## From Clocks to Chaos: The Rhythms of LifeIn an important new contribution to the literature of chaos, two distinguished researchers in the field of physiology probe central theoretical questions about physiological rhythms. Topics discussed include: How are rhythms generated? How do they start and stop? What are the effects of perturbation of the rhythms? How are oscillations organized in space? Leon Glass and Michael Mackey address an audience of biological scientists, physicians, physical scientists, and mathematicians, but the work assumes no knowledge of advanced mathematics. Variation of rhythms outside normal limits, or appearance of new rhythms where none existed previously, are associated with disease. One of the most interesting features of the book is that it makes a start at explaining "dynamical diseases" that are not the result of infection by pathogens but that stem from abnormalities in the timing of essential functions. From Clocks to Chaos provides a firm foundation for understanding dynamic processes in physiology. |

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### Contents

Introduction The Rhythms of Life | 3 |

12 Mathematical Models for Biological Oscillators | 8 |

13 Perturbing Physiological Rhythms | 10 |

14 Spatial Oscillations | 13 |

15 Dynamical Disease | 16 |

Notes and References | 17 |

Steady States Oscillations and Chaos in Physiological Systems | 19 |

22 Steady States | 21 |

62 Phase Resetting in Integrate and Fire Models | 102 |

63 Phase Resetting of Limit Cycle Oscillations | 104 |

64 Phase Resetting in Diverse Systems | 109 |

65 Practical Problems with Application of the Topological Theory | 113 |

66 Summary | 116 |

Notes and References | 117 |

Periodic Stimulation of Biological Oscillators | 119 |

72 Mathematical Concepts | 123 |

23 Limit Cycles and the Phase Plane | 22 |

24 Local Stability Bifurcations and Structural Stability | 25 |

25 Bifurcation and Chaos in Finite Difference Equations | 26 |

26 Summary | 34 |

Noise and Chaos | 36 |

32 Noise versus Chaos | 42 |

33 Identification of Chaos | 47 |

34 Strange Attractors Dimension and Liapunov Numbers | 50 |

35 Summary | 55 |

Mathematical Models for Biological Oscillations | 57 |

42 Central Pattern Generators | 63 |

43 Mutual Inhibition | 64 |

44 Sequential Disinhibition | 66 |

45 Negative Feedback Systems | 68 |

46 Oscillations in Mixed Feedback Systems with Time Delays | 72 |

47 Summary | 78 |

Initiation and Termination of Physiological Rhythms | 82 |

52 Soft Excitation | 84 |

53 Hard Excitation | 90 |

The Black Hole | 93 |

55 Summary | 95 |

Notes and References | 96 |

Single Pulse Perturbation of Biological Oscillators | 98 |

73 Periodic Forcing of Integrate and Fire Models | 128 |

74 Entrainment of Limit Cycle Oscillators | 132 |

75 Phase Locking of Rhythms in Humans | 135 |

76 Summary | 141 |

Spatial Oscillations | 144 |

82 Wave Propagation in a Ring of Tissue | 155 |

83 Waves and Spirals in Two Dimensions | 157 |

84 Organizing Centers in Three Dimensions | 159 |

85 Fibrillation and Other Disorders | 160 |

86 Summary | 167 |

Dynamical Diseases | 172 |

92 Formulation of Mathematical Models for Dynamical Diseases | 174 |

93 Development of Biological Models for Dynamical Diseases | 175 |

94 Diagnosis and Therapy | 176 |

95 Summary | 179 |

Afterthoughts | 182 |

Mathematical Appendix | 183 |

A2 Finite Difference Equations | 194 |

A3 Problems | 205 |

Notes and References | 208 |

References | 211 |

243 | |