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An Introduction to the Mathematics of Neurons: Modeling in the Frequency Domain

Frank C. Hoppensteadt
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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This book is about using mathematical models and signal processing to understand neural networks. The author’s goal is to study the frequency and timing of neuron firing to understand how these interact in networks of neurons to carry information and control biological processes.

Around 1978, computer studies of a model of neural activity based on the van der Pol equation identified a rich structure of phase-locking behavior. Experiments motivated by this work followed up by investigating forced rhythms in the axons of a squid; these showed remarkably similar phase-locking behavior. The author and his collaborators then devised an approach that led to the Voltage Controlled Oscillator Neuron model. Much of the current book is devoted to a study of that model and its consequences. Such models have fairly broad applicability; they have also proved to be more amenable to mathematical analysis than earlier models.

The book begins with a short background section on electric circuits leading up to a discussion of voltage controlled oscillators and phased-locked loops. The next chapter, curiously enough, is about clocks. Clocks defined by physiological processes are important, often not simple, and can have very intriguing timing devices. For example, a stable limit cycle in a neurophysiological system can define a timer where progress around the cycle is like movement of the hand of a clock.

The core of the book begins with mathematical models of neurons. First there is a little background in neurophysiology. The author then describes two famous earlier neuron models due to Hodgkin-Huxley and FitzHugh-Nagumo. After describing frequency domain modeling of neurons, he develops the voltage controlled oscillator model in some detail, and investigates its frequency and phase response.

Some of the most interesting material in the book arises in the applications. Control of respiration during exercise — breathing in, breathing out and inhibitory feedback from pulmonary stretch receptors — can be modeled with a small network of neurons. Numerical simulations show similar patterns to those of resting and running humans. Another example describes a model of how small mammals govern their activity and rest cycles during the year, including the seasonal switch between nocturnal and diurnal activity. Large network models are also considered. One of the most interesting is a model of the “thalamic searchlight”, a paradigm for focusing attention.

The book provides a good introduction at an accessible level to neural modeling. This is the second edition of a book first published in 1986. It was revised and updated for a 1997 printing, but there have been a lot of developments in neuroscience in the last fifteen years. While the book continues to have real value, a reader would be advised to review the more recent literature as well.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. Some useful electrical circuits
2. A theory of simple clocks
3. Some mathematical models of neurons
4. Signal processing in phase-locked systems
5. Small physiological control networks
6. Memory, phase change, and synchronization
7. Attention and other brain phenomena