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.

A comparison of manual neuronal reconstruction from biocytin histology or 2-photon imaging: morphometry and computer modeling

Accurate 3D reconstruction of neurons is vital for applications linking anatomy and physiology. Reconstructions are typically created using Neurolucida after biocytin histology (BH). An alternative inexpensive and fast method is to use freeware such as Neuromantic to reconstruct from fluorescence imaging (FI) stacks acquired using 2-photon laser-scanning microscopy during physiological recording. We compare these two methods with respect to morphometry, cell classification, and multicompartmental modeling in the NEURON simulation environment. Quantitative morphological analysis of the same cells reconstructed using both methods reveals that whilst biocytin reconstructions facilitate tracing of more distal collaterals, both methods are comparable in representing the overall morphology: automated clustering of reconstructions from both methods successfully separates neocortical basket cells from pyramidal cells but not BH from FI reconstructions. BH reconstructions suffer more from tissue shrinkage and compression artifacts than FI reconstructions do. FI reconstructions, on the other hand, consistently have larger process diameters. Consequently, significant differences in NEURON modeling of excitatory post-synaptic potential (EPSP) forward propagation are seen between the two methods, with FI reconstructions exhibiting smaller depolarizations. Simulated action potential backpropagation (bAP), however, is indistinguishable between reconstructions obtained with the two methods. In our hands, BH reconstructions are necessary for NEURON modeling and detailed morphological tracing, and thus remain state of the art, although they are more labor intensive, more expensive, and suffer from a higher failure rate due to the occasional poor outcome of histological processing. However, for a subset of anatomical applications such as cell type identification, FI reconstructions are superior, because of indistinguishable classification performance with greater ease of use, essentially 100% success rate, and lower cost.

Mossy cell dendritic structure quantified and compared with other hippocampal neurons labeled in rats in vivo

Mossy cells are likely to contribute to normal hippocampal function and to the pathogenesis of neurologic disorders that involve the hippocampus, including epilepsy. Mossy cells are the least well-characterized excitatory neurons in the hippocampus. Their somatic and dendritic morphology has been described qualitatively but not quantitatively. In the present study rat mossy cells were labeled intracellularly with biocytin in vivo. Somatic and dendritic structure was reconstructed three-dimensionally. For comparison, granule cells, CA3 pyramidal cells, and CA1 pyramidal cells were labeled and analyzed using the same approach. Among the four types of hippocampal neurons, granule cells had the smallest somata, fewest primary dendrites and dendritic branches, and shortest total dendritic length. CA1 pyramidal cells had the most dendritic branches and longest total dendritic length. Mossy cells and CA3 pyramidal cells both had large somata and similar total dendritic lengths. However, mossy cell dendrites branched less than CA3 pyramidal cells, especially close to the soma. These findings suggest that mossy cells have dendritic features that are not identical to any other type of hippocampal neuron. Therefore, electrotonic properties that depend on soma-dendritic structure are likely to be distinct in mossy cells compared to other neurons.

Principal component and cluster analysis of layer V pyramidal cells in visual and non-visual cortical areas projecting to the primary visual cortex of the mouse

The long-distance corticocortical connections between visual and nonvisual sensory areas that arise from pyramidal neurons located within layer V can be considered as a subpopulation of feedback connections. The purpose of the present study is to determine if layer V pyramidal neurons from visual and nonvisual sensory cortical areas that project onto the visual cortex (V1) constitute a homogeneous population of cells. Additionally, we ask whether dendritic arborization relates to the target, the sensory modality, the hierarchical level, or laterality of the source cortical area. Complete 3D reconstructions of dendritic arbors of retrogradely labeled layer V pyramidal neurons were performed for neurons of the primary auditory (A1) and somatosensory (S1) cortices and from the lateral (V2L) and medial (V2M) parts of the secondary visual cortices of both hemispheres. The morphological parameters extracted from these reconstructions were subjected to principal component analysis (PCA) and cluster analysis. The PCA showed that neurons are distributed within a continuous range of morphologies and do not form discrete groups. Nevertheless, the cluster analysis defines neuronal groups that share similar features. Each cortical area includes neurons belonging to several clusters. We suggest that layer V feedback connections within a single cortical area comprise several cell types.

