Synchronization of delayed coupled neurons in presence of inhomogeneity

In principle, two directly coupled limit cycle oscillators can overcome mismatch in intrinsic rates and match their frequencies, but zero phase lag synchronization is just achievable in the limit of zero mismatch, i.e., with identical oscillators. Delay in communication, on the other hand, can exert phase shift in the activity of the coupled oscillators. In this study, we address the question of how phase locked, and in particular zero phase lag synchronization, can be achieved for a heterogeneous system of two delayed coupled neurons. We have analytically studied the possibility of inphase synchronization and near inphase synchronization when the neurons are not identical or the connections are not exactly symmetric. We have shown that while any single source of inhomogeneity can violate isochronous synchrony, multiple sources of inhomogeneity can compensate for each other and maintain synchrony. Numeric studies on biologically plausible models also support the analytic results.     

Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.     

Factors affecting phase synchronization in integrate-and-fire oscillators

Step changes in input current are known to induce partial phase synchrony in ensembles of leaky integrate-and-fire neurons operating in the oscillatory or “regular firing” regime. An analysis of this phenomenon in the absence of noise is presented based on the probability flux within an ensemble of generalized integrate-and-fire neurons. It is shown that the induction of phase synchrony by a step input can be determined by calculating the ratio of the voltage densities obtained from fully desynchronized ensembles firing at the pre and post-step firing rates. In the limit of low noise and in the absence of phase synchrony, the probability density as a function of voltage is inversely proportional to the time derivative along the voltage trajectory. It follows that the magnitude of phase synchronization depends on the degree to which a change in input leads to a uniform multiplication of the voltage derivative over the range from reset to spike threshold. This analysis is used to investigate several factors affecting phase synchronization including high firing rates, inputs modeled as conductances rather than currents, peri-threshold sodium currents, and spike-triggered potassium currents. Finally, we show that without noise, the equilibrium ensemble density is proportional to the phase response curve commonly used to analyze oscillatory systems. Action Editor: John Rinzel     

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.     

Spatial coupling in cyclic population dynamics: models and data

We use a dynamic random field to model a spatial collection of coupled oscillators with discrete time stochastic dynamics. At each time step the phase of each cyclic local population is subject to random noise, incremented by a common dynamic, and pulled by a coupling force in the direction of some collective mean phase. We define asynchrony and derive expressions for its measurement in this model. We describe robust methods for phase estimation of cyclic population time series, for estimating strength of coupling between local populations, and for measuring variance of locally acting noise from field data. Proposed methods allow intermittently acting phase synchronizing events operating over large spatial scales to be distinguished from more continuous and possibly locally acting coupling, both of which could result in elevated levels of phase synchronization. We demonstrate the utility of this approach by applying it to classical spatial time series data of Canadian lynx. Analysis confirms findings of previous studies and reveals evidence to suggest that interpopulation coupling was weaker over the 20th century than for the 1800s. Analysis supports the notion that synchrony in these populations is maintained by a continuous and locally acting coupling between adjacent regions with large phase adjustments occurring only infrequently. When this coupling is absent, asynchrony develops between populations.     

Collective Cell Movement Promotes Synchronization of Coupled Genetic Oscillators

Collective cell movement is a crucial component of embryonic development. Intercellular interactions regulate collective cell movement by allowing cells to transfer information. A key question is how collective cell movement itself influences information flow produced in tissues by intercellular interactions. Here, we study the effect of collective cell movement on the synchronization of locally coupled genetic oscillators. This study is motivated by the segmentation clock in zebrafish somitogenesis, where short-range correlated movement of cells has been observed. We describe the segmentation clock tissue by a Voronoi diagram, cell movement by the force balance of self-propelled and repulsive forces between cells, the dynamics of the direction of self-propelled motion, and the synchronization of genetic oscillators by locally coupled phase oscillators. We find that movement with a correlation length of about 2 ∼ 3 cell diameters is optimal for the synchronization of coupled oscillators. Quantification of cell mixing reveals that this short-range correlation of cell movement allows cells to exchange neighbors most efficiently. Moreover, short-range correlated movement strongly destabilizes nonuniform spatial phase patterns, further promoting global synchronization. Our theoretical results suggest that collective cell movement may enhance the synchronization of the segmentation clock in zebrafish somitogenesis. More generally, collective cell movement may promote information flow in tissues by enhancing cell mixing and destabilizing spurious patterns.     

