Mathematical modelling of transformations of asymmetrically distributed biological data: an application to a quantitative classification of spiny neurons of the human putamen.

Many measurements in biology follow distributions that can be approximated well by the normal distribution. The normal distribution plays an extremely important role in probability theory. However, some of the experimental data in biology are distributed asymmetrically. In order to transform such an asymmetrical distribution into a normal distribution, for which the standard statistical tables can be used for probability analysis of the available data, one must choose suitable transformation functions. We have met this problem when we qualitatively classified the spiny neurons in the adult human putamen. But, if one tries to test a qualitative classification of neurons quantitatively, a considerable class overlap between cells occurs as well as asymmetry often appears in the distributions of the data. We have already offered a method to overcome the overlapping problem when the data distributions are normal. In order to resolve the asymmetry problem in data distribution, we transformed our asymmetrically distributed data into an approximately normal distribution using a family of simple power functions and on a basis of appropriate probability analysis we propose a more acceptable classification scheme for the spiny neurons. The significance of our results in terms of current classifications of neurons in the adult human putamen is discussed.

On the classification of normally distributed neurons: an application to human dentate nucleus.

One of the major goals in cellular neurobiology is the meaningful cell classification. However, in cell classification there are many unresolved issues that need to be addressed. Neuronal classification usually starts with grouping cells into classes according to their main morphological features. If one tries to test quantitatively such a qualitative classification, a considerable overlap in cell types often appears. There is little published information on it. In order to remove the above-mentioned shortcoming, we undertook the present study with the aim to offer a novel method for solving the class overlapping problem. To illustrate our method, we analyzed a sample of 124 neurons from adult human dentate nucleus. Among them we qualitatively selected 55 neurons with small dendritic fields (the small neurons), and 69 asymmetrical neurons with large dendritic fields (the large neurons). We showed that these two samples are normally and independently distributed. By measuring the neuronal soma areas of both samples, we observed that the corresponding normal curves cut each other. We proved that the abscissa of the point of intersection of the curves could represent the boundary between the two adjacent overlapping neuronal classes, since the error done by such division is minimal. Statistical evaluation of the division was also performed.

Quantitative analysis of dendritic morphology of the α and δ retinal ganglion cells in the rat: a cell classification study.

Type I retinal ganglion cells in the rat have been classified into several groups based on the cell body size and dendritic morphology. Considerable overlap and heterogeneity within groups have been reported, which is especially obvious for the morphology of the dendritic tree. For that purpose, we analysed quantitatively the dendritic morphology of the alpha and delta rat retinal ganglion cells, using parameters which provide information on the dendritic field size, shape of the dendritic tree and dendritic branching complexity. We show that the alpha and delta cells have significantly different dendritic field sizes. Taking into account the level of stratification of the dendritic tree, we found a difference in the properties of the dendritic morphology between alpha inner and alpha outer cells, while the opposite result was obtained for the delta inner and delta outer delta cells. In this study we also call attention to the relationship between morphological parameters and retinal eccentricity. The significance of our quantitative results in terms of present alpha and delta rat retinal ganglion cell classification is discussed.

Mathematical modelling of neuronal dendritic branching patterns in two dimensions: application to retinal ganglion cells in the cat and rat.

Sholl's analysis has been used for about 50 years to study neuron branching characteristics based on a linear, semi-log or log-log method. Using the linear two- dimensional Sholl's method, we call attention to a relationship between the number of intersections of neuronal dendrites with a circle and the numbers of branching points and terminal tips encompassed by the circle, with respect to the circle radius. For that purpose, we present a mathematical model, which incorporates a supposition that the number of dendritic intersections with a circle can be resolved into two components: the number of branching points and the number of terminal tips within the annulus of two adjoining circles. The numbers of intersections and last two sets of data are also presented as cumulative frequency plots and analysed using a logistic model (Boltzmann's function). Such approaches give rise to several new morphometric parameters, such as, the critical, maximal and mean values of the numbers of intersections, branching points and terminal tips, as well as the abscissas of the inflection points of the corresponding sigmoid plots, with respect to the radius. We discuss these parameters as an additional tool for further morphological classification schemes of vertebrate retinal ganglion cells. To test the models, we apply them first to three groups of morphologically different cat's retinal ganglion cells (the alpha, gamma and epsilon cells). After that, in order to quantitatively support the classification of the rat's alpha cells into the inner and outer cells, we apply our models to two subgroups of these cells grouped according to their stratification levels in the inner plexiform layer. We show that differences between most of our parameters calculated for these subgroups are statistically significant. We believe that these models have the potential to aid in the classification of biological images.

