Department of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, UK

Department of Experimental Neurophysiology, Vrije Universiteit, De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands

Abstract

Background

The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other.

Methods

A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described.

Results

Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development.

Conclusion

A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity.

Background

A highly distinctive feature of neurons is their morphology. Neurons exhibit long processes, or neurites, that are fundamental to the formation of the connected networks of neurons that constitute a nervous system. One neurite, the axon, forms the output electrical signalling pathway of a neuron. A typical axon has a main trunk from which shorter side branches, or collaterals, emerge to form points of contact with appropriate target neurons. Axons may be extremely long, up to about one metre in humans, for example. The remaining neurites of a neuron are dendrites, which form complex tree-like structures and are the recipients of most synaptic contacts from the axons of other neurons. Different types of neuron can be distinguished by the structure of their dendrites, which can be characterized in terms of segment lengths and diameters, the number of terminals (unbranched tips), the number of branch points, and topological factors such as symmetry

Much of the research in the field of computational neuroscience has been directed at understanding the electrical signalling properties of neurons, with a particular emphasis on the impact of complex neuronal morphology on signal integration (e.g.

However, this research into electrical signalling ignores the fascinating problem of how a neuron's complex morphology is created during nervous system development. Mathematical modelling and numerical simulation are invaluable tools to help us unravel the processes underlying morphological development. The sorts of model that have been investigated vary widely and are typically aimed at a particular aspect, such as target finding by axons, rather than describing the entire development of a neuron. Consequently, no uniform mathematical or numerical techniques have yet emerged to lead to the building of the sort of user-friendly software available for modelling electrical signalling. In this review we consider a range of modelling endeavours exploring aspects of neuronal morphological development. In conclusion we suggest the features of a computer simulation package that could ease the pain of creating new models of neuronal development and eventually allow the exploration of whole neuron development.

Methods

Overview

This review is largely about methods for modelling neuronal development, rather than the latest research results. We will, however, point out the major findings of various models along the way. The review is not exhaustive of all theoretical work on neuronal development (for a comprehensive overview see

Neurite initiation and differentiation

The first problem concerns how neurites form from an initially spherical neuronal cell. Then, subsequent to initiation, there is a process of differentiation in which one of the neurites becomes the axon, and the remaining neurites form the dendrites. These stages of growth are illustrated in Figure

Neurite initiation and differentiation

**Neurite initiation and differentiation**. (a) Cell surface inhomogeneities cause instabilities that lead to the formation of broad lamellipodia that condense into short neurites. (b) Competition for a growth-permitting chemical produced in the cell body can lead to the fastest growing neurite becoming the only outgrowing neurite, characteristic of axonal differentiation.

The model was simulated using a cellular automata approach in which the cell interior is divided into 1 ^{2 }blocks and the cell surface is described by a linked list of 1 _{0}, and in each of _{1 }to _{n}, and the lengths of each neurite, _{1 }to _{n}. The equations are:

where _{i }is the transfer rate of chemical from the soma to a neurite tip, _{0 }is the volume of the soma, _{i }is the volume of a neurite tip, and _{i}, and chemical is consumed in the tip at a rate proportional (

A "winner-take-all" dynamic instability can emerge if one neurite has a slightly larger initial growth rate. This leads to greater consumption of the chemical at this neurite tip and the subsequent capturing of more chemical by this neurite than by the others. This quickly leads to rapid growth by this one neurite, with all other neurites having only very slow growth. This is characteristic of the differential growth of the neurite that becomes the axon in real neurons. This model is able to reproduce the results of a number of axotomy experiments in which severing of the nascent axon at different lengths, relative to the other neurites, results in either the axon reestablishing itself or another neurite becoming the new axon

Neurite elongation

Following initiation, neurites elongate and branch, reaching lengths of tens to hundreds of micrometres for dendrites, to over a metre or more for long axons. Modelling has been used to explore just how neurites elongate and the biophysical constraints on the lengths that may be achieved. This work has centred around the dynamics of cytoskeleton construction, particularly of the microtubules that form the major supporting scaffold within neurites

Neurite elongation

**Neurite elongation**. Construction of the microtubule cytoskeleton is a rate-limiting factor in neurite elongation. The rate of construction is determined by the production and transport of tubulin along a neurite, and the assembly of tubulin onto the ends of microtubules at the neurite tip.

The free tubulin concentration at a point _{0}_{0 }results in a flux of tubulin across the boundary at _{l }and return flux _{l}) and a change in length (assembly rate _{g }and disassembly _{g}).

