`UK Nonlinear News`,
`November 2000`

Springer-Verlag 1998

0-387-98381-3

766 pages

£53.50 (£47.63 from Amazon).

It is widely thought that whereas the 20th century saw a massive expansion in the applications of nonlinear mathematics to the physical and engineering sciences, in the 21st century we will see a similar development in its applications to biology. This is something of great potential benefit to both disciplines: on one hand the life sciences are increasingly facing fundamental problems that require mathematical skills for their solution, and one the other such problems often stimulate the development of novel mathematical ideas. Despite this, there are still relatively few mathematicians working in this area, particularly in the UK. As someone who acts as a first point of contact for biologists looking for mathematical help, I am constantly faced with the problem of far more enquiries from biologists than of mathematical colleagues willing or able to help. This is also reflected in the shortage of books in this area, compared to say the vast range that is now available in the field of nonlinear dynamics in general.

Physiology is an ideal subject for the application of mathematical techniques, in that it seeks to integrate the behaviour of entities at the molecular or cellular level to understand the behaviour of whole tissues or organisms. This is in comparison to, for instance, molecular biology, which tends to focus much more on at most the interaction of one or two molecules in isolation. Furthermore, physiological systems are almost always inherently nonlinear (if only because most physiological processes have to saturate at some stage or another). Despite these attractions, there are as far as I am aware no texts at any level dealing with the application of mathematics to physiology. This is perhaps partly due to the current fashion for molecular methods in biology, and hence the relative lack of exposure of physiology to those outside of the field.

The appearance of this monumental volume on Mathematical Physiology is thus extremely welcome. It is written by two international experts in the field, and covers an enormous range of physiological applications. It is well written, and well structured, beginning with basic techniques such as reaction kinetics and membrane electrophysiology, which are then repeatedly applied to more complex biological systems, including the nervous system, the circulation, the kidney, the eye and the ear, and many others. Each topic is introduced with a brief description of the biology, a detailed derivation of an appropriate model, and then a thorough analysis of the resulting equations. The mathematical tools are on the whole straightforward, consisting mainly of low dimensional ordinary differential equations, with an occasional partial differential equation. The volume should thus be easily accessible to most graduate students in nonlinear dynamics, as well as many final year undergraduates in applied mathematics. At UCL we have already put it at the top of the list of books we recommend to incoming students on our new interdisciplinary 4 year PhD programme.

My only possible criticism is the heavy emphasis on mathematics at the expense of biology. Often, having developed and analysed a model there is little attempt to draw biological conclusions, or to suggest new biological hypotheses or experiments. Thus, whilst the book shows very well how biology motivates the development of interesting mathematics, there is much less indication of how mathematics can help solve fundamental biological problems. Furthermore, the material is presented in such a way that I doubt many physiologists (even those whose background is originally in physics or mathematics) would be able to make much headway with it. Of course, such criticisms can be applied equally well to most current work in mathematical biology, and it is not even clear that in a massive text such as this it is possible to achieve a more equal balance. Certainly, including more biological material would make the volume even longer, and might ruin the clear and logical exposition that is one of the books great strengths.

Overall, I therefore think that this is an immensely impressive work, which should be of interest to a wide audience. I am sure it will serve to get many young mathematicians excited about this developing field, and hence in writing it the authors have done the community a valuable service. I thoroughly recommend anyone with any interest in the interaction of mathematics and the life sciences to read it.

A listing of books reviewed in `UK Nonlinear News`
is available.

`UK Nonlinear News` thanks
Springer-Verlag for
providing a review copy of this book.

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