A confirmation of Rexed's laminar hypothesis using the Sholl linear method complemented by nonparametric statistics.

Images of Golgi-impregnated neurons from laminae I to VI in the dorsal horn of the cat spinal cord were subjected to the linear Sholl analysis of concentric circles to support Rexed's hypothesis on the laminar organization of spinal gray matter in mammals. Since Rexed's determination of the laminae is based upon size, location, and grouping of cell bodies, neglecting one of the principal morphologic attributes of the neuron-the dendritic tree, the purpose of the present study was to evaluate Rexed's hypothesis testing the structure of dendritic arborization patterns of neurons. The differences in the complexity of dendritic trees between the groups of neurons from different laminae were evaluated by nonparametric statistics. Data obtained using Sholl's method is not always subjected to complete statistical analysis. The problem becomes particularly apparent in the quantitative examination of dendritic structures. Our aim was also to perform a careful analysis of our data for normality, in order to choose the appropriate statistical method for data processing. In the linear Sholl analysis, it is important to properly represent and interpret the frequency functions. The objective of this study was also to investigate the problems of determining the frequency functions, plotting the corresponding lines of regression, and measuring the degree of fluctuation of experimental data points around these lines. The main result of our testing is a confirmation of Rexed's laminar scheme: we have proved that there are 6 out of 10 possible pairs of samples where one member significantly differs from the other, i.e. one lamina is significantly distinguishable from the other.

Application of fractal analysis to neuronal dendritic arborisation patterns of the monkey dentate nucleus.

The deep nuclei of the cerebellar cortex have not yet received adequate exploratory attention. An exception is represented by the pioneering work of Chan-Palay, published in 1977, on the dentate nucleus morphology. She has classified each individual cell in the dentatus of the monkey into one of six types. Although fractal analysis is presently the most prominent quantitative method for morphometric neuronal studies, no article referring to applications of this method to the analysis of cell types of the dentate nucleus has so far been published. In the present study we apply fractal analysis to this unsolved problem and calculate the fractal dimension for each dendritic arbour of a neuron. We will hereby prove that by application of fractal analysis to the dendritic arbours of these cells whilst ignoring other neuronal attributes allows for clear discrimination of only three cell types.

Application of modified Sholl analysis to neuronal dendritic arborization of the cat spinal cord.

The drawings of Golgi-impregnated neurons from laminae I to VI in dorsal horn of the cat spinal cord were analysed morphometrically with a modified Sholl method of concentric circles. In order to advance the Sholl analysis of neuronal dendritic arborization patterns, we developed a new method of data presentation using polynomial regression and defining three parameters: the critical value of the circle radius (which defines the place of a possible circle intersecting maximum number of dendrites), the maximum number of dendritic intersections with the circles (counted for consecutive circles placed starting at the cell body to the border of the dendritic tree), and the mean value of the fitted polynomial function (which describes an average property concerning numbers of branches of dendritic tree over the whole region occupied by the dendritic arbor). For that purpose we also used the Sholl regression coefficient as well as the Schoenen ramification index. As an illustration of our model, we demonstrate that proposed modification of the Sholl method can successfully discriminate neuronal populations among different laminae of the cat spinal cord.

Fractality of dendritic arborization of spinal cord neurons.

Skeletonized images of Golgi impregnated neurons from the human, monkey, cat and rat dorsal horns were subjected to fractal analysis. These neurons have sparse branching of dendrite arbors. It is noticed that, in certain neuronal samples, some authors report that scaling range of experimentally declared fractals is extremely limited and spanned approximately between 0.5 and 2.0 decades. In order to retain our hypothesis that neurons with dendrites of uncomplicated shapes can be considered fractal over three decades of scale, we conducted four procedures: (i) we used the box-counting method, (ii) we scaled the box sizes as a power of 2, (iii) we chose the coefficient of correlation, measuring the "goodness of fit" of experimental data points to regression straight line, to be equal to or larger than 0.995, and (iv) we pointed out that all the neurons analyzed have a single fractal dimension measuring a global fractality showing no linear regions. As a control, we used some cerebellar Purkinje cells whose dendrite trees show much more complex structure and profuseness of branching. Since, generally, the neuronal structure is among the most complex of all cellular morphologies, we believe that supporting this hypothesis we advance the neuroscience and fractal theory.

