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Dealing with dynamics of processes that repeat themselves regularly, this revised and updated edition extends the thread from 1980 to the present day, concentrating on areas of interest where there will be much activity in the future. This involves going through spatial biochemical, electrophysiological, and organismic dynamical systems and patterns that were discovered by pursuing the theme of phase singularities introduced in the original book. In particular the work on excitability in cell membranes will be thoroughly updated as will the references throughout the book.

From cell division to heartbeat, clocklike rhythms pervade the activities of every living organism. The cycles of life are ultimately biochemical in mechanism but many of the principles that dominate their orchestration are essentially mathematical.

The Geometry of Biological Time describes periodic processes in living systems and their non-living analogues in the abstract terms of nonlinear dynamics. Enphasis is given in phase singularities, waves, and mutual synchronization in tissues composed of many clocklike units. Also provided are descriptions of the best-studied experimental systems such as chemical oscillators, pacemaker neurons, circadian clocks, and excitable media organized into biochemical and bioelectrical wave patterns in two and three dimensions. No theoretical background is assumed; the required notions are introduced through an extensive collection of pictures and easily understood examples. This extensively updated new edition incorporates the fruits of two decades' further exploration guided by the same principles. Limit cycle theories of circadian clocks are now applied to human jet lag and are understood in terms of the molecular genetics of their recently discovered mechanisms. Supercomputers reveal the unforeseen architecture and dynamics of three-dimensional scroll waves in excitable media. Their role in life-threatening electrical aberrations of the heartbeat is exposed by laboratory experiments and corroborated in the clinic. These developments trace back to three basic mathematical ideas.

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1980 Preface 1999 Preface 1999 Acknowledgements Introduction 1 Circular Logic 2 Phase Singularities (Screwy Results of Circular Logic) 3 The Rules of the Ring 4 Ring Populations 5 Getting Off the Ring 6 Attracting Cycles and Isochrons 7 Measuring the Trajectories of a Circadian Clock 8 Populations of Attractor Cycle Oscillators 9 Excitable Kinetics and Excitable Media 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways 11 The Firefly Machine 12 Energy Metabolism in Cells 13 The Malonic Acid Reagent ('Sodium Geometrate') 14 Electrical Rhythmicity and Excitability in Cell Membranes 15 The Aggregation of Slime Mold Amoebae 16 Numerical Organizing Centers 17 Electrical Singular Filaments in the Heart Wall 18 Pattern Formation in the Fungi 19 Circadian Rhythms in General 20 The Circadian Clocks of Insect Eclosion 21 The Flower of Kalanchoe 22 The Cell Mitotic Cycle 23 The Female Cycle References Index of Names Index of Subjects

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