Competitive interactions during dendritic growth: a simple stochastic growth algorithm

Mathematical models of neuronal growth

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites-axons and dendrites-to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

Generation of Granule Cell Dendritic Morphologies by Estimating the Spatial Heterogeneity of Dendritic Branching

Biological realism of dendritic morphologies is important for simulating electrical stimulation of brain tissue. By adding point process modeling and conditional sampling to existing generation strategies, we provide a novel means of reproducing the nuanced branching behavior that occurs in different layers of granule cell dendritic morphologies. In this study, a heterogeneous Poisson point process was used to simulate branching events. Conditional distributions were then used to select branch angles depending on the orthogonal distance to the somatic plane. The proposed method was compared to an existing generation tool and a control version of the proposed method that used a homogeneous Poisson point process. Morphologies were generated with each method and then compared to a set of digitally reconstructed neurons. The introduction of a conditionally dependent branching rate resulted in the generation of morphologies that more accurately reproduced the emergent properties of dendritic material per layer, Sholl intersections, and proximal passive current flow. Conditional dependence was critically important for the generation of realistic granule cell dendritic morphologies.

A generative growth model for thalamocortical axonal branching in primary visual cortex

Axonal morphology displays large variability and complexity, yet the canonical regularities of the cortex suggest that such wiring is based on the repeated initiation of a small set of genetically encoded rules. Extracting underlying developmental principles can hence shed light on what genetically encoded instructions must be available during cortical development. Within a generative model, we investigate growth rules for axonal branching patterns in cat area 17, originating from the lateral geniculate nucleus of the thalamus. This target area of synaptic connections is characterized by extensive ramifications and a high bouton density, characteristics thought to preserve the spatial resolution of receptive fields and to enable connections for the ocular dominance columns. We compare individual and global statistics, such as a newly introduced length-weighted asymmetry index and the global segment-length distribution, of generated and biological branching patterns as the benchmark for growth rules. We show that the proposed model surpasses the statistical accuracy of the Galton-Watson model, which is the most commonly employed model for biological growth processes. In contrast to the Galton-Watson model, our model can recreate the log-normal segment-length distribution of the experimental dataset and is considerably more accurate in recreating individual axonal morphologies. To provide a biophysical interpretation for statistical quantifications of the axonal branching patterns, the generative model is ported into the physically accurate simulation framework of Cx3D. In this 3D simulation environment we demonstrate how the proposed growth process can be formulated as an interactive process between genetic growth rules and chemical cues in the local environment.

Stochastic Modeling of Radiation-induced Dendritic Damage on in silico Mouse Hippocampal Neurons

Cognitive dysfunction associated with radiotherapy for cancer treatment has been correlated to several factors, one of which is changes to the dendritic morphology of neuronal cells. Alterations in dendritic geometry and branching patterns are often accompanied by deficits that impact learning and memory. The purpose of this study is to develop a novel predictive model of neuronal dendritic damages caused by exposure to low linear energy transfer (LET) radiation, such as X-rays, γ-rays and high-energy protons. We established in silico representations of mouse hippocampal dentate granule cell layer (GCL) and CA1 pyramidal neurons, which are frequently examined in radiation-induced cognitive decrements. The in silico representations are used in a stochastic model that describes time dependent dendritic damage induced by exposure to low LET radiation. Changes in morphometric parameters, such as total dendritic length, number of branch points and branch number, including the Sholl analysis for single neurons are described by the model. Our model based predictions for different patterns of morphological changes based on energy deposition in dendritic segments (EDDS) will serve as a useful basis to compare specific patterns of morphological alterations caused by EDDS mechanisms.

