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Spikes, decisions, and actions. The dynamical foundations of neuroscience. (English) Zbl 1027.92008
New York, NY: Oxford University Press. xii, 307 p. (1999).
The main purpose of this book is to provide advanced undergraduates and graduates in physiological psychology and neurosciences with basic ideas and techniques on how mathematics can be used for modelling in neurobiology. The prerequisites for a reader are basic knowledge of neurobiology, calculus, elementary linear algebra and differential equations. However, a comprehensive undergraduate course in nonlinear differential equations and dynamical systems would be very helpful since the presentation is aimed primarily at a non-mathematically oriented audience and several important mathematical details did not receive enough attention (it suffices to have a look at Figure 3.3 on p. 38, where typical phase plane trajectories for linear systems are presented, to understand to what extent the presentation is simplified).
There are sixteen chapters in the book, an appendix with brief information on MatLab\(^{\text{TM}}\) and the MatLab scripts which are collected on a floppy provided with the text, and an exhaustive bibliography. After the first chapter introducing the reader to the subject, the following three chapters discuss first order linear differential equations, second-order linear differential equations, higher order linear differential equations, and methods used to study these. In particular, variation of constants, stability and phase space techniques for second-order systems, and the Routh-Hurwitz criterion are examined. Chapter 5 deals with approximation techniques for differential equations. Here Euler and Runge-Kutta methods are explained, with the focus on the latter one. Chapter 6 is devoted to nonlinear differential equations arising in neurodynamics. It introduces important concepts like bifurcation, hysteresis, memory, catastrophe, etc. Neural networks in which each neuron inhibits all other neurons except itself, known as winner-takes-all networks, are studied in Chapter 7. These systems model very interesting phenomena including, for instance, retinal light adaptation or visual hallucinations. Chapter 8 is concerned with oscillations in neural systems. Two important issues related to limit cycles are briefly discussed: Poincaré-Bendixson theory and Hopf bifurcations. The models studied in this chapter include Wilson-Cowan network oscillators, FitzHugh-Nagumo equations, and two-neuron winner-takes-all networks with adaptation and perceptual reversals. Equations for action potential generation related to Hodgkin-Huxley equations are considered in Chapter 9, where it is shown that a periodic spike train is actually a limit cycle. Interesting issues like stochastic and subharmonic resonance, or saddle-node bifurcation, are briefly commented on. The study of FitzHugh-Nagumo equations continues in Chapter 10, where two complex phenomena, spike rate adaptation and bursting, are explained by examining the simplest dynamical models.
Chapter 11 provides a brief introduction to neural chaos. Interactions between neurons that are synaptically interconnected are studied in Chapter 12 where the neural swimming system of the mollusk Clione is simulated and analyzed, leading to the prediction that mutual inhibition is the most effective method for synchronizing neurons. The chapter is concluded with the analysis of inhibitory coupling responsible for synchronizing bursting thalamic neurons during deep sleep. Based on the analysis of phase locking and synchronization, control of swimming by neural circuits in the spinal cord of the lamprey, a primitive aquatic vertebrate, is discussed in Chapter 13, where generation of smooth travelling waves of neural activity propagating along the body is examined. The chapter closes with a more detailed simulation of swimming for the marine mollusk tritonia, where neurons are modelled at the level of ion currents and individual spikes. The direct Lyapunov method is addressed in Chapter 14, and its application to the study of long-term memory networks is discussed. Chapter 15 deals with partial differential equations which are used to model the spread of a post-synaptic potential along a dendrite to the cell body and propagation of an action potential along an unmyelinated axon. The last chapter provides useful guidelines on the use of nonlinear dynamics in research along with a brief survey of links between the nonlinear dynamics in neural problems and related issues in other areas.
All chapters, except the first and the last, conclude with helpful sets of exercises, and comments on MatLab\(^{\text{TM}}\) scripts written by the author for numerical exploration of problems discussed in the book are provided whenever necessary. The book is a valuable source of information at the level accessible to well-trained undergraduate students and can be successfully used by specialists interested in mathematical modelling in neuroscience.

92C20 Neural biology
34C60 Qualitative investigation and simulation of ordinary differential equation models
92-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology
37N25 Dynamical systems in biology