A fractal analysis of pyramidal neurons in mammalian motor cortex

The Morphology of Brain Neurons: The Box-Counting Method in the Quantitative Analysis of 2D Images

This chapter begins by showing the difference between fractal geometry and fractal analysis. The text shows the difference between mathematical and natural fractals and how they are best defined by explaining the concept of fractal analysis. Furthermore, the text presents the most famous technique of fractal analysis: the box-counting method. Defining this method and showing the methodology that leads to the precise value of the fractal (i.e., the box) dimension is done by demonstrating the images of human dentate neurons. A more detailed explanation of the methodology was presented in the previous version of this chapter.This version promotes the notion of monofractal analysis and shows how three types of the same neuronal images can quantify four image properties. The results showed that monofractal parameters successfully quantified four image properties in three nuclei of the cerebellum. Finally, the author discusses the results of this chapter and previously published conclusions. The results show how the monofractal parameters discriminate images of neurons from the three nuclei of the human cerebrum. These outcomes are discussed along with the results of previous studies.

Fractals in Neuroanatomy and Basic Neurosciences: An Overview

The introduction of fractal geometry to the neurosciences has been a major paradigm shift over the last decades as it has helped overcome approximations and limitations that occur when Euclidean and reductionist approaches are used to analyze neurons or the entire brain. Fractal geometry allows for quantitative analysis and description of the geometric complexity of the brain, from its single units to the neuronal networks.As illustrated in the second section of this book, fractal analysis provides a quantitative tool for the study of the morphology of brain cells (i.e., neurons and microglia) and its components (e.g., dendritic trees, synapses), as well as the brain structure itself (cortex, functional modules, neuronal networks). The self-similar logic which generates and shapes the different hierarchical systems of the brain and even some structures related to its "container," that is, the cranial sutures on the skull, is widely discussed in the following chapters, with a link between the applications of fractal analysis to the neuroanatomy and basic neurosciences to the clinical applications discussed in the third section.

Emergence of lump-like solitonic waves in Heimburg-Jackson biomembranes and nerves fractal model

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev-Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

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.

Development of gerbil medial superior olive: integration of temporally delayed excitation and inhibition at physiological temperature

The sensitivity of medial superior olive (MSO) neurons to tens of microsecond differences in interaural temporal delay (ITD) derives in part from their membrane electrical characteristics, kinetics and timing of excitatory and inhibitory inputs, and dendrite structure. However, maturation of these physiological and structural characteristics are little studied, especially in relationship to the onset of auditory experience. We showed, using brain slices at physiological temperature, that MSO neurons exhibited sensitivity to simulated temporally delayed (TD) EPSCs (simEPSC), injected through the recording electrode, by the initial phase of hearing onset at P10, and TD sensitivity was reduced by block of low threshold potassium channels. The spike generation mechanism matured between P10 and P16 to support TD sensitivity to adult-like excitatory stimuli (1-4 ms duration) by P14. IPSP duration was shorter at physiological temperature than reported for lower temperatures, was longer than EPSP duration at young ages, but approached the duration of EPSPs by P16, when hearing thresholds neared maturity. Dendrite branching became less complex over a more restricted time frame between P10 and P12. Because many physiological and structural properties approximated mature values between P14 and P16, we studied temporal integration of simEPSCs and IPSPs at P15. Only a narrow range of relative onset times (< 1 ms) yielded responses showing sensitivity to TD. We propose that shaping of excitatory circuitry to mediate TD sensitivity can begin before airborne sound is detectable, and that inhibitory inputs having suboptimal neural delays may then be pruned by cellular mechanisms activated by sensitivity to ITD.

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.

Neuropil reactivity, distribution and morphology of NADPH diaphorase type I neurons in the barrel cortex of the adult mouse

The mouse, like a few other rodent and marsupial species, displays a striking modular architecture in its primary somatosensory cortex (SI). These modules, known as barrels, are mostly defined by the peculiar arrangement of granule cells and thalamic axons in layer IV. In the present work, we studied both the distribution and morphology of neurons stained for NADPH diaphorase (NADPH-d) and neuropil reactivity in the posteromedial barrel subfield (PMBSF), which represents the mystacial whiskers. We then compared our results with previous descriptions of NADPH-d distribution in both neonatal and young mice. We found two types of neurons in the PMBSF: type I neurons, which have large cell bodies and are heavily stained by the NADPH-d reaction; and type II neurons, characterized by relatively small and poorly stained cell bodies. The distribution of type I cells in the PMBSF was not homogenous, with cells tending to concentrate in septa between barrels. Moreover, the cells found in septal region possess both a larger and more complex dendritic arborization than cells located inside barrels. Our findings are at variance with results from other groups that reported both an absence of type II cells and a homogeneous distribution of type I cells in the PMBSF of young animals. In addition, our results show a distribution of type I cells which is very similar to that previously described for the rat's barrel field.

