*(English)*Zbl 0705.92004

As stated by the authors, “This book deals with the applications of mathematics to normal and pathological physiological rhythms. It is directed toward an audience of biological scientists, physicians, physical scientists, and mathematicians who wish to read about biological rhythms from a theoretical perspective.” Relatively little mathematics appears in the main text of the book. Consequently it can be read by anyone who understands basic calculus and is willing to accept some of the concepts and equations on the authority of the authors. A mathematical appendix provides a good explanation of these concepts and equations for readers who are acquainted with the methods of applied mathematics, but it is too compact to benefit many of the biological scientists and physicians in the intended audience. All readers, whether mathematically sophisticated or not, will find the discussion of much of the research which is presented to be too brief for full comprehension. However the text serves as an excellent introduction to a well chosen selection of cited literature.

The authors assert that “Although there are frequent cross references between chapters, the chapters are largely independent of one another and do not have to be read in the sequence presented.” This review adheres to the chapter organization of the book. The words of the authors are used wherever possible.

Chapter 1 presents a brief outline of the book and summarizes its themes by giving several physiological examples. Chapter 2 introduces the key mathematical concepts which are to be developed: the steady state, oscillations, and chaos. Deterministic differential equations and finite difference equations are treated. Bifurcation of solutions leading to chaos is illustrated.

Chapter 3 contrasts random noise with deterministic chaos. The Poisson process and the analysis of inter-interval histograms are discussed. A number of techniques which help to identify chaos in the presence of noise are presented. These include power spectra, the Poincaré map, the observation of period-doubling bifurcations, the existence of strange attractors, and the calculation of Lyapunov numbers.

Chapter 4 summarizes the main classes of mechanisms that have been proposed for biological oscillators and illustrates them with representative data. The major classes are pacemakers and central pattern generators. The latter class includes neural-network interaction models and other systems displaying mutual inhibition, sequential disinhibition, time delay, negative feedback and mixed positive and negative feedback. Chaotic behavior can be demonstrated in some of these systems.

Chapter 5 discusses experimentally observed transitions between oscillatory and nonoscillatory dynamics and offers hypotheses to account for these transitions. Four methods are described for turning oscillations on and off: (1) Subthreshold oscillations can be elevated to superthreshold levels. (2) Soft excitation. As a parameter increases, oscillations build up gradually. (3) Hard excitation. As a parameter increases, large-amplitude oscillations are abruptly observed. (4) A single pulse causes a transition between a stable limit cycle and a stable steady state.

Chapter 6 describes experimental and theoretical results concerning the phase resetting effects of single stimuli delivered to biological oscillators. Integrate-and-fire models as well as limit cycle oscillations are treated. Chapter 7 presents the consequences of periodic stimulation of oscillatory systems. Periodic forcing of integrate-and- fire models and entrainment of limit cycle oscillators are discussed. Illustrations of phase locking of rhythms in humans are given.

Chapter 8 concerns spatial oscillations. Physiological rhythms are ordered in space as well as in time. Wave propagation in one, two, and three dimensions is considered. Several of the illustrations are based on abnormal cardiac dynamics. Chapter 9 proposes that diseases characterized by abnormal temporal organization be called dynamical diseases. Identification and diagnosis of dynamical diseases are discussed. An examination of mathematical and biological models of these diseases indicates that it is relatively easy to generate a model which will mimic the temporal description of a disease, but it is often difficult to show that the model correctly represents the origin of the disease.

##### MSC:

92C30 | Physiology (general) |

92-02 | Research monographs (appl. to natural sciences) |

92C50 | Medical applications of mathematical biology |