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Multicylinder models for synaptic and gap-junctional integration. 2nd ed. (English) Zbl 1091.92015
Reeke, G. N. (ed.) et al., Modeling in the neurosciences. From biological systems to neuromimetic robotics. Boca Raton, FL: Taylor & Francis (ISBN 0-415-32868-3/hbk). 117-177 (2005).
From the introduction: One objective of neuronal modeling is to construct a mathematical model of synaptic spread and summation which agrees with existing measurements to within a specified accuracy and can be used with confidence to predict future observations and behaviour. Thus, to pursue neuronal modeling successfully (as in other areas of mathematical modeling), two contrasting but complimentary skills are needed, namely, the ability to formulate the problem in appropriate mathematical terms, and the possession of sufficient knowledge (or techniques) to obtain useful information from that mathematical model. The skill in formulation lies in finding a model that is simple enough to give useful information easily, but that is complex enough to give all the information with sufficient accuracy. In attempting to construct a model, judgment has to be made about which features to include and which to neglect. If an important feature is omitted, then the model will not describe the observed phenomenon accurately enough or the model may not be self-consistent. If unnecessary features are included then the model will be more difficult to solve because of its increased complexity. Thus it is advisable to adopt a simple approach with a minimum of features included at the first attempt, and additional features added, if necessary, one by one. It should then be possible to estimate the effect of each additional feature on the results of the original model. It is this bottom-up (rather than top-down) approach that has been followed in the development of neuron models, where the passive properties of neurons need to be investigated first before proceeding to models that include active properties.
We discuss both passive and active neuron models and concentrate on the recent development of analytical solutions for these models rather than their numerical solutions by, for example, segmental cable analysis or compartmental analysis. Analytical results are of value not only in their own right to reveal trends and provide simple rules of thumb in certain situations, but also as benchmarks for confirming the results from numerical approaches, which are often of unknown accuracy.
For the entire collection see [Zbl 1107.92301].

92C20 Neural biology
92C05 Biophysics
78A70 Biological applications of optics and electromagnetic theory
35Q92 PDEs in connection with biology, chemistry and other natural sciences