Fractal analysis of the laminar organization of spinal cord neurons

Images of Golgi impregnated neurons from different laminae of the human and rat dorsal horns were subjected to a quantitative analysis to support the Rexed's laminar scheme in mammals. Four methods of fractal analysis were performed in the proceedings: box-counting, mass-radius, cumulative intersection, and vectorized intersection. The results show that the box-counting method is more precise than the other fractal methods performed, and offers support for the conclusion that fractal analysis can successfully discriminate the neuron populations among different laminae. The analysis supports the concept of Rexed's cytoarchitectonic lamination of the dorsal horn.

AUTOMATED MORPHOMETRIC ANALYSIS OF THE CAT RETINAL α/Y, β/X AND δ GANGLION CELLS USING WAVELET STATISTICAL MOMENT AND CLUSTERING ALGORITHMS

Computational morphological analysis comprises the development of measures (indicators) that describe different form attributes of a neuron and provides additional parameters for classification algorithms. Our work addressed the problem of small group sizes often encountered in neuromorphological and neurophysiological research, automated classification tasks (unsupervised learning) and introduced a new morphological measure: the wavelet statistical moment. We analysed cat alpha/Y, beta/X and delta Golgi-stained retinal ganglion cells using six different shape features (circularity, 2(nd) statistical moment and entropy of Gaussian blurred images, wavelet statistical moment, number of terminations and the fractal dimension). This allowed us to compare the sensitivity of the methods in uniquely describing morphological attributes of these cells.

Fractal Analysis of the Vascular Tree in the Human Retina

The retinal circulation of the normal human retinal vasculature is statistically self-similar and fractal. Studies from several groups present strong evidence that the fractal dimension of the blood vessels in the normal human retina is approximately 1.7. This is the same fractal dimension that is found for a diffusion-limited growth process, and it may have implications for the embryological development of the retinal vascular system. The methods of determining the fractal dimension for branching trees are reviewed together with proposed models for the optimal formation (Murray Principle) of the branching vascular tree in the human retina and the branching pattern of the human bronchial tree. The limitations of fractal analysis of branching biological structures are evaluated. Understanding the design principles of branching vascular systems and the human bronchial tree may find applications in tissue and organ engineering, i.e., bioartificial organs for both liver and kidney.

Fractal dimension in human cerebellum measured by magnetic resonance imaging

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 +/- 0.01. No significant difference in the CB fractal dimension was observed between men and women.

Characterization of Melanophore Morphology by Fractal Dimension Analysis

Fractal or focal dimension (FD) analysis is a valuable tool to identify physiologic stimuli at the cellular and tissue levels that allows for quantification of cell perimeter complexity. The FD analysis was determined on fluorescence images of caffeine- or epinephrine-treated (or untreated control) killifish Fundulus heteroclitus (Linneaus) melanophores in culture. Cell perimeters were indicated by rhodamine-phalloidin labeling of cortical microfilaments using box-counting FD analysis. Caffeine-treated melanophores displayed dispersed melanosomes in cells with less serrated edges and reduced FD and complexity. Complexity in epinephrine-treated cells was significantly higher than the caffeine-treated cells or in the control. Cytoarchitectural variability of the cell perimeter is expected because cells change shape when cued with agents. Epinephrine-treated melanophores demonstrated aggregated melanosomes in cells with more serrated edges, significantly higher FD and thus complexity. Melanophores not treated with caffeine or epinephrine produced variable distributions of melanosomes and resulted in cells with variably serrated edges and intermediate FD with a larger SE of the regression and greater range of complexity. Dispersion of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centrifugal distribution of melanosomes throughout the cell and to the periphery. The loading of melanosomes onto cortical microfilaments may provide a less complex cell contour, with the even distribution of the cytoskeleton and melanosomes. Aggregation of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centripetal distribution of melanosomes. The aggregation of melanosomes may contribute to centripetal retraction of the cytoskeleton and plasma membrane. The FD analysis is, therefore, a convenient method to measure contrasting morphologic changes within stimulated cells.