Synergistic effects of 3D ECM and chemogradients on neurite outgrowth and guidance: a simple modeling and microfluidic framework

During nervous system development, numerous cues within the extracellular matrix microenvironment (ECM) guide the growing neurites along specific pathways to reach their intended targets. Neurite motility is controlled by extracellular signal sensing through the growth cone at the neurite tip, including chemoattractive and repulsive cues. However, it is difficult to regenerate and restore neurite tracts, lost or degraded due to an injury or disease, in the adult central nervous system. Thus, it is important to evaluate the dynamic interplay between ECM and the concentration gradients of these cues, which would elicit robust neuritogenesis. Such information is critical in understanding the processes involved in developmental biology, and in developing high-fidelity neurite regenerative strategies post-injury, and in drug discovery and targeted therapeutics for neurodegenerative conditions. Here, we quantitatively investigated this relationship using a combination of mathematical modeling and in vitro experiments, and determined the synergistic role of guidance cues and ECM on neurite outgrowth and turning. Using a biomimetic microfluidic system, we have shown that cortical neurite outgrowth and turning under chemogradients (IGF-1 or BDNF) within 3D scaffolds is highly regulated by the source concentration of the guidance cue and the physical characteristics of the scaffold. A mechanistic-driven partial differential equation model of neurite outgrowth has been proposed, which could also be used prospectively as a predictive tool. The parameters for the chemotaxis term in the model are determined from the experimental data using our microfluidic assay. Resulting model simulations demonstrate how neurite outgrowth was critically influenced by the experimental variables, which was further supported by experimental data on cell-surface-receptor expressions. The model results are in excellent agreement with the experimental findings. This integrated approach represents a framework for further elucidation of biological mechanisms underlying neuronal responses of specialized cell types, during various stages of development, and under healthy or diseased conditions.

Stochastic continuous time neurite branching models with tree and segment dependent rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p(γ,t) for finding a tree γ at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number μ(f(n),t) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p(n,t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.

Parsimonious modelling allows generation of the dendrograms of primate striatal medium spiny and pallidal type II neurons using a stochastic algorithm

Data from quantitative three-dimensional analysis of primate striatal medium spiny neurons (MSNs) and pallidal type I and type II neurons were used to search for possible rules underlying the dendritic architecture of these cells. Branching and terminating probabilities per unit length of dendrite were computed from all available measurement points. In the three neuronal groups, terminating probabilities were found to be exponentially increasing functions of the path distance to soma. MSNs and type II branching probabilities could be accurately modelled with decreasing functions of both the metrical (exponential functions) and topological (power functions of the centrifugal branch order) distances to soma. Additionally, type II branching also slightly depended on the distance to the proximal tip of the supporting branches. Type I branching probabilities did not follow these rules accurately. Embedding the modelled probability functions in a stochastic algorithm allowed generation of dendrograms close to those of the real MSNs and pallidal type II neurons, while the algorithm failed to simulate type I dendrites. MSN and pallidal type II neuron branching and terminating probabilities are thus highly dependent on the position in the dendritic arbor. This relationship can be modelled with simple functions and has a strong incidence on the dendrogram structure of the cells concerned. The additional dependence of the branching probability on the within-branch position led us to propose an extension of a previous modelling study by Nowakowski and co-workers which could account for a large range of topological and metrical (length) dendritic tree structures.

Non-parametric algorithmic generation of neuronal morphologies

Generation algorithms allow for the generation of Virtual Neurons (VNs) from a small set of morphological properties. The set describes the morphological properties of real neurons in terms of statistical descriptors such as the number of branches and segment lengths (among others). The majority of reconstruction algorithms use the observed properties to estimate the parameters of a priori fixed probability distributions in order to construct statistical descriptors that fit well with the observed data. In this article, we present a non-parametric generation algorithm based on kernel density estimators (KDEs). The new algorithm is called KDE-NEURON: and has three advantages over parametric reconstruction algorithms: (1) no a priori specifications about the distributions underlying the real data, (2) peculiarities in the biological data will be reflected in the VNs, and (3) ability to reconstruct different cell types. We experimentally generated motor neurons and granule cells, and statistically validated the obtained results. Moreover, we assessed the quality of the prototype data set and observed that our generated neurons are as good as the prototype data in terms of the used statistical descriptors. The opportunities and limitations of data-driven algorithmic reconstruction of neurons are discussed.

Self-organizing mechanism for development of space-filling neuronal dendrites

Neurons develop distinctive dendritic morphologies to receive and process information. Previous experiments showed that competitive dendro-dendritic interactions play critical roles in shaping dendrites of the space-filling type, which uniformly cover their receptive field. We incorporated this finding in constructing a new mathematical model, in which reaction dynamics of two chemicals (activator and suppressor) are coupled to neuronal dendrite growth. Our numerical analysis determined the conditions for dendritic branching and suggested that the self-organizing property of the proposed system can underlie dendritogenesis. Furthermore, we found a clear correlation between dendrite shape and the distribution of the activator, thus providing a morphological criterion to predict the in vivo distribution of the hypothetical molecular complexes responsible for dendrite elongation and branching.