This model is a generalisation of previous ODE and algebraic treatments of this scenario

Steady-state analysis has revealed that growth can proceed in three different regimes, as determined by the relative proportions of

Aspects of this model have been the subject of more detailed treatments. Active transport of tubulin is here assumed to proceed with a flux proportional to the local concentration. In reality, tubulin dimers undergo periods of active transport when they are attached to the molecular motors, interspersed with periods of free diffusion. A PDE reaction-diffusion-transport model that covers this scenario, and considers both uni- and bidirectional motors for the active transport of intracellular organelles has been investigated by Smith and Simmons

Modelling has also been directed at examining the detailed process of microtubule assembly. Monte Carlo simulation of the reaction and diffusion of individual tubulin dimers near the tip of a microtubule has been used to investigate the outgrowth dynamics of a microtubule and the sources of variability that might underly the observed dynamic instability in which a microtubule may switch from elongation to retraction and vice versa

The growth cone at the tip of a neurite also plays a significant part in neurite elongation and is not explicitly included in the microtubule assembly models. However, microtubules extend into the growth cone and interactions between the growth cone and microtubules are significant for neurite outgrowth. Microtubules exhibit so-called dynamic instability, the random alternation between microtubule growth and shrinkage. This process plays an important role in the motility of growth cones and their finger-like protrusions, the filopodia. The volume of a growth cone is so small that growing or shrinking microtubules are expected to cause fluctuations in the concentration of free tubulin. Monte Carlo simulations have shown that these fluctuations have a significant effect on microtubule dynamic instability

The growth cone exerts tension on the trailing neurite which influences microtubule assembly rates. Precisely how tension affects assembly has been the subject of a number of modelling studies (reviewed in

where _{g }is the unmodified assembly rate and _{g }is the disassembly rate (assumed not to be a affected by tension). Tension from the growth cone can relieve this elastic tension (lowering

The growth cone itself contains a complex actin cytoskeleton. It has been proposed that this cytoskeleton, when supporting lamellipodia at the leading edge of the cone, undergoes a caterpillar like movement to propel the growth cone along

Axon pathfinding

A fundamental feature of the outgrowth of an axon is its ability to follow a path to an appropriate target. This requires the axon's growth cone to sense environmental cues and to be able to turn towards the desired direction, or away from an incorrect location. Modelling efforts have been directed at the growth cone's ability to sense external chemical gradients

The most fully worked example of axonal elongation and direction finding is that of Aeschlimann

She considers the outgrowth of a single, unbranched axon in a two-dimensional external environment containing chemical gradients and glial cells (Figure

Axonal pathfinding

**Axonal pathfinding**. Detection of chemoattractants in the external environment by filopodia produces tension on the growth cone in particular directions. The growth cone will turn towards and grow along the dominant direction. If similar forces are exerted on opposite sides of the cone, the tension may be enough to split the cone into two, leading to the formation of daughter branches.

Maskery et al

Models of how groups of growing axons form bundles, or fascicles, have been explored by Hentschel and van Ooyen

The two dimensional implementation of this scenario involves treating each target cell and each growth cone as a point source for attractant or repellant. The quasi-steady-state concentration gradient of each chemical from each source is calculated and summed. The change in position, **r**_{α}, of each growth cone,

where ∇_{x }is the concentration gradient of growth cone (_{x }is the associated growth rate constant.

Segev and Ben-Jacob

Neurite branching and dendritic shape formation

Neurites do not just elongate, they also make repeated branches. Branch formation is either due to a bifurcation of the growth cone, or the interstitial formation of a new branch part way along an existing neurite. Growth cone bifurcation seems to largely underly the formation of many different types of dendrite

Models of branching due to external influences consider tension on the growth cone due to filopodia sensing cues in different directions in the environment

Theoretical studies indicate that dendritic branching angles may follow a principle of neurite volume minimisation

The basic model considers whether the production and transport of an unspecified branch-determining substance imparts constraints on branching

More sophisticated models describe both elongation and the branching rate explicitly as functions of tubulin and MAP2 concentrations at a terminal (Figure

Neurite branching

**Neurite branching**. Construction and stability of the microtubule cytoskeleton may determine both elongation and branching rates. Microtubule bundles are cross-linked by microtubule-associated proteins, such as MAP2. Dephosphorylated MAP2 stabilises the bundles and promotes microtubule assembly, and hence elongation. Phosphorylated MAP2 loses its cross-linking ability, destabilising the bundles and thus increasing the likelihood of neurite branching.

where _{l }is the concentration of free tubulin at a neurite tip, _{l }is the concentration of dephosphorylated and microtubule-bound MAP2, _{l }is the concentration of phosphorylated MAP2 and _{b}, is the probability of branching.

