Showing papers by "Steven H. Strogatz published in 1990"

Synchronization of pulse-coupled biological oscillators

Abstract: A simple model for synchronous firing of biological oscillators based on Peskin's model of the cardiac pacemaker (Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975, pp. 268-278) is studied. The model consists of a population of identical integrate-and-fire oscillators. The coupling between oscillators is pulsatile: when a given oscillator fires, it pulls the others up by a fixed amount, or brings them to the firing threshold, whichever is less. The main result is that for almost all initial conditions, the population evolves to a state in which all the oscillators are firing synchronously. The relationship between the model and real communities of biological oscillators is discussed; examples include populations of synchronously flashing fireflies, crickets that chirp in unison, electrically synchronous pacemaker cells, and groups of women whose menstrual cycles become mutually synchronized.

Amplitude death in an array of limit-cycle oscillators

Abstract: We analyze a large system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. We prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance, the system undergoes “amplitude death”-the oscillators pull each other off their limit cycles and into the origin, which in this case is astable equilibrium point for the coupled system. We determine the region in couplingvariance space for which amplitude death is stable, and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system.

Phase diagram for the collective behavior of limit-cycle oscillators.

Abstract: We analyze a large dynamical system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. Depending on the choice of coupling strength and the spread of natural frequencies, the system exhibits frequency locking, amplitude death, and incoherence, as well as novel unsteady behavior characterized by periodic, quasiperiodic, or chaotic evolution of the system's order parameter. The phase boundaries between several of these states are obtained analytically.

Interpreting the human phase response curve to multiple bright-light exposures.

Abstract: Czeisler and his colleagues have recently reported that bright light can induce strong (Type O) resetting of the human circadian pacemaker. This surprising result shows that the human clock is more responsive to light than has been previously thought. The interpretation of their results is subtle, however, because of an unconventional aspect of their experimental protocol: They measured the phase shift after three cycles of the bright-light stimulus, rather than after the usual single pulse. A natural question is whether the apparent Type O response could reflect the summation of three weaker Type 1 responses to each of the daily light pulses. In this paper I show mathematically that repeated Type 1 resetting cannot account for the observed Type O response. This finding corroborates the strong resetting reported by Czeisler et al., and supports their claim that bright light induces strong resetting by crushing the amplitude of the circadian pacemaker. Furthermore, the results indicate that back-to-back light pulses can have a cooperative effect different from that obtained by simple iteration of a phase response curve (PRC). In this sense the resetting response of humans is similar to that of Drosophila, Kalanchoe, and Culex, and is more complex than that predicted by conventional PRC theory. To describe the way in which light resets the human circadian pacemaker, one needs a theory that includes amplitude resetting, as pioneered by Winfree and developed for humans by Kronauer.

Jump bifurcation and hysteresis in an infinite-dimensional dynamical system of coupled spins

Abstract: This paper studies an infinite-dimensional dynamical system that models a collection of coupled spins in a random magnetic field. A state of the system is given by a self-map of the unit circle. Critical states for the dynamical system correspond to equilibrium configurations of the spins. Exact solutions are obtained for all the critical states and their stability is characterized by analyzing the second variation of the system’s potential function at those states. It is proven rigorously that the system exhibits a jump bifurcation and hysteresis as the coupling between spins is varied.

Amplitude Death in an Array of

Abstract: We analyze a large system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies We prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance, the system undergoes "amplitude death"--the oscillators pull each other off their limit cycles and into the origin, which in this case is a stable equilibrium point for the coupled system We determine the region in couplingvariance space for which amplitude death is stable, and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system