Comparison of the topology and growth rules of motoneuronal dendrites

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.

Somato-dendritic morphology and dendritic signal transfer properties differentiate between fore- and hindlimb innervating motoneurons in the frog Rana esculenta

Background

The location specific motor pattern generation properties of the spinal cord along its rostro-caudal axis have been demonstrated. However, it is still unclear that these differences are due to the different spinal interneuronal networks underlying locomotions or there are also segmental differences in motoneurons innervating different limbs. Frogs use their fore- and hindlimbs differently during jumping and swimming. Therefore we hypothesized that limb innervating motoneurons, located in the cervical and lumbar spinal cord, are different in their morphology and dendritic signal transfer properties. The test of this hypothesis what we report here.

Results

Discriminant analysis classified segmental origin of the intracellularly labeled and three-dimensionally reconstructed motoneurons 100% correctly based on twelve morphological variables. Somata of lumbar motoneurons were rounder; the dendrites had bigger total length, more branches with higher branching orders and different spatial distributions of branch points. The ventro-medial extent of cervical dendrites was bigger than in lumbar motoneurons. Computational models of the motoneurons showed that dendritic signal transfer properties were also different in the two groups of motoneurons. Whether log attenuations were higher or lower in cervical than in lumbar motoneurons depended on the proximity of dendritic input to the soma. To investigate dendritic voltage and current transfer properties imposed by dendritic architecture rather than by neuronal size we used standardized distributions of transfer variables. We introduced a novel combination of cluster analysis and homogeneity indexes to quantify segmental segregation tendencies of motoneurons based on their dendritic transfer properties. A segregation tendency of cervical and lumbar motoneurons was detected by the rates of steady-state and transient voltage-amplitude transfers from dendrites to soma at all levels of synaptic background activities, modeled by varying the specific dendritic membrane resistance. On the other hand no segregation was observed by the steady-state current transfer except under high background activity.

Conclusions

We found size-dependent and size-independent differences in morphology and electrical structure of the limb moving motoneurons based on their spinal segmental location in frogs. Location specificity of locomotor networks is therefore partly due to segmental differences in motoneurons driving fore-, and hindlimbs.

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.

Growth and electrophysiological properties of rat embryonic cardiomyocytes on hydroxyl- and carboxyl-modified surfaces

Biodegradable scaffolds such as poly(lactic acid) (PLA), poly(lactic-co-glycolic acid) (PLGA) or poly(glycolic acid) (PGA) are commonly used materials in tissue engineering. The chemical composition of these scaffolds changes during degradation which provides a differential environment for the seeded cells. In this study we have developed a simple and relatively high-throughput method in order to test the physiological effects of this varying chemical environment on rat embryonic cardiac myocytes. In order to model the different degradation stages of the scaffold, glass coverslips were functionalized with 11-mercaptoundecanoic acid (MUA) and 11-mercapto-1-undecanol (MUL) as carboxyl- and hydroxyl-groups presenting surfaces, and with trimethoxysilylpropyldiethylenetriamine (DETA) and (3-aminopropyl)triethoxysilane (APTES) as controls. Embryonic cardiac myocytes formed beating islands on all tested surfaces, but the number of attached cells and beating patches was significantly lower on MUL compared to any of the other functionalized surfaces. Moreover, whole-cell patch-clamp experiments showed that the average length of action potentials generated by the beating-cardiac myocytes were significantly longer on MUL compared to the other surfaces. Our results, using our simple test system, are in basic agreement with earlier observations that utilized a complex 3D biodegradable scaffold. Thus, surface functionalization with self-assembled monolayers combined with histological/physiological testing could be a relatively high throughput method for biocompatibility studies and for the optimization of the material/tissue interface in tissue engineering.

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.

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.

Mathematical foundations of the dendritic growth models

At present two growth models describe successfully the distribution of size and topological complexity in populations of dendritic trees with considerable accuracy and simplicity, the BE model (Van Pelt et al. in J. Comp. Neurol. 387:325-340, 1997) and the S model (Van Pelt and Verwer in Bull. Math. Biol. 48:197-211, 1986). This paper discusses the mathematical basis of these models and analyzes quantitatively the relationship between the BE model and the S model assumed in the literature by developing a new explicit equation describing the BES model (a dendritic growth model integrating the features of both preceding models; Van Pelt et al. in J. Comp. Neurol. 387:325-340, 1997). In numerous studies it is implicitly presupposed that the S model is conditionally linked to the BE model (Granato and Van Pelt in Brain Res. Dev. Brain Res. 142:223-227, 2003; Uylings and Van Pelt in Network 13:397-414, 2002; Van Pelt, Dityatev and Uylings in J. Comp. Neurol. 387:325-340, 1997; Van Pelt and Schierwagen in Math. Biosci. 188:147-155, 2004; Van Pelt and Uylings in Network. 13:261-281, 2002; Van Pelt, Van Ooyen and Uylings in Modeling Dendritic Geometry and the Development of Nerve Connections, pp 179, 2000). In this paper we prove the non-exactness of this assumption, quantify involved errors and determine the conditions under which the BE and S models can be separately used instead of the BES model, which is more exact but considerably more difficult to apply. This study leads to a novel expression describing the BE model in an analytical closed form, much more efficient than the traditional iterative equation (Van Pelt et al. in J. Comp. Neurol. 387:325-340, 1997) in many neuronal classes. Finally we propose a new algorithm in order to obtain the values of the parameters of the BE model when this growth model is matched to experimental data, and discuss its advantages and improvements over the more commonly used procedures.

