the cat spinal cord (unpublished material). 4.2. The linear Sholl method When using the linear Sholl method for data presentation (number of intersections versus radius—see Fig. 3,
right scatters and Fig. 4), the authors usually choose groups of neurons of the same type, (identified cells) and of similar dendritic organization. The network of concentric circles, traced with radii increasing at regular steps, is superimposed on the drawing of a neuron from a given type and the intersections with the circles of increasing radii (r) are counted (N). The same procedure is applied to the next neuron of the same type etc. Finally, the means of the numbers of intersections for all cells are calculated for each value of the radius on the network and the data obtained are presented as a frequency histogram or a graph on the r, N-plane of a Descartes coordinate system (Sholl, 1953; Schoenen, 1982; Neale et al., 1993; Galhardo and Lima, 1999; Duan et al., 2003; Jelinek et al., 2005). Using such figure, for an arborisation pattern of a corresponding type of a neuron one can estimate the following parameters: (i) the critical value (rc) of the radius r which defines the place of a possible circle intersecting maximum number of dendrites, (ii) the maximum of the function N(r) (maximum of the diagram), denoted as Nm, which measures the maximum density of dendritic arbor specified by the distance (rc) from the cell body, where this maximum density could be expected (iii) the Schoenen ramification index (Schoenen, 1982) calculating the quotient of the maximum number of dendritic intersections and the number of primary dendrites of the cell (the branches arising directly from the cell’s perikaryon). It is seen that the logarithmic Sholl methods provide different data (e.g. the Sholl regression coefficient, coefficient of determination, etc.) as compared to the linear Sholl one. In view of that, it is to be noted that these methods are complementary: in order to achieve the entire results using the Sholl analysis it is necessary to perform both the linear and the logarithmic Sholl method. From the results of applying the linear Sholl analysis to neuronal dendritic trees, the following conclusions can generally be drawn: (i) Use of linear Sholl analysis proves that similar plots of the relationship between the mean number of intersections and the radius of the circle can be found for every type of cells; (ii) Most of the authors apply the Sholl method to pyramidal cells (Elston, 2001; Cook and Wellman, 2004; Vega et al., 2004; Kheirandish et al., 2005; Martinez-Tellez et al., 2005) and reported differences in their Sholl coefficients depending on the experimental set-up (Vega et al. 2004; Kheirandish et al., 2005; MartinezTellez et al., 2005; Nelson et al., 2005); (iii) The main field of interest is the central nervous system, such as, the frontal cortex (Kheirandish et al., 2005) and prefrontal cortex (Cook and Wellman, 2004; Vega et al., 2004; MartinezTellez et al., 2005). 4.3. Applicability of the log–log Sholl method to neurons Some islet neurons in substantia gelatinosa of the cat spinal cord could be analysed using the log–log method (Gobel, 1978). Analysing one of them we have showed that RSL ¼ 0.950 and RLL ¼ 0.973, so that the determination
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ratio is D ¼ 0.953. Islet cells are found in small clusters in layer II. Their dendrites and axons are largely confined in that layer. One of the main characteristics of dendritic branching patterns is the extent to which the dendrites fill the spatial domain of their arborization. The dendrites of dendritic arbor lie on a continuum of values. At one extreme, a dendrite connects the cell body to a single remote target. An olfactory sensory cell is used to illustrate this being an example of a selective arborization (Fiala and Harris, 2001). Such a neuron can be successfully analysed using the log–log method. Also, using the same method the Lugaro and bipolar cells of the cortex, having two dendrites emerging from opposite poles of the cell’s perikaryon and few branches, can be effectively treated. Retinal horizontal cell is also an example where dendrites of approximately equal lengths radiate from the cell’s perikaryon in all directions within a thin domain. Dendrites of pallidal and reticular neurons ramify from a central soma in a thick disk-shaped domain (Fiala and Harris, 2001). All of these cells could attain better results when analysing using the log–log method. In fact, all of them have sparse, equally sized and poorly ramified dendritic trees. By inspecting the drawings of the cells on the cat retina (Wa¨ssle et al., 1987) we noticed that most of them satisfy the proposed conditions for cells to be successfully analysed with the log–log method. They have straight-lined, equally sized and very long dendrites (Wa¨ssle et al., 