Mathematical modeling and computational analysis of neuronal cell images: Application to dendritic arborization of Golgi-impregnated neurons in dorsal horns of the rat spinal cord

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Neurocomputing 69 (2006) 403–423 www.elsevier.com/locate/neucom

Mathematical modeling and computational analysis of neuronal cell images: Application to dendritic arborization of Golgi-impregnated neurons in dorsal horns of the rat spinal cord D. Ristanovic´a,, B.D. Stefanovic´b, N.T. Milosˇ evic´a, M. Grgurevic´c, J.B. Stankovic´a a

Department of Biophysics, School of Medicine, University of Belgrade, Visˇegradska 26, 11000 Belgrade, Serbia and Montenegro b Department of Histology, School of Medicine, University of Belgrade, Belgrade, Serbia and Montenegro c Department of Physiology, School of Medicine, University of Belgrade, Belgrade, Serbia and Montenegro Received 12 October 2004; received in revised form 23 March 2005; accepted 11 April 2005 Available online 6 September 2005 Communicated by A. Borst

Abstract Neurons of the rat spinal cord have been stained using the Golgi impregnation method. Successfully impregnated neurons from laminae I to VI were subjected to a computational analysis for complexity of dendritic tree structure. The analysis was performed using ruler-counting and circle-counting techniques. Our analysis aimed to support quantitatively the general concept of Rexed’s laminar scheme of the dorsal horn of mammals. For that purpose, we have developed two mathematical models of neuronal arborization patterns, whose solutions yielded the inverse powerlaw and generalized power-law scaling. The latter comprises two main parameters: (i) the anfractuosity, characterizing the degree of dendritic complexity and (ii) an estimate of the total length of arbor dendrites. The anfractuosity can distinguish among the sets of drawings over all six laminae. r 2005 Elsevier B.V. All rights reserved. Keywords: Dendritic arborization; Golgi method; Mathematical modeling; Rat spinal cord; Rexed’s scheme Corresponding author. Tel: +381 11 3615767; fax: +381 11 3615775.

E-mail address: [email protected] (D. Ristanovic´). 0925-2312/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2005.04.007

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1. Introduction One of the major goals in cellular neurobiology is meaningful quantitative analysis of neuronal arborization patterns. Sholl’s [41] consecutive-circle (cumulative intersection) method specifies dendrite geometry, ramification richness, and branching patterns. This analysis, both manual [40,41] and computer-assisted [6,15,31,35], has long been widely applied and used for quantitative morphometric studies of Golgi-impregnated cells [8]. Mandelbrot’s [33] fractal theory analyses the neuronal arbors for fractality [8,9,35,43,49], and classifies neurons according to their fractal dimensions [8,16,25,26,35,50]. In order to analyze the overall shape complexity of neuronal cells, the wavelets and multi-scale curvature methods were used [9]. The characterization of neurons by means of semi-automated generation of dendrograms, describing the branched structures of neurons in terms of the lengths, average thickness, and ‘‘branching energy’’ of each of the dendrite segments, is also described [10]. This ‘‘energy’’ quantifies the complexity of the shape and can be used to characterize the spatial coverage of the arborization. The spectral method for estimating geometric structures of contours and the use of exact dilations to estimate geometric measures were recently offered [11]. For the quantification of neuronal shapes, the segmentation of neuronal objects from con-focal digitized images is also performed [49]. Various computer software for digitization and neuron analysis have been designed. Special attention was paid to the dendrite reconstruction and its visualization on screen mainly after applying the method of Golgi impregnation. Outputs are given either to a color plotter or to laser printers in graphical form as a summary of various metrical and topological parameters [24,31]. However, most of the authors use direct-measurement techniques. Usually, the quantitative descriptions of neuronal arborization, as offered by many authors, are almost entirely based on the analysis of somata sizes, their location and grouping, dendrite arborization sizes and bifurcation ratios, total dendrite areas, number of branching points, etc. [17,21,24,32,41]. Mathematical modeling, as a quantitative method, is also applied to neurobiological studies. For example, the mathematical modeling approach to determination of how the morphology of the dendrite population affects the velocity of RNA transport in neurons is successfully used [14]. The geometrical branching pattern of the dendrites is also analyzed adopting a mathematical model that incorporates random and deterministic effects [48]. The mathematical QS-model of dendrite growth as a sequence of branching events at randomly selected segments is presented. It is shown that the variation of the number of dendrite segments can be accounted for by assuming that new branches during neuronal outgrowth tend to be formed randomly at terminal segments [36,37]. Besides that, growth processes can be successfully modeled as Lindenmayer systems [23]. MicroMod is web-based software that allows models to be easily built using a comprehensive set of adjustable parameters [23]. Jelinek et al. [23] suggest some of the numerous applications for this approach to neuronal modeling. A suitable polynomial function is used as a fitting mathematical model to estimate some morphologic parameters characterizing

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dendritic arbor complexity [39]. Mathematical and computational modeling, including L-systems, has recently emerged as a powerful approach to the quantitative anatomical characterization of dendrite morphology [3]. Burke and Marks [7] investigate the relationship between dendritic structure and its morphologic patterns, and neuronal packing in spinal motoneurons. Analysis of the effects of dendrite complexity on network connectivity is oriented toward the issues of neuronal shape characterization and cell geometry at the individual cell level. This analysis is carried out by means of critical percolation probability [12]. Other models concerning neuronal arborization have also been developed [29,30]. In qualitative analysis of dendritic arborization patterns, neurons are mostly studied in relation to their localization in the nervous system [52], particularly in certain regions and laminae [40,52]. It seems that each lamina possesses a comparatively distinct dendrite organization. The laminar structure of the spinal cord is related to the morphological classification of spinal neurons and dendrite tree geometry. Recent studies mostly analyze the morphology of neurons in lamina I [13,18,52]. As for the dendritic arborization of cells in the lumbar spinal cord of the rat, little analytical evidence is available on neurons in relation to their laminar organization. In most studies, the morphology of cells in the rat dorsal horn is considered and analyzed only over lamina I [27,28,34,51]. Neurons extending over laminae II and III (the substantia gelatinosa) are also described [5]. To our knowledge, there is little published information on the applications of quantitative methods to the laminar classification of neurons and dendrite arborization patterns in the rat. In the present study, we develop two mathematical models whose solutions are used for the morphometric computational analysis of dendritic organization of neuronal arbors in dorsal horns of the rat spinal cord. Our analysis aims primarily to support quantitatively the concept of Rexed’s lamination of the dorsal horns of mammals. Through the analysis of such laminar organization, we show that the proposed method is sensitive in practice, enabling a rather precise discrimination of neurons over the laminae of the dorsal horn.

