Biological Oscillators: Their Mathematical Analysis introduces the main features of the dynamic properties of biological oscillators and the mathematical techniques necessary for their investigation. It is not a comprehensive description of all known biological oscillators, since this would require a much bigger volume as well as a different type of expertise. Instead certain classes of biological oscillators are described, and then only in as much detail as required for the study of their dynamics.
The opening chapter reviews fundamental mathematical concepts and techniques which will be used in the remainder of the book. These include phase plane techniques; asymptotic techniques of Krylov, Bogoliubov, and Mitopolski; and the describing function. Subsequent chapters discuss examples of biological oscillators; phase shifts and phase response curves; the entrainment of oscillators by external inputs; the dynamics of circadian oscillators; effects of changing environment on the dynamics of biological oscillators; the features peculiar to populations of interacting oscillators; and biological phenomena attributable to populations of oscillators.
Preface Acknowledgments Chapter 1. Fundamentals of the Mathematical Theory of Oscillators 1.1 Introduction 1.2 Phase Plane Techniques 1.3 An Example of Application of Phase Plane Techniques 1.4 State Space 1.5 Asymptotic Techniques 1.6 Describing Function 1.7 Conservative Systems Chapter 2. Examples of Biological Rhythms 2.1 Introduction 2.2 Circadian Rhythms 2.3 Circadian Rhythms in an Insect Population 2.4 Circadian Rhythms in Cell Populations 2.5 Biochemical Oscillators 2.6 Neural Oscillators 2.7 Oscillations in Cultures of Fungi Chapter 3: Phase Shifts and Phase Response Curves 3.1 Introduction 3.2 Timing Sequences, Isochrones, Phase Response Curves, and Phase Transition Curves 3.3 Experimental Phase Response Curves 3.4 Analysis of Phase Response Curves 3.5 Isochrones and Phase Response Curves of a van der Pol Oscillator 3.6 Phase Response Curves of a Biochemical Oscillator 3.7 Bibliographical Notes Chapter 4. Entrainment of Oscillators by External Inputs 4.1 Introduction 4.2 Entrainment of a van der Pol Oscillator 4.3 Entrainment of Circadian Clocks by Light and Temperature Cycles 4.4 Fringe Entrainment 4.5 Entrainment of Oscillators by Pulses 4.6 Entrainment of a Circadian Rhythm by Light Pulses 4.7 Subharmonic Entrainment 4.8 Bibliographical Notes Chapter 5. The Dynamics of Circadian Oscillators 5.1 Introduction 5.2 Dependence of the Period of an Oscillator on Constant Environmental Factors 5.3 Mathematical Formulation of a Model for Biological Oscillators 5.4 The Independence between the Phase Response Curve and the Effects of Light on the Free-Run Period 5.5 Computer Simulation of Models for the Circadian Clock 5.6 Starting and Stopping the Clock 5.7 Bibliographical Notes Chapter 6. Effects of Changing Environment on the Dynamics of Biolo
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- © Academic Press 1973
- 28th January 1973
- Academic Press
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