Collective Cell Movement Promotes Synchronization of Coupled Genetic Oscillators

Collective cell movement is a crucial component of embryonic development. Intercellular interactions regulate collective cell movement by allowing cells to transfer information. A key question is how collective cell movement itself influences information flow produced in tissues by intercellular interactions. Here, we study the effect of collective cell movement on the synchronization of locally coupled genetic oscillators. This study is motivated by the segmentation clock in zebrafish somitogenesis, where short-range correlated movement of cells has been observed. We describe the segmentation clock tissue by a Voronoi diagram, cell movement by the force balance of self-propelled and repulsive forces between cells, the dynamics of the direction of self-propelled motion, and the synchronization of genetic oscillators by locally coupled phase oscillators. We find that movement with a correlation length of about 2 ∼ 3 cell diameters is optimal for the synchronization of coupled oscillators. Quantification of cell mixing reveals that this short-range correlation of cell movement allows cells to exchange neighbors most efficiently. Moreover, short-range correlated movement strongly destabilizes nonuniform spatial phase patterns, further promoting global synchronization. Our theoretical results suggest that collective cell movement may enhance the synchronization of the segmentation clock in zebrafish somitogenesis. More generally, collective cell movement may promote information flow in tissues by enhancing cell mixing and destabilizing spurious patterns.     

Coupling-induced synchronization in multicellular circadian oscillators of mammals

In mammals, circadian rhythms are controlled by the neurons located in the suprachiasmatic nucleus (SCN) of the hypothalamus. Each neuron in the SCN contains an autonomous molecular clock. The fundamental question is how the individual cellular oscillators, expressing a wide range of periods, interact and assemble to achieve phase synchronization. Most of the studies carried out so far emphasize the crucial role of the periodicity imposed by the light-dark cycle in neuronal synchronization. However, in natural conditions, the interaction between the SCN neurons is non-negligible and coupling between cells in the SCN is achieved partly by neurotransmitters. In this paper, we use a model of nonidentical, globally coupled cellular clocks considered as Goodwin oscillators. We mainly study the synchronization induced by coupling from an analytical way. Our results show that the role of the coupling is to enhance the synchronization to the external forcing. The conclusion of this paper can help us better understand the mechanism of circadian rhythm.     

Synchronization of coupled biological oscillators under spatially heterogeneous environmental forcing

Spatially heterogeneous intensities of environmental signals are common in nature, being caused, e.g., by rugged or curved surfaces leading to varying angles of incidence and intensities. In this work, we perform numerical studies of one-dimensional arrays of coupled phase oscillators driven by a periodic signal with spatially heterogeneous amplitude, considering both random and gradual amplitude distributions of the driving. We compare the effects of global and next-neighbor interactions, respectively, on the mutual and forced synchronization in the array. Weak global coupling leads to full mutual synchronization for all studied driving configurations. The degree of external synchronization follows a majority rule, depending on the number of externally entrained oscillators in the uncoupled case. The effects of next-neighbor coupling depend on the spatial distribution of the driving amplitude. For random distributions, local interactions show the same qualitative effects as global coupling. In contrast, for gradual distributions and large driving heterogeneities, next-neighbor coupling is detrimental to both mutual and external synchronization. We discuss these observations with respect to fundamental aspects of heterogeneity and variability of dynamical systems, as well as the intercellular synchronization of circadian oscillators.     

