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Reviewer: W. Richard Stark

The mathematical dynamics, computer science, and biological motivation of neural networks are discussed in this beautifully written book. In particular, it is devoted to the theory of weakly connected neural networks, with applications to mathematical neuroscience. For n neurons, the local dynamics d dt x i =f i x i + e i g i x 1 ,&ldots;,x n is weakly connected if e i <<1 for each i=1,&ldots;,n . Thus, the term “weakly connected” is not used in the graph-theoretic sense. Much of the exposition focuses on critical regimes for continuous dynamic systems and, in the last third of the book, applications to various networks to determine what they can and cannot do. The book emphasizes mathematics, computational dynamics, and biological paradigms. The authors orchestrate a wide variety of topics—including canonical models, Wilson-Cowan models, Hodgkin-Huxley models, Hopfield nets, Lyapunov functions, saddle nodes, cusps, pitchfork bifurcations, Andronov-Hopf bifurcations and periodic activity, Bogdanov-Takens bifurcations, Poincare´ mappings, nonhyperbolic nets, the Kuramoto-Tsuzuki equation, Malkin's theorem, neural oscillators, the hippocampus, the olfactory bulb, brain dynamics, and learning rules—into a cohesive story. Both classical and new material are covered. The presentation is appropriate for first-year graduate students, mature investigators, and everyone in between. The mathematics is developed with great clarity and skill. Students of, and researchers in, such diverse areas as automata networks, self-stabilizing systems, Boolean nets, and asynchronous packet-switching networks may find useful philosophies, models, and mathematical tools. For example, early in the book, the authors discuss communication and bifurcation. Local behavior in weakly connected and equivalent uncoupled systems is the same unless at least one neuron is near a bifurcation point. Neurons whose states are far from their threshold are not necessary for the network's dynamics. This immediately suggests a way of viewing communication and local redundancy which, although not new for neural nets, is potentially elegant and profound in the context of asynchronous packet-switching networks. Practitioners of discrete models will notice that state changes are continuous, so issues of serial and asynchronous activity do not arise. This attractive monograph has razor-sharp print, 173 illustrations, and 12 pages of references. The text has been carefully written and edited. Neither “Hopfield” nor “Hodgkin-Huxley” appear in the index, though they are in the text, suggesting that the index may be a bit sketchy. All things considered, this is my favorite presentation of the mathematics of neural networks.