Recurrent fractal neural networks: a strategy for the exchange of local and global information processing in the brain

The regulation of biological networks relies significantly on convergent feedback signaling loops that render a global output locally accessible. Ideally, the recurrent connectivity within these systems is self-organized by a time-dependent phase-locking mechanism. This study analyzes recurrent fractal neural networks (RFNNs), which utilize a self-similar or fractal branching structure of dendrites and downstream networks for phase-locking of reciprocal feedback loops: output from outer branch nodes of the network tree enters inner branch nodes of the dendritic tree in single neurons. This structural organization enables RFNNs to amplify re-entrant input by over-the-threshold signal summation from feedback loops with equivalent signal traveling times. The columnar organization of pyramidal neurons in the neocortical layers V and III is discussed as the structural substrate for this network architecture. RFNNs self-organize spike trains and render the entire neural network output accessible to the dendritic tree of each neuron within this network. As the result of a contraction mapping operation, the local dendritic input pattern contains a downscaled version of the network output coding structure. RFNNs perform robust, fractal data compression, thus coping with a limited number of feedback loops for signal transport in convergent neural networks. This property is discussed as a significant step toward the solution of a fundamental problem in neuroscience: how is neuronal computation in separate neurons and remote brain areas unified as an instance of experience in consciousness? RFNNs are promising candidates for engaging neural networks into a coherent activity and provide a strategy for the exchange of global and local information processing in the human brain, thereby ensuring the completeness of a transformation from neuronal computation into conscious experience.

The effect of type 1 astrocytes on neuronal complexity: a fractal analysis

Embryonic, ventral spinal cord neurons were grown on poly(d-lysine) (PDL) or on a monolayer of type 1 astrocytes. At various times from 6 h to 2 weeks postplating, cells were fluorescently labeled and fixed with 4% paraformaldehyde. The cell surface immunoreaction allowed visualization of neurons in their entirety, namely, cell bodies and various membranous extensions that included lamellipodia, growth cones, axons, and dendrites. Outlines were drawn for individual neurons and their fractal dimension (D) was calculated. Neurons on poly(d-lysine) reached a peak D at 3 days in vitro, 1 day later than neurons on astrocytes (2 days in vitro). The maximum D was greater for cells on poly(d-lysine) when compared with neurons on astrocytes. In a second experiment the maximum D was similar for neurons on both surfaces but neurons on PDL maintained a higher D for a much longer period than neurons on astrocytes. An examination of fluorescent images revealed that neurons on poly(d-lysine) exhibited lamellipodia and large growth cones for several days and these structures were likely responsible for the high D seen in these cells. These structures were rarely observed in neurons plated on astrocytes. Interestingly, D on both surfaces decreased to a similar value at between 1 and 2 weeks in vitro. The trend for D in these cultures, an initial increase to a peak value followed by a decrease to a stable value, is discussed in light of the chemical nature of the two surfaces and synapse formation and stabilization.

Use of fractal theory in neuroscience: methods, advantages, and potential problems

Fractal analysis has already found widespread application in the field of neuroscience and is being used in many other areas. Applications are many and include ion channel kinetics of biological membranes and classification of neurons according to their branching characteristics. In this article we review some practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and estimation of the fractal dimensions from the data points, as well as the advantages and problems of fractal geometric analysis, are discussed.

Morphology of reactive microglia in the injured cerebral cortex. Fractal analysis and complementary quantitative methods

The present study focuses on application of quantitative methods measuring differences between particular morphological types of microglial cells as well as between their proliferating and non-proliferating examples. On the basis of subjective classification, microglial cells of three morphological types (ramified, hypertrophied and bushy) were selected from the neocortex of injured rat brain. Thereafter, the morphological complexity of each cell was assessed by calculation its fractal dimension as well as its form factor, convexity, ramification factor and solidity. The fractal dimension seemed a good parameter for detecting small changes in the space-filing capacity of cells, for example, it shows differences between ramified cells from control and injured brains. This measure seemed insensitive to some aspects of cell morphology. To obtain precise quantification of observed changes other morphological parameters had to be applied. Proliferating and non-proliferating microglial cells displayed significant differences in their solidity and ramification factors, but not in fractal dimension and convexity. The results indicated that proliferating microglia were more massive and less-ramified but they did not reduce their spatial complexity.