Non-homogeneous stereological properties of the rat hippocampus from high-resolution 3D serial reconstruction of thin histological sections

Integrating hippocampal anatomy from neuronal dendrites to whole system may help elucidate its relation to function. Toward this aim, we digitally traced the cytoarchitectonic boundaries of the dentate gyrus (DG) and areas CA3/CA1 throughout their entire longitudinal extent from high-resolution images of thin cryostatic sections of adult rat brain. The 3D computational reconstruction identified all isotropic 16 μm voxels with appropriate subregions and layers (http://krasnow1.gmu.edu/cn3/hippocampus3d). Overall, DG, CA3, and CA1 occupied comparable volumes (15.3, 12.2, and 18.8 mm(3), respectively), but displayed substantial rostrocaudal volumetric gradients: CA1 made up more than half of the posterior hippocampus, whereas CA3 and DG were more prominent in the anterior regions. The CA3/CA1 ratio increased from ∼0.4 to ∼1 septo-temporally because of a specific change in stratum radiatum volume. Next we virtually embedded 1.8 million neuronal morphologies stochastically resampled from 244 digital reconstructions, emulating the dense packing of granular and pyramidal layers, and appropriately orienting the principal dendritic axes relative to local curvature. The resulting neuropil occupancy reproduced recent electron microscopy data measured in a restricted location. Extension of this analysis across each layer and subregion over the whole hippocampus revealed highly non-homogeneous dendritic density. In CA1, dendritic occupancy was >60% higher temporally than septally (0.46 vs. 0.28, s.e.m. ∼0.05). CA3 values varied both across subfields (from 0.35 in CA3b/CA3c to 0.50 in CA3a) and layers (0.48, 0.34, and 0.27 in oriens, radiatum, and lacunosum-moleculare, respectively). Dendritic occupancy was substantially lower in DG, especially in the supra-pyramidal blade (0.18). The computed probability of dendrodendritic collision significantly correlated with expression of the membrane repulsion signal Down syndrome cell adhesion molecule (DSCAM). These heterogeneous stereological properties reflect and complement the non-uniform molecular composition, circuit connectivity, and computational function of the hippocampus across its transverse, longitudinal, and laminar organization.

Spatial heterogeneity of passive electrical transfer properties of neuronal dendrites due to their metrical asymmetry

The complex and diverse geometry of neuronal dendrites determines the different morphological types of neurons and influences the generation of complex and diverse discharge patterns at the cell output. The recent finding that each temporal pattern has its spatial signature in the form of a combination of high- and low-depolarization states of asymmetrical dendritic branches with active membrane properties raises the question of the nature of such characteristic spatial heterogeneity of electrical states. To answer this, we consider passive dendrites as a conventional reference case using the known current transfer functions, which we complete by corresponding parametric sensitivity functions. These functions for metrically asymmetrical bifurcations of different sizes, as the simplest elements constituting arborizations of arbitrary geometry, are analyzed under different membrane conductivity conditions related to the intensity of activation of ion channels. Characteristic relationships are obtained on the one hand among the size (branch lengths), metrical asymmetry (difference between sister branches in length and/or diameter), and membrane conductivity, and on the other hand, for the difference between the branches in their current transfer effectiveness as an indicator of their electrical asymmetry (heterogeneity). These relationships (i) allow the introduction of a biophysically based criterion for the electrical distinction between metrically asymmetrical branches, (ii) show how the difference first increases and then decreases with increasing membrane conductivity, and (iii) show that the greatest electrical heterogeneity occurs in a lower or higher range of conductivity, corresponding to larger or smaller bifurcation size. As a consequence, the characteristic low-, medium-, and high-conductance states are derived such that metrically asymmetrical parts of simple and complex trees are electrically distinct when the membrane conductivity lies in the size-related medium range, and indistinct otherwise.

Introduction to the fractality principle of consciousness and the sentyon postulate

Recently, consciousness research has gained much attention. Indeed, the question at stake is significant: why is the brain not just a computing device, but generates a perception from within? Ambitious endeavors trying to simulate the entire human brain assume that the algorithm will do the trick: as soon as we assemble the brain in a computer and increase the number of operations per time, consciousness will emerge by itself. I disagree with this simplistic representation. My argument emerges from the "atomism paradox": the irreducible space of the consciously perceived world, the endospace is incompatible with the reducible and decomposable architecture of the brain or a computer. I will first discuss the fundamental challenges in current consciousness models and then propose a new model based on the fractality principle: "the whole is in each of its parts". This new model copes with the atomism paradox by implementing an iterative mapping of information from higher order brain structures to smaller scales on the cellular and molecular level, which I will refer to as "fractalization". This information fractalization gives rise to a new form of matter that is conscious ("bright matter"). Bright matter is composed of conscious particles or units named "sentyons". The internal fractality of these sentyons will close a loop (the "psychic loop") in a recurrent fractal neural network (RFNN) that allows for continuous and complete information transformation and sharing between higher order brain structures and the endpoint substrate of consciousness at the molecular level.

