Fractal analysis of the laminar organization of spinal cord neurons

Fractal Resonance: Can Fractal Geometry Be Used to Optimize the Connectivity of Neurons to Artificial Implants?

In parallel to medical applications, exploring how neurons interact with the artificial interface of implants in the human body can be used to learn about their fundamental behavior. For both fundamental and applied research, it is important to determine the conditions that encourage neurons to maintain their natural behavior during these interactions. Whereas previous biocompatibility studies have focused on the material properties of the neuron-implant interface, here we discuss the concept of fractal resonance - the possibility that favorable connectivity properties might emerge by matching the fractal geometry of the implant surface to that of the neurons.To investigate fractal resonance, we first determine the degree to which neurons are fractal and the impact of this fractality on their functionality. By analyzing three-dimensional images of rat hippocampal neurons, we find that the way their dendrites fork and weave through space is important for generating their fractal-like behavior. By modeling variations in neuron connectivity along with the associated energetic and material costs, we highlight how the neurons' fractal dimension optimizes these constraints. To simulate neuron interactions with implant interfaces, we distort the neuron models away from their natural form by modifying the dendrites' fork and weaving patterns. We find that small deviations can induce large changes in fractal dimension, causing the balance between connectivity and cost to deteriorate rapidly. We propose that implant surfaces should be patterned to match the fractal dimension of the neurons, allowing them to maintain their natural functionality as they interact with the implant.

A Self-Similarity Logic May Shape the Organization of the Nervous System

From the morphological point of view, the nervous system exhibits a fractal, self-similar geometry at various levels of observations, from single cells up to cell networks. From the functional point of view, it is characterized by a hierarchical organization in which self-similar structures (networks) of different miniaturizations are nested within each other. In particular, neuronal networks, interconnected to form neuronal systems, are formed by neurons, which operate thanks to their molecular networks, mainly having proteins as components that via protein-protein interactions can be assembled in multimeric complexes working as micro-devices. On this basis, the term "self-similarity logic" was introduced to describe a nested organization where, at the various levels, almost the same rules (logic) to perform operations are used. Self-similarity and self-similarity logic both appear to be intimately linked to the biophysical evidence for the nervous system being a pattern-forming system that can flexibly switch from one coherent state to another. Thus, they can represent the key concepts to describe its complexity and its concerted, holistic behavior.

Investigating Fractal Analysis as a Diagnostic Tool That Probes the Connectivity of Hippocampal Neurons

Many of nature’s fractal objects benefit from the favorable functionality that results from their pattern repetition at multiple scales. Our recent research focused on the importance of fractal scaling in establishing connectivity between neurons. Fractal dimension D_A of the neuron arbors was shown to relate to the optimization of competing functional constraints—the ability of dendrites to connect to other neurons versus the costs associated with building the dendrites. Here, we consider whether pathological states of neurons might affect this fractal optimization and if changes in D_A might therefore be used as a diagnostic tool in parallel with traditional measures like Sholl analyses. We use confocal microscopy to obtain images of CA1 pyramidal neurons in the coronal plane of the dorsal rat hippocampus and construct 3-dimensional models of the dendritic arbors using Neurolucida software. We examine six rodent groups which vary in brain condition (whether they had lesions in the anterior thalamic nuclei, ATN) and experience (their housing environment and experience in a spatial task). Previously, we showed ATN lesions reduced spine density in hippocampal CA1 neurons, whereas enriched housing increased spine density in both ATN lesion and sham rats. Here, we investigate whether ATN lesions and experience also effect the complexity and connectivity of CA1 dendritic arbors. We show that sham rats exposed to enriched housing and spatial memory training exhibited higher complexity (as measured by D_A) and connectivity compared to other groups. When we categorize the rodent groups into those with or without lesions, we find that both categories achieve an optimal balance of connectivity with respect to material cost. However, the D_A value used to achieve this optimization does not change between these two categories, suggesting any morphological differences induced by the lesions are too small to influence the optimization process. Accordingly, we highlight considerations associated with applying our technique to publicly accessible repositories of neuron images with a broader range of pathological conditions.