2. Methods 2.1. Materials Tissue samples of lumbar spinal cord of adult Wister rats weighting 200–250 g were used in the present study. Formaldehyde tissue fixation and Golgi staining techniques were performed as previously described [39,45]. A total of 15 spinal cords with satisfactory impregnations of dendritic trees were used. Excised spinal blocks, 3–5 mm long, were further processed. Sectioning was enabled by immersing whole blocks in liquid paraffin at 56 1C. Prepared tissue blocks were fixed on a microtome and sectioning was done at 90–120 mm thickness. The sections were placed on glass slides, covered in DPX resin (Fluka) and analyzed using the light microscope. The cord was studied mainly in sagital sections. Twenty-three representative and

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successfully impregnated neurons were selected and traced under a magnification of 500
with the aid of a camera lucida attachment (Zeiss drawing tube), at an objective magnification of 40
. The magnification scale was indicated by bars corresponding to 10 and 25 mm lengths. Incomplete impregnation can influence the quality of measurements. To overcome this, special care was taken in the choice of representative neurons—such where by our judgment these problems were minimal. In accordance with other studies, impregnated dorsal neurons have shown mainly fusiform, stellate–fusiform, and stellate-pyramidal constellations of cell bodies and primary dendrites. The present study is concerned with the morphology of laminae I–VI of the dorsal horns. The neurons were classified according to the Rexed laminar concept. For that purpose, all neurons were pooled in five groups according to their laminar position: (i) four neurons from spinal lamina I (the marginal layer), (ii) seven neurons occupying spinal laminae II and III (the substantia gelatinosa), (iii) five neurons from lamina IV, (iv) three neurons from lamina V, and (v) four neurons from lamina VI. This set of neurons was taken from our previously published article [39] and other official sources [19,20]. The multi-polar organization of lamina IV cells contrasts with the gelatinous elements above (laminae I–III), representing sensory (afferent) neurons, and with the interneuron of deeper laminae (laminae V and VI), conducting impulses from sensory to motor (efferent) neurons. The neurons over lamina IV represent a mix of the two. 2.2. Mathematical modeling In the present study, we consider all images of dendrites as their silhouettes, or as their ‘‘skeleton’’ tracings. In other words, dendrites are reduced to their skeletons, i.e., to a one-pixel-wide black image on a white background. Euclidean lines of finite lengths along the dendrite axes are also analyzed. The following quantitative features are found to be characteristic of a neuron from the dorsal horn laminae: (i) its length and (ii) its anfractuosity, i.e., a numerical quantifier of its complexity. 2.2.1. Length of a curve In Euclidean geometry, the distance between two points is axiomatically introduced as the length of a straight-line segment joining these points. On the other hand, in differential geometry, where the basic objects are mainly smooth curved lines, the length of a differentiable continuous curve joining two points is defined by the well-known definite integral. The condition to calculate this length (to perform the curve rectification) is to know the equation of the curve, on condition that this equation satisfies some additional properties. The problem of defining the length of a curve joining two points when the equation of the curve is not known (almost all skeleton images of dendrites are of that kind) is not yet completely solved in calculus. To solve the problem of measuring the length of any curved line, or, to ‘‘rectify’’ any curve, a polygonal line of equal side lengths should be inscribed on the curve. The length of this polygonal

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line is surely less than the quantity we wish to call the ‘‘length’’ of the curve. In any case, the lesser the side length of an inscribed polygon, the closer the length of the polygonal line to that of the real curve. If a scale (e.g., a ruler) of length r is used to measure the length of a curve (for that purpose a divider can be used), and if the measurement is performed in such manner that the ruler is moved along the curve N times until the curve is completely traversed, then the approximate length of the curve is given by the number of ruler steps (N) multiplied by the ruler length (r): L ¼ Nr. This procedure is in essence the same as inscribing a polygonal line of equal side lengths on a curve and measuring its length by multiplying the number of polygonal sides with the polygonal side length. It is known that this approximate length of the curve and the ruler length should be in inverse proportion; therefore, we say that such estimation of the curve length scales with the ruler (scale) length. So far, the law of length decrease of these polygonal lines with the ruler length increase has not yet been completely investigated. Following the methodology of mathematical modeling, we have at first undertaken to explore this problem experimentally. 2.2.2. Model of scale-counting We chose by chance the drawing of a neuron from our pool of drawings. For several ruler (scale) lengths r (say, between 2 and 25 mm) we found, by traversing the ruler across the imagined longitudinal axes of dendrites, the corresponding numbers of steps N(r) for all dendrites of the neuron and obtained the results presented in Fig. 1A by open circles. The graph in this figure fitting the data seems to demonstrate a hyperbolic decrease of N with r. This is one of the results of our experiments. It is known that any natural process can often be described starting by a rule that tells what happens in part of the process. If one is assured that the same rule always applies to all parts of the process, it is possible to derive the course of the entire process [2,42]. This is the first step in mathematical modeling. By successive inscribing of polygonal lines of equal segments each on a curve (or, of measuring the curve length traversing a ruler across it) and counting the numbers of segments (or, steps) N for a given segment (ruler) length r, it is obvious that we need to specify two things: (i) the rule that governs the construction process over a small change of length r of a polygonal line segment and (ii) the assurance that the same rule holds over a large number of different construction processes. If both conditions are always fulfilled, the starting rule could become a mathematical model. Let us assume that r is the length of a ruler and N the corresponding number of steps (Fig. 1A). If r decreases by a small value Dr, the number of steps N increases by a small value DN. It is obvious that DN is the change in N produced by the change Dr in r. In accordance with our numerous measurements carried out on this (and other) drawing, the first issue noted above can be specified by stating the following rule that most probably governs the whole process of construction: the ratio F of the relative change of the number of ruler steps, DN/N, to the corresponding relative

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Fig. 1. Derivation of the model of scale-counting. The number of ruler (scale) steps N is plotted against the ruler length r (A), and the ratio DN/N is plotted against the ratio Dr/r on Cartesian axes (B). The plot (A) is obtained by fitting (3) to data points (open circles), and the plot (B) is drawn using (1) (filled circles). R is the coefficient of correlation. The proportionality factor between real dendrite size and traced drawing: 25 mm ¼ 19 mm.

change of the ruler length, Dr/r, is constant and given by DN Dr ¼ F , N r

(1)

where F is a constant. To show generally that the rule, given by (1), applies over a long interval of ruler lengths (not just between 2 and 25 mm), we calculate by using the data in Fig. 1A as shown by open circles, the values in (1), i.e., Y ¼ DN/N and X ¼ Dr/r, and the corresponding points present in Fig. 1B by filled circles. These data are fitted by a straight line (1) Y ¼ FX, whose (constant) slope 1.032 is the ratio F for this neuron. If N ¼ N(r) is the equation of the fitting plot graphically shown in Fig. 1A, and if Dr-0, then DN-0 and (1) can be rewritten as an ordinary differential equation: dN dr ¼ F , N r

(2)

representing the model of scale-counting. The general solution to this equation is ln N ¼ F ln r þ ln B ¼ lnðBrF Þ, where B is an arbitrary constant of integration. Finally, N ¼ BrF .