Modeling sharp wave-ripple complexes through a CA3-CA1 network model with chemical synapses

The hippocampus, and particularly the CA3 and CA1 areas, exhibit a variety of oscillatory rhythms that span frequencies from the slow theta range (4-10 Hz) up to fast ripples (200 Hz). Various computational models of different complexities have been developed in an effort to simulate such population oscillations. Nevertheless the mechanism that underlies the so called Sharp Wave-Ripple complex (SPWR), observed in extracellular recordings in CA1, still remains elusive. We present here, the combination of two simple but realistic models of the rat CA3 and CA1 areas, connected together in a feedforward scheme mimicking Schaffer collaterals. Both network models are computationally simple one-dimensional arrays of excitatory and inhibitory populations interacting only via fast chemical synapses. Connectivity schemes and postsynaptic potentials are based on physiological data, yielding a realistic network topology. The CA3 model exhibits quasi-synchronous population bursts, which give rise to sharp wave-like deep depolarizations in the CA1 dendritic layer accompanied by transient field oscillations at ≈ 150-200 Hz in the somatic layer. The frequency and synchrony of these oscillations is based on interneuronal activity and fast-decaying recurrent inhibition in CA1. Pyramidal cell spikes are sparse and come from a subset of cells receiving stronger than average excitatory input from CA3. The model is shown to accurately reproduce a large number of basic characteristics of SPWRs and yields a new mechanism for the generation of ripples, offering an interpretation to a range of neurophysiological observations, such as the ripple disruption by halothane and the selective firing of pyramidal cells during ripples, which may have implications for memory consolidation during SPWRs.

Corticosterone reduces dendritic complexity in developing hippocampal CA1 neurons

Although prolonged stress and corticosteroid exposure induce morphological changes in the hippocampal CA3 area, the adult CA1 area is quite resistant to such changes. Here we addressed the question whether elevated corticosteroid hormone levels change dendritic complexity in young, developing CA1 cells. In organotypic cultures (prepared from P5 rats) that were 14-21 days cultured in vitro, two doses of corticosterone (30 and 100 nM) were tested. Dendritic morphology of CA1 neurons was established by imaging neurons filled with the fluorescent dye Alexa. Application of 100 nM corticosterone for 20 minutes induced atrophy of the apical dendritic tree 1-4 hours later. Fractal analysis showed that total neuronal complexity was reduced twofold when compared with vehicle-treated neurons. Exposing organotypic slices to 30 nM corticosterone reduced apical length in a more delayed manner: only neurons examined more than 2 hours after exposure to corticosterone showed atrophy of the apical dendritic tree. Neither dose of corticosterone affected the length of basal dendrites or spine density. Corticosterone was ineffective in changing morphology of the apical dendrites when tested in the presence of the glucocorticoid receptor antagonist RU38486. These results suggest that high physiological levels of corticosterone, via activation of the glucocorticoid receptor, can, during the course of only a few hours, reduce the dendritic complexity of CA1 pyramidal neurons in young, developing hippocampal tissue. These findings suggest that it is relevant to maintain plasma corticosterone levels low during hippocampal development.

L-Measure: a web-accessible tool for the analysis, comparison and search of digital reconstructions of neuronal morphologies

L-Measure (LM) is a freely available software tool for the quantitative characterization of neuronal morphology. LM computes a large number of neuroanatomical parameters from 3D digital reconstruction files starting from and combining a set of core metrics. After more than six years of development and use in the neuroscience community, LM enables the execution of commonly adopted analyses as well as of more advanced functions. This report illustrates several LM protocols: (i) extraction of basic morphological parameters, (ii) computation of frequency distributions, (iii) measurements from user-specified subregions of the neuronal arbors, (iv) statistical comparison between two groups of cells and (v) filtered selections and searches from collections of neurons based on any Boolean combination of the available morphometric measures. These functionalities are easily accessed and deployed through a user-friendly graphical interface and typically execute within few minutes on a set of approximately 20 neurons. The tool is available at http://krasnow.gmu.edu/cn3 for either online use on any Java-enabled browser and platform or download for local execution under Windows and Linux.

