Mathematical Foundations of Neuroscience

G. Bard Ermentrout and David H. Terman. Springer: 2010. 422 pp. $74.95

9780387877075

Until recently, biologists treated theory as a reward, claimed after a lifetime of labour in experiment and observation. Yet, within just a few generations, theorists in neuroscience have begun to resemble their cousins in physics, choosing to specialize in theory early in their careers. The focus of theoretical neuroscience has shifted in that time towards complexity: from models of nerve conduction to an emphasis on the dynamics of nonlinear neural interactions. Bard Ermentrout, a biophysicist, had much to do with that transformation, which is highlighted in his and mathematician David Terman's textbook, Mathematical Foundations of Neuroscience.

Terman and Ermentrout share an interest in the failure modes of neural systems. Nonlinear dynamic aspects are often only revealed when neural systems are pushed to the edges of their performance abilities. Migraines, strobe lights and drug intoxication can all cause geometric hallucinations: Ermentrout studies these as well as the illusions produced by viewing moving images during electric retinal stimulation. Terman's model of image segmentation fragments noisy images (such as television static) in a manner that is reminiscent of these visual effects. However, aside from geometric hallucinations, the wilder sides of Ermentrout and Terman's research interests are not emphasized in the book, which is directed at a broad interdisciplinary audience.

The visual disturbances seen by some people with migraines can be modelled mathematically. Credit: M. RULE

The traditional material on membrane biophysics, cable theory and neural-spike generation models is presented first. The latter part of the book — covering the nonlinear dynamics of neural interactions — takes a balanced approach, describing models in which the correct timing of individual neural spikes is crucial, and population models based on the firing rates of an ensemble of neurons. Rapidly evolving topics such as neural synchronization and spatially extended models are included.

Should maths be conveyed separately to students who show theoretical aptitude, or mixed in as digressions to a lecture series?

Ermentrout and Terman go deeper into the mathematics of neural activity than say, Hugh Wilson's 1999 textbook Spikes, Decisions and Actions (Oxford University Press). But, unlike Wilson, they pass up most opportunities to connect the mathematics to its cognitive and perceptual consequences.

They emphasize the mathematical basics even over exciting developments in theory. For example, a strong chapter on neural noise neglects stochastic resonance — a phenomenon of nonlinear systems in which a weak signal can be amplified and optimized by noise — and its role in promoting neural pattern formation. Similarly, Terman omits his own model when describing oscillatory neural synchronization, a process that may perceptually bind together the disparate parts of a stimulus.

This tight focus raises the question of how mathematical skills should be taught across science subjects. Should they be conveyed separately to students who show theoretical aptitude, or mixed in as digressions to a science-based lecture series? Mathematical Foundations of Neuroscience falls somewhere in between: it is a good substitute for a lengthy regime of abstract maths classes, but it is also well integrated into the field of neuroscience. Ermentrout and Terman's book conveys much of the advanced mathematics used in theoretical neuroscience today.