(3)

This solution to model (2), i.e., the inverse proportionality relationship between the number of scale (ruler) steps N and the scale side length r, determines one of the basic laws in calculus concerning the problem of curve rectification, which can be thought of as the inverse power law. It is to be noted that this relationship should hold for every unformed curved line and therefore for every skeleton image of a neuron.

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2.2.3. Ratio F The constant ratio F is a transcendental number. One of the properties of such numbers is that they have an infinite number of non-periodically arranged decimals. Such numbers are p ¼ 3.14y and e ¼ 2.71y There are algorithms for calculating the approximate values of these two numbers. To solve this problem with the ratio F, we inscribe a class of regular polygons up to the polygon of 64 equal side lengths in a circle of unit radius and calculate, using formulae of elementary plane geometry, the corresponding side lengths of these regular polygons. Eq. (3) can be transported in a straight-line form: Y ¼ FX þ b,

(4)

where Y ¼ log N, X ¼ log r, and b ¼ log B. Applying the least-squares method to (4), we have shown from data obtained on the circle, by using the calculated side lengths r of the polygons and the corresponding numbers of polygonal sides N, that the ratio F ¼ 1.042y the correlation coefficient being R40:999. But, if the value of F is constant for any neuron, it cannot be used as a parameter for discrimination between neurons. Taking two drawings of neurons from our pool of drawings, we have measured the number of steps N for a series of ruler lengths r and then calculated the total lengths L of their dendrites. The data are plotted on log–log axes in Fig. 2 by open circles. On the other hand, according to (4), it is seen that (3) on log–log axes gives straightline plots (Fig. 2, full lines), but the relation for L ¼ Nr, using the same data, demonstrates no linearity (Fig. 2, dashed lines, filled circles). On such axes, these fitting graphs represent convex-upward curves. 2.2.4. Model of scaling The length of neuronal dendrites (say, that shown in Fig. 1) is measured by using a ruler (a divider) and data points are plotted in a Cartesian coordinate system

Fig. 2. Number of ruler steps N (left axes, full lines) and length of dendrites L (right axes, dashed lines) of two neurons, plotted against the ruler length r on log–log axes. Full lines are obtained by fitting (4) and dashed lines by fitting (7) to data points. F is the constant ratio for an inserted neuron. Proportionality factor between real dendrite size and traced drawings: 25 mm ¼ 45 mm (A) and 25 mm ¼ 46 mm (B).

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(Fig. 3A). A graph fitting these data is also drawn. It seems that the length (L) of all dendrites of a neuron decreases with r in an exponential manner. Indeed, since the total length of all dendrites of a neuron must be finite, the curve must cut the vertical axis at a point C, being at finite distance from the origin of the coordinate system. The distance C between the origin and point C could be estimated by graphical extrapolation of the fitting curve in Fig. 3A. Let r be the ruler length chosen to measure the length of the dendrites of a neuron, and L the corresponding length of the dendrites (Fig. 3A). If r reduces for Dr, the length L increases for DL. We define then the relative change of L as the ratio of DL to a difference between the estimate of the starting length (C) of the graph and the value of L, i.e., DL/(CL). From Fig. 3A, it is obvious that, with the increasing of r, this ratio decreases. Actually, for a constant Dr, the value DL decreases while CL, which is in denominator of this ratio, increases. The relative change of ruler length r at its change for Dr is Dr/r. On the basis of many measurements carried out on this (and other) neuron we have stated the following rule: If the approximate length (L) of a curve (e.g., of a cell’s dendrite) is given by the number of ruler steps N multiplied by the ruler length r, between the relative change of L and the relative change of r there is a linear relationship: DL Dr ¼ A , CL r

(5)

where A should be a constant. Indeed, if for all chosen r in Fig. 3A, we apply the values from the fitting curve in Fig. 3A, instead of experimentally obtained data, as new data points (given by open circles), it can be seen that the straight line in Fig. 3B fits all the data obtained from (5), with the correlation coefficient R ¼ 0:999. It should be noted that real experimental data, shown in Fig. 3A by open circles, treated in the same way, exhibit a bit more pronounced scatter than the data points

Fig. 3. Derivation of the model of scaling. The length of dendrites L of a neuron is plotted against the ruler length r (A), and the ratio DL/(CL) is plotted against the ratio Dr/r (B). The first plot (A) is obtained by fitting (7) to data points (open circles), and the second plot (B) is drawn using (5) (open circles). Proportionality factor between real dendrite size and both traced drawings: 25 mm ¼ 19 mm.

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in Fig. 3B because such measuring method is very sensitive to accompanying experimental errors during manual measurement (e.g., when using a divider). If we solve the corresponding differential equation (the model of scaling) dL dr ¼ A , CL r we obtain

(6)

lnðC  LÞ ¼ ðA ln r þ ln BÞ ¼ lnðBrA Þ, where B is an arbitrary constant of integration. Finally, the general solution of (6) is L ¼ C  BrA ,

(7)

where L is the approximate length of an image (e.g., all dendrites of a neuron), r is the ruler length, C is the length estimate of the image (since from r ¼ 0 and A6¼0 it follows that L ¼ C), B is a constant, and A is the anfractuosity, a quantifier measuring the degree of image deviations from a straight line. Therefore, the solution for the model, given by (7), can be thought of as the generalized power-law scaling. The main usefulness of this law is the possibility to measure at the same time two independent but very important features of a neuronal dendritic tree: (i) the anfractuosity (A) of its dendrites and (ii) the total length (C) of neuronal dendrites. 2.3. Image processing methods 2.3.1. Image preparation In order to enable computer examination of the material, the drawings of the neurons were converted into digitized images by using a scanner (Mustek, OpticPro 9630P) with a resolution of 600 dpi (Fig. 4A). To avoid the problems associated with large measuring elements [22], we have ensured that the images printed on paper (with the same resolution) by a printer (Hewlett-Packard 5L), were no larger than A5 size (i.e., 15
21 cm2). In addition, a proportionality factor derived for scaling between a real dendritic size and sketched drawings was calculated for each image, by means of which the length (in millimeters) of a printed dendrite was scaled with its real length (in micrometers). Such images of neurons can be thought of as the original (real) images [22]. The next step in image preparation was importing scanned images into a special software package for image processing—ImageJ (NIH Image). Using the corresponding tools, neuronal axons (if present) and soma were removed digitally from the drawing and each dendrite was filled with pixels (Fig. 4B). This drawing represents a silhouette of dendrites. To simplify the manipulation, we have performed arbitrary numeration of all the dendrites of the neuron and their segments. We did not catalogue the dendrites, since we search for the complete length of all the dendrites of a neuron. This length represents the sum of the lengths of all dendrites. It is to be noted that it is important to catalogue the dendrites and their segments when using Sholl’s [41] method. After that, the software reduced images of dendrites to their skeletons (Fig. 4C) providing binary images, thus