How neurons exploit fractal geometry to optimize their network connectivity

We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three-dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal-like behavior well-described by an ‘effective’ fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D-dependent variations in inter-neuron connectivity along with the associated costs, we propose that their D values reflect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes).

The histology, physiology, neurochemistry and circuitry of the substantia gelatinosa Rolandi (lamina II) in mammalian spinal cord

The substantia gelatinosa Rolandi (SGR) was first described about two centuries ago. In the following decades an enormous amount of information has permitted us to understand - at least in part - its role in the initial processing of pain and itch. Here, I will first provide a comprehensive picture of the histology, physiology, and neurochemistry of the normal SGR. Then, I will analytically discuss the SGR circuits that have been directly demonstrated or deductively envisaged in the course of the intensive research on this area of the spinal cord, with particular emphasis on the pathways connecting the primary afferent fibers and the intrinsic neurons. The perspective existence of neurochemically-defined sets of primary afferent neurons giving rise to these circuits will be also discussed, with the proposition that a cross-talk between different subsets of peptidergic fibers may be the structural and functional substrate of additional gating mechanisms in SGR. Finally, I highlight the role played by slow acting high molecular weight modulators in these gating mechanisms.

Multiscale Analysis of Neurite Orientation and Spatial Organization in Neuronal Images

The spatial organization of neurites, the thin processes (i.e., dendrites and axons) that stem from a neuron's soma, conveys structural information required for proper brain function. The alignment, direction and overall geometry of neurites in the brain are subject to continuous remodeling in response to healthy and noxious stimuli. In the developing brain, during neurogenesis or in neuroregeneration, these structural changes are indicators of the ability of neurons to establish axon-to-dendrite connections that can ultimately develop into functional synapses. Enabling a proper quantification of this structural remodeling would facilitate the identification of new phenotypic criteria to classify developmental stages and further our understanding of brain function. However, adequate algorithms to accurately and reliably quantify neurite orientation and alignment are still lacking. To fill this gap, we introduce a novel algorithm that relies on multiscale directional filters designed to measure local neurites orientation over multiple scales. This innovative approach allows us to discriminate the physical orientation of neurites from finer scale phenomena associated with local irregularities and noise. Building on this multiscale framework, we also introduce a notion of alignment score that we apply to quantify the degree of spatial organization of neurites in tissue and cultured neurons. Numerical codes were implemented in Python and released open source and freely available to the scientific community.

CMEIAS JFrad: a digital computing tool to discriminate the fractal geometry of landscape architectures and spatial patterns of individual cells in microbial biofilms

Image analysis of fractal geometry can be used to gain deeper insights into complex ecophysiological patterns and processes occurring within natural microbial biofilm landscapes, including the scale-dependent heterogeneities of their spatial architecture, biomass, and cell-cell interactions, all driven by the colonization behavior of optimal spatial positioning of organisms to maximize their efficiency in utilization of allocated nutrient resources. Here, we introduce CMEIAS JFrad, a new computing technology that analyzes the fractal geometry of complex biofilm architectures in digital landscape images. The software uniquely features a data-mining opportunity based on a comprehensive collection of 11 different mathematical methods to compute fractal dimension that are implemented into a wizard design to maximize ease-of-use for semi-automatic analysis of single images or fully automatic analysis of multiple images in a batch process. As examples of application, quantitative analyses of fractal dimension were used to optimize the important variable settings of brightness threshold and minimum object size in order to discriminate the complex architecture of freshwater microbial biofilms at multiple spatial scales, and also to differentiate the spatial patterns of individual bacterial cells that influence their cooperative interactions, resource use, and apportionment in situ. Version 1.0 of JFrad is implemented into a software package containing the program files, user manual, and tutorial images that will be freely available at http://cme.msu.edu/cmeias/. This improvement in computational image informatics will strengthen microscopy-based approaches to analyze the dynamic landscape ecology of microbial biofilm populations and communities in situ at spatial resolutions that range from single cells to microcolonies.