Modified Richardson's method versus the box-counting method in neuroscience.

BACKGROUND: The morphology of dendrites, including apical dendrites of pyramidal neurons, is already well-known. However, the quantification of their complexity still remains open. Fractal analysis has proven to be a valuable method of analyzing the degree of complexity of dendrite morphology. NEW METHOD: Richardson's method is a technique of measuring the fractal dimension of open and closed lines of objects. This method was modified in order to measure the fractal dimension of neuronal arborization. The focus of this experiment was on the apical dendrites of superficial and deep pyramidal neurons in the rat cerebral cortex. RESULTS: Apical dendrites of superficial cortical pyramidal neurons have a higher mean value of the fractal dimension as compared to deep pyramidal neurons. COMPARISON WITH EXISTING METHOD: Using the modified Richardson's method we showed that the mean value of the fractal dimension of apical dendrites in superficial pyramidal neurons is highly statistically significant as compared to the value of the fractal dimension in deep pyramidal neurons. On the other hand, the mean values of the fractal dimension between the same groups of apical dendrites measured by the most popular box-counting method showed merely a statistically significant difference. CONCLUSION: The modified Richardson's method of fractal analysis is an efficient mathematical method for calculating the fractal dimension of dendrites and could be used in order to calculate the complexity of dendrite arborization.

Fractal dimension of apical dendritic arborization differs in the superficial and the deep pyramidal neurons of the rat cerebral neocortex.

Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones.

Fractal analysis of dendrite morphology of rotated neuronal pictures: the modified box counting method.

The fractal dimension of a non-stellate neuron changes continuously with rotation of the neuronal picture. For a stellate neuron such changes cannot be noticed. During preprocessing for the box counting, non-stellate neurons should be arranged so that the major diameters of their dendrite fields are parallel. It was shown that a non-stellate neuronal picture had the smallest box dimension when the angle between the horizontal or vertical axis and its major diameter was about 45 degrees. The box counting method which uses ImageJ software does not consider the position of a picture on the computer's screen. A dispersion of the box dimension values of a sample is generally rather large so that their mean value is with larger standard deviation. Modified box counting method partly diminishes these findings. To improve a dependence on neuronal rotation for the box counting dimension of nonstellate neurons, prior to applying the box counting, the non-stellate neurons should be arranged so that the major diameters of their dendrite fields are parallel.

Fractal analysis of dendrite morphology using modified box-counting method.

The box-counting dimension of a non-stellate neuron changes continuously with its rotation. During preprocessing for box-counting, non-stellate neurons should be arranged so that the major diameters of their dendrite fields are parallel. A non-stellate neuronal picture should have the smallest fractal dimension when the angle between the horizontal axis and its major diameter is about 45°. The box-counting method does not consider the position of a picture on the computer screen. Therefore a dispersion of the box dimension values of a neuronal sample is rather large and their mean value is with larger variance. Modified box-counting method partly diminishes these findings. To improve a dependence of neuronal rotation on the box-counting dimension of non-stellate neurons, prior to applying box-counting method, non-stellate neurons should be arranged so that the major diameters of their dendrite fields are parallel.

Mathematical model of neuronal morphology: prenatal development of the human dentate nucleus.

The aim of the study was to quantify the morphological changes of the human dentate nucleus during prenatal development using mathematical models that take into account main morphometric parameters. The camera lucida drawings of Golgi impregnated neurons taken from human fetuses of gestational ages ranging from 14 to 41 weeks were analyzed. Four morphometric parameters, the size of the neuron, the dendritic complexity, maximum dendritic density, and the position of maximum density, were obtained using the modified Scholl method and fractal analysis. Their increase during the entire prenatal development can be adequately fitted with a simple exponential. The three parameters describing the evolution of branching complexity of the dendritic arbor positively correlated with the increase of the size of neurons, but with different rate constants, showing that the complex development of the dendritic arbor is complete during the prenatal period. The findings of the present study are in accordance with previous crude qualitative data on prenatal development of the human dentate nucleus, but provide much greater amount of fine details. The mathematical model developed here provides a sound foundation enabling further studies on natal development or analyzing neurological disorders during prenatal development.