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Fig. 4. Preparation of camera lucida drawings. A drawing is converted into digital image (A) and imported into the ImageJ program. Axons and soma are digitally removed and each dendrite is filled with pixels (B). The program reduces the image to a skeleton (C). For dendrite branches (if exist), pixels connecting the branches to the dendrite stems are removed. The segments 3, 4, and 7 are broken off from their main stems 2, 2, and 6, respectively, by removing pixels denoted by arrows (D). If two dendrites intersect, two pixels from one of them, joining the other, are removed (E, dendrite 5). The removed pixels are shown by two arrows (E).

reducing object thickness to one pixel. In such a computer-generated image, each dendrite was presented by a curve whose thickness was one pixel. For a dendrite branch (if present), a pixel connecting the branch to the dendrite stem was removed. In Fig. 4D, the segments 3, 4, and 7 were broken off from their main stems 2, 2, and 6, respectively, by removing the pixels denoted by arrows. If two dendrites crossed, two pixels of one of the dendrites at the crossing area were removed. In Fig. 4E, dendrite 5 crosses dendrite 2; the positions of removed pixels from dendrite 5 are also shown by arrows. Using the same software, the total length of all processed dendrites and their parts or branches (LIJ) was measured. After that, the image of the neuron was saved and printed again on a paper no larger than A5 size. Such a digital pattern of the neuron, printed on paper, can be thought of as the skeleton image. Each dendrite or its branch was cropped, saved in a separate file, and all parts of the neighboring dendrites were digitally removed. In the case of two intersecting

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dendrites, such as those shown in Fig. 4E, the dendrite with no break (Fig. 4E, dendrite 2) was taken first. After that, both parts of the other crossing dendrite (Fig. 4E, dendrite 5) were imported into a new file and a pixel-to-pixel line was used to connect its breaking ends. Using the same program, a set of ordered pairs of coordinates (X, Y) of each pixel of a dendrite image were obtained, arranged and saved in a separate file. The content of a folder consisting of all files of dendrites relating to a given neuron can be thought of as the neuronal analytic image. 2.3.2. Methods of image analysis Original and skeleton images, plotted on paper, have been analyzed using the ruler-counting method [4,39,44]. This method relies upon the technique of curve rectification, well known from calculus. For that purpose, in each measurement we employed a divider, keeping a fixed distance between the two divider ends. The measurement of the number of divider steps (N) and the dendrite length (L) were carried out around the imagined longitudinal axis of a dendrite in an original image, or, around a curve in a skeleton image. For a chosen ruler length r we have counted the number of steps (N) of the divider first, and then obtaining the scaled length (L) of the dendrite as the number (N) of divider steps multiplied by the distance (r) between the divider ends. The smallest span to enable reliable results using the divider was 2 mm. Measurements of lengths of all dendrites drawn on a paper were carried out using the ruler spans between 2 mm and up to 25 mm. The approximate total length of all dendrites was the sum of the lengths of each dendrite and all the dendrite branches. An analytic image of a neuron was analyzed using the circle-counting method. This is one of the techniques for measuring the length of dendrites used in this study. The analytic image, i.e., the ordered set of coordinates (X, Y) of all the pixels of dendrites shown as a skeleton, was input into our proprietary software for drawing circles, written in Visual Basic (rMicrosoft), along with 25 different values of the radii of circles to be drawn. Using the coordinates of the dendrite pixels, the software has reproduced a dendritic tree image on the screen. After that, the software for circles drawing was applied. Firstly, the software has registered the coordinates of a pixel at the very beginning of a chosen dendrite. At this terminal pixel, used as the center of a circle, the software has drawn on a computer screen a one-pixel-wide circle of the first chosen radius. After that, the software has performed a search for the pixel which corresponded to the point of intersection of the dendrite and the circle. The software has then graphed the next circle of the same radius with the center at this intersecting point, and so forth. If the circle was generally graphed with the center at any point of the dendrite, this circle intersected, as a rule, the drawing on two points. The program ignores already used points as centers of new circles. This procedure has continued until the drawing of the dendrite was traversed and completely covered with circles of the chosen radius. The program has inscribed the number of these circles. The rest of the dendrite—enclosed by the last circle—was transformed into an approximate length (R) measuring this length using the number of enclosed pixels and their sizes. The whole length of the dendrite, scaled to this radius, was L ¼ Nr þ R.

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After processing a chosen dendrite with 25 values of the circle radii, the software has continued with the next dendrite (or, its branch or segment) and performed the same procedure until the end. The final result was the total number of steps (N) being equal to the total number of traced circles, and the total dendrite length (L). After that, the software has fitted (4) to the results obtained and found the estimate of the ratio F. In order to fit three-parameter function (7) to experimental data, the function was first transformed into a straight-line form: Y ¼ AX þ b,

(8)

where Y ¼ log(C—L), X ¼ log(r), and b ¼ log(B). This algorithm can readily be implemented on a computer. To prepare the software for calculation of unknown values of the parameters A, B, and C by an iterative procedure, we have chosen an interval of possible values for C and used the first estimate of C at the beginning of this interval. Then we have applied the leastsquares method to (8) to calculate the values of the other two parameters. These estimates of A and B enabled, further, to calculate—from the same equation—the new estimate of C. This new estimate could prove to be better or worse than its starting value. For the next iteration, a better estimate of C should be accepted. The criterion of this decision was the value of the coefficient of variation (CV) of (8) from the experimental data, which should be as small as possible.