Simulation of motoneuron morphology in three dimensions. I. Building individual dendritic trees

We have developed a computational method that accurately reproduces the three-dimensional (3D) morphology of individual dendritic trees of six cat alpha motoneurons. The first step was simulation of trees with straight branches based on the branch lengths and topology of actual trees. A second step introduced the meandering, or wandering, trajectories observed in natural dendritic branches into the straight-branch tree simulations. These two steps each required only two parameters, one extracted from the data on actual motoneuron dendrites and the other adjusted by comparing simulated and observed trees, using measurements that were independent of the model specifications (i.e., emergent properties). The results suggest that: 1) there is a somatofugal "tropism" (a bias introduced by the environment that affects the trajectory of dendritic branches) that tends to constrain the lateral expansion of alpha motoneuron dendrites; and 2) that most of the meandering of natural dendritic branches can be described by assuming that they are fractal objects with an average fractal dimension D of about 1.05. When analyzed in the same way, the dendrites of gamma motoneurons showed no evidence of a similar tropism, although they had the same fractal dimension of branch meandering.

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.

Deterministic and Stochastic Elements of Axonal Guidance

An enormous literature has been developed on investigations of the growth and guidance of axons during development and after injury. In this review, we provide a guide to this literature as a resource for biomedical investigators. We first review briefly the molecular biology that is known to regulate migration of the growth cone and branching of axonal arbors. We then outline some important fundamental considerations that are important to the modeling of the phenomenology of these guidance effects and of what is known of their underlying internal mechanisms. We conclude by providing some thoughts on the outlook for future biomedical modeling in the field.

Abl tyrosine kinase promotes dendrogenesis by inducing actin cytoskeletal rearrangements in cooperation with Rho family small GTPases in hippocampal neurons

Nonreceptor tyrosine kinase Abl is an actin-binding protein and a key regulator of neuronal axonal development. Although Abl family kinases also are localized in dendrites and are implicated in postsynaptic functions, it is not clear how Abl kinases regulate dendritic morphogenesis. Using a developing hippocampal culture as a model, we found that the inhibition of Abl kinases by STI571 leads to a remarkable simplification of dendritic branching similar to the phenotype caused by an increased activity of small GTPase RhoA. Time-lapse microscopic imaging reveals a prominent reduction of dendritic branching. In contrast, neurons expressing a constitutively active v-abl construct (CA-Abl) show an exuberant microtubule-associated protein 2-positive (MAP2-positive) dendrite outgrowth, suggesting that Abl modulates dendritic growth. Biochemical assays using a glutathione S-transferase pull-down method to determine GTP-bound active Rho GTPases demonstrate that Abl inhibition increases RhoA activity but has no effect on the activity of Rac1 or Cdc42. At the cellular level the alteration of Abl also changes actin organization consistent with RhoA inhibition. Suppression of the RhoA downstream effector Rho kinase reverses STI571-induced dendritic simplification, demonstrating that activity of the Rho pathway is responsible for the Abl-induced changes in dendrogenesis. Furthermore, CA-Abl-induced neurite outgrowth is blocked by the expression of a constitutively active RhoA construct. The CA-Abl phenotype is not affected by destabilization of microtubules but is reversed partially when actin filaments are stabilized with jasplakinolide. Together, these studies support a critical role for Abl kinases in regulating dendrogenesis by inducing actin cytoskeletal rearrangements in cooperation with Rho GTPases.

Transport limited effects in a model of dendritic branching

A variety of stochastic models of dendritic growth in developing neurons have been formulated previously. Such models indicate that the probability of a new branch forming in a growing tree may be modulated by factors such as the number of terminals in the tree and their centrifugal order. However, these models cannot identify any underlying biophysical mechanisms that may cause such dependencies. Here, we explore a new model in which branching depends on the concentration of a branch-determining substance in each terminal segment. The substance is produced in the cell body and is transported by active transport and diffusion to the terminals. The model reveals that transport-limited effects may give rise to the same modulation of branching as indicated by the stochastic models. Different limitations arise if transport is dominated by active transport or by diffusion.