1987; Fig. 6). On the other hand, some of these cells have symmetrically distributed and densely packed dendrites of nearly equal sizes so that all of them can be surrounded by a circle of a large radius (Wa¨ssle et al., 1987; Fig. 7). A theoretical reason why to insist on an importance to analyse dendritic trees using the log–log Sholl method could be the abovementioned analogy between the Sholl and fractal analysis. In the special case shown in Fig. 1 the prefactor of log r in Eq. (2) is 2, but that in Eq. (6), which follows from an inverse power law (Eq. (5)) of the fractal theory, is (D1) where D is the fractal dimension. It is interesting to emphasize that the corresponding prefactors of equations related to Eq. (2), calculated for the mentioned neurons in the present study (Fig. 3B and D, as well as Fig. 4) are around 2 (more precisely, between 1.38 and 3.13). In this respect, we also analysed a Gamma retinal cell taken from Jelinek and Spence (1997) with the Sholl analysis and found that RLL ¼ 0.960 and RSL ¼ 0.987 so that D ¼ 1.058. When the determination ratio is approximately equal to 1, both methods can be successfully used. The reason for that is a very long dendrite of that neuron (Jelinek and Spence, 1997; Fig. 2). It should be noticed that D ¼ 1.266 (Fig. 3B) for the alpha neuron, thus much larger than that for the gamma neuron considered. 4.4. The diameter exponent Mandelbrot (2004) has argued that trees cannot be self similar but they involve a parameter to be called diameter
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exponent d. When in special cases the tree is self similar, the d coincides with the fractal dimension D of the branch tips. Otherwise, D and d are separate characteristics so that doD. If d is a diameter of the trunk before bifurcation and d1 and d2 these after bifurcation (dichotomous branching is the main mode of dendritic bifurcation), these quantities satisfy the relation d d ¼ d d1 þ d d2 .
(7)
If, for example, two branches of a trunk, put together, are equal in thickness to the trunk before branching, it is easy to point out that d 2. For the river width, when one river empts into the other, the d is also equal to 2. For lung’s bronchial tree the d 3, for arteries d 2.7, etc. Our inspection of photographs of neurons presented by Peichl (1989) and Wa¨ssle et al. (1987) clearly show that the diameters of dendrites, measured from the cell body to their tips, slowly diminish so that it is expectable that d should be larger than 2. There is little published information on use of the bifurcation diameter in analysing neuronal branching patterns. Therefore, we cannot offer at present any definite conclusion dealing with this problem since we tested with the Sholl analysis only skeletonized neurons. References Bassingthwaighte, J.B., Liebovitch, L.S., West, B.J., 1994. Fractal Physiology. Oxford University Press, New York, Oxford. Caserta, F., Eldred, W.D., Ferna´ndez, E., Hausman, R.E., Stanford, L.R., Bulderev, S.V., et al., 1995. Determination of fractal dimension of physiologically characterized neurons in two and three dimensions. J. Neurosci. Meth. 56, 133–144. Cohen, D.H., Sherman, S.M., 1988. The nervous system. In: Berne, R.M., Levy, M.N. (Eds.), Physiology, second ed. The C.V. Mosby Co., St. Louis, Washington DC, Toronto, pp. 69–311. Cook, S.C., Wellman, C.L., 2004. Chronic stress alters dendritic morphology in rat medial prefrontal cortex. J. Neurobiol., 236–248. Duan, H., Wearne, S.L., Rocher, A.B., Macedo, A., Morrison, J.H., Hof, P.R., 2003. Age-related dendritic and spine changes in corticocortically projecting neurons in macaque monkeys. Cereb. Cortex 13, 950–961. Elston, G.N., 2001. Interlaminar differences in the pyramidal cell phenotype in cortical areas 7m and STP (the superior temporal polysensory area) of the macaque monkey. Exp. Brain Res. 138, 141–152. Elston, G.N., Zietsch, B., 2005. Fractal analysis as a tool for studying specialization in neuronal structure: the study of the evolution of the primate cerebral cortex and human intellect. Adv. Complex Syst. 8, 217–227. Ferna´ndez, E., Bolea, J.A., Ortega, G., Louis, E., 1999. Are neurons multifractals? J. Neurosci. Meth. 89, 151–157. Ferna´ndez, E., Jelinek, F., 2001. Use of fractal theory in neuroscience: mathods, advantages and potintialproblems. Methods 24, 309–321. Fiala, J.C., Harris, K.M., 2001. Dendrite structure. In: Stuart, G., Spruston, N., Ha¨usser, M. (Eds.), Dendrites. Oxford University Press, Oxford, New York, pp. 1–34. Galhardo, V., Lima, D., 1999. Structural characteristics of marginal (lamina I) spinal cord neurons in the cat: a Golgi study. J. Comp. Neurol. 414, 315–333. Gao, W.J., Zheng, Z.H., 2004. Target-specific differences in somatodendritic morphology of Layer V pyramidal neurons in rat motor cortex. J. Comp. Neurol. 476, 174–175.
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