Fractal analysis of dendrites morphology using modified Richardson's and box counting method.

Fractal analysis has proven to be a useful tool in analysis of various phenomena in numerous naturel sciences including biology and medicine. It has been widely used in quantitative morphologic studies mainly in calculating the fractal dimension of objects. The fractal dimension describes an object's complexity: it is higher if the object is more complex, that is, its border more rugged, its linear structure more winding, or its space more filled. We use a manual version of Richardson's (ruler-based) method and a most popular computer-based box-counting method applying to the problem of measuring the fractal dimension of dendritic arborization in neurons. We also compare how these methods work with skeletonized vs. unskeletonized binary images. We show that for dendrite arborization, the mean box dimension of unskeletonized images is significantly larger than that of skeletonized images. We also show that the box-counting method is sensitive to an object's orientation, whereas the ruler-based dimension is unaffected by skeletonizing and orientation. We show that the mean fractal dimension measured using the ruler-based method is significantly smaller than that measured using the box-counting method. Whereas the box-counting method requires defined usage that limits its utility for analyzing dendritic arborization, the ruler-based method based on Richardson's model presented here can be used more liberally. Although this method is rather tedious to use manually, an accessible computer-based implementation for the neuroscientist has not yet been made available.

Fractal analysis: methodologies for biomedical researchers.

Fractal analysis has become a popular method in all branches of scientific investigations including biology and medicine. Although there is a growing interest in the application of fractal analysis in biological sciences, questions about the methodology of fractal analysis have partly restricted its wider and comprehensible application. It is a notable fact that fractal analysis is derived from fractal geometry, but there are some unresolved issues that need to be addressed. In this respect, we discuss several related underlying principles for fractal analysis and establish the meaningful relationship between fractal analysis and fractal geometry. Since some concepts in fractal analysis are determined descriptively and/or qualitatively, this paper provides their exact mathematical definitions or explanations. Another aim of this study is to show that nowadays fractal analysis is an independent mathematical and experimental method based on Mandelbrot's fractal geometry, Euclidean traditiontal geometry and Richardson's coastline method.

Neuronal images of the putamen in the adult human neostriatum: a revised classification supported by a qualitative and quantitative analysis.

A qualitative analysis of the morphology of human putamen nerve cells involves a detailed description of the structure and features of neurons and, accordingly, their classification into already defined classes and types. In our sample of 301 neurons, 64.78 % (195) were spiny and 35.22 % (106) aspiny cells. By analyzing cell bodies and dendritic trees, we subdivided spiny cells into two types (I and II) and aspiny cells into three types (III, IV and V). Our sample of neurons, classified according to the previously described scheme, consisted of 80 type I, 115 type II, 16 type III, 42 type IV and 48 type V nerve cells. In the present study, after qualitative analysis of microscopic images of the Golgi impregnated neurons of the putamen, we measured/quantified five morphological properties, i.e., the sizes of the soma and dendritic field, shape of the neuron, straightness of individual dendrites and the branching complexity of the dendritic tree, using eight morphometric parameters. Hence, we identify five types of nerve cells in the human putamen: type I-small spiny neurons; type II-large spiny neurons; type III-large aspiny neurons; type IV-neurons with a large soma and a medium dendritic field; and type V-small aspiny neurons. By performing an adequate statistical analysis on these parameters, we point out that the proposed types differ enough in their morphology to warrant our qualitative classification.

The morphology and classification of α ganglion cells in the rat retinae: a fractal analysis study.

Rat retinal ganglion cells have been proposed to consist of a varying number of subtypes. Dendritic morphology is an essential aspect of classification and a necessary step toward understanding structure-function relationships of retinal ganglion cells. This study aimed at using a heuristic classification procedure in combination with the box-counting analysis to classify the alpha ganglion cells in the rat retinae based on the dendritic branching pattern and to investigate morphological changes with retinal eccentricity. The cells could be divided into two groups: cells with simple dendritic pattern (box dimension lower than 1.390) and cells with complex dendritic pattern (box dimension higher than 1.390) according to their dendritic branching pattern complexity. Both were further divided into two subtypes due to the stratification within the inner plexiform layer. In the present study we have shown that the alpha rat RCGs can be classified further by their dendritic branching complexity and thus extend those of previous reports that fractal analysis can be successfully used in neuronal classification, particularly that the fractal dimension represents a robust and sensitive tool for the classification of retinal ganglion cells. A hypothesis of possible functional significance of our classification scheme is also discussed.