Petilla terminology: nomenclature of features of GABAergic interneurons of the cerebral cortex

Neuroscience produces a vast amount of data from an enormous diversity of neurons. A neuronal classification system is essential to organize such data and the knowledge that is derived from them. Classification depends on the unequivocal identification of the features that distinguish one type of neuron from another. The problems inherent in this are particularly acute when studying cortical interneurons. To tackle this, we convened a representative group of researchers to agree on a set of terms to describe the anatomical, physiological and molecular features of GABAergic interneurons of the cerebral cortex. The resulting terminology might provide a stepping stone towards a future classification of these complex and heterogeneous cells. Consistent adoption will be important for the success of such an initiative, and we also encourage the active involvement of the broader scientific community in the dynamic evolution of this project.

Successes and rewards in sharing digital reconstructions of neuronal morphology

The computer-assisted three-dimensional reconstruction of neuronal morphology is becoming an increasingly popular technique to quantify the arborization patterns of dendrites and axons. The resulting digital files are suitable for comprehensive morphometric analyses as well as for building anatomically realistic compartmental models of membrane biophysics and neuronal electrophysiology. The digital tracings acquired in a lab for a specific purpose can be often re-used by a different research group to address a completely unrelated scientific question, if the original investigators are willing to share the data. Since reconstructing neuronal morphology is a labor-intensive process, data sharing and re-analysis is particularly advantageous for the neuroscience and biomedical communities. Here we present numerous cases of "success stories" in which digital reconstructions of neuronal morphology were shared and re-used, leading to additional, independent discoveries and publications, and thus amplifying the impact of the "source" study for which the data set was first collected. In particular, we overview four main applications of this kind of data: comparative morphometric analyses, statistical estimation of potential synaptic connectivity, morphologically accurate electrophysiological simulations, and computational models of neuronal shape and development.

Current trends in shape and texture analysis in neurology: aspects of the morphological substrate of volume and wiring transmission

Morphological and morphometrical studies of neural shape and texture are becoming more and more important in the field of neurosciences due to the recognized close link between shape and function at molecular, cellular and tissutal level. Indeed, some different morphological classes of neurons are known to be correlated to well defined functional classes; several neurological pathologies are associated with modification of neuronal shape; during a neural development, cells impose geometrical and physical constrains at one another. The understanding of these fundamental processes requires morphological/morphometrical analysis. In addition, the geometric properties at the individual level plays a relevant role in defining the actual and the potential global connectivity of the system. From this standpoint it will be important to study the relationship between the shape descriptors illustrated in this paper and the potential global connectivity of the system. In such a context, the classical multivariate statistical tools of analysis (and probably new ones) will become necessary to correctly utilize huge set of information provided from shape and texture descriptors. In this review, avoiding to consider some common measures of shape, such as area, perimeter, perimeter/area ratio, eccentricity, we considered only shape and texture analysis methods classified within the set of scalar transform techniques, discussing their advantages and limitations, especially regarding their application to neuromorphometry.

Adaptive reconfiguration of fractal small-world human brain functional networks

Brain function depends on adaptive self-organization of large-scale neural assemblies, but little is known about quantitative network parameters governing these processes in humans. Here, we describe the topology and synchronizability of frequency-specific brain functional networks using wavelet decomposition of magnetoencephalographic time series, followed by construction and analysis of undirected graphs. Magnetoencephalographic data were acquired from 22 subjects, half of whom performed a finger-tapping task, whereas the other half were studied at rest. We found that brain functional networks were characterized by small-world properties at all six wavelet scales considered, corresponding approximately to classical delta (low and high), , alpha, beta, and gamma frequency bands. Global topological parameters (path length, clustering) were conserved across scales, most consistently in the frequency range 2-37 Hz, implying a scale-invariant or fractal small-world organization. Dynamical analysis showed that networks were located close to the threshold of order/disorder transition in all frequency bands. The highest-frequency gamma network had greater synchronizability, greater clustering of connections, and shorter path length than networks in the scaling regime of (lower) frequencies. Behavioral state did not strongly influence global topology or synchronizability; however, motor task performance was associated with emergence of long-range connections in both beta and gamma networks. Long-range connectivity, e.g., between frontal and parietal cortex, at high frequencies during a motor task may facilitate sensorimotor binding. Human brain functional networks demonstrate a fractal small-world architecture that supports critical dynamics and task-related spatial reconfiguration while preserving global topological parameters.

