Synchronization Threshold Research Articles

Enhancing synchrony in chaotic oscillators by dynamic relaying

In a chain of mutually coupled oscillators, the coupling threshold for synchronization between the outermost identical oscillators decreases when a type of impurity (in terms of parameter mismatch) is introduced in the inner oscillator(s). The outer oscillators interact indirectly via dynamic relaying, mediated by the inner oscillator(s). We confirm this enhancing of critical coupling in the chaotic regimes of the Lorenz system, in the Rössler system in the absence of coupling delay, and in the Mackey-Glass system with delay coupling. The enhancing effect is experimentally verified in the electronic circuit of Rössler oscillators.

Collective synchronization in the presence of reactive coupling and shear diversity

We analyze the synchronization dynamics of a model obtained from the phase reduction of the mean-field complex Ginzburg-Landau equation with heterogeneity. We present exact results that uncover the role of dissipative and reactive couplings on the synchronization transition when shears and natural frequencies are independently distributed. As it occurs in the purely dissipative case, an excess of shear diversity prevents the onset of synchronization, but this does not hold true if coupling is purely reactive. In this case, the synchronization threshold turns out to depend on the mean of the shear distribution, but not on all the other distribution's moments.

Synchronization of period-doubling oscillations in vascular coupled nephrons

The mechanisms by which the individual functional unit (nephron) of the kidney regulates the incoming blood flow give rise to a number of nonlinear dynamic phenomena, including period-doubling bifurcations and intra-nephron synchronization between two different oscillatory modes. Interaction between the nephrons produces complicated and time-dependent inter-nephron synchronization patterns. In order to understand the processes by which a pair of vascular coupled nephrons synchronize, the paper presents a detailed analysis of the bifurcations that occur at the threshold of synchronization. We show that, besides infinite cascades of saddle-node bifurcations, these transitions involve mutually connected cascades of torus and homoclinic bifurcations. To illustrate the broader range of occurrence of this bifurcation structure for coupled period-doubling systems, we show that a similar structure arises in a system of two coupled, non-identical Rössler oscillators.

On the role of astrocytes in synchronization of two coupled neurons: a mathematical perspective

Based on recent findings, astrocytes, a subtype of glial cells, dynamically regulate the synaptic transmission of neuronal networks. In this research, a biologically inspired neuronal network model is constructed by connecting two Morris-Lecar neuron models. In this minimal network model, neuron-astrocyte interactions are considered in a functional-based procedure. Utilizing the developed model and according to the theoretical analysis carried out in the article, it is confirmed that, the astrocyte increases the threshold value of synchronization and provides appropriate feedback control in regulating the neural activities. Therefore, the healthy astrocyte has the potential to desynchronize the synchrony between two coupled neurons. Next, we investigate malfunction of the astrocyte in the regulatory feedback loop. Mathematically, we verify that pathologic astrocyte is no longer able to increase the synchronization threshold and therefore, it cannot compensate excessive increase in the excitation level. The main reason behind this is the fact that healthy astrocyte can optimally increase the input current of the individual neurons, while the so-called pathological astrocyte is unable to modify correctly the amount of this current. Consequently, disruptions of the signaling function of astrocyte initiate the hypersynchronous firing of neurons. In other words, reduction in neuron-astrocyte cross-talk will lead to synchronized firing of neurons. Therefore, our results propose that the astrocyte could have a key role in stabilizing neural activity.

Isochronal chaos synchronization of delay-coupled optoelectronic oscillators

We study experimentally chaos synchronization of nonlinear optoelectronic oscillators with time-delayed mutual coupling and self-feedback. Coupling three oscillators in a chain, we find that the outer two oscillators always synchronize. In contrast, isochronal synchronization of the mediating middle oscillator is found only when self-feedback is added to the middle oscillator. We show how the stability of the isochronal solution of any network, including the case of three coupled oscillators, can be determined by measuring the synchronization threshold of two unidirectionally coupled systems. In addition, we provide a sufficient condition that guarantees global asymptotic stability of the synchronized solution.

Detecting the temporal structure of intermittent phase locking

This study explores a method to characterize the temporal structure of intermittent phase locking in oscillatory systems. When an oscillatory system is in a weakly synchronized regime away from a synchronization threshold, it spends most of the time in parts of its phase space away from the synchronization state. Therefore characteristics of dynamics near this state (such as its stability properties and Lyapunov exponents or distributions of the durations of synchronized episodes) do not describe the system's dynamics for most of the time. We consider an approach to characterize the system dynamics in this case by exploring the relationship between the phases on each cycle of oscillations. If some overall level of phase locking is present, one can quantify when and for how long phase locking is lost, and how the system returns back to the phase-locked state. We consider several examples to illustrate this approach: coupled skewed tent maps, the stability of which can be evaluated analytically; coupled Rössler and Lorenz oscillators, undergoing through different intermittency types on the way to phase synchronization; and a more complex example of coupled neurons. We show that the obtained measures can describe the differences in the dynamics and temporal structure of synchronization and desynchronization events for the systems with a similar overall level of phase locking and similar stability of the synchronized state.