The effects of congenital deafness on auditory nerve synapses and globular bushy cells in cats

It is well known that auditory deprivation affects the structure and function of the central nervous system. Congenital deafness represents one form of deprivation, and in the adult white cat, it has been shown to have a clear effect upon the synaptic interface between endbulbs of Held and spherical bushy cells. It is not known, however, whether all primary synapses are affected and/or whether they are affected in the same way and to the same extent. Thus, we studied a second neuronal circuit in the deaf white cat involving modified (small) endbulbs and globular bushy cells. Compared to normal hearing cats, modified endbulbs of congenitally deaf cats were 52.2% smaller but unchanged in structural complexity. There was also a striking loss of extracellular space between ending and cell body. The somata of postsynaptic globular bushy cells were 13.4% smaller and had enlarged postsynaptic densities. These data reveal that axosomatic synapses demonstrate abnormal structure as a consequence of deafness and that the extent of the abnormalities can vary with respect to the circuits involved. The implication of these observations is that synaptic anomalies would introduce differential delays within separate circuits, thereby desynchronizing neural activity from sound stimuli. This loss of synchronization could in turn disrupt temporal processing and compromise a host of related functions, including language comprehension.

Single unit recordings in the auditory nerve of congenitally deaf white cats: morphological correlates in the cochlea and cochlear nucleus

It is well known that experimentally induced cochlear damage produces structural, physiological, and biochemical alterations in neurons of the cochlear nucleus. In contrast, much less is known with respect to the naturally occurring cochlear pathology presented by congenital deafness. The present study attempts to relate organ of Corti structure and auditory nerve activity to the morphology of primary synaptic endings in the cochlear nucleus of congenitally deaf white cats. Our observations reveal that the amount of sound-evoked spike activity in auditory nerve fibers influences terminal morphology and synaptic structure in the anteroventral cochlear nucleus. Some white cats had no hearing. They exhibited severely reduced spontaneous activity and no sound-evoked activity in auditory nerve fibers. They had no recognizable organ of Corti, presented >90% loss of spiral ganglion cells, and displayed marked structural abnormalities of endbulbs of Held and their synapses. Other white cats had partial hearing and possessed auditory nerve fibers with a wide range of spontaneous activity but elevated sound-evoked thresholds (60-70 dB SPL). They also exhibited obvious abnormalities in the tectorial membrane, supporting cells, and Reissner's membrane throughout the cochlear duct and had complete inner and outer hair cell loss in the base. The spatial distribution of spiral ganglion cell loss correlated with the pattern of hair cell loss. Primary neurons of hearing-impaired cats displayed structural abnormalities of their endbulbs and synapses in the cochlear nucleus which were intermediate in form compared to normal and totally deaf cats. Changes in endbulb structure appear to correspond to relative levels of deafness. These data suggest that endbulb structure is significantly influenced by sound-evoked auditory nerve activity.

Neurons and fractals: how reliable and useful are calculations of fractal dimensions?

In the past 15 years it has become possible to determine the fractal dimension (Df) of complex objects, including neurons, by automated image analysis methods. However, there are many unresolved issues that need to be addressed. In this paper we discuss how the Df calculated by different methods may vary and how fractal analysis may be of use for retinal ganglion cell characterization. The goal of this work was to acknowledge inherent sources of variation during measurement and evaluate current fractal analysis methods for describing structure. Our results show that different algorithms and even the same algorithm performed by different computer programs and/or experimenters may give different but consistent numerical values. All described methods demonstrated their suitability for classifying cat retinal ganglion cells into distinct groups. Our results reinforce the idea that comparison of measurements of different profiles using the same measurement method may be useful and valid even if an exact numeric value of the dimension is not realised in practice.

