Biological Oscillators: Their Mathematical Analysis

Biological Oscillators: Their Mathematical Analysis

1st Edition - January 28, 1973

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  • Editor: Theodosios Pavlidis
  • eBook ISBN: 9780323159821

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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.

Table of Contents

  • Preface


    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 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 Notes

    Chapter 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 Notes

    Chapter 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



Product details

  • No. of pages: 222
  • Language: English
  • Copyright: © Academic Press 1973
  • Published: January 28, 1973
  • Imprint: Academic Press
  • eBook ISBN: 9780323159821

About the Editor

Theodosios Pavlidis

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