3. Results 3.1. Comparison of methods In the present study, our first step was to test the obtained data with solution (7) of mathematical model (6) for differences among the three methods: (i) the rulercounting method applied to original images, (ii) the ruler-counting method applied to skeleton images, and (iii) the circle-counting method applied as well to skeleton images. The aim was to examine whether all methods give rise to identical values for the tested parameters. In this respect, we conducted tests in all three techniques on a neuronal image chosen at random from our pool of drawings. The results are presented in Fig. 5. As a control, for the same neuron we measured the total length (LIJ) of the dendritic tree using the ImageJ software. In the first case, using the divider, the measurement of dendrite lengths was done around imagined longitudinal axes of the dendrites. After fitting (7) to obtained data, it is seen from inset in Fig. 5 that the estimate of dendrite length C( ¼ 437.9 mm) is larger for about 15% as compared to the control (LIJ) (Fig. 5, squares). The corresponding anfractuosity (A) is rather small. In the second case, the divider was also used to measure the dendrite lengths, but this time around skeleton tracings of dendrites, i.e., around the curves being embodiments of the previously imagined longitudinal axes of the dendrites. The estimate C( ¼ 410.9 mm) is a bit closer to the control value and the anfractuosity A is larger (Fig. 5, filled circles). The difference between the length estimate C and the control length LIJ is now less than

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Fig. 5. Comparison of the three methods used in neuronal images processing. Total length L of the dendrites of a neuron is plotted against the ruler length r. All the plots are obtained using (7). The result of measuring the real drawing using the divider is shown by a dotted plot (squares), that of skeleton image by a dashed line (filled circles), and that of the circle-counting method, by a full line (open circles). Arrows show the position of intersection of the plots with the vertical axis. The subscript IJ denotes ImageJ. The values of anfractuosity A and dendrite length C estimated from (7) are presented in the inset.

8%. The third method consists of applying our software to the same procedure. The value of the length estimate C( ¼ 388.3 mm) is larger with respect to the control unit only some 2% (Fig. 5, open circles). From these results the conclusion can be drawn that the circle-counting method is the most reliable and confidence should be placed in the estimates of the parameters A, B, and C of (7) obtained using the third method. It should be noted that there is an inverse proportionality between the estimate of C and calculating values of A. We have applied the last action to seven neurons from the substantia gelatinosa (Fig. 6B) and the results are summarized in Table 1. From the table it can be seen that the comparison of the circle-counting method (Method 3) versus ruler-counting (manual) methods (Methods 1 and 2) shows that the analytical images (Method 3) give rise to smaller values of the length estimate C than the other two methods, and are closer to the control value (LIJ ¼ 398 mm). At the same time, the mean value of the anfractuosity A is significantly different from the others. In this respect, the values of the anfractuosity A obtained using the first and second method are unreliable; therefore, further research should be directed toward the use of the third (circle-counting) method. 3.2. Morphometric analysis of rat spinal cord Using skeleton images of neurons from the five laminar regions, we have measured: (i) the anfractuosity (A) of neuronal dendritic arbor structures, (ii) the total length (C) of arbor dendrites, and (iii) the value of the ratio F. By means of the ImageJ software we have also measured the dendrite lengths (LIJ), which were used as controls for the appraisal of values of the length C.

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Table 1 The ratio (F), anfractuosity (A), estimate of length of dendrites (C), length of dendrites measured by ImageJ (LIJ), and percentage of differences between C and LIJ (d) for neurons from laminae II and III Method

F

A

C (mm)

LIJ (mm)

d (%)

1 2 3

1.05470.020 1.05370.024 1.05270.014

0.21070.012*(2) 0.2487 ¼ 0.123*(3) 0.36670.053**(1)

451 433 403

398

11.8 8.1 1.2

Note: Each value in columns F and A represents the mean7SD and in rest of the columns, the mean. Significance marks *po0.05 and **po0.01 put on the values that are significantly different from those obtained by the method marked in brackets.

Fig. 6. Silhouettes of drawings of Golgi-impregnated lamina I (A) and laminae II–III (B) neurons from dorsal horns of the rat spinal cord. Vertical axes (left and right) show the anfractuosity A and the vertical axis in the middle shows the ratio F. Scale bars: 25 mm for neurons 1 and 3, 10 mm for neurons 2 and 4 (A), and 10 mm for all neurons (B).

The silhouettes of the original images of all neurons, taken from the laminar regions of the spinal cord, are shown in Figs. 6 and 7, along with the values of the anfractuosity (A) calculated according to (7), and those of the ratio F obtained using (4). The tracings in Figs. 6 and 7 have ranged from complex to simple (top to bottom). The means of these two groups of values for sets of neurons are shown in Table 2. We have applied a one-way analysis of variance to five neuronal samples shown in Figs. 6 and 7 to investigate whether the differences between their means are statistically significant. It is shown that Fisher’s statistic F(4,18) ¼ 5.08, po0:01, calculated for the anfractuosity, is highly significant. From Figs. 6 and 7 it can be seen that the sample sizes are unequal. There are few suitable post hoc tests for

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Fig. 7. Silhouettes of drawings of Golgi-impregnated laminae IV (A), laminae V (B), and laminae VI (C) neurons from dorsal horns of the rat spinal cord. Vertical axes (left and right) show the anfractuosity A and two vertical axes (middle) show the ratio F. Scale bars: 25 mm for neuron 1, 10 mm for neurons 2–5 (A), and 25 mm for all neurons (B, C). Table 2 Anfractuosity (A) estimated by fitting (7) to data, and ratio (F) calculated according to (4) for all laminae of the rat spinal cord Case

Lamina

F

A

1 2 3 4 5

I II–III IV V VI

1.08370.020 1.05270.014 1.04470.007 1.03470.015 1.05570.004

0.24070.089**(2) 0.36670.049 0.29370.071*(4) 0.18670.009**(2) 0.22370.067**(2)

Mean7SD

I–VI

1.05070.011

0.26270.070

Note: Each value represents the mean7SD. Significance marks *po0.05 and **po0.01 put on the values that are significantly different from the cases marked in brackets.

unequal sample sizes. Whenever unequal numbers of variations occur in the samples and the F-value indicates significance, there is no simple way to decide which pairs of sample means differ significantly [1]. Statisticians usually recommend that one must test all possible pairs individually, by means of the t-test, using each time the appropriate number of variations for the samples being compared [1]. Such a post hoc test shows that the means of four pairs of the anfractuosity A in Table 2 are significantly different. These pairs are shown by means of standard significance marks. The same procedure shows that the pairs of means of the ratio F are not significantly different (Table 2). Indeed, in that case F(4,18) ¼ 2.51, p40:05, being less than tabular value F0.05 ¼ 2.93. These values remain relatively constant

