Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Snowy tree crickets synchronize their chirps by responding to the preceding chirp of their neighbors. If a neighbor's chirp precedes his own, a cricket shortens his chirp and the following interval. If it follows his own, he lengthens his chirp interval and sometimes the following chirp. A single response of the first type may advance his phase of chirping 160 degrees and one of the second type may retard it 200 degrees . <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> The regular appearance of large NADH oscillations in mixtures of equal amounts of two suspensions of S. carlsbergensis originally oscillating approximately 180 ° out of phase is highly suggestive of the existence of an extracellular synchronizer of the glycolytic oscillation in this yeast. ::: ::: Two types of synchronization are usually observed: (a) both the phase and frequency of the mixed population oscillation are synchronized with one of the parent populations and (b) the frequency only is synchronized, i.e., the phase of the oscillation in the mixture is not close to either of the two parent suspensions. The large number of sustained oscillations at higher pH as compared to pH 4.5 suggests the possibility that damping may be due to a desynchronization process which is less effective at higher pH. ::: ::: In the absence of aldehyde traps such as KCN, Girard's reagent P, semicarbazide, or a mixture of NADH and ADH, the oscillations in the mixed suspension are highly damped and the frequency changes continuously. This suggests that acetaldehyde, secreted by the cells, may have a role as a desynchronizer of the oscillations. ::: ::: Mg2+ ion has the property of increasing the frequency of NADH oscillations in S. carlsbergensis. The synchronization of metabolism between individual yeast cells, which is described here, may be important as an example of a phenomenon common to many different cell types. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> A model of electrically coupled sinus node cells was used to investigate pacemaker coordination and conduction. Individual cells were simulated using differential equations describing transmembrane ionic currents. Intrinsic cycle lengths (periods) were adjusted by applying constant depolarizing or hyperpolarizing bias current, and cells were coupled through ohmic resistances to form two-dimensional arrays. Activation maps of 81-225 coupled cells showed an apparent wavefront conducting from a leading pacemaker region to the rest of the matrix even though the pattern actually resulted from mutual entrainment of all spontaneously beating cells. Apparent conduction time increased with increasing intercellular resistance. Appropriate selection of pacemaker cycle lengths and intercellular resistances permitted the accurate simulation of the activation sequence seen experimentally for the rabbit sinus node. Furthermore, a simulated acetylcholine pulse applied to a randomly selected 20% of the cells in this model produced a pacemaker shift that lasted several beats. These results support the hypothesis that sinus node synchronization occurs through a "democratic" process resulting from the phase-dependent interactions of thousands of pacemakers. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Synchronized flashing by males of some firefly species involves a capacity for visually coordinated, rhythmically coincident, inter-individual behavior that is apparently unique in the animal kingdom except for a few other arthropods and for man. This paper reviews (1) diverse communicative interactions that have evolved from elementary photic signals, (2) physiological mechanisms of synchronism, and (3) theories about its biological meaning. Work of the past 20 years shows that flash synchrony is widespread geographically and taxonomically, appears in an astonishing range of spectacular display types, utilizes several neural flash-control mechanisms and is pervasively but enigmatically involved in courtship. No proposed function for synchrony has been fully established but theory and physiology concur in indicating that synchrony aids male orientation toward the female, female recognition of male flashing, or both. Increased mate choice for the female is one likely ultimate benefit. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Abstract Much can be deduced about the behavior of chains of oscillators under minimal assumptions about the nature of the oscillators or the coupling. This paper reviews work on such chains, and provides a framework within which implications may be drawn about the neural networks that govern undulatory locomotion in lower vertebrates. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> We describe a new model for synchronization of neuronal oscillators that is based on the observation that certain species of fireflies are able to alter their free-running period. We show that by adding adaptation to standard oscillator models it is possible to observe the frequency alteration. One consequence of this is the perfect synchrony between coupled oscillators. Stability and some analytic results are included along with numerical simulations. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> It is numerically shown that there may be a new type of ordered state (in some sense glassy) in far-from-equilibrium systems which can be identified with a large population of coupled limit-cycle oscillators, provided couplings are not only random but also frustrated. It is characterized by quasientrainment and algebraic relaxation. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Abstract 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. