by Wulfram Gerstner and Werner Kistler

Cambridge University Press, Cambridge, 2002, xiv + 479 pp., ISBN 0-521-89079-9, paperback, £24.95 (also hardback, ISBN 0-521-81384-0, £65.00)

Developments in the use of artificial neural nets, in the last few decades, have encouraged the view of the neuron as a continuous device with a sigmoid response function. Such a view is required for both the backpropagation method of learning and the one due to Hopfield. The transmission of information, at least in peripheral nerves, appears to depend on pulse frequency coding and hence a continuous signalling function. This is in contrast to the earlier theory of McCulloch and Pitts, which has been the basis of much speculation in cybernetics, where the all-or-none character of neural excitation was associated with the true-or-false of formal logic. The McCulloch-Pitts model was put forward as a basis for discussion and not as a detailed representation of real neural activity.

Real neurons (except some in the mammalian retina) do "fire" or "pike" and a variety of neural phenomena can only be explained by taking this into account. It is easy to show that neural channels conveying continuous data by frequency modulation do not account for observed behaviour because the reception of an analogue value requires a certain time of integration, and the responses of people and animals are faster than this would allow. A faster response is possible if the integration is instantaneous over many parallel channels with appropriate randomness, and the analysis of this requires consideration of spiking neurons.

Another demonstration that the spiking behaviour is significant is in connection with stereophony, where location of a source of sound depends on extremely accurate estimation of the difference in times of arrival of a sound at the two ears, or phase differences in a sustained input. For barn owls the time difference can certainly be estimated within 5 microseconds.

The treatment is divided into the three parts indicated in the second part of the title. In the first part the structure of a neuron is described, and the nature of its excitability is explained in terms of the well-known Hodgkin-Huxley equations that refer to flows of different ions through the membrane. Since the equations have four variables they are intractable for incorporation in population models, and several more manageable approximations are considered. Synaptic transmission is treated in considerable detail, with accounts of the changes in permeability of postsynaptic membrane. It is acknowledged that neurons may have to be considered as compartmented models, since for example an inhibitory synapse can be at the base of a particular branch of the dendritic tree and it is then more effective in nullifying excitation arising in that branch than in countering excitation generally. Possible sources of noise affecting the output of the neuron are discussed.

In the second part of the book a number of approaches to the modelling and analysis of neuron populations are treated, and it is interesting that early papers by Wilson and Cowan are still highly relevant, where Jack Cowan worked at one time with Warren McCulloch. It is shown that the output of an entirely deterministic net can appear highly irregular, and also that information can be transmitted through an active net much more rapidly than would be expected from the time courses of individual neural responses. This can be attributed to the fact that at any moment there are neurons about to fire and whose firing can be precipitated by the input. Special attention is given to oscillatory behaviour and synchronisation, where one reason for special interest is the theory developed by W. Singer and reviewed by Andrew (1995) which suggests that precise synchronisation of impulses is part of the means by which sensory stimuli are grouped or ôboundö to allow recognition of objects. A theory of complex reverberatory behaviour, first analysed in connection with interactions of fireflies, is shown to be applicable also to neural populations.

The third part of the book, on plasticity and hence possible mechanisms of learning, is specially interesting. The mechanism at the single-cell level is assumed to be essentially as postulated by Hebb but its implications in neural structures and populations are developed in considerable detail. It is shown how "learning to be fast" can occur, so that a response is triggered by the earliest of the events associated with it, an effect that is illustrated by the classical conditioned reflex. A less obvious effect that can also be accounted for is "learning to be precise" where the response comes to be triggered by the event with least time variability. In this part there is discussion of binaural sound localisation and of the localisation abilities of electric fish, and other special features of perception.

The book is essentially mathematical and there are few pages that are free of equations. The authors insist that only fairly elementary mathematics is used and that a diligent reader or researcher should not be deterred. Each chapter ends with a summary of its main points and a discussion of the relevant literature. The latter is specially welcome in some of the chapters where it is easy to get the impression that the emphasis is on the mathematical model, and wider reading is needed to appreciate the biological relevance. It is difficult to see how this could have been avoided while keeping the volume manageable in size.

Despite all the good work that has been done, in the "decade of he brain" and earlier, it is clear that the nervous system has yielded only some of its secrets. Further advances will certainly depend on accurate modelling of neural activity, for which the review provided here is a much-needed basis.