Biological Oscillators: Their Mathematical Analysis

PrefaceAcknowledgmentsChapter 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 SystemsChapter 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 FungiChapter 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 NotesChapter 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 NotesChapter 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 NotesChapter 6. Effects of Changing Environment on the Dynamics of Biological Oscillators 6.1 Introduction 6.2 State Variables and Parameters 6.3 Temperature Compensation in the Drosophila Pseudoobscura Eclosion Rhythm 6.4 Temperature Compensation in Unicellular Organisms 6.5 Effects of Heavy Water 6.6 Bibliographical NotesChapter 7. Populations of Interacting Oscillators 7.1 Introduction 7.2 Structural Stability 7.3 Monofrequency Oscillations in Systems of Coupled Oscillators Studied by Asymptotic Techniques 7.4 Strategies for the Study of Populations of Oscillators 7.5 Synchronization of Populations of Oscillators: The Weakly Nonlinear Case 7.6 Synchronization of Populations of Oscillators: The General Case 7.7 Concluding Remarks 7.8 Bibliographical NotesChapter 8. Biological Phenomena Attributable to Populations of Oscillators 8.1 Introduction 8.2 Frequency Doubling in Circadian Rhythms 8.3 A System of Coupled Biochemical Oscillators 8.4 Spatial Organization of Populations of Oscillators and an Interpretation of the Pattern of Zonation of Fungi Cultures 8.5 The Relation between Frequency and Tissue Size 8.6 Synchronization of Populations of Light-Emitting Organisms 8.7 Nonmonotonic Period Transients 8.8 Bibliographical Notes ReferencesIndex