Efficient processing of fluorescence images using directional multiscale representations

Recent advances in high-resolution fluorescence microscopy have enabled the systematic study of morphological changes in large populations of cells induced by chemical and genetic perturbations, facilitating the discovery of signaling pathways underlying diseases and the development of new pharmacological treatments. In these studies, though, due to the complexity of the data, quantification and analysis of morphological features are for the vast majority handled manually, slowing significantly data processing and limiting often the information gained to a descriptive level. Thus, there is an urgent need for developing highly efficient automated analysis and processing tools for fluorescent images. In this paper, we present the application of a method based on the shearlet representation for confocal image analysis of neurons. The shearlet representation is a newly emerged method designed to combine multiscale data analysis with superior directional sensitivity, making this approach particularly effective for the representation of objects defined over a wide range of scales and with highly anisotropic features. Here, we apply the shearlet representation to problems of soma detection of neurons in culture and extraction of geometrical features of neuronal processes in brain tissue, and propose it as a new framework for large-scale fluorescent image analysis of biomedical data.

Fractal dimension as a measure of altered actin cytoskeleton in MC3T3-E1 cells under simulated microgravity using 3-D/2-D clinostats

Osteoblasts, the bone-forming cells, respond to various mechanical forces, such as stretch and fluid shear force in essentially similar ways. The cytoskeleton, as the load-bearing architecture of the cell, is sensitive to altered inertial forces. Disruption of the cytoskeleton will result in alteration of cellular structure and function. However, it is difficult to quantitatively illustrate cytoskeletal rearrangement because of the complexity of cytoskeletal structure. Usually, the morphological changes in actin organization caused by external stimulus are basically descriptive. In this study, fractal dimensions (D) analysis was used to quantify the morphological changes in the actin cytoskeleton of osteoblast-like cells (MC3T3-E1) under simulated microgravity using 3-D/2-D clinostats. The ImageJ software was used to count the fractal dimension of actin cytoskeleton by box-counting methods. Real-time PCR and immunofluroscent assays were used to further confirm the results obtained by fractal dimension analysis. The results showed significant decreases in D value of actin cytoskeleton, β-actin mRNA expression, and the mean fluorescence intensity of F-actin in osteoblast-like cells after 24 or 48 h of incubation under 3-D/2-D clinorotation condition compared with control. The findings indicate that 3-D/2-D clinorotation affects both actin cytoskeleton architecture and mRNA expression, and fractal may be a promising approach for quantitative analysis of the changes in cytoskeleton in different environments.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

Adhesamine, a new synthetic molecule, accelerates differentiation and prolongs survival of primary cultured mouse hippocampal neurons

Attachment to the substrate is essential for both survival and differentiation of various kinds of cells, such as neurons and epithelial cells. We recently found a small synthetic molecule, adhesamine, which boosts adhesion and growth of mammalian cells. In the present study, we applied adhesamine to primary cultured hippocampal neuronal cells and compared its effects with those of PLL (poly-L-lysine), which is widely used as a substrate for cell cultures. Neurons grown on adhesamine-coated coverslips survived for up to 1 month without a feeder layer of glial cells, and had greater viability than cells grown on PLL-coated coverslips. Morphological analysis revealed that neurons cultured with adhesamine exhibited earlier differentiation, i.e. earlier axonal outgrowth and dendritic maturation with enhanced neurite branching, than neurons cultured with PLL. Synaptic formation and postsynaptic responses were evident as early as 4 days in cells cultured with adhesamine. Acceleration of differentiation is mediated by earlier activation of the signalling pathways from heparan sulfate in the extracellular matrix to both FAK (focal adhesion kinase) and MAPK (mitogen-activated protein kinase). Improved survival rates and accelerated maturation of neurons exposed to adhesamine suggest that this completely synthetic molecule may be a useful reagent for culturing neuronal cells.