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.

Progress and perspectives in computational neuroanatomy

The tremendous increase in processing power of personal computers has recently allowed the construction of highly sophisticated models of neuronal function and behavior. Anatomy plays a fundamental role in supporting and shaping nervous system activity, yet to date most details of such a role have escaped the efforts of experimental and theoretical neuroscientists, mainly because of the problem's complexity. When accurate cellular morphologies are included in electrophysiological computer simulations, quantitative and qualitative effects of dendritic structure on firing properties can be extensively characterized. Complete models of dendritic morphology can be implemented to allow the computer generation of virtual neurons that model the anatomical characteristics of their real counterparts to a great degree of approximation. From a restricted and already available experimental database, stochastic and statistical algorithms can create an unlimited number of non-identical virtual neurons within several mammalian morphological classes, storing them in a compact and parsimonious format. When modeled neurons are distributed in three-dimensional and biologically plausible rules governing axonal navigation and connectivity are added to the simulations, entire portions of the nervous system can be "grown" as anatomically realistic neural networks. These computational constructs are useful to determine the influence of local geometry on system neuroanatomy, and to investigate systematically the mutual interactions between anatomical parameters and electrophysiological activity at the network level. A detailed computer model of a "virtual brain" that was truly equivalent to the biological structure could in principle allow scientists to carry out experiments that could not be performed on real nervous systems because of physical constraints. The computational approach to neuroanatomy is just at its beginning, but has a great potential to enhance the intuition of investigators and to aid neuroscience education. Anat Rec (New Anat): 257:195-207, 1999.

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.

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.

Dendritic orientation of thalamocortical projection neurons in the ventrobasal complex of macaques

This study describes the orientation of dendritic arbors from intracellularly labeled thalamocortical projection (TCP) neurons of Macaca fascicularis or Macaca mulatta. All neurons were located in the ventrobasal complex and responded to non-noxious stimuli. Each neuron was composed of dendrites that varied considerably in size and each dendrite tended to occupy a particular region of the perisomatic space with minimal overlap with other dendrites. Quantitative and qualitative analysis of the dendritic arbors of the labeled neurons showed they had an asymmetric distribution so that some regions of the perisomatic space contained more of the dendritic tree than others. Eleven of the thirteen reconstructed neurons had a larger percentage of the dendritic tree projecting into the medial portion of the perisomatic space. These results show that the dendritic arbors of macaque TCP neurons are not organized in a radially symmetric pattern as previously described and the asymmetric distribution of dendrites may be related to synaptic input.

Quantitative analyses of intracellularly characterized and labeled thalamocortical projection neurons in the ventrobasal complex of primates

This study describes the architecture of neurons and individual dendritic arbors of thirteen intracellularly labeled thalamocortical projection neurons that respond to non-noxious stimuli from the primate (Macaca fascicularis or Macaca mulatta) ventrobasal complex (VB). The neurons compose a homogeneous morphological class with total dendritic lengths from 10,169 microns to 21,711 microns (mean 17,615 microns +/- 3,705). The labeled neurons were remarkably similar in most measured parameters including the number of dendrites (7.5 +/- 1.2), percentage of dichotomous branching (89.8% +/- 3.4), and contribution of terminal branches to total dendritic length (88.4% +/- 2.0). The individual dendrites ranged in total length from 443 microns to 7,657 microns with a mean of 2,346 microns (+/- 137, n = 98). There was a positive correlation between stem dendrite diameter and total dendrite length, making it possible to estimate the total size of an individual dendrite by measuring the stem dendrite diameter. There was only a small increase in mean path distance with increasing dendritic size at the whole neuron and individual dendritic levels, so that for individual dendrites the mean path distance of a dendrite consisting of only two segments was 199 microns, while the mean path distance for a dendrite with eight segments was only 45 microns longer. Analysis of dendrite diameter, segment order, and path distance shows that dendritic diameter is not reliable for determining the location of synaptic contacts viewed by electron microscopy onto dendritic trees. The small variation of measured parameters between these neurons presents a powerful tool for future developmental, plasticity and comparative studies of VB neurons.