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Biomedical subjects

S H Strogatz

Publications and source records attributed to S H Strogatz.

15 recordsLinked to original sources
PubMed2001

Are randomly grown graphs really random?

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at delta=1/8. At the transition, the average component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second-order phase transition at delta=1/4, and the average component size diverges there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph-older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.

Journal Article↗
PubMed2001

Random graphs with arbitrary degree distributions and their applications.

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Journal Article↗
PubMed2001

Phase diagram for the Winfree model of coupled nonlinear oscillators.

In 1967 Winfree proposed a mean-field model for the spontaneous synchronization of chorusing crickets, flashing fireflies, circadian pacemaker cells, or other large populations of biological oscillators. Here we give the first bifurcation analysis of the model, for a tractable special case. The system displays rich collective dynamics as a function of the coupling strength and the spread of natural frequencies. Besides incoherence, frequency locking, and oscillator death, there exist hybrid solutions that combine two or more of these states. We present the phase diagram and derive several of the stability boundaries analytically.

Journal Article↗
PubMed2001

Exploring complex networks.

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

Models, Theoretical↗
PubMed2000

Network robustness and fragility: percolation on random graphs.

Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.

Algorithms↗
PubMed1998

Collective dynamics of 'small-world' networks.

Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

Animals↗
PubMed1997

Cellular construction of a circadian clock: period determination in the suprachiasmatic nuclei.

The circadian clock in the suprachiasmatic nuclei is composed of multiple, single-cell circadian oscillators (clock cells). We now test the hypothesis that the circadian period in behavior is determined by the mean period that arises from the coupling of clock cells with diverse circadian periods. For these studies, we monitored firing rate rhythms of individual suprachiasmatic nuclei neurons on fixed multielectrode plates and exploited the altered circadian periods expressed by heterozygous and homozygous tau mutant hamsters. The results show that circadian period in the whole animal is determined by averaging widely dispersed periods of individual clock cells. The data also demonstrate that the tau mutation affects circadian function in a cell-autonomous manner.

Animals↗
PubMed1990

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

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.

Circadian Rhythm↗
PubMed1987

Human sleep and circadian rhythms: a simple model based on two coupled oscillators.

We propose a model of the human circadian system. The sleep-wake and body temperature rhythms are assumed to be driven by a pair of coupled nonlinear oscillators described by phase variables alone. The novel aspect of the model is that its equations may be solved analytically. Computer simulations are used to test the model against sleep-wake data pooled from 15 studies of subjects living for weeks in unscheduled, time-free environments. On these tests the model performs about as well as the existing models, although its mathematical structure is far simpler.

Body Temperature↗
PubMed1987

Circadian pacemaker interferes with sleep onset at specific times each day: role in insomnia.

The human circadian pacemaker modulates our desire and ability to fall asleep at different times of day. To study this circadian component of sleep tendency, we have analyzed the sleep-wake patterns recorded from 15 free-running subjects in whom the sleep-wake cycle spontaneously desynchronized from the circadian rhythm of body temperature. The analysis indicates that the distribution of sleep onsets during free run is bimodal, with one peak at the temperature trough and, contrary to previous reports, a second peak 9-10 h later. Furthermore, there are two consistent zones in the circadian temperature cycle during which normal subjects rarely fall asleep. We hypothesize that this bimodal rhythm of sleep tendency, revealed under free-running conditions, maintains the same fixed phase relation to the circadian temperature cycle during 24-h entrainment. This would imply that normally entrained individuals should experience a peak of sleep tendency in the midafternoon and a zone of minimal sleep tendency approximately 1-3 h before habitual bedtime. Our temporal isolation data thereby account quantitatively for the timing of the afternoon siesta and suggest that malfunctions of the phasing of the circadian pacemaker may underlie the insomnia associated with sleep-scheduling disorders.

Chronobiology Phenomena↗
PubMed1986

Bright light resets the human circadian pacemaker independent of the timing of the sleep-wake cycle.

Human circadian rhythms were once thought to be insensitive to light, with synchronization to the 24-hour day accomplished either through social contacts or the sleep-wake schedule. Yet the demonstration of an intensity-dependent neuroendocrine response to bright light has led to renewed consideration of light as a possible synchronizer of the human circadian pacemaker. In a laboratory study, the output of the circadian pacemaker of an elderly woman was monitored before and after exposure to 4 hours of bright light for seven consecutive evenings, and before and after a control study in ordinary room light while her sleep-wake schedule and social contacts remained unchanged. The exposure to bright light in the evening induced a 6-hour delay shift of her circadian pacemaker, as indicated by recordings of body temperature and cortisol secretion. The unexpected magnitude, rapidity, and stability of the shift challenge existing concepts regarding circadian phase-resetting capacity in man and suggest that exposure to bright light can indeed reset the human circadian pacemaker, which controls daily variations in physiologic, behavioral, and cognitive function.

Aged↗
PubMed1986

Circadian regulation dominates homeostatic control of sleep length and prior wake length in humans.

During prolonged temporal isolation in caves or windowless rooms, human subjects often develop complicated sleep-wake patterns. Seeking lawful structure in these patterns, we have reanalyzed the spontaneous timing of 359 sleep-wake cycles recorded from 15 internally desynchronized human subjects. The observed sleep-wake patterns obey a simple rule: The phase of the circadian temperature rhythm at bedtime determines the lengths of both prior wake (alpha) and subsequent sleep (rho). From this rule we derive an average alpha:rho relationship that depends on circadian phase. The relationship reconciles the established negative alpha:rho correlation observed in synchronized subjects with the positive alpha:rho correlation found in desynchronized subjects. Our most surprising result concerns the residual deviations of alpha and rho from their circadian phase-adjusted mean values. We report that there is no significant positive correlation between the residuals of alpha and rho, contrary to the prediction of restorative models of sleep duration. Our findings illuminate the mechanisms underlying sleep regulation and provide much-needed tests of mathematical models of the sleep-wake cycle.

Adult↗
PubMedDate not supplied

Organizing centres for three-dimensional chemical waves.

Excitable media propagate periodic waves of chemical activity. In three dimensions, they resemble scrolls rotating about axes where reactions are not periodic. These axes close in rings which may be linked and knotted, in ways limited by an exclusion principle.

Animals↗