Three people can synchronize as coupled oscillators during sports activities

We experimentally investigated the synchronized patterns of three people during sports activities and found that the activity corresponded to spatiotemporal patterns in rings of coupled biological oscillators derived from symmetric Hopf bifurcation theory, which is based on group theory. This theory can provide catalogs of possible generic spatiotemporal patterns irrespective of their internal models. Instead, they are simply based on the geometrical symmetries of the systems. We predicted the synchronization patterns of rings of three coupled oscillators as trajectories on the phase plane. The interactions among three people during a 3 vs. 1 ball possession task were plotted on the phase plane. We then demonstrated that two patterns conformed to two of the three patterns predicted by the theory. One of these patterns was a rotation pattern (R) in which phase differences between adjacent oscillators were almost 2π/3. The other was a partial anti-phase pattern (PA) in which the two oscillators were anti-phase and the third oscillator frequency was dead. These results suggested that symmetric Hopf bifurcation theory could be used to understand synchronization phenomena among three people who communicate via perceptual information, not just physically connected systems such as slime molds, chemical reactions, and animal gaits. In addition, the skill level in human synchronization may play the role of the bifurcation parameter.     

Phase computations and phase models for discrete molecular oscillators

Background

Biochemical oscillators perform crucial functions in cells, e.g., they set up circadian clocks. The dynamical behavior of oscillators is best described and analyzed in terms of the scalar quantity, phase. A rigorous and useful definition for phase is based on the so-called isochrons of oscillators. Phase computation techniques for continuous oscillators that are based on isochrons have been used for characterizing the behavior of various types of oscillators under the influence of perturbations such as noise.

Results

In this article, we extend the applicability of these phase computation methods to biochemical oscillators as discrete molecular systems, upon the information obtained from a continuous-state approximation of such oscillators. In particular, we describe techniques for computing the instantaneous phase of discrete, molecular oscillators for stochastic simulation algorithm generated sample paths. We comment on the accuracies and derive certain measures for assessing the feasibilities of the proposed phase computation methods. Phase computation experiments on the sample paths of well-known biological oscillators validate our analyses.

Conclusions

The impact of noise that arises from the discrete and random nature of the mechanisms that make up molecular oscillators can be characterized based on the phase computation techniques proposed in this article. The concept of isochrons is the natural choice upon which the phase notion of oscillators can be founded. The isochron-theoretic phase computation methods that we propose can be applied to discrete molecular oscillators of any dimension, provided that the oscillatory behavior observed in discrete-state does not vanish in a continuous-state approximation. Analysis of the full versatility of phase noise phenomena in molecular oscillators will be possible if a proper phase model theory is developed, without resorting to such approximations.     

Quantifying Neural Oscillatory Synchronization: A Comparison between Spectral Coherence and Phase-Locking Value Approaches

Synchronization or phase-locking between oscillating neuronal groups is considered to be important for coordination of information among cortical networks. Spectral coherence is a commonly used approach to quantify phase locking between neural signals. We systematically explored the validity of spectral coherence measures for quantifying synchronization among neural oscillators. To that aim, we simulated coupled oscillatory signals that exhibited synchronization dynamics using an abstract phase-oscillator model as well as interacting gamma-generating spiking neural networks. We found that, within a large parameter range, the spectral coherence measure deviated substantially from the expected phase-locking. Moreover, spectral coherence did not converge to the expected value with increasing signal-to-noise ratio. We found that spectral coherence particularly failed when oscillators were in the partially (intermittent) synchronized state, which we expect to be the most likely state for neural synchronization. The failure was due to the fast frequency and amplitude changes induced by synchronization forces. We then investigated whether spectral coherence reflected the information flow among networks measured by transfer entropy (TE) of spike trains. We found that spectral coherence failed to robustly reflect changes in synchrony-mediated information flow between neural networks in many instances. As an alternative approach we explored a phase-locking value (PLV) method based on the reconstruction of the instantaneous phase. As one approach for reconstructing instantaneous phase, we used the Hilbert Transform (HT) preceded by Singular Spectrum Decomposition (SSD) of the signal. PLV estimates have broad applicability as they do not rely on stationarity, and, unlike spectral coherence, they enable more accurate estimations of oscillatory synchronization across a wide range of different synchronization regimes, and better tracking of synchronization-mediated information flow among networks.     

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.     