How novel is too novel? Stream community thresholds at exceptionally low levels of catchment urbanization

Novel physical and chemical conditions of many modern ecosystems increasingly diverge from the environments known to have existed at any time in the history of Earth. The loss of natural land to urbanization is one of the most prevalent drivers of novel environments in freshwaters. However, current understanding of aquatic community response to urbanization is based heavily upon aggregate indicators of community structure and linear or wedge-shaped community response models that challenge ecological community theory. We applied a new analytical method, threshold indicator taxa analysis (TITAN), to a stream biomonitoring data set from Maryland to explicitly evaluate linear community response models to urbanization that implicitly assume individual taxa decline or increase at incrementally different levels of urbanization. We used TITAN (1) to identify the location and magnitude of greatest change in the frequency and abundance of individual taxa and (2) to assess synchrony in the location of change points as evidence for stream community thresholds in response to percent impervious cover in catchments. We documented clear and synchronous threshold declines of 110 of 238 macroinvertebrate taxa in response to low levels of impervious cover. Approximately 80% of the declining taxa did so between 0.5% and 2% impervious cover, whereas the last 20% declined sporadically from 2% to 25% impervious cover. Synchrony of individual responses resulted in distinct community-level thresholds ranging from < or = 0.68% (mountains), 1.28% (piedmont), and 0.96% (coastal plain) impervious cover. Upper limits (95% confidence intervals) of community thresholds were < 2% cover in all regions. Within distinct physiographic classes, higher-gradient, smaller catchments required less impervious cover than lower gradient, larger catchments to elicit community thresholds. Relatively few taxa showed positive responses to increasing impervious cover, and those that did gradually increased in frequency and abundance, approximating a linear cumulative distribution. The sharp, synchronous declines of numerous taxa established a consistent threshold response at exceptionally low levels of catchment urbanization, and uncertainty regarding the estimation of impervious cover from satellite data was mitigated by several corroborating lines of evidence. We suggest that threshold responses of communities to urban and other novel environmental gradients may be more prevalent than currently recognized.

Microscopic interactions lead to mutual synchronization in a network of networks

This Letter proposes a stochastic coupling scheme to study the collective dynamics in a network that comprises random Boolean networks. Based on microscopic interactions, which are understood as the exchange of information among nodes, mutual synchronization can be achieved when the product of the assigning probability and influence probability exceeds a critical threshold. A mean field model is developed to approximate the dynamical behaviors of the original system. The effect of finite system size can be further mimicked by incorporating a noise term into the model. The dependence of the synchronization threshold on the degrees of connectivity and coupling configuration is analyzed.

Adaptations in biceps brachii motor unit activity after repeated bouts of eccentric exercise in elbow flexor muscles

The purpose of this study was to examine changes in motor unit activity in the biceps brachii muscle after an initial (Bout 1) and repeated (Bout 2) session of eccentric exercise separated by 1 wk. Eight subjects (aged 22 ± 2 yr) participated in experimental assessments of neuromuscular function obtained before, immediately after, 24 h after, and 7 days after each exercise bout. Each experimental session involved assessments of elbow-flexor force and biceps and triceps brachii electromyography during maximum voluntary isometric contractions (MVCs) and constant-force isometric contractions at five contraction intensities (5-50% MVC), along with indicators of muscle damage (muscle pain and passive tension). In addition, motor unit recordings were obtained before exercise, 7 days after Bout 1, and 24 h after Bout 2 to assess motor unit synchronization and recruitment thresholds. Following a single eccentric exercise session that elicited significant indicators of muscle damage, we found a 57% increase in motor unit synchronization 7 days later compared with before exercise, despite the recovery of maximal strength, soreness, and relaxed elbow-joint angle at this time. Furthermore, a second bout of the same eccentric exercise resulted in reduced indicators of muscle damage and a decline in the strength of motor unit synchronization (24 h after Bout 2) toward levels observed before both exercise sessions. In contrast, no changes in motor unit recruitment thresholds were observed 7 days after Bout 1 or 24 h after Bout 2 compared with before exercise. The increased motor unit synchronization 7 days after a single eccentric exercise session provides new evidence of changes in motor unit activity during the putative repair and regeneration phase following eccentric muscle damage.