Maximization of the connectivity repertoire as a statistical principle governing the shapes of dendritic arbors

The shapes of dendritic arbors are fascinating and important, yet the principles underlying these complex and diverse structures remain unclear. Here, we analyzed basal dendritic arbors of 2,171 pyramidal neurons sampled from mammalian brains and discovered 3 statistical properties: the dendritic arbor size scales with the total dendritic length, the spatial correlation of dendritic branches within an arbor has a universal functional form, and small parts of an arbor are self-similar. We proposed that these properties result from maximizing the repertoire of possible connectivity patterns between dendrites and surrounding axons while keeping the cost of dendrites low. We solved this optimization problem by drawing an analogy with maximization of the entropy for a given energy in statistical physics. The solution is consistent with the above observations and predicts scaling relations that can be tested experimentally. In addition, our theory explains why dendritic branches of pyramidal cells are distributed more sparsely than those of Purkinje cells. Our results represent a step toward a unifying view of the relationship between neuronal morphology and function.

Assessment and comparison of neural morphology through metrical feature extraction and analysis in neuron and neuron-glia cultures

The morphology of dissociated single cerebellar Purkinje cells obtained from wild-type P1 CD1 mice was assessed in the absence and in the presence of glia. A dedicated noninvasive technique based on optical microscopy was developed. Image processing algorithms were implemented to extract metrical features characterizing cell structure and dendritic arborization. The morphological features were analyzed in order to identify quantitative differences in Purkinje cell morphology due to interactions with astrocytes.

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.

Fractal and image analysis of morphological changes in the actin cytoskeleton of neonatal cardiac fibroblasts in response to mechanical stretch

Cardiac fibroblasts are the most numerous cells in the heart and are critical in the formation and normal functioning of the organ. Cardiac fibroblasts are firmly attached to and surrounded by extracellular matrix (ECM). Mechanical forces transmitted through interaction with the ECM can result in changes of overall cellular shape, cytoskeletal organization, proliferation, and gene expression of cardiac fibroblasts. These responses may be different in the normally functioning heart, when compared with various pathological conditions, including inflammation or hypertrophy. It is apparent that cellular phenotype and physiology, in turn, are affected by multiple signal transduction pathways modulated directly by the state of polymerization of the actin cytoskeleton. Morphological changes in actin organization resulting from response to adverse conditions in fibroblasts and other cell types are basically descriptive. Some studies have approached quantifying changes in actin cytoskeletal morphology, but these have involved complex and difficult procedures. In this study, we apply image analysis and non-Euclidian geometrical fractal analysis to quantify and describe changes induced in the actin cytoskeleton of cardiac fibroblasts responding to mechanical stress. Characterization of these rapid responses of fibroblasts to mechanical stress may provide insight into the regulation of fibroblasts behavior and gene expression during heart development and disease.

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.

The Sholl analysis of neuronal cell images: Semi-log or log–log method?

Although the Sholl analysis is a quantitative method for morphometric neuronal studies and its application provides many benefits to neurobiology since it is obvious, common and meaningful, there are many unresolved theoretical issues that need to be addressed. Nevertheless, it can be practiced without much background or sophistication. The two different methods of the Sholl analysis--log-log and semi-log--have been applied previously without a clear basis as to what to use. To make an adequate choice of the method, one should try and accept that one which gives a better result. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be found in basic mathematics. In order to compare the results of applications of the semi-log and log-log methods to the same neuron, we introduce the concept of determination ratio as the ratio of the coefficient of determination for the semi-log method and that for the log-log method. If the semi-log method is better as related to the log-log method, the value of this parameter is larger than 1. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method is more frequently utilizable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log-log method. We also compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns and found that between the Sholl and fractal analysis exist various and important analogies.

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.