2,208 research outputs found
Perturbation in Population of Pulse-Coupled Oscillators Leads to Emergence of Structure
A new synchronization model based on pulse-coupled oscillators is proposed. A population of coupled oscillators is represented as a cellular automaton. Each cell periodically enters a firing state. Firing of a cell is sensed by other cells in a neighborhood of radius R. As a result the sensing cell may change its firing rate. The interaction strength between a firing and a sensing cell decreases with the squared distance between the two cells. For most starting conditions waves of synchronized firing cells emerge. Simulations indicate that for certain parameter values the emergence of synchronization waves occurs only if there is dispersion in the intrinsic firing frequencies of the cells. Emergence of synchronization waves is an important feature of the model
Two-cluster bifurcations in systems of globally pulse-coupled oscillators
For a system of globally pulse-coupled phase-oscillators, we derive
conditions for stability of the completely synchronous state and all possible
two-cluster states and explain how the different states are naturally connected
via bifurcations. The coupling is modeled using the phase-response-curve (PRC),
which measures the sensitivity of each oscillator's phase to perturbations. For
large systems with a PRC, which turns to zero at the spiking threshold, we are
able to find the parameter regions where multiple stable two-cluster states
coexist and illustrate this by an example. In addition, we explain how a
locally unstable one-cluster state may form an attractor together will its
homoclinic connections. This leads to the phenomenon of intermittent,
asymptotic synchronization with abating beats away from the perfect synchrony.Comment: 12 pages. 6 figure
Synchronization in complex networks
Synchronization processes in populations of locally interacting elements are
in the focus of intense research in physical, biological, chemical,
technological and social systems. The many efforts devoted to understand
synchronization phenomena in natural systems take now advantage of the recent
theory of complex networks. In this review, we report the advances in the
comprehension of synchronization phenomena when oscillating elements are
constrained to interact in a complex network topology. We also overview the new
emergent features coming out from the interplay between the structure and the
function of the underlying pattern of connections. Extensive numerical work as
well as analytical approaches to the problem are presented. Finally, we review
several applications of synchronization in complex networks to different
disciplines: biological systems and neuroscience, engineering and computer
science, and economy and social sciences.Comment: Final version published in Physics Reports. More information
available at http://synchronets.googlepages.com
Macroscopic Models and Phase Resetting of Coupled Biological Oscillators
This thesis concerns the derivation and analysis of macroscopic mathematical models for coupled biological oscillators. Circadian rhythms, heart beats, and brain waves are all examples of biological rhythms formed through the aggregation of the rhythmic contributions of thousands of cellular oscillations. These systems evolve in an extremely high-dimensional phase space having at least as many degrees of freedom as the number of oscillators. This high-dimensionality often contrasts with the low-dimensional behavior observed on the collective or macroscopic scale. Moreover, the macroscopic dynamics are often of greater interest in biological applications.
Therefore, it is imperative that mathematical techniques are developed to extract low-dimensional models for the macroscopic behavior of these systems. One such mathematical technique is the Ott-Antonsen ansatz. The Ott-Antonsen ansatz may be applied to high-dimensional systems of heterogeneous coupled oscillators to derive an exact low-dimensional description of the system in terms of macroscopic variables. We apply the Ott-Antonsen technique to determine the sensitivity of collective oscillations to perturbations with applications to neuroscience.
The power of the Ott-Antonsen technique comes at the expense of several limitations which could limit its applicability to biological systems. To address this we compare the Ott-Antonsen ansatz with experimental measurements of circadian rhythms and numerical simulations of several other biological systems. This analysis reveals that a key assumption of the Ott-Antonsen approach is violated in these systems. However, we discover a low-dimensional structure in these data sets and characterize its emergence through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate the structure's emergence in networks of noisy heterogeneous oscillators with complex network connectivity. We show how this structure may be applied as an ansatz to derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system.