Phase-locking swallows in coupled oscillators with delayed feedback

We show that a nonlinear coupling with delayed feedback between two limit-cycle oscillators can lead to phase-locked, periodically modulated, and chaotic phase synchronization as well as to desynchronization. Parameter regions with stable phase-locked states attain the well-known form of the swallows or shrimps found and studied for nonlinear maps. We demonstrate that the swallow regions can be accompanied by a different bifurcation scenario where the periodic orbits of the phase-locked states undergo a torus bifurcation instead of a previously reported period-doubling bifurcation. This property has an impact on the spatial organization of the swallows in the parameter space. The swallow regions contribute to the synchronization domain of the considered system, and we analytically approximate the parameter synchronization threshold.

Considerations for analyzing ecological community thresholds in response to anthropogenic environmental gradients

The goal of this paper is to help managers better understand implications of using aggregate community metrics, such as taxon richness or Indices of Biotic Integrity (IBI), for detecting threshold responses to anthropogenic environmental gradients. To illustrate, we offer an alternative analytical approach, Threshold Indicator Taxa ANalysis (TITAN), geared toward identifying synchronous changes in the distribution of multiple taxa as evidence of an ecological community threshold. Our approach underscores the fundamental reality that which taxa are affected by stressors is important, both from a conservation standpoint and because taxon-specific life-history traits help us understand relevant mechanisms. First, we examine macroinvertebrate community response to an impervious cover gradient using a well-studied biomonitoring data set to show that representative community metrics are relatively insensitive to synchronous threshold declines of numerous individual taxa. We then reproduce these response relationships using a simulated community data set with similar properties to demonstrate that linear or wedge-shaped responses of community metrics to anthropogenic gradients can occur as an artifact of aggregating multiple taxa into a single value per sampling unit, despite strong nonlinearity in community response. Our findings do not repudiate the use of community metrics or multimetric indices, but they challenge assumptions that such metrics are capable of accurately reflecting community thresholds across a broad range of anthropogenic gradients. We recommend an alternative analysis framework that begins with characterization of the responses of individual taxa and uses aggregation only after distinguishing the magnitude, direction, and uncertainty in the responses of individual members of the community.

Effect of tumor resection on the characteristics of functional brain networks

Brain functioning such as cognitive performance depends on the functional interactions between brain areas, namely, the functional brain networks. The functional brain networks of a group of patients with brain tumors are measured before and after tumor resection. In this work, we perform a weighted network analysis to understand the effect of neurosurgery on the characteristics of functional brain networks. Statistically significant changes in network features have been discovered in the beta (13-30 Hz) band after neurosurgery: the link weight correlation around nodes and within triangles increases which implies improvement in local efficiency of information transfer and robustness; the clustering of high link weights in a subgraph becomes stronger, which enhances the global transport capability; and the decrease in the synchronization or virus spreading threshold, revealed by the increase in the largest eigenvalue of the adjacency matrix, which suggests again the improvement of information dissemination.

Effect of time delay on the onset of synchronization of the stochastic Kuramotomodel

We consider the Kuramoto model of globally coupled phase oscillatorswith time-delayed interactions, that is subject to Ornstein–Uhlenbeck(Gaussian) colored or non-Gaussian colored noise. We investigate numericallythe interplay between the influences of the finite correlation time of noiseτ and thetime delay τd on the onset of the synchronization process. The cases for identical and nonidentical oscillatorshave both been considered. Among the obtained results for identical oscillators is a large increaseof the synchronization threshold as a function of time delay for the colored non-Gaussiannoise compared to the case of the colored Gaussian noise at low noise correlation timeτ. However, the difference reduces remarkably for large noise correlation times. For the caseof nonidentical oscillators, the incoherent state may become unstable around the maximumvalue of the threshold (as a function of time delay) even at lower coupling strength valuesin the presence of colored noise as compared to the noiseless case. We have studiedthe dependence of the critical value of the coupling strength (the threshold ofsynchronization) on given parameters of the stochastic Kuramoto model in great detailand presented results for possible cases of colored Gaussian and non-Gaussiannoises.

Synchronization time in a hyperbolic dynamical system with long-range interactions

We show that the threshold of complete synchronization in a lattice of coupled non-smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace. We examine carefully the synchronization time and show that an inadequate observation of the system evolution leads to wrong results. We present both careful numerical experiments and a rigorous mathematical explanation confirming this fact, allowing for a generalization involving hyperbolic coupled map lattices.

GLOBAL DYNAMICS IN NETWORKS OF 1D ELEMENTS WITH TIME DELAYED MEAN FIELD COUPLING SUBJECT TO NOISE: SYNCHRONIZATION THRESHOLD, MULTISTABILITY AND HYSTERESIS

We determine the boundary of the synchronization domain of a large number of one-dimensional continuous stochastic elements with time delayed nonhomogeneous mean-field coupling. The exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker–Planck equation. Here the synchronization threshold is found by solving this BVP using the continuation technique (AUTO). Approximate analytics is obtained using expansion into eigenfunctions of the stationary Fokker–Planck operator. Multistability and hysteresis are demonstrated for the case of bistable elements with a polynomial potential.

