## Chaos in Biological SystemsHans Degn, Arunn V. Holden, Lars Folke Olsen In recent years experimental and numerical studies have shown that chaos is a widespread phenomenon throughout the biological hierarchy ranging from simple enzyme reactions to ecosystems. Although a coherent picture of the fundamental mechanisms responsible for chaotic dynamics has started to appear it is not yet clear what the implications of such dynamics are for biological systems in general. In some systems it appears that chaotic dynamics are associated with a pathological condi tion. In other systems the pathological condition has regular periodic dynamics whilst the normal non-pathological condition has chaotic dyna mics. Since chaotic behaviour is so ubiquitous in nature and since the phenomenon raises some fundamental questions about its implications for biology it seemed timely to organize an interdisciplinary meeting at which leading scientists could meet to exchange ideas, to evaluate the current state of the field and to stipulate the guidelines along which future research should be directed. The present volume contains the contributions to the NATO Advanced Research Workshop on "Chaos in Biological Systems" held at Dyffryn House, St. Nicholas, Cardiff, U. K. , December 8-12, 1986. At this meeting 38 researchers with highly different backgrounds met to present their latest results through lectures and posters and to discuss the applica tions of non-linear techniques to problems of common interest. . In spite of their involvement in the study of chaotic dynamics for several years many of the participants met here for the first time. |

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### Contents

HEALTH OR DISEASE? | 1 |

PATTERNS OF ACTIVITY IN A REDUCED IONIC MODEL OF A CELL FROM THE RABBIT SINOATRIAL NODE | 5 |

PROBLEM OF A MATHEMATICALLY CORRECT MODEL | 13 |

BIFURCATIONS IN A MODEL OF THE PLATELET REGULATORY SYSTEM | 19 |

CHAOS IN A SYSTEM OF INTERACTING NEPHRONS | 23 |

PARTICIPANTS IN AND PRODUCTS OF A TWO PARAMETER DISSIPATIVE MEASURE PRESERVING SMOOTH DYNAMICAL SYSTEM IN ... | 33 |

INTERPLAY BETWEEN TWO PERIODIC ENZYME REACTIONS AS A SOURCE FOR COMPLEX OSCILLATORY BEHAVIOUR | 49 |

DYNAMICS OF CONTROLLED METABOLIC NETWORK AND CELLULAR BEHAVIOUR | 59 |

CHAOTIC AND IRREGULAR BURSTING PATTERN | 167 |

PANCREATIC BCELL AS AN EXAMPLE | 173 |

BIFURCATIONS IN THE ROSEHINDMARCH MODEL AND THE CHAY MODEL | 183 |

DENDRITIC BRANCHING PATTERNS | 191 |

CHAOS AND NEURAL NETWORKS | 195 |

DATA REQUIREMENTS FOR RELIABLE ESTIMATION OF CORRELATION DIMENSIONS | 207 |

A HEURISTIC OUTLINE | 221 |

CHAOS IN ECOLOGY AND EPIDEMIOLOGY | 233 |

PERIODIC FORCING OF A BIOCHEMICAL SYSTEM WITH MULTIPLE MODES OF OSCILLATORY BEHAVIOUR | 67 |

PERIODIC BEHAVIOUR AND CHAOS IN THE MECHANISM OF INTERCELLULAR COMMUNICATION GOVERNING AGGREGATION OF D... | 79 |

TURBULENT MORPHOGENESIS OF A PROTOTYPE MODEL REACTIONDIFFUSION SYSTEM | 91 |

PERIODIC SOLUTIONS AND GLOBAL BIFURCATIONS FOR NERVE IMPULSE EQUATIONS | 97 |

HOMOCLINIC AND PERIODIC SOLUTIONS OF NERVE IMPULSE EQUATIONS | 105 |

HIGH SENSITIVITY CHAOTIC BEHAVIOUR IN SINUSOIDALLY DRIVEN HODGKINHUXLEY EQUATIONS | 113 |

FORCED OSCILLATIONS AND ROUTES TO CHAOS IN THE HODGKINHUXLEY AXONS AND SQUID GIANT AXONS | 121 |

QUANTIFICATION OF CHAOS FROM PERIODICALLY FORCED SQUID AXONS | 133 |

CHAOS PHASE LOCKING AND BIFURCATION IN NORMAL SQUID AXONS | 143 |

CHAOS IN MOLLUSCAN NEURON | 157 |

LOW DIMENSIONAL STRANGE ATTRACTORS IN EPIDEMICS OF CHILDHOOD DISEASES IN COPENHAGEN DENMARK | 249 |

NEW TOOLS FOR THE STUDY OF ATTRACTORS | 255 |

POPULATIONS UNDER PERIODICALLY AND RANDOMLY VARYING GROWTH CONDITIONS | 267 |

BIFRACTAL BASIN BOUNDARIES IN INVERTIBLE SYSTEMS | 279 |

HOMOCLINIC BIFURCATIONS IN ORDINARY DIFFERENTIAL EQUATIONS | 285 |

CHARACTERIZATION OF ORDER AND DISORDER IN SPATIAL PATTERNS | 295 |

FRACTALS INTERMITTENCY AND MORPHOGENESIS | 305 |

TEMPERATURE STABILITY OF DAVYDOV SOLITONS | 315 |

321 | |

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Acad action potentials Aihara amplitude analysis attractor bifurcation diagram bifurcation points Biol biological birhythmicity burst solution cAMP cell chaotic behaviour chaotic dynamics chaotic responses complex constant correlation dimension correlation integral corresponding coupling CR mice current pulses curves described dimensional dU/dt dynamical systems electrical activity embedding dimension entropy example experimental feedback Figure flip bifurcation flow fractal frequency function Goldbeter Hodgkin-Huxley homoclinic orbits Hopf bifurcation hydrophobic hydrophobic mass energy initial conditions intermittency interval irregular Lett limit cycle linear Lyapunov exponent Math Matsumoto membrane potential mode nephron neuron Nicolis nonlinear observed obtained one-dimensional oscillatory pacemaker parameter space parameter values period-doubling bifurcations periodic orbits periodic solution phase portraits phase-locked Phys Physica Physiol plane Poincare potential responses protein random receptor region return map scale sequence shown in Fig shows signal spike squid giant axons stable stimulation stroboscopic structure Theor trajectories unstable variable W.M. Schaffer