We first apply our ansatz to study the impact of light on the mammalian circadian system. To begin we derive a low-dimensional macroscopic model for the core circadian clock in mammals. Significantly, the variables and parameters in our model have physiological interpretations and may be compared with experimental results. We focus on the effect of four key factors which help shape the mammalian phase response to light: heterogeneity in the population of oscillators, the structure of the typical light phase response curve, the fraction of oscillators which receive direct light input and changes in the coupling strengths associated with seasonal day-lengths. We find these factors can explain several experimental results and provide insight into the processing of light information in the mammalian circadian system.
In a second application of our ansatz we derive a pair of low-dimensional models for human circadian rhythms. We fit the model parameters to measurements of light sensitivity in human subjects, and validate these parameter fits with three additional data sets. We compare our model predictions with those made by previous phenomenological models for human circadian rhythms. We find our models make new predictions concerning the amplitude dynamics of the human circadian clock and the light entrainment properties of the clock. These results could have applications to the development of light-based therapies for circadian disorders.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138766/1/khannay_1.pd
Synchronization of Kuramoto Oscillators in Scale-Free Networks
In this work, we study the synchronization of coupled phase oscillators on
the underlying topology of scale-free networks. In particular, we assume that
each network's component is an oscillator and that each interacts with the
others following the Kuramoto model. We then study the onset of global phase
synchronization and fully characterize the system's dynamics. We also found
that the resynchronization time of a perturbed node decays as a power law of
its connectivity, providing a simple analytical explanation to this interesting
behavior.Comment: 7 pages and 4 eps figures, the text has been slightly modified and
new references have been included. Final version to appear in Europhysics
Letter
Phaselocked patterns and amplitude death in a ring of delay coupled limit cycle oscillators
We study the existence and stability of phaselocked patterns and amplitude
death states in a closed chain of delay coupled identical limit cycle
oscillators that are near a supercritical Hopf bifurcation. The coupling is
limited to nearest neighbors and is linear. We analyze a model set of discrete
dynamical equations using the method of plane waves. The resultant dispersion
relation, which is valid for any arbitrary number of oscillators, displays
important differences from similar relations obtained from continuum models. We
discuss the general characteristics of the equilibrium states including their
dependencies on various system parameters. We next carry out a detailed linear
stability investigation of these states in order to delineate their actual
existence regions and to determine their parametric dependence on time delay.
Time delay is found to expand the range of possible phaselocked patterns and to
contribute favorably toward their stability. The amplitude death state is
studied in the parameter space of time delay and coupling strength. It is shown
that death island regions can exist for any number of oscillators N in the
presence of finite time delay. A particularly interesting result is that the
size of an island is independent of N when N is even but is a decreasing
function of N when N is odd.Comment: 23 pages, 12 figures (3 of the figures in PNG format, separately from
TeX); minor additions; typos correcte
Echo Phenomena in Populations of Chemical Oscillators
The emergence of collective behavior has been observed in all levels of biological systems, for example, the aggregation of slime mold, swarm motion of insects and the collective motion in schools of sh. Synchronization is one of the most important collective behaviors and can play a pivotal role in maintaining the normal function of a living system, such as pacemaker cells in the heart, circadian rhythms, and insulin release from pancreatic cells. Synchronization typically arises as a result of the interaction of large ensembles of oscillators. Studies of the Belousov-Zhabotinsky (BZ) chemical oscillators have shown a variety of collective dynamical behaviors, such as phase clusters, dynamical quorum sensing and chimera states. The discovery of echo phenomena in large populations of coupled Kuramoto oscillators motivates us to study this dynamical behavior using photosensitive BZ oscillators.;In this thesis, we examine echo behavior experimentally and mathematically. The experiments are carried out with a BZ micro-oscillator system. A large system of micro-oscillators is achieved by the design of a large oscillator array (LOA), which permits coupling of over 1000 oscillators. The dimensionless Zhabotinsky, Buchholtz, Kiyatkin and Epstein (ZBKE) mathematical model is used to investigate the behavior. The experiment and numerical results illustrate that if a BZ system of oscillators is subject to two perturbations, separated by time tau, then at the time tau after second perturbation, the oscillators show a measurable response in their collective signal. Factors such as noise and size of the perturbation impacting the magnitude of the echo are examined and a theoretical calculation of the magnitude of the echo as a function of the size of the perturbation exhibits good agreement with simulation results using the ZBKE model
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