Synchronized Anti-Phase and In-Phase Oscillations of Intracellular Calcium Ions in Two Coupled Hepatocytes System

We study the dynamic behaviour of two intracellufar calcium oscillators that are coupled through gap junctions both to Ca^2＋ and inositol（1,4,5）-trisphosphate （IP3）. It is found that synchronized anti-phase and in-phase oscillations of cytoplasmic cadcium coexist in parameters space. Especially, synchronized anti-phase oscillations only occur near the onset of a Hopf bifurcation point when the velocity of IP3 synthesis is increased. In addition, two kinds of coupling effects, i.e., the diffusions of Ca^2＋ and IP3 among cells on synchronous behaviour, are considered. We fnd that small coupling of Ca^2＋ and large coupling of IP3 facilitate the emergence of synchronized anti-phase oscillations. However, the result is contrary for the synchronized in-phase case. Our findings may provide a qualitative understanding about the mechanism of synchronous behaviour of intercellular calcium signalling.     

Bifurcation mechanisms of regular and chaotic network signaling in brain astrocytes

Bifurcation mechanisms underlying calcium oscillations in the network of astrocytes are investigated. Network model includes the dynamics of intracellular calcium concentration and intercellular diffusion of inositol 1,4,5-trisphosphate through gap junctions. Bifurcation analysis of underlying nonlinear dynamical system is presented. Parameter regions and principle bifurcation boundaries have been delineated and described. We show how variations of the diffusion rate can lead to generation of network calcium oscillations in originally nonoscillating cells. Different scenarios of regular activity and its transitions to chaotic dynamics have been obtained. Then, the bifurcations have been associated with statistical characteristics of calcium signals showing that different bifurcation scenarios yield qualitative changes in experimentally measurable quantities of the astrocyte activity, e.g., statistics of calcium spikes.     

细胞钙振荡对周期信号的响应和胞间同步

钙离子是细胞中一种很重要的第二信使,通常它以细胞质钙离子浓度振荡的方式转导多种生理学信息,影响细胞分化、成熟和凋亡等各种生理过程.在有些组织如肝脏中,虽然不同的肝细胞个体可以具有非常不同的本征振荡频率,钙振荡却往往以扫过整个肝小叶的周期波的同步形式出现,表明细胞钙振荡在某种耦合因素作用下互相锁相.对一个典型的细胞钙振荡数学模型,通过数值模拟,研究了振荡对外界周期性刺激信号的响应,以及不同细胞之间通过细胞间接头耦合的同步特性.     

Effects of reduced discrete coupling on filament tension in excitable media

Wave propagation in the heart has a discrete nature, because it is mediated by discrete intercellular connections via gap junctions. Although effects of discreteness on wave propagation have been studied for planar traveling waves and vortexes (spiral waves) in two dimensions, its possible effects on vortexes (scroll waves) in three dimensions are not yet explored. In this article, we study the effect of discrete cell coupling on the filament dynamics in a generic model of an excitable medium. We find that reduced cell coupling decreases the line tension of scroll wave filaments and may induce negative filament tension instability in three-dimensional excitable lattices.     

Effects of charge fluctuation on two-membrane instability and fusion

Membrane fusion is fundamental to diverse biological processes ranging from intercellular and intracellular transport to egg fertilization. We study the effects of coupling between membrane undulation and