Ultrastructural analysis of primary endings in deaf white cats: morphologic alterations in endbulbs of Held

Changes in structure and function of the auditory system can be produced by experimentally manipulating the sensory environment, and especially dramatic effects result from deprivation procedures. An alternative deprivation strategy utilizes naturally occurring lesions. The congenitally deaf white cat represents an animal model of sensory deprivation because it mimics a form of human deafness called the Scheibe deformity and permits studies of how central neurons react to early-onset cochlear degeneration. We studied the synaptic characteristics of the endbulb of Held, a prominent auditory nerve terminal in the cochlear nucleus. Endbulbs arise from the ascending branch of the auditory nerve fiber and contact the cell body of spherical bushy cells. After 6 months, endbulbs of deaf white cats exhibit alterations in structure that are clearly distinguishable from those of normal hearing cats, including a diminution in terminal branching, a reduction in synaptic vesicle density, structural abnormalities in mitochondria, thickening of the pre- and postsynaptic densities, and enlargement of synapse size. The hypertrophied membrane densities are suggestive of a compensatory response to diminished transmitter release. These data reveal that early-onset, long-term deafness produces unambiguous alterations in synaptic structure and may be relevant to rehabilitation strategies that promote aural/oral communication.

Fractal methods and results in cellular morphology--dimensions, lacunarity and multifractals

This paper discusses the concepts of fractal geometry in a cellular biological context. It defines the concept of the fractal dimension. D, as a measure of complexity and illustrates the two different general ways of quantitatively measuring D by length-related and mass-related methods. Then, these several Ds are compared and contrasted. A goal of the paper is to find methods other than length-related measures that can distinguish between two objects that have the same D but are structurally different. The mass-related D is shown potentially to be such a measure. The concept of lacunarity, L, is defined and methods of measuring L are illustrated. L is also shown to be a potentially distinguishing measure. Finally, the notion of multifracticality is defined and illustrated to exist in certain individual nerve and glial cells.

Complexity analysis of oligodendroglial processes expressing myelin-associated glycoprotein

Oligodendroglia synthesize myelin in the mammalian central nervous system. Mature oligodendroglia have been identified in culture by two criteria; the expression of molecules characteristic of myelin, such as galactocerebroside (galC) and myelin-associated glycoprotein (MAG), and the elaboration of complex processes. Myelin gene expression can be documented by the binding of specific antibodies and antisera to the myelin-specific molecules; process complexity can be described by the fractal dimension, D. In this study, anti-MAG antisera was used to document MAG expression in the processes of oligodendroglia. Eighty percent of the galC+ oligodendroglia bound anti-MAG antiserum. With time in culture, MAG immunoreactivity seemed to extend from the cell soma into the oligodendroglial processes. To quantify this observation, fractal dimensions were calculated using either galC or MAG immunoreactivity to visualize oligodendroglial processes. A fractal dimension of 1.5 was calculated for O1+ processes by day 4 of culture; this value for D remained constant over the course of 1 month in culture. The fractal dimension calculated for MAG+ processes increased from 1.2 to 1.5 over the course of 28 days in culture. This change in fractal dimension confirms our visual impression that galC-containing processes acquire MAG slowly over the course of several weeks in culture.

Complexity and scaling properties of amacrine, ganglion, horizontal, and bipolar cells in the turtle retina

In the present study we have evaluated the complexity and scaling properties of the morphology of retinal neurons using fractal dimension as a quantitative parameter. We examined a large number of cells from Pseudemys scripta and Mauremys caspica turtles that had been labeled using Golgi-impregnation techniques, intracellular injection of Lucifer Yellow followed by photooxidation, intracellular injection of rhodamine conjugated horseradish peroxidase, or intracellular injection of Lucifer Yellow or horseradish peroxidase alone. The fractal dimensions of two-dimensional projections of the cells were calculated using a box counting method. Discriminant analysis revealed fractal dimension to be a significant classification parameter among several other parameters typically used for placing turtle retinal neurons in different cell classes. The fractal dimension of amacrine cells was significantly correlated with dendritic field diameters, while the fractal dimensions of ganglion cells did not vary with dendritic field span. There were no significant differences between the same cell types in two different turtle species, or between the same types of neurons in the same species after labeling with different techniques. The application of fractal dimension, as a quantitative measure of complexity and scaling properties and as a classification criterion of neuronal types, appears to be useful and may have wide applicability to other parts of the central nervous system.