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irrespective of the laminar position of a neuron in the dorsal horn. It seems that the anfractuosity can be used as a reliable parameter, yielding a satisfactory discrimination among the five sets of drawings, and as a correct quantitative means for classifying neurons according to their laminar position. In contrast, such an inference cannot be drawn as to the ratio F. This, further, asserts our finding that this quantity is much closer to be a constant for all neurons, than to be a discriminating parameter. In Table 3, the values of the estimated length C and measured lengths LIJ for the neurons are presented and compared. The measurements show that the estimates of the total dendrite length (C) and lengths measured by the ImageJ software (LIJ) are about the same. More precisely, the difference is less than 2%. The number of neurons used in this study could be considered as comparatively small. However, we find that there is ground for believing that all drawings of the neurons used in the present study are representative of each particular lamina and that these statistical inferences could be adopted with confidence. In fact, in spinal cords, all types of these cells and their arborization patterns have already been identified, described and classified by other workers. Although it is known that the staining method is selective [41]—since most cells, as noted by many others, have been successfully stained, and most of these neurons were also seen in our experiments, there is no reason to suppose that bias can be introduced in our results. Such a criterion for a cell to be representative has been supported by Sholl [41]. For example, all neurons of substantia gelatinosa (laminae II and III) are small and multipolar, with highly ramified dendrites. Stellate and fusiform neurons are the most commonly seen type. The dendrite trees of these cells are extremely arborized. Three main types of neurons, as to the shape of their dendrite arborization, at horizontal and sagital sections, are representative for substantia gelatinosa of the rat spinal cord: the stalk cells (Fig. 6B, drawing 1), the most commonly found islet cells (Fig. 6B, drawing 6) being characteristic for their ‘‘recurrent’’ dendrites, and arboreal cells (Fig. 6B, drawing 4). At the transversal section of the dorsal horn the cells seen are described as limitrophe cells (Fig. 6B, drawing 7), bipolar cells with dorsal dendrite branching (Fig. 6B, drawing 3), as well as multi-polar (Fig. 6B, drawing 2) and bipolar (fusiform) neurons (Fig. 6B, drawing 5).

Table 3 Estimates of lengths of the images (C), the lengths of the dendrites measured by means of the software ImageJ (LIJ), and the percentage of differences between C and LIJ (d) Case

Lamina

C (mm)

LIJ (mm)

d (%)

1 2 3 4 5

I II–III IV V VI

462 403 581 962 408

454 398 576 956 405

1.73 1.24 0.86 0.62 0.74

Note: Each value represents the mean of values obtained.

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4. Discussion We have pointed out that the anfractuosity (A) of a neuronal border was proven to be a useful descriptor of a cell’s complexity. This raises the question of what aspect of the cell’s ‘‘complexity’’ is the anfractuosity measuring: what does ‘‘complexity’’ actually mean, apart from being a notion opposite to ‘‘simple’’? There have been attempts to define this notion exactly [44,47]. In the present study, the anfractuosity (A) has proven to be a useful tool in quantifying the degree of global or average dendrite aberrance from straight lines. Although the mechanism and basic nature of neuronal Golgi impregnation have been established only recently [45], it is generally accepted that Golgi impregnation satisfactory reveals naturally occurring neuronal structures and dendrites branching. One of the predictions of this finding is that the Golgi impregnation method permits an acceptance of the anfractuosity (A) in neuronal studies. Besides, outcomes from our preceding studies [39] enable an adoption of the anfractuosity as one of the basic parameters in neurocomputational investigations. To test the properties of dendrite drawings with mathematical model (7) for differences among five groups of neurons, we have used, in essence, one of the fractal length-related techniques known as the trace (ruler-counting) method [4,8,16,22,39,44]. This authentic method is based on the use of a divider for dendrite lengths measuring. It has been noted that the major drawback of this method was inaccuracy in processing the neuronal images; some other length-related techniques seem to be superior [16,43,44]. To measure the dendrite lengths by means of a divider, we have used a rather broad interval (from 2 to 25 mm) for the divider spans. Although only a part of this data is presented in Fig. 5, all of it is used for fitting model (7). If we mathematically extrapolate this graph to the section with the ordinate axis, unreal (extremely large) values for length estimate C are obtained (see the inset in Fig. 5). This is another shortcoming of the ruler-counting method. Besides, despite the use of such a long interval of divider spans, the starting dendrite length L (of 345 mm, see the graph in Fig. 5) is abbreviated only for 10%. In this respect, we can consider that this length is approximately constant. Under this assumption, it follows from model (7) that the length estimate C is inversely proportional to the anfractuosity A. Indeed, if the values of A are equal to unity, the graph of the model is a decreasing straight line. For A41, the graph is a convexupward parabola with the smaller value of C. But, if these values of A are less than 1 (being just our case), the graphs are concave-upward curves, so that the values of C must be larger than previously. Therefore, the small (unreal) value for A obtained when the ruler-counting method is used seems to be the consequence of the (also unreal) value for C estimated using this manual technique for length measuring. The use of the software for circles drawing eliminates all these shortcomings. The laminar organization of the spinal grey matter, described in the cat by Rexed [38], has been rapidly adopted by most neuroanatomists because it avoids the problems of varied nomenclatures of classical anatomic descriptors. But, one could question whether data obtained in the cat can be generalized to other animals, because it has been noticed that Rexed’s laminar scheme can be found in all mammals but not

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in other vertebrates [40]. Our quantitative morphologic analysis of the rat spinal cord favors the Rexed [38] concept on laminar cytoarchitectonic scheme, since each lamina I–VI of the rat testifies to a quite specific dendritic organization. This is in full agreement with the qualitative observations made in the cat [38,46] and human [40]. We cannot directly compare our results for complexity of dendrite arbors with those of others, but all main types of substantia gelatinosa neurons (‘‘islet’’ and ‘‘type III stalk’’ cells) have short longitudinal dendrite arbors [5]. This result is in accordance with our finding that the total lengths of arbor dendrites extending over the substantia gelatinosa are the smallest as compared to these of neurons from other laminae (Table 3). Although the number of neurons used in this study could be considered as comparatively small, with our statistical analysis of groups of cells of different morphologic types, the significance of differences in the anfractuosity A for different types of cells is exactly determined. The methods of statistical analysis used here have demonstrated that the six laminae we have analyzed differ in their anfractuosity. These values of the A support the concept that these cell groups represent populations of different neurons that can be clearly discriminated from one another. Our conclusions concerning the analysis of dendritic complexity of neurons over these six laminae are given within precisely determined statistical limits, with levels of significance of p o0:05 and p o0:01. This conclusion emphasizes our impression that further effort in neuroscience should be directed toward the continuation of these investigations.

