Compartmental Modelling of Developing Neurons

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 morphology 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. This uses so-called “compartmental” models of neurons which are based upon solving a well understood partial differential equation (PDE) for membrane voltage over space and time, where space is one dimensional along the length of neurites. Powerful numerical techniques for integrating this equation and for coping with the branched structures of neurons have been developed. Sophisticated computer simulation packages, such as NEURON (www.neuron.yale.edu) and GENESIS (www.genesis-sim.org), make the creation of electrical signalling models a near user-friendly experience. Such packages provide graphical interfaces for both model construction and for the display of results. However, this research into electrical signalling ignores the fascinating and fundamental problem of how a neuron’s complex morphology arises during nervous system development. Understanding of this process is vital if we are to develop therapies for nervous system repair following injury, such as promoting axon regrowth, and the use of stem cells to develop into replacement neurons. 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 so far 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. Such software is required for the expansion of this research field.