Preserved features of thalamocortical projection neuron dendritic architecture in the somatosensory thalamus of the rat, cat and macaque

A number of studies have shown that the organization of the mammalian somatosensory thalamus varies between species. As differences in cellular and synaptic thalamic organization would be expected to influence neuronal dendritic architecture, we compared somatosensory thalamocortical projection (TCP) neurons from the rat, cat and macaque. The results show that key features of the dendritic branching pattern remain unchanged despite large differences in the size of TCP neurons between the species. The features examined were: (i) ratio of the length of terminal branches to the length of the entire dendritic tree; (ii) the percentage of branch points that gave rise to two daughter branches as opposed to those that gave rise to three or more daughter branches; (iii) the proportional sum of absolute deviations (a measure of branching symmetry), and (iv) the mean branch order of the terminal segments. The present study provides evidence that somatosensory TCP neurons in these species comprise a homogeneous class and share a common dendritic architecture that is conserved across species despite changes in other aspects of thalamic circuitry. This suggests that TCP neuronal form is based on relatively stable genetic blueprint and that epigenetic factors (e.g. synaptic input) resulting from evolutionary changes in thalamic organization have had less influence on dendritic architecture.

Are there three types of horizontal cell in the human retina?

Golgi-impregnated horizontal cells (HCs) as viewed in whole mount human retinas have been studied by light microscopic (LM) techniques. Impregnated HCs have been drawn by camera lucida and by the Eutectics neuron tracing method to provide quantitative data on dendritic tree sizes, dendritic tree shapes, and dendritic terminals for statistical treatment and cluster analysis. In addition, fractal analyses of HC dendritic branching patterns have been performed. Three significantly different HCs can be classified on both subjective and objective morphological criteria in central and peripheral human retina. In the fovea all HCs are so small that it is difficult to achieve a clear separation of the subtypes, although they can be distinguished by the experienced observer. HI types are the classic HCs of Polyak (The Retina, Chicago: University of Chicago Press, 1941) with distinct dendritic terminal clusters going to cones and a fan-shaped axon terminal consisting of large numbers of rod-destined terminals. HII cells have profusely branched, overlapping dendrites, with poorly defined terminals going to cones and a short curled axon bearing small terminals also going to cones. The HIII types exhibit larger diameter, more asymmetrically shaped dendritic trees and 30% more dendritic terminal clusters than HI cells at any location on the retina. Many HIII cells appear to emit a process from the cell body in the inner nuclear layer (INL) that descends into the outer strata of the inner plexiform layer (IPL). The axon of the HIII cell may end in a loosely organized, sprawling arborization. Fractal dimensions of the horizontal cells also show significant differences between the three groups. HII cells exhibit the highest fractal dimension followed by HI and HIII cells with lower and lowest fractal dimensions, respectively. The fractal dimension of HII cells of rhesus monkey, as determined from drawings by other authors in other publications, are the same as HII cells of human retina.

Dendritic architecture of rat somatosensory thalamocortical projection neurons

This study examines dendrites from physiologically characterized and intracellularly labelled thalamocortical projection (TCP) neurons from the rat ventrobasal complex (VB) and posterior nucleus (POm). The goals were to provide quantitative descriptions of TCP neuron dendrites, examine underlying design principles of dendritic morphology, and determine correlations between dendritic size parameters. Forty-four dendrites from seven VB neurons and 21 dendrites from three POm TCP neurons that responded to low-threshold mechanical stimuli were reconstructed and quantitatively analyzed at the light microscopic level. The dendritic architecture of the neurons was remarkably similar in most parameters studied, including the percentage of dichotomous branching, contribution of terminal branches to total dendritic length, and branching symmetry. There was a positive correlation between stem dendrite diameter and the length of the entire dendrite arbor, making it possible to estimate the total length of a dendritic arbor by measuring the stem dendrite diameter. The correlations of the VB and POm dendrites had different slopes. The path distance (the distance from the soma to a dendritic end point) of individual dendrites showed only a small variation with large differences in the total dendritic length of an arbor. The constant diameter of distal dendrites shows that dendrite diameter is a poor predictor of synaptic location on the dendritic tree. Although the morphology of neurons and their individual dendrites varied considerably in overall size and qualitative appearance, when examined qualitatively, many aspects of dendritic structure were similar within and between groups. We suggest that the rat somatosensory TCP neurons have a stereotyped dendritic architecture and present data which provide a base for future comparative, developmental, and plasticity studies.