The evolution of density-dependent dispersal in a noisy spatial population model

It is well-known that dispersal is advantageous in many different ecological situations, e.g. to survive local catastrophes where populations live in spatially and temporally heterogeneous habitats. However, the key question, what kind of dispersal strategy is optimal in a particular situation, has remained unanswered. We studied the evolution of density-dependent dispersal in a coupled map lattice model, where the population dynamics are perturbed by external environmental noise. We used a very flexible dispersal function to enable evolution to select from practically all possible types of monotonous density-dependent dispersal functions. We treated the parameters of the dispersal function as continuously changing phenotypic traits. The evolutionary stable dispersal strategies were investigated by numerical simulations. We pointed out that irrespective of the cost of dispersal and the strength of environmental noise, this strategy leads to a very weak dispersal below a threshold density, and dispersal rate increases in an accelerating manner above this threshold. Decreasing the cost of dispersal increases the skewness of the population density distribution, while increasing the environmental noise causes more pronounced bimodality in this distribution. In case of positive temporal autocorrelation of the environmental noise, there is no dispersal below the threshold, and only low dispersal below it, on the other hand with negative autocorrelation practically all individual disperses above the threshold. We found our results to be in good concordance with empirical observations.     

A method to determine the mean pollen dispersal of individual plants growing within a large pollen source

Pollen dispersal has been recently focused on as a major issue in the risk assessment of transgenic crop plants. The shape of the pollen dispersal of individual plants is hard to determine since a very large number of plants must be monitored in order to track rare longdistance dispersal events. Conversely, studies using large plots as a pollen source provide a pollen distribution that depends on the shape of the source plot. We report here on a method based on the use of Fourier transforms by which the pollen dispersal of a single, average individual can be obtained from data using large plots as pollen sources, thus allowing the estimation of the probability of long-distance dispersal for single plants. This method is subsequently implemented on simulated data to test its susceptibility to random noise and edge effects. Its conditions of application and value for use in ecological studies, in particular risk assessment of the deliberate release of transgenic plants, are discussed.     

Density-dependent dispersal in integrodifference equations

Many species exhibit dispersal processes with positive density- dependence. We model this behavior using an integrodifference equation where the individual dispersal probability is a monotone increasing function of local density. We investigate how this dispersal probability affects the spreading speed of a single population and its ability to persist in fragmented habitats. We demonstrate that density-dependent dispersal probability can act as a mechanism for coexistence of otherwise non-coexisting competitors. We show that in time-varying habitats, an intermediate dispersal probability will evolve. Analytically, we find that the spreading speed for the integrodifference equation with density-dependent dispersal probability is not linearly determined. Furthermore, the next-generation operator is not compact and, in general, neither order-preserving nor monotonicity-preserving. We give two explicit examples of non-monotone, discontinuous traveling-wave profiles.      

Modeling species dispersal with occupancy urn models

Models for species dispersal make various simplifications to facilitate analysis, such as ignoring spatial correlations or assuming equal probability of colonization among all sites within a dispersal neighborhood. Here we introduce a variation of the basic contact process (BCP) which allows us to separate the number of offspring produced from the neighborhood size, which are confounded in the original BCP. We then use classical results arising from probability models involving placing balls in urns to study our modified BCP, obtaining bounds for the critical value of the survival probability needed for the population to persist. We also use the probability urn calculations with a local-dispersal mean-field approximation to estimate equilibrium population density. These methods are able to include features such as unequal dispersal probabilities to different sites in the neighborhood, e.g., as would arise when dispersers have a fixed rate of mortality per distance traveled from the parent site. We also show how urn models allow one to generalize these results to two species competing for space.     

Synchronization of coupled nonidentical genetic oscillators

The study of the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using a control theory approach, we provide a theoretical method for analysing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.     

Coupling and internal noise sustain synchronized oscillation in calcium system

In this work, the effects of coupling on two calcium subsystems were investigated, the cooperation between coupling and internal noise was also considered. When two non-identical subsystems are in steady state, coupling can induce oscillations, and distinctly enlarge the oscillatory region in bifurcation diagram. Besides, coupling can make the two non-identical oscillators synchronized. With the increment of the coupling strength, the cross-correlation time of the two oscillators firstly increases and then decreases to be constant, showing the synchronization without tuning coupling strength. When internal noise is considered, similar phenomena can also be obtained under the cooperation between coupling and internal noise.