Selection of in-phase or out-of-phase synchronization in a model based on global coupling of cells undergoing metabolic oscillations

Dynamical analysis of a periodically forced chaotic chemical oscillator

We present a comprehensive dynamical analysis of a chaotic chemical model referred to as the autocatalator, when subject to a periodic administration of one substrate. Our investigation encompasses the dynamical characterization of both unforced and forced systems utilizing isospikes and largest Lyapunov exponents-based parameter planes, bifurcation diagrams, and analysis of complex oscillations. Additionally, we present a phase diagram showing the effect of the period and amplitude of the forcing signal on the system's behavior. Furthermore, we show how the landscapes of parameter planes are altered in response to forcing application. This analysis contributes to a deeper understanding of the intricate dynamics induced by the periodic forcing of a chaotic system.

Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as ‘multi-synchronization’.

Emergence of multistability and strongly asymmetric collective modes in two quorum sensing coupled identical ring oscillators

The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally, resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressor genes in bacteria, creating the synthetic genetic oscillator known as the Repressilator. We consider the main collective modes produced when two identical Repressilators are mean-field-coupled via the quorum-sensing mechanism. In-phase and anti-phase oscillations of the coupled oscillators emerge from two Andronov-Hopf bifurcations of the homogeneous steady state. Using the rate of the repressor's production and the value of coupling strength as the bifurcation parameters, we performed one-parameter continuations of limit cycles and two-parameter continuations of their bifurcations to show how bifurcations of the in-phase and anti-phase oscillations influence the dynamical behaviors for this system. Pitchfork bifurcation of the unstable in-phase cycle leads to the creation of novel inhomogeneous limit cycles with very different amplitudes, in contrast to the well-known asymmetrical limit cycles arising from oscillation death. The Neimark-Sacker bifurcation of the anti-phase cycle determines the border of an island in two-parameter space containing almost all the interesting regimes including the set of resonant limit cycles, the area with stable inhomogeneous cycle, and very large areas with chaotic regimes resulting from torus destruction and period doubling of resonant cycles and inhomogeneous cycles. We discuss the structure of the chaos skeleton to show the role of inhomogeneous cycles in its formation. Many regions of multistability and transitions between regimes are presented. These results provide new insights into the coupling-dependent mechanisms of multistability and collective regime symmetry breaking in populations of identical multidimensional oscillators.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

The bird circadian clock: insights from a computational model

The circadian timekeeping system appears more complex in birds than in mammals. In mammals, the main pacemaker is centralized in the suprachiasmatic nuclei, whereas in birds, the pacemaker involves the interplay between the pineal and hypothalamic oscillators. In order to investigate the consequence of this complex mechanism, we propose here a mathematical model for the bird circadian clock. The model is based on the internal resonance between the pineal and hypothalamic oscillators, each described by Goodwin-like equations. We show that, consistently with experimental observations, self-sustained oscillations can be generated by mutual inhibitory coupling of the 2 clocks, even if individual oscillators present damped oscillations. We study the effect of constant and periodic administrations of melatonin, which, in intact birds, acts as the coupling variable between the pineal and the hypothalamus, and compare the prediction of the model with the experiments performed in pinealectomized birds. We also assess the entrainment properties when the system is subject to light-dark cycles. Analyses of the entrainment range, resynchronization time after jet lag, and entrainment phase with respect to the photoperiod lead us to formulate hypotheses about the physiological advantage of the particular architecture of the avian circadian clock. Although minimal, our model opens promising perspectives in modeling and understanding the bird circadian clock.

Synchronization of stochastic oscillators in biochemical systems

We investigate the synchronization of stochastic oscillations in biochemical models by calculating the complex coherence function within the linear noise approximation. The method is illustrated on a simple example and then applied to study the synchronization of chemical concentrations in social amoeba. The degree to which variation of rate constants in different cells and the volume of the cells affects synchronization of the oscillations is explored and the phase lag calculated. In all cases the analytical results are shown to be in good agreement with those obtained through numerical simulations.

Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh-Nagumo model

We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of-phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.

Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system

We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatory cluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatory chemical reactions with other types of non-local coupling.

Synchronization ability of coupled cell-cycle oscillators in changing environments

BACKGROUND

The biochemical oscillator that controls periodic events during the Xenopus embryonic cell cycle is centered on the activity of CDKs, and the cell cycle is driven by a protein circuit that is centered on the cyclin-dependent protein kinase CDK1 and the anaphase-promoting complex (APC). Many studies have been conducted to confirm that the interactions in the cell cycle can produce oscillations and predict behaviors such as synchronization, but much less is known about how the various elaborations and collective behavior of the basic oscillators can affect the robustness of the system. Therefore, in this study, we investigate and model a multi-cell system of the Xenopus embryonic cell cycle oscillators that are coupled through a common complex protein, and then analyze their synchronization ability under four different external stimuli, including a constant input signal, a square-wave periodic signal, a sinusoidal signal and a noise signal.