This complex model of elongation and branching has been implemented as a coupled system of ODEs, with their numerical solution following a 'compartmental" approach, as has been used for solving the voltage equation for simulating electrical activity. In this approach, a neurite is divided into a number of short compartments, with chemical concentrations being calculated for the volume of each compartment. Chemicals move between compartments due to bulk diffusion and active transport. As a neurite elongates, new compartments are added when needed. New compartments are also created when a branching event occurs. Various strategies for when and where new compartments are added have been investigated

Results and Discussion

Modelling results and experimental data

All of the models described here are aimed at understanding the biophysics of various processes underlying the morphological development of a neuron. They seek to support explanations for the rate and direction of outgrowth, segment lengths, branching angles and the branching structure of dendrites and axons. Different models have been successful at matching experimental data for some but not all of these features at the same time.

Much of the work draws its motivation from statistical models of neurite outgrowth, with the aim of providing biophysical explanations for statistical phenomena. Dendrites exhibit correlations between segment diameters, lengths and segment branching probabilities

Predictions have been made about the requirements and limitations of extracellular signals for axon guidance and bundling

In summary, biophysically-based models are making predictions about the nature of intracellular and extracellular constraints that could determine the neuronal morphologies and network topologies seen experimentally. Such models consider the dynamics of cytoskeleton construction, the effect of tension on neurite outgrowth, and the temporal and spatial requirements for chemotactic signals.

Modelling techniques

A major feature of models of neuronal development is the necessity to calculate quantities that vary over space as well as time, with the significant complication that the spatial domain itself varies with time. The natural mathematical specification of such problems is the partial differential equation (PDE). The formulation and numerical solution of PDE models for which the spatial domain changes with time are of the moving boundary type and are more difficult to deal with than models with a fixed spatial domain. Nonetheless a PDE model for neurite elongation has been developed

A two dimensional PDE model of an external environment of arbitrary geometry and containing various point sources of chemical gradients has been formulated and solved numerically

While the mathematical rigour provided by the PDE approach is highly desirable, the flexibility of a "compartmental" approach using a system of coupled ODEs is also potentially very useful. This is particularly evident in simulation software for modelling electrical activity in neurons (e.g. NEURON and GENESIS) in which nonlinear models of ion channel activity are easily incorporated and coupled to the underlying membrane voltage equation. Spatially inhomogeneous distributions of ion channels are easily handled, as are "point processes", such as synapses, that only occur at particular locations. The natural extension of this approach to the scenario of a neuron undergoing morphological development is the addition or deletion of new membrane "compartments" over time

Towards a unified modelling environment for neuronal development

From both a conceptual and a mathematical point of view it is possible to envisage a unified scenario that encompasses many of the individual models described above. Most of the models lie within the framework of a two-dimensional external environment coupled with a one-dimensional intracellular space. The external environment contains either point sources or continuous chemical gradients of attractors and repellors. Physical barriers, such as other cells, could also be present. The intracellular space describes the concentration of various chemicals along the length of neurites as determined by their sites of production and their transport by diffusion or active motors. Exceptions to this description are the two-dimensional intracellular space used to model neurite initiation

Appropriate numerical schemes for implementing the required 2D external environment

Thus an appropriate unified modelling environment needs to encompass both varying spatial and temporal scales and inhomogeneities in the components of a given model. Appropriate and diverse mathematical techniques are required for calculating quantities at different scales of interest. These techniques are available individually, but a numerical simulation package that incorporates all required techniques to implement the unified modelling environment outlined here remains to be built.

Conclusion

Mathematical modelling and numerical simulation are powerful tools to help us understand neuroscientific experimental data and ultimately the operation of the nervous system. User-friendly simulation software has led to an enormous body of work on modelling electrical activity in morphologically complex neurons and networks of neurons. Mathematical models of aspects of nervous system development are more disparate in nature but there is some commonality of themes and mathematical techniques, as reviewed here. Powerful simulation environments are beginning to be developed and this will hopefully lead to an expansion in the use of mathematical modelling to understand development of the nervous system.

Acknowledgements

This work has been funded by EPSRC grant GR/R89769/01 to BPG.

This article has been published as part of