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> In the early universe and in supernovae, the flavor evolution of massive neutrinos is nonlinear. Previously, numerical simulations have explored these conditions and have sometimes found collective, synchronized neutrino oscillations. Here these coherent phenomena are studied in the simplest possible system, an isotropic gas of two-flavor neutrinos. An analytical method is used to study the stability of the incoherent state. It is found that the incoherent state has neutral stability. That is, a steady state synchronization can exist for all nonzero neutrino densities, but the amount depends on the initial conditions. This result is verified by numerical simulation, but it is shown that numerical simulations are accurate for only a limited time. In more complicated neutrino systems, the incoherent state could be stable or unstable. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Research was supported in part by the ONR under Contract No. N00014-J- 1257, NATO Collaborative Research Grant No. CRG- 950282, NSF Grants Nos. DMS-9057433 and DMS- 9500948, and DGICYT ~Spain! Grant No. PB94-1167. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> We investigate an XY spin-glass model in which both spins and interactions (or couplings) evolve in time, but with widely separated time-scales. For large times this model can be solved using replica theory, requiring two levels of replicas, one level for the spins and one for the couplings. We define the relevant order parameters, and derive a phase diagram in the replica-symmetric approximation, which exhibits two distinct spin-glass phases. The first phase is characterized by freezing of the spins only, whereas in the second phase both spins and couplings are frozen. A detailed stability analysis also leads to two distinct corresponding de Almeida-Thouless lines, each marking continuous replica-symmetry breaking. Numerical simulations support our theoretical study. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> We report on a series of measurements aimed to characterize the development and the dynamics of the rhythmic applause in concert halls. Our results demonstrate that while this process shares many characteristics of other systems that are known to synchronize, it also has features that are unexpected and unaccounted for in many other systems. In particular, we find that the mechanism lying at the heart of the synchronization process is the period doubling of the clapping rhythm. The characteristic interplay between synchronized and unsynchronized regimes during the applause is the result of a frustration in the system. All results are understandable in the framework of the Kuramoto model. <s> BIB012 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> The emergence of a unified cognitive moment relies on the coordination of scattered mosaics of functionally specialized brain regions. Here we review the mechanisms of large-scale integration that counterbalance the distributed anatomical and functional organization of brain activity to enable the emergence of coherent behaviour and cognition. Although the mechanisms involved in large-scale integration are still largely unknown, we argue that the most plausible candidate is the formation of dynamic links mediated by synchrony over multiple frequency bands. <s> BIB013 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Synchronization is a common phenomenon in physical and biological systems. We examine the synchronization of two (and more) metronomes placed on a freely moving base. The small motion of the base couples the pendulums causing synchronization. The synchronization is generally in-phase, with antiphase synchronization occurring only under special conditions. The metronome system provides a mechanical realization of the popular Kuramoto model for synchronization of biological oscillators, and is excellent for classroom demonstrations and an undergraduate physics lab. <s> BIB014 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Coupled oscillator models use a single phase variable to approximate the voltage oscillation of each neuron during repetitivefiring where the behavior of the model depends on the connectivityand the interaction function chosen to describe the coupling. Weintroduce a network model consisting of a continuum of theseoscillators that includes the effects of spatially decaying coupling and axonal delay. We derive equations for determining the stability of solutions and analyze the network behavior for two different interaction functions. The first is a sine function, and the second is derived from a compartmental model of a pyramidal cell.In both cases, the system of coupled neural oscillators can undergo a bifurcation from synchronous oscillations to waves.The change in qualitative behavior is due to the axonal delay,which causes distant connections to encourage a phase shift between cells. We suggest that this mechanism could contribute to the behavior observed in several neurobiological systems. <s> BIB015 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> We present a planar agent-based flocking model with a distance-dependent communication weight. We derive a sufficient condition for the asymptotic flocking in terms of the initial spatial and heading-angle diameters and a communication weight. For this, we employ differential inequalities for the spatial and phase diameters together with the Lyapunov functional approach. When the diameter of the agent's initial heading-angles is sufficiently small, we show that the diameter of the heading-angles converges to the