RESULTS

Through bifurcation analysis and numerical simulations, we obtain synchronization intervals of the sensitive parameters in the individual oscillator and the coupling parameters in the coupled oscillators. Then, we analyze the effects of these parameters on the synchronization period and amplitude, and find interesting phenomena, e.g., there are two synchronization intervals with activation coefficient in the Hill function of the activated CDK1 that activates the Plk1, and different synchronization intervals have distinct influences on the synchronization period and amplitude. To quantify the speediness and robustness of the synchronization, we use two quantities, the synchronization time and the robustness index, to evaluate the synchronization ability. More interestingly, we find that the coupled system has an optimal signal strength that maximizes the synchronization index under different external stimuli. Simulation results also show that the ability and robustness of the synchronization for the square-wave periodic signal of cyclin synthesis is strongest in comparison to the other three different signals.

CONCLUSIONS

These results suggest that the reaction process in which the activated cyclin-CDK1 activates the Plk1 has a very important influence on the synchronization ability of the coupled system, and the square-wave periodic signal of cyclin synthesis is more conducive to the synchronization and robustness of the coupled cell-cycle oscillators. Our study provides insight into the internal mechanisms of the cell cycle system and helps to generate hypotheses for further research.

Dynamics of coupled repressilators: the role of mRNA kinetics and transcription cooperativity

Oscillatory regulatory networks have been discovered in many cellular pathways. An especially challenging area is studying dynamics of cellular oscillators interacting with one another in a population. Synchronization is only one of and the simplest outcome of such interaction. It is suggested that the outcome depends on the structure of the network. Phase-attractive (synchronizing) and phase-repulsive coupling structures were distinguished for regulatory oscillators. In this paper, we question this separation. We study an example of two interacting repressilators (artificial regulatory oscillators based on cyclic repression). We show that changing the cooperativity of transcription repression (Hill coefficient) and reaction timescales dramatically alter synchronization properties. The network becomes birhythmic-it chooses between the in-phase and antiphase synchronization. Thus, the type of synchronization is not characteristic for the network structure. However, we conclude that the specific scenario of emergence and stabilization of synchronous solutions is much more characteristic.

Electrical bursting, calcium oscillations, and synchronization of pancreatic islets

Oscillations are an integral part of insulin secretion and are ultimately due to oscillations in the electrical activity of pancreatic beta-cells, called bursting. In this chapter we discuss islet bursting oscillations and a unified biophysical model for this multi-scale behavior. We describe how electrical bursting is related to oscillations in the intracellular Ca(2+) concentration within beta-cells and the role played by metabolic oscillations. Finally, we discuss two potential mechanisms for the synchronization of islets within the pancreas. Some degree of synchronization must occur, since distinct oscillations in insulin levels have been observed in hepatic portal blood and in peripheral blood sampling of rats, dogs, and humans. Our central hypothesis, supported by several lines of evidence, is that insulin oscillations are crucial to normal glucose homeostasis. Disturbance of oscillations, either at the level of the individual islet or at the level of islet synchronization, is detrimental and can play a major role in type 2 diabetes.

Synchronization and clustering of synthetic genetic networks: a role for cis-regulatory modules

The effect of signal integration through cis-regulatory modules (CRMs) on synchronization and clustering of populations of two-component genetic oscillators coupled with quorum sensing is investigated in detail. We find that the CRMs play an important role in achieving synchronization and clustering. For this, we investigate six possible cis-regulatory input functions with AND, OR, ANDN, ORN, XOR, and EQU types of responses in two possible kinds of cell-to-cell communications: activator-regulated communication (i.e., the autoinducer regulates the activator) and repressor-regulated communication (i.e., the autoinducer regulates the repressor). Both theoretical analysis and numerical simulation show that different CRMs drive fundamentally different cellular patterns, such as complete synchronization, various cluster-balanced states and several cluster-nonbalanced states.

Introduction to Focus Issue: synchronization in complex networks

Synchronization in large ensembles of coupled interacting units is a fundamental phenomenon relevant for the understanding of working mechanisms in neuronal networks, genetic networks, coupled electrical and laser networks, coupled mechanical systems, networks in social sciences, and others. It relates to mathematical and computational analysis of the existence of different states and its stability, clustering, bifurcations and chaos, robustness and sensitivity analysis, etc., at the intersection between synchronization and pattern formation in complex networks. This interdisciplinary oriented Focus Issue presents recent progress in this area with contributions on generic methods, specific model studies, and applications.