Comparative fractal analysis of cultured glia derived from optic nerve and brain demonstrate different rates of morphological differentiation

O-2A progenitor cells derived from neonatal rat cerebral hemispheres or optic nerves, were induced to differentiate in culture into either oligodendrocytes or type 2 astrocytes. The fractal dimensions, a measure of morphological complexity, of the differentiating glial cells were measured over time. Analysis of the changes in fractal dimension (D) with respect to time revealed specific rates of growth for each glial phenotype and a specific final D. The time course of these changes is well fit by a simple mathematical model. While brain-derived oligodendrocytes matured faster than the astrocytes, they ultimately attained comparable levels of complexity, with similar maximum fractal dimensions. Oligodendrocytes from nerve also matured faster than nerve derived astrocytes, in contrast, however, they attained a greater morphological complexity than nerve astrocytes. While the brain-derived oligodendrocytes showed a faster rate of maturation than their optic nerve counterparts, astrocytes from both regions had similar rates of morphological differentiation. Self-similarity, a defining property of fractal objects was investigated, by determining the fractal dimension of cells over a range of magnifications. The calculated fractal dimension remained constant over a 10-fold range in optical magnification, illustrating that cultured glial cells exhibit this important characteristic of fractal objects. In addition, we analyzed the branching patterns of glial processes by the Sholl method and found that the results were not as interpretable or meaningful as those of fractal analysis.

The application of fractal geometric analysis to microscopic images

Fractal geometry is a relatively new tool for the quantitative microscopist that is a more valid way of measuring dimensions of complex irregular objects than the integer-dimensional geometries (such as Euclidean geometry). This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket. The problems of describing the dimensions of these objects are discussed and the concept of fractal dimensionality is introduced. Methods for measuring fractal dimensions are discussed, including their implementation on microcomputer-based image analysis systems . The advantages and problems of fractal geometric analysis are discussed and current applications in the field of microscopy are reviewed.

Quantitative phylogenetic constancy of cerebellar Purkinje cell morphological complexity

Golgi-stained material of cerebellar cortices from 17 species was examined by measuring the fractal dimensions of the borders of Purkinje cells, which is a quantitative, objective measure of morphological complexity. Nine species (from birds to man) were chosen for a comparison with ANOVA and no statistically significant differences were found in their fractal dimensions. In contrast, a wide range of differences was found in the membrane areas across species lines. The Sholl coefficient, a measure of branch formation and termination away from the soma, showed no consistent pattern for each cell. We interpret our results as indicating a constancy in morphological cellular complexity of Purkinje cells during late evolutionary time.

Early dendrite development in spinal cord cell cultures: a quantitative study

Neurons in dissociated cell culture provide a favorable system for the quantitative analysis of structural changes and the examination of structure-function relationships during development. Fragment C of tetanus toxin was used to label neurons in murine spinal cord cell cultures and dendrite outgrowth was monitored by a number of measures. The dissociated neurons increased in morphologic complexity from approximate spheres to highly branched structures during the first week in culture. Much of the structural complexity of the dendrite arbor, as quantified by fractal dimension, was established within 48 hr after plating, i.e., prior to the development of interneuronal contacts. During the first few days in culture, dendrite branching complexity increased more rapidly than dendrite size, whereas after 4 days, fractal dimension remained relatively constant while dendrites continued to grow. Fractal analysis has provided data which suggest that the early development of dendrite branching complexity is determined intrinsically. Fractal dimension, as an effective index of morphologic complexity, should be a useful tool for the further study of extrinsic signals which might modify the generation or stabilization of dendrite form.

A comparative fractal analysis of various mammalian astroglial cell types

Camera-lucida drawings of Golgi-impregnated astroglial cells and their processes are described by the fractal dimension of their borders, which is an objective, quantitative measure of morphological complexity. Protoplasmic astrocytes from human neocortex have fractal dimensions (D) that are larger than those of fibrous astrocytes from the cat optic nerve. Marginal astrocytes from monkey cerebropontile angle have two kinds of processes: (1) short, thick processes with endfeet abutting the pial surface, with relatively high D's, and (2) very long, thin processes extending into the neuronal tissue, with very low D's. These data indicate that short astrocytic processes may have a complex surface (and have a high D), whereas long processes are rather smooth (and have a low D). A comparison between transmission electron microscopy morphometry and measures of D at the light microscopic level, performed on different parts of rabbit retinal Müller glial cells, suggests that D is strongly correlated to the surface-to-volume ratio which, in part, determines the length constant of a cable for core-conductance of currents. We provide data supporting the hypothesis that astroglial cell geometry is adjusted to allow for sufficient spatial buffering K+ currents, even through very long processes.