Effects of beta-catenin on dendritic morphology and simulated firing patterns in cultured hippocampal neurons

Beta-catenin is an intracellular signaling molecule that has been shown to be important in activity-dependent dendritic morphogenesis. Here, we investigate the detailed morphological changes elicited in dendritic arbors of cultured hippocampal neurons by overexpression of beta-catenin, and we simulate the electrophysiological consequences of these changes. Compared to control neurons, cells overexpressing beta-catenin have dendritic arbors with significantly greater surface area and more branches, as well as different topological characteristics. To investigate possible effects of beta-catenin expression on the electrophysiological properties of neurons, we converted confocal images of neurons expressing beta-catenin into computational simulator formats using parameters that evenly distributed voltage-dependent channels across the cells' membranes. In simulated current clamp experiments, somata were injected with a normalized current such that the observed electrophysiological differences in the neurons would be due only to morphological differences. We found that the morphology of beta-catenin-expressing neurons contributes to significantly smaller action potential amplitude and greater sensitivity than seen in control neurons. As a consequence, beta-catenin-expressing neurons tended to exhibit higher spike rates and needed less excitation to induce firing. These findings show that beta-catenin, by modifying dendritic arborization, could have profound influences on the electrophysiological behavior of neurons.

Mobilizing the base of neuroscience data: the case of neuronal morphologies

Despite the explosive growth of bioinformatics, data sharing has not yet become routine in neuroscience, possibly because of several broad-spanning issues, from data heterogeneity to privacy regulations. We present the case of neuronal morphology as an ideal example of shareable data. Drawing from recent experience, we argue that the tremendous research potential of existing (and largely unused) digital reconstructions should diffuse any reticence to sharing this type of data.

Spatial embedding of neuronal trees modeled by diffusive growth

The relative importance of the intrinsic and extrinsic factors determining the variety of geometric shapes exhibited by dendritic trees remains unclear. This question was addressed by developing a model of the growth of dendritic trees based on diffusion-limited aggregation process. The model reproduces diverse neuronal shapes (i.e., granule cells, Purkinje cells, the basal and apical dendrites of pyramidal cells, and the axonal trees of interneurons) by changing only the size of the growth area, the time span of pruning, and the spatial concentration of 'neurotrophic particles'. Moreover, the presented model shows how competition between neurons can affect the shape of the dendritic trees. The model reveals that the creation of complex (but reproducible) dendrite-like trees does not require precise guidance or an intrinsic plan of the dendrite geometry. Instead, basic environmental factors and the simple rules of diffusive growth adequately account for the spatial embedding of different types of dendrites observed in the cortex. An example demonstrating the broad applicability of the algorithm to model diverse types of tree structures is also presented.

Constitutive Ras activity induces hippocampal hypertrophy and remodeling of pyramidal neurons in synRas mice

The small G protein Ras, which is involved critically in neurotrophic signal transduction, has been implicated in neuronal plasticity of both the developing and the adult nervous systems. In the present study, the cumulative effects of constitutive Ras activity from early in postnatal development into the adult upon the morphology of hippocampal pyramidal neurons were investigated in synRas mice overexpressing Val12-Ha-Ras postmitotically under the control of the rat synapsin I promoter. In synRas mice, stereologic investigations revealed hypertrophy of the hippocampus associated with an increase in perikaryal size of pyramidal neurons within the CA2/CA3 region and the gyrus dentatus. Morphometric analyses of Lucifer Yellow-filled CA1 pyramidal neurons, in addition, demonstrated considerable expansion of dendritic arbors. The increase in basal dendritic size was caused primarily by alterations of intermediate and distal segments and was associated with an enlarged dendritic surface. Apical dendrites showed similar but more moderate changes, which were attributed mainly to elongation of terminal segments. Sholl analyses illustrated higher complexity of both basal and apical trees. Despite significant morphologic alterations, dendritic arbors preserve their major design principles. The synaptic density within the stratum radiatum of CA1 remained unchanged; however, increases in the total hippocampal volume and in apical dendritic size imply an increment in the absolute number of synaptic contacts. The data presented here suggest a critical involvement of Ras dependent signaling in morphoregulatory processes during the maturation and in the maintenance of hippocampal pyramidal neurons.