Evolution of the brainstem orofacial motor system in primates: a comparative study of trigeminal, facial, and hypoglossal nuclei

The trigeminal motor (Vmo), facial (VII), and hypoglossal (XII) nuclei of the brainstem comprise the final common output for neural control of most orofacial muscles. Hence, these cranial motor nuclei are involved in the production of adaptive behaviors such as feeding, facial expression, and vocalization. We measured the volume and Grey Level Index (GLI) of Vmo, VII, and XII in 47 species of primates and examined these nuclei for scaling patterns and phylogenetic specializations. Allometric regression, using medulla volume as an independent variable, did not reveal a significant difference between strepsirrhines and haplorhines in the scaling of Vmo volume. In addition, correlation analysis using independent contrasts did not find a relationship between Vmo size or GLI and the percent of leaves in the diet. The scaling trajectory of VII volume, in contrast, differed significantly between suborders. Great ape and human VII volumes, furthermore, were significantly larger than predicted by the haplorhine regression. Enlargement of VII in these taxa may reflect increased differentiation of the facial muscles of expression and greater utilization of the visual channel in social communication. The independent contrasts of VII volume and GLI, however, were not correlated with social group size. To examine whether the human hypoglossal motor system is specialized to control the tongue for speech, we tested human XII volume and GLI for departures from nonhuman haplorhine prediction lines. Although human XII volumes were observed above the regression line, they did not exceed prediction intervals. Of note, orang-utan XII volumes had greater residuals than humans. Human XII GLI values also did not differ from allometric prediction. In sum, these findings indicate that the cranial orofacial motor nuclei evince a mosaic of phylogenetic specializations for innervation of the facial muscles of expression in the context of a generally conservative scaling relationship with respect to medulla size.

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.

Morphological analysis and modeling of neuronal dendrites

Morphological data on two classes of neurons from mammalian midbrain have quantitatively been analyzed for dendritic shape parameters. Their frequency distributions were used to optimize the parameters of a dendritic growth model which describes dendritic morphology by a stochastic growth process of segment branching. The model assumes randomness with respect to both the selection of the branching segment out of the tree segments and the occurrence of the branching event in time. Model-generated trees have shape properties closely matching the observed ones. The dendritic trees of each of the two classes of neurons are represented by a specific set of growth model parameters, thus achieving morphological data compression.

Quantitative morphological analysis of the motoneurons innervating muscles involved in tongue movements of the frogRana esculenta

We give an account of an effort to make quantitative morphological distinctions between motoneurons of the frog innervating functionally different groups of muscles involved in the movements of the tongue. The protractor, retractor, and inner muscles of the tongue were considered on the basis of their major action during the prey-catching behavior of the frog. Motoneurons were selectively labeled with cobalt lysin through the nerves of the individual muscles, and dendritic trees of successfully labeled neurons were reconstructed. Each motoneuron was characterized by 15 quantitative morphological parameters describing the size of the soma and dendritic tree and 12 orientation variables related to the shape and orientation of the dendritic field. The variables were subjected to multivariate discriminant analysis to find correlations between form and function of these motoneurons. According to the morphological parameters, the motoneurons were classified into three functionally different groups weighted by the shape of the perikaryon, mean diameter of stem dendrites, and mean length of dendritic segments. The most important orientation variables in the separation of three groups were the ellipses describing the shape of dendritic arborization in the horizontal, frontal, and sagittal planes of the brainstem. These findings indicate that characteristic geometry of the dendritic tree may have a preference for one array of fibers over another.

Relationship between morphoelectrotonic properties of motoneuron dendrites and their trajectory

The distribution and geometry of the dendritic trees of spinal motoneurons obey several well-established rules. Some of these rules are based on systematic relationships between quantitative geometrical features (e.g., total dendritic length) and the three-dimensional trajectory followed by dendrites from their origin to their termination. Because dendritic geometry partially determines the transmission of current and voltage signals generated by synapses on the dendritic tree, our goal was to compare the efficacy of signal transmission by dendritic trajectories that followed different directions. To achieve this goal, we constructed detailed compartmental models of the dendritic trees of three intracellularly stained biventer cervicis/complexus (BCCM) motoneurons and calculated the electronic properties of 361 dendritic paths. Each trajectory was classified according to its orientation, e.g., rostral, rostral-dorsal-lateral. The attenuation of current and voltage signals en route to the soma was strongly related to trajectory orientation. Trajectories with similar attenuation factors formed functional subunits that were arranged in distinct domains within the ventral horn. Changes in R(m) or R(i) had little effect on which trajectories belonged to each functional subunit. However, differences in the efficacy of signal transmission between subunits increased during high network activity (mimicked by decreases in R(m)). The most efficient subunit delivered two times more current and four times more voltage to the soma than the least efficient subunit. These results indicate that the input-output properties of motoneurons depend on the direction of the path taken by dendrites from their origin at the cell body to their terminals.