References [1] H.L. Alder, E.B. Roesler, Introduction to Probability and Statistics, W.H. Freeman and Co., San Francisco, 1972. [2] Y.G. Antomonov, The Modelling of Biological Systems (Handbook), Naukovaja ‘‘Dunka’’, Kiev, 1977 (in Russian). [3] G.A. Ascoli, Neuroanatomical algorithms for dendritic modelling, Network 13 (2002) 247–260. [4] J.B. Bassingthwaighte, L.S. Liebovitch, B.J. West, Fractal Physiology, Oxford University Press, New York and Oxford, 1994. [5] J.A. Beal, K.N. Nandi, D.S. Knight, Neurons with asymmetrical dendritic arbors in the substantia gelatinosa of the rat spinal cord, Exp. Brain Res. 83 (1990) 225–227. [6] B.S. Bregman, W.L. Cruce, Normal dendritic morphology of frog spinal motoneurons: a Golgi study, J. Comp. Neurol. 193 (1980) 1035–1045. [7] R.E. Burke, W.B. Marks, Some approaches to quantitative dendritic morphology, in: G.A. Ascoli (Ed.), Computational Neuroanatomy—Principles and Methods, Humana Press, Totowa, 2002, pp. 27–48. [8] F. Caserta, W.D. Eldred, E. Ferna´ndez, R.E. Hausman, L.R. Stanford, S.V. Bulderev, et al., Determination of fractal dimension of physiologically characterized neurons in two and three dimensions, J. Neurosci. Meth. 56 (1995) 133–144. [9] R.M. Cesar Jr., LdaF. Costa, Neural cell classification by wavelets and multiscale curvature, Biol. Cybern. 79 (1998) 347–360. [10] R.M. Cesar, LdaF. Costa, Computer-vision-based extraction of neural dendrograms, J. Neurosci. Meth. 93 (1999) 121–131. [11] LdaF. Costa, A.G. Campos, E.T.M. Manoel, An integrated approach to shape analysis: results and perspectives, in: Proceedings of the International Conference of Quality Control by Artificial Vision, Le Creusot, 2001, pp. 23–34.

ARTICLE IN PRESS D. Ristanovic´ et al. / Neurocomputing 69 (2006) 403–423

421

[12] LdaF. Costa, E.T. Manoel, A percolation approach to neural morphometry and connectivity, Neuroinformatics 1 (1) (2003) 65–80. [13] A.D. Craig, Distribution of brainstem projections from spinal lamina I neurons in the cat and the monkey, J. Comp. Neurol. 361 (1995) 225–248. [14] L. Davis, M. Burger, G.A. Banker, Dendritic transport: quantitative analysis of the time course of somatodendritic transport of recently synthesized RNA, J. Neurosci. 10 (1990) 3056–3068. [15] H. Duan, S.L. Wearne, J.H. Morrison, P.R Hof, Quantitative analysis of the dendritic morphology of corticocortical projection neurons in the macaque monkey association cortex, Neuroscience 114 (2002) 349–359. [16] E. Ferna´ndez, H.F. Jelinek, Use of fractal theory in neuroscience: methods, advantages, and potential problems, Methods 24 (2001) 309–321. [17] Y. Fukunishi, Y. Nagase, A. Yoshida, M. Moritani, S. Honma, Y. Hirose, et al., Quantitative analysis of the dendritic architectures of cat hypoglossal motoneurons stained intracellularly with horseradish peroxidase, J. Comp. Neurol. 405 (1999) 345–358. [18] V. Galhardo, D. Lima, Structural characterization of marginal (Lamina I) spinal cord neurons in the cat: a Golgi study, J. Comp. Neurol. 414 (1999) 315–333. [19] M. Grgurevic-Nedeljkov, Golgi study of dorsal horn neurons in the rat spinal cord, M.S. Thesis, School of Medicine, University of Belgrade, 1988 (in Serbian). [20] M. Grgurevic-Nedeljkov, Golgi study of neurons over II and III laminae of the spinal cord, Ph.D. Thesis, School of Medicine, University of Belgrade, 1997 (in Serbian). [21] O.K. Hassani, C. Francois, J. Yelnik, J. Feger, Evidence for a dopaminergic innervation of the subthalamic nucleus in the rat, Brain Res. 749 (1997) 88–94. [22] H.F. Jelinek, E. Ferna´ndez, Neurons and fractals: how reliable and useful are calculations of fractal dimensions?, J. Neurosci. Meth. 81 (1998) 9–18. [23] H. Jelinek, A. Karperien, D. Cornforth, R.M. Cesar Jr., JdeJ.G. Leandro, MicroMod—an L-system approach to neuron modelling, in: The Sixth Australasia–Japan Workshop ANU, Canberra, 2002, pp. 1–8. [24] H.F. Jelinek, D.J. Maddalena, I. Spence, Application of artificial neural networks to the cat retinal ganglion cell categorization, in: Proceedings of the Fifth Australian Conference on Neural Networks, Sidney, 1994, pp. 177–180. [25] C.L. Jones, H.F. Jelinek, Wavelet packet fractal analysis of neuronal morphology, Methods 24 (2001) 347–358. [26] H. Kolb, E. Ferna´ndez, J. Schouten, P. Anelt, K.A. Lindberg, S.K. Fisher, Are there three types of horizontal cells in the human retina?, J. Comp. Neurol. 343 (1994) 370–386. [27] D. Lima, A. Coimbra, A Golgi study of the neuronal population of the marginal zone (lamina I) of the rat spinal cord, J. Comp. Neurol. 244 (1986) 53–71. [28] D. Lima, A. Coimbra, The spinothalamic system of the rat: structural types of retrogradely labelled neurons in the marginal zone (lamina I), Neuroscience 27 (1988) 215–230. [29] K.A. Lindsay, J.M. Ogden, J.R. Rosenberg, Equivalence transformations for dendritic Y-junctions: a new definition of dendritic sub-unit, Math. Biosci. 170 (2001) 133–154. [30] K.A. Lindsay, J.R. Rosenberg, G. Tucker, From Maxwell’s equations to the cable equation and beyond, Prog. Biophys. Mol. Biol. 85 (2004) 71–146. [31] M. Lowndes, D. Stanford, M.G. Stewart, A system for the reconstruction and analysis of dendritic fields, J. Neurosci. Meth. 31 (1990) 235–245. [32] M. Madarasz, T. Tombol, F. Hajdu, G. Somogyi, Quantitative histological study on the thalamic ventro-basal complex of the cat. Numerical aspects of the transfer from medial lemniscal fibers to cortical relay, Anat. Embryol. (Berl.) 166 (1983) 291–308. [33] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982. [34] S.S. Mokha, G.E. Goldsmith, R.F. Hellon, R. Puri, Hypothalamic control of nocireceptive and other neurons in the marginal layer of the dorsal horn of the medulla (trigenimal nucleus caudalis) in the rat, Exp. Brain Res. 65 (1987) 427–436. [35] E.A. Neale, L.M. Bowers, T.G. Smith Jr., Early dendrite development in spinal cord cell cultures: a quantitative study, J. Neurosci. Res. 34 (1993) 54–66.