[Quantitative analysis of the change in neuronal numerical density of the human nucleus dentatus within development].

BACKGROUND/AIM: The role of the dentate nucleus is to coordinate input information coming from the lower olivary complex and various parts of the brainstem of the spinal marrow with the output information from the cerebellar cortex. To better understand functions and relations of the dentate nucleus it is highly important to study its development process. The aim of this study was to determine a possible mathematical model of decrease in neuronal numerical density of the human nucleus dentatus at different stages of development. METHODS: This study included 25 fetal brains of different age (12.5-31 weeks of gestational age and one brain of a 6-day-old newborn). The brains were fixed in 10% formalin-alcohol solution and embedded in paraffin. Sections were cut at a thickness of 6, 15, and 30 microm and stained with cresyl violet. Each fifth section was analyzed using a light microscope, and numerical density of dentate nucleus neurons was established using the M42 Weibel's grid system. RESULTS: The obtained results revealed a constant decrease in numerical density value. The changes of numerical densities at different stages of development correspond with Boltzmann function principles. The first, almost perpendicular part of Boltzmann function corresponds with the development of the dorsomedial lamina and the appearance of ventrolateral lamina primordium. The second, more or less horizontal part of Boltzmann function corresponds with the development of both laminae. CONCLUSION. The obtained results indicate that Boltzmann function can be considered a mathematical model of change in neuronal numerical density of dentate nucleus at different stage of development.

Morphology and cell classification of large neurons in the adult human dentate nucleus: a quantitative study.

The dentate nucleus represents the most lateral of the four cerebellar nuclei that serve as a major relay centres for fibres coming from the cerebellar cortex. Although many relevant findings regarding to the three-dimensional structure, the neuronal morphology and the cytoarchitectural development of the dentate nucleus have been presented so far, very little quantitative information has been collected to further explain several types of large neurons in the dentate nucleus. In this study we quantified the morphology of the large dentate neurons in the adult human taking, into account seven morphometric parameters that describe the main properties of the cell soma, the dendritic field and the dendritic branching pattern. Since the lateral cerebellar nucleus in the cat and other lower mammals is homologous to the dentate nucleus in primates and man, we have classified our sample of large neurons in accordance with the shape of the cell body, the dendritic arborization and their location within the dentate nucleus. By performing the appropriate statistical analysis, we have proved that our sample of human dentate neurons can be classified into four distinct types. In that sense, our quantitative analysis verifies the validity of previous qualitative conclusions concerning the large neurons in the developing human dentate nucleus. Furthermore, the present study represents the first attempt to perform a quantitative analysis and cell classification of the large projection neurons in the adult human dentate nucleus.

[Quantitative analysis of dendritic branching pattern of large neurons in human cerebellum].

BACKGROUND/AIM: Dentate nucleus (nucleus dentatus) is the most distant of the cerebellar nuclei and the major system for information transfer in the cerebellum. So far, dendritic branches of four different kinds of large neurons of dentate nucleus, have been considered mainly qualitatively with no quantification of their morphological features. The aim of the study was to test the qualitative hypothesis that the human dentate nucleus is composed of various types of the large neurons by quantitative analysis of their dendritic branching patterns. METHODS: Series of horizontal sections of the dentate nuclei were taken from 15 adult human brains, free of diagnosed neurological disorders. The 189 Golgi-impregnated images of large neurons were recorded by a digital camera connected to a light microscope. Dendritic branching patterns of digitized neuronal images were analyzed by modified Sholl and fractal analyses. RESULTS: The number of intersections (N(m)), critical radius (r(c)) and fractal dimension (D) of dendritic branching pattern for four types of the large neurons were calculated, statistically evaluated and analyzed. The results show that there is a significant difference between four neuronal types in one morphometric parameter at least. CONCLUSION: The present study is the first attempt to analyze quantitatively the dendritic branching pattern of neurons from the dentate nucleus in the human. The hypothesis that the four types of the large neurons exist in this part of human cerebellum is successfully supported.