A computational model of dendrite elongation and branching based on MAP2 phosphorylation

We introduce a new computational model of dendritic development in neurons. In contrast to previous models, our model explicitly includes cellular mechanisms involved in dendritic development. It is based on recent experimental data which indicates that the phosphorylation state of microtubule-associated protein 2 (MAP2) may play a key role in controlling dendritic elongation and branching (Audesirk et al., 1997). Dephosphorylated MAP2 favours elongation by promoting microtubule polymerization and bundling, whilst branching is more likely to occur when MAP2 is phosphorylated and microtubules are spaced apart. In the model, the rate of elongation and branching is directly determined by the ratio of phosphorylated to dephosphorylated MAP2. This is regulated by calmodulin-dependent protein kinase II (CaMKII) and calcineurin, which are both dependent on the intracellular calcium concentration. Results from computer simulations of the model suggest that the wide variety of branching patterns observed among different cell types may be generated by the same underlying mechanisms and that elongation and branching are not necessarily independent processes. The model predicts how the branching pattern will change following manipulations with calcium, CaMKII and MAP2 phosphorylation.

The problem of the morphological noise in reconstructed dendritic arborizations

For technical, instrumental and operator-related reasons, three-dimensional (3D) reconstructions of neurons obtained from intracellularly stained neuronal pieces scattered in serial sections are blurred by some morphological noise. This noise may strongly invalidate conclusions drawn from models built using the 3D reconstructions and it must be taken into account when retrieving digitized neurons from available databases. We analyse on several vertebrate neurons examples the main noise-generating sources and the consequences of the noise on the 'quality' of the data. We show how the noise can be detected and evaluated in any database, if sufficient information is presented in this database.

Expansion of the dendritic tree of motoneurons innervating neck muscles of the adult cat after permanent axotomy

The morphologic characteristics of neck motoneurons with intact axons were compared with those of neck motoneurons that had been permanently axotomized for 11 to 17 weeks. Motoneurons were identified antidromically, intracellularly stained with horseradish peroxidase (HRP) and examined after reconstructions of their entire dendritic tree. Axotomized motoneurons differed qualitatively and quantitatively from motoneurons with intact axons. The distal branches of axotomized motoneurons exhibited two novel features: some gave rise to tangled appendages that exhibited growth cone-like specializations resembling lamellipodia and filopodia; others followed a meandering path and had unusually large diameters. These branches showed a discontinuous pattern of staining that was similar to the appearance of myelinated axons stained intra-axonally with HRP. A quantitative analysis of the dendritic trees of 13 completely reconstructed dendritic trees (five axotomized motoneurons and eight motoneurons with intact axons) showed that total dendritic surface area, total dendritic length, and total number of branches increased 38, 34, and 215%, respectively, after axotomy. These measurements were confirmed by comparing the sizes of a larger number of motoneurons (16 axotomized and 21 intact), calculated on the basis of correlations between dendritic tree size and proximal dendritic diameter. We conclude, therefore, that neck motoneurons, in contrast to other types of motoneurons, expand their dendritic trees after axotomy. It is suggested that this expansion is a consequence of two mechanisms: one involves dendritic growth, possibly leading to new synaptic connections; the other causes a conversion of some dendrites into axons.

Growth cone dynamics and activity-dependent processes in neuronal network development

Many structural and functional properties of neuronal networks find their origin in the dynamic behavior of growth cones during development. The variation in dendritic morphologies can be traced back to random branching of growth cones. Segment length characteristics arise under random branching and steady growth cone propagation. Delayed outgrowth, as a result of competition between growth cones after splitting, is hypothesized to explain different lengths of paired terminal segments in Purkinje cells. The implications of activity-dependent neurite outgrowth were studied using an outgrowth function based on the theory of Kater et al. (1988, 1990). This theory embodies a homeostatic principle, according to which a neuron adapts its neuritic field so as to maintain a certain level bioelectric activity. It is shown that such homeostasis has many implications for neuromorphogenesis and network formation, as it may underlie phenomena such as overshoot during development, size differences among cells, differentiation between excitatory and inhibitory cells and compensatory sprouting. Finally, function-dependent regulation of development involves physiological as well as morphological variables. For instance, activity dependent regulation of ionic conductances such as to stabilize functional activity can result in a differentiation of certain neurons into, respectively, bursting and regular firing sub-types (Abbot et al., 1993; LeMasson et al., 1993). Similarly, the GABAergic phenotype comes fully to expression in hindbrain (cerebellar) and forebrain (neocortical) networks only if the level of ongoing excitatory activity during development is sufficiently high, whereas chronically intensified activity leads to a compensatory hypertrophy of inhibitory mechanisms (for review, see Corner 1994). Many of these results could only have been obtained by the use of mathematical models which allow rigorous analysis of the consequences of basic assumptions in the dynamics of neurite outgrowth. All in all, the findings further emphasize the role of spontaneous bioelectric activity during early development in neuronal network formation, the importance of which was first established in cultures of developing neural tissue.