ARTICLE IN PRESS 422

D. Ristanovic´ et al. / Neurocomputing 69 (2006) 403–423

[36] J. van Pelt, Effect of pruning on dendritic tree topology, J. Theor. Biol. 186 (1997) 17–32. [37] J. van Pelt, A.E. Dityatev, H.B.M. Uyling, Natural variability in the number of dendritic segments: model-based inferences about branching during neurite outgrowth, J. Comp. Neurol. 387 (1997) 325–340. [38] B. Rexed, The cytoarchitectontic organization of the spinal cord in the cat, J. Comp. Neurol. 96 (1952) 414–495. [39] D. Ristanovic´, V. Nedeljkov, B.D. Stefanovic´, N.T. Milosˇ evic´, M. Grgurevic´, V. Sˇtulic´, Fractal and nonfractal analysis of cell images: comparison and application to neuronal dendritic arborization, Biol. Cybern. 87 (2002) 278–288. [40] J. Schoenen, The dendritic organization of the human spinal cord: the dorsal horn, Neuroscience 7 (1982) 2057–2087. [41] D.A. Sholl, Dendritic organization of the neurons of the visual and motor cortices of the cat, J. Anat. 87 (1953) 387–406. [42] W. Simon, Mathematical Techniques for Physiology and Medicine, Academic Press, New York and London, 1972. [43] T.G. Smith Jr., G.D. Lange, W.B. Marks, Fractal methods and results in cellular morphology— dimensions, lacunarity and multifractals, J. Neurosci. Meth. 69 (1996) 123–136. [44] T.G. Smith Jr., W.B. Marks, D.G. Lange, W.H. Sheriff Jr., E.A. Neale, A fractal analysis of cell images, J. Neurosci. Meth. 27 (1989) 173–180. [45] B.D. Stefanovic´, D. Ristanovic´, D. Trpinac, V. ŒorXevic´-Cˇamba, V. Lacˇkovic´, V. Bumbasˇ irevic´, et al., The acidophilic nature of neuronal Golgi impregnation, Acta Histochem. 100 (1998) 217–227. [46] J. Szentagothai, M. Rethely, Cyto- and neuropil architecture of the spinal cord, in: J.E. Desmedt (Ed.), New Developments in Electromyography and Neurophysiology, Vol. 1, Karger, Basel, 1973, pp. 20–37. [47] T. Takeda, A. Ishikawa, K. Ohtomo, Y. Kobayashi, T. Matsuoka, Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development, Neurosci. Res. 13 (1992) 19–31. [48] E. Uemura, A. Carriquiry, W. Kliemann, J. Goodwin, Mathematical modeling of dendritic growth in vitro, Brain Res. 671 (1995) 187–194. [49] H.B.M. Uylings, J. van Pelt, Measures for quantifying dendritic arborizations, Network: Comput. Neural Syst. 13 (2002) 397–414. [50] R.J.T. Wingate, T. Fitzgibbon, I.D. Thompson, Lucifer yellow retrograde tracers and fractal analysis characterise adult ferret retinal ganglion cells, J. Comp. Neurol. 323 (1992) 449–474. [51] C.J. Woolf, M. Fitzgerald, The properties of neurons recorded in the superficial dorsal horn of the rat spinal cord, J. Comp. Neurol. 221 (1983) 313–328. [52] E.T. Zhang, A.D. Craig, Morphology and distribution of spinothalamic lamina I neurons in the monkey, J. Neurosci. 17 (1997) 3274–3284.

Dusˇ an Ristanovic´ (www.ristanovic.com/dusan) received the diploma in physics from the Faculty of Sciences, University of Belgrade (Serbia), in 1957, his M.Sc. degree in theoretical physics in 1967, and his Ph.D. degree in biology, in 1972, both from the same University. Since that year, he has been Assistant Professor of physics, Associated Professor of biophysics, and Professor of medical biophysics with the Department of Biophysics at the School of Medicine in Belgrade. He has been the Head and Director of the Department of Biophysics, staff scientist at the Institute for Biological Research, and staff scientist at the Institute for Mental Health. He has published more than 320 articles and books which are cited about 130 times in international journals. He was editor-in-chief and associate editor in a few journals. He was the leader in five supported projects and member of many scientific organizations. His research interest covers mathematical modeling in biology, neuroscience, compartmental analysis, computational analysis and programming, fractal analysis, and imaging in medicine.

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Bratislav D. Stefanovic was born in Belgrade (Serbia) in 1953, accomplished his M.D. studies in 1978, obtained his M.Sc. in molecular neurobiology in 1983, and earned his Ph.D. in medical neuroscience and neurohistology in 1990. He is employed at the Medical School in Belgrade as research associate (1980–1990), assistant professor (1991–1995), associated professor (1996–2000), and professor since 2001. His principal scientific interests were mRNAs and endoribonucleases in combination with the brain postnatal development, sensory nerve corpuscles, and human nerves texture, as well as the basic nature of Golgi impregnation of central (CNS) neurons. In time, he became interested in mathematical aspects of spatial characteristics of complex neuronal dendritic arborization. So far, participated and/or leaded in five grants. He is a member of several Serbian scientific societies and European Microscopical Society. His general future interest concerns a possible relationship between Golgi impregnation, neuronetwork reorganization, and memory patterns consolidation.

Nebojsˇ a T. Milosˇ evic´ studied physics at the University of Belgrade, Serbia. He received his B.Sc. degree in general physics in 1996. Since 1997, he has been a research associate with the Department of Biophysics at the School of Medicine. He obtained his M.Sc. in experimental nuclear physics form the University of Belgrade, in 2000. He is currently pursuing his Ph.D. degree in medical physics at the University of Novi Sad (Serbia). He is a member of the Image Analysis Group with the Department of Biophysics. His research interests also include mathematical modeling, medical imaging, dosimetry, radiation protection, and quality assurance in radiotherapy.

Jovan B. Stankovic´ graduated 1983 with a B.Sc. in physical chemistry. He awarded his M.Sc. in 1988 and his Ph.D. in physical chemistry, in 1995, at Belgrade University. Presently, he is employed as assistant professor of biophysics with the Department of Biophysics at School of Medicine, University of Belgrade. Beside radiation biophysics and radiobiology modeling, his research interest is oriented toward quantification of hystopathological findings by application of mathematical modeling.