average value of the initial heading-angles exponentially fast. As an application of flocking estimates, we also show that the Kuramoto model with a connected communication topology on the regular lattice for identical oscillators exhibits a complete-phase-frequency synchronization, when coupled oscillators are initially distributed on the half circle. <s> BIB016 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Synchronization on the nanoscale has a wide range of applications ranging from timing, navigation, signal processing, microwave communication and novel computing and memory concepts. Existing coupled micromechanical oscillators suffer from limited range, neighborhood restriction and non-configurable coupling which limit the control, physical size and possible topologies of complex oscillator networks [1]. Here, we demonstrate the synchronization of two micromechanical oscillators, which are actuated and coupled by optical radiation field. We show that the coupling between the two oscillators can be tuned continuously from uncoupled to maximally coupled. These results pave a path towards massive and long-range synchronized oscillator networks. <s> BIB017 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Abstract In this report we recall the famous Huygens’ experiment which gave the first evidence of the synchronization phenomenon. We consider the synchronization of two clocks which are accurate (show the same time) but have pendula with different masses. It has been shown that such clocks hanging on the same beam can show the almost complete (in-phase) and almost antiphase synchronizations. By almost complete and almost antiphase synchronization we defined the periodic motion of the pendula in which the phase shift between the displacements of the pendula is respectively close (but not equal) to 0 or π . We give evidence that almost antiphase synchronization was the phenomenon observed by Huygens in XVII century. We support our numerical studies by considering the energy balance in the system and showing how the energy is transferred between the pendula via oscillating beam allowing the pendula’s synchronization. Additionally we discuss the synchronization of a number of different pendulum clocks hanging from a horizontal beam which can roll on the parallel surface. It has been shown that after a transient, different types of synchronization between pendula can be observed; (i) the complete synchronization in which all pendula behave identically, (ii) pendula create three or five clusters of synchronized pendula. We derive the equations for the estimation of the phase differences between phase synchronized clusters. The evidence, why other configurations with a different number of clusters are not observed, is given. <s> BIB018 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in sciences: <s> Previously, we showed using a computational agent-based model that a group of animals moving together can make a collective decision on direction of motion, even if there is a conflict between the directional preferences of two small subgroups of “informed” individuals and the remaining “uninformed” individuals have no directional preference. The model requires no explicit signaling or identification of informed individuals; individuals merely adjust their steering in response to socially acquired information on relative motion of neighbors. In this paper, we show how the dynamics of this system can be modeled analytically, and we derive a testable result that adding uninformed individuals improves stability of collective decision making. We first present a continuous-time dynamic model and prove a necessary and sufficient condition for stable convergence to a collective decision in this model. The stability of the decision, which corresponds to most of the group moving in one of two alternative preferred directions, depends explicitly on the magnitude of the difference in preferred directions; for a difference above a threshold the decision is stable and below that same threshold the decision is unstable. Given qualitative agreement with the results of the previous simulation study, we proceed to explore analytically the subtle but important role of the uninformed individuals in the continuous-time model. Significantly, we show that the likelihood of a collective decision increases with increasing numbers of uninformed individuals. <s> BIB019
The coupled oscillator model (1) and its generalization (4) appear in the study of biological synchronization and rhythmic phenomena. Example systems include pacemaker cells in the heart BIB003 , circadian cells in the brain BIB008 , coupled cortical neurons BIB015 , Hodgkin-Huxley neurons  , brain networks BIB013 ), yeast cells BIB002 , flashing fireflies BIB004 BIB006 , chirping crickets BIB001 , central pattern generators for animal locomotion BIB005 , particle models mimicking animal flocking behavior BIB016 , and fish schools , among others. The coupled oscillator model (1) also appears in physics and chemistry in modeling and analysis of spin glass models BIB007 BIB011 , flavor evolution of neutrinos BIB009 , coupled Josephson junctions BIB010 , coupled metronomes BIB014 ), Huygen's coupled pendulum clocks  BIB018 , micromechanical oscillators with optical BIB017 or mechanical  coupling, and in the analysis of chemical oscillations   . Finally, oscillator networks of the form (1) also serve as phenomenological models for synchronization phenomena in social networks, such as rhythmic applause BIB012 , opinion dynamics (Pluchino et al., 2006a,b) , pedestrian crowd synchrony on London's Millennium bridge , and decision making in animal groups BIB019 .