Quantitative morphometry of hippocampal pyramidal cells: Differences between anatomical classes and reconstructing laboratories

The dendritic trees of hippocampal pyramidal cells play important roles in the establishment and regulation of network connectivity, synaptic plasticity, and firing dynamics. Several laboratories routinely reconstruct CA3 and CA1 dendrites to correlate their three-dimensional structure with biophysical, electrophysiological, and anatomical observables. To integrate and assess the consistency of the quantitative data available to the scientific community, we exhaustively analyzed 143 completely reconstructed neurons intracellularly filled and digitized in five different laboratories from 10 experimental conditions. Thirty morphometric parameters, including the most common neuroanatomical measurements, were extracted from all neurons. A consistent fraction of parameters (11 of 30) was significantly different between CA3 and CA1 cells. A considerably large number of parameters was also found that discriminated among neurons within the same morphological class, but reconstructed in different laboratories. These interlaboratory differences (8 of 30 parameters) far outweighed the differences between experimental conditions within a single lab, such as aging or preparation method (at most two significant parameters). The set of morphometrics separating anatomical regions and that separating reconstructing laboratories were almost entirely nonoverlapping. CA3 and CA1 neurons could be distinguished by global quantities such as branch order and Sholl distance. Differences among laboratories were largely due to local variables such as branch diameter and local bifurcation angles. Only one parameter (a ratio of branch diameters) separated both morphological classes and reconstructing laboratories. Compartmental simulations of electrophysiological activity showed that both differences between anatomical classes and reconstructing laboratories could dramatically affect the firing rate of these neurons under different experimental conditions.

Development of rat CA1 neurones in acute versus organotypic slices: role of experience in synaptic morphology and activity

Despite their wide use, the physiological relevance of organotypic slices remains controversial. Such cultures are prepared at 5 days postnatal. Although some local circuitry remains intact, they develop subsequently in isolation from the animal and hence without plasticity due to experience. Development of synaptic connectivity and morphology might be expected to proceed differently under these conditions than in a behaving animal. To address these questions, patch-clamp techniques and confocal microscopy were used in the CA1 region of the rat hippocampus to compare acute slices from the third postnatal week with various stages of organotypic slices. Acute slices prepared at postnatal days (P) 14, 17 and 21 were found to be developmentally equivalent to organotypic slices cultured for 1, 2 and 3 weeks, respectively, in terms of development of synaptic transmission and dendritic morphology. The frequency of inhibitory and excitatory miniature synaptic currents increased in parallel. Development of dendritic length and primary branching as well as spine density and proportions of different spine types were also similar in both preparations,at these corresponding stages. The most notable difference between organotypic and acute slices was a four- to five-fold increase in the absolute frequency of glutamatergic (but not GABAergic)miniature postsynaptic currents in organotypic slices. This was probably related to an increase in complexity of higher order dendritic branching in organotypic slices, as measured by fractal analysis, resulting in an increased total synapse number. Both increased excitatory miniature synaptic current frequency and dendritic complexity were already established during the first week in culture. The level of complexity then stayed constant in both preparations over subsequent stages, with synaptic frequency increasing in parallel. Thus, although connectivity was greater in organotypic slices, once this was established, development continued in both preparations at are markably similar rate. We conclude that, for the parameters studied, changes seem to be preprogrammed by 5 days and their subsequent development is largely independent of environment.

Development of a mathematical method for classifying and comparing tree architecture using parameters from a topological model of a trifurcating botanical tree

This paper describes a model for the topological mapping of trifurcating botanical trees. The model was based on a system of modular units that represented the interconnectivity of shoot meristems (terminal segments) and internodes (internal segments) within whole plant canopies, organized with increasing centrifugal ordering. The model was capable of describing the dynamics of plant growth as expressed by changes in topological parameters over time. Preliminary calculations for experimental trees indicated that the model represents growth in a biologically sound manner. Methods are described for the calculation of the architecture parameters size, size-complexity, structural complexity, and tree asymmetry index (TAI). Parameter calculations were based on the mathematical principles developed for the classification of bifurcating dendrite trees, and were designed to both extract structural information, and to enable statistical comparison between trees of different size. Parameters were mathematically adjusted for trifurcation, and appeared to be able to represent quantitatively the architectural properties of tree structures. In addition to the calculation of the TAI for trifurcating trees, new methods were developed to enable comparisons to be made of the architectural complexity of trifurcating trees of differing size. These were based on the principle of the pair-wise comparison of the mean centrifugal order number (MCON) with respect to segments against highest order number. We argue and illustrate that this principle can be more informative than that of pair-wise comparison of the MCON against tree degree (topological size). Further improvements to this method were made by examining branching points (vertices) rather than segments (links) to calculate the MCON.