A lower bound for the length of the shortest carefully synchronizing words

We introduce the notion of careful synchronization for partial finite automata as a natural generalization of the synchronization notion for complete finite automata. We obtain a lower bound for the careful synchronization threshold for automata with a given number of states.

Influence of passive elements on the dynamics of oscillatory ensembles of cardiac cells

In this paper we focus on the influence of passive elements on the collective dynamics of oscillatory ensembles. Two major effects considered are (i) the influence of passive elements on the synchronization properties of ensembles of coupled nonidentical oscillators and (ii) the influence of passive elements on the wave dynamics of such systems. For the first effect, it is demonstrated that the introduction of passive elements may lead to both an increase or decrease in the global synchronization threshold. For the second effect, it is also demonstrated that the steady state of the passive element is a key parameter which defines how this passive element affects the wave dynamics of the oscillatory ensemble. It was shown that for different values of this parameter, one can observe increase or decrease in wave propagation velocity and increase or decrease in synchronization frequency in oscillatory ensembles with the growth of influence of passive elements. The results are obtained for the models of cardiac cells dynamics as well as for the Bonhoeffer-Van der Pol model and are compared with data of real biological experiments.

Synchronization of chaotic semiconductor lasers by optical injection with random phase modulation

It is shown that identical synchronization of two chaotic semiconductor lasers can be achieved by injection of a common optical signal with randomly varying phase. An optical signal with randomly modulated phase is injected into two semiconductor lasers which have chaotic oscillations due to optical feedback. Strong correlation between complex intensity oscillations of the two lasers is observed even though the intensity of the common injection signal is constant. Characteristic properties of this type of synchronization are shown, in particular, the dependence of the synchronization threshold on the injection strength and the rate of phase modulation, and the dependence of the intensity correlation on the difference in phase of optical feedback.

Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling

We study synchronization as a means of control of collective behavior of an ensemble of coupled stochastic units in which oscillations are induced merely by external noise. For a large number of one-dimensional continuous stochastic elements coupled non-homogeneously through the mean field with delay we developed an approach to find a boundary of synchronization domain and the frequency of the mean-field oscillations on it. Namely, the exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker–Planck equation. Here the synchronization threshold is found by solving this BVP numerically. Approximate analytics is obtained by expanding the solution of the linearized Fokker–Planck equation into a series of eigenfunctions of the stationary Fokker–Planck operator. Bistable systems with a polynomial and piece-wise linear potential are considered as examples. Multistability and hysteresis in the mean-field behavior are observed in the stochastic network at finite noise intensities. In the limit of small noise intensities the critical coupling strength is shown to remain finite, provided that the delay in the coupling function is not infinitely small. Delay in the coupling term can be used as a control parameter that manipulates the location of the synchronization threshold.

Optimal Sequential Frame Synchronization

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We consider the “one-shot frame synchronization problem,” where a decoder wants to locate a sync pattern at the output of a memoryless channel on the basis of sequential observations. The sync pattern of length <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis> starts being emitted at a random time within some interval of size <emphasis><formula formulatype="inline"> <tex>$A$</tex></formula></emphasis>, where <emphasis><formula formulatype="inline"> <tex>$A$</tex></formula></emphasis> characterizes the asynchronism level. We show that a sequential decoder can optimally locate the sync pattern, i.e., exactly, without delay, and with probability approaching one as <emphasis><formula formulatype="inline"><tex>$N \rightarrow \infty$</tex></formula></emphasis>, if the asynchronism level grows as <emphasis><formula formulatype="inline"> <tex>$O(e^{N\alpha})$</tex></formula></emphasis>, with <emphasis><formula formulatype="inline"><tex>$\alpha$</tex></formula></emphasis> below the <emphasis emphasistype="boldital">synchronization threshold</emphasis>, a constant that admits a simple expression depending on the channel. If <emphasis><formula formulatype="inline"><tex>$\alpha$</tex></formula></emphasis> exceeds the synchronization threshold, any decoder, sequential or nonsequential, locates the sync pattern with an error that tends to one as <emphasis><formula formulatype="inline"> <tex>$N\rightarrow \infty$</tex></formula></emphasis>. Hence, a sequential decoder can locate a sync pattern as well as the (nonsequential) maximum-likelihood decoder that operates on the basis of output sequences of maximum length <emphasis><formula formulatype="inline"><tex>$A+N-1$</tex></formula></emphasis>, but with far fewer observations. </para>