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.

Morphological characterization of in vitro neuronal networks

We use in vitro neuronal networks as a model system for studying self-organization processes in the nervous system. We follow the neuronal growth process, from isolated neurons to fully connected two-dimensional networks. The mature networks are mapped into connected graphs and their morphological characteristics are measured. The distributions of segment lengths, node connectivity, and path length between nodes, and the clustering coefficient of the networks are used to characterize network morphology and to demonstrate that our networks fall into the category of small-world networks.

Generation, description and storage of dendritic morphology data

It is generally assumed that the variability of neuronal morphology has an important effect on both the connectivity and the activity of the nervous system, but this effect has not been thoroughly investigated. Neuroanatomical archives represent a crucial tool to explore structure-function relationships in the brain. We are developing computational tools to describe, generate, store and render large sets of three-dimensional neuronal structures in a format that is compact, quantitative, accurate and readily accessible to the neuroscientist. Single-cell neuroanatomy can be characterized quantitatively at several levels. In computer-aided neuronal tracing files, a dendritic tree is described as a series of cylinders, each represented by diameter, spatial coordinates and the connectivity to other cylinders in the tree. This 'Cartesian' description constitutes a completely accurate mapping of dendritic morphology but it bears little intuitive information for the neuroscientist. In contrast, a classical neuroanatomical analysis characterizes neuronal dendrites on the basis of the statistical distributions of morphological parameters, e.g. maximum branching order or bifurcation asymmetry. This description is intuitively more accessible, but it only yields information on the collective anatomy of a group of dendrites, i.e. it is not complete enough to provide a precise 'blueprint' of the original data. We are adopting a third, intermediate level of description, which consists of the algorithmic generation of neuronal structures within a certain morphological class based on a set of 'fundamental', measured parameters. This description is as intuitive as a classical neuroanatomical analysis (parameters have an intuitive interpretation), and as complete as a Cartesian file (the algorithms generate and display complete neurons). The advantages of the algorithmic description of neuronal structure are immense. If an algorithm can measure the values of a handful of parameters from an experimental database and generate virtual neurons whose anatomy is statistically indistinguishable from that of their real counterparts, a great deal of data compression and amplification can be achieved. Data compression results from the quantitative and complete description of thousands of neurons with a handful of statistical distributions of parameters. Data amplification is possible because, from a set of experimental neurons, many more virtual analogues can be generated. This approach could allow one, in principle, to create and store a neuroanatomical database containing data for an entire human brain in a personal computer. We are using two programs, L-NEURON and ARBORVITAE, to investigate systematically the potential of several different algorithms for the generation of virtual neurons. Using these programs, we have generated anatomically plausible virtual neurons for several morphological classes, including guinea pig cerebellar Purkinje cells and cat spinal cord motor neurons. These virtual neurons are stored in an online electronic archive of dendritic morphology. This process highlights the potential and the limitations of the 'computational neuroanatomy' strategy for neuroscience databases.

Dendritic structure of single hippocampal neurons according to sex and hemisphere of origin in middle-aged and elderly human subjects

The organization of basilar dendritic patterns in the CA1 hippocampal region obtained from 13 middle-aged and elderly human subjects was assessed using the Golgi method. Neurons were classified according to hemisphere of origin and the sex of the respective subjects. Three parameters were measured: total dendritic length (TDL), number of dendritic segments (NDS) and average segment length (ASL, which is TDL divided by NDS). Dendritic segments were classified into proximal (first to third order) and distal (fourth order and above). Sex differences were found in distal TDL and in proximal and distal NDS, neurons belonging to males having larger values than those belonging to females. In addition, a hemispheric difference was detected in distal TDL, in which neurons of the left hemisphere had larger values than those of the right hemisphere.

NEURON: a tool for neuroscientists

NEURON is a simulation environment for models of individual neurons and networks of neurons that are closely linked to experimental data. NEURON provides tools for conveniently constructing, exercising, and managing models, so that special expertise in numerical methods or programming is not required for its productive use. This article describes two tools that address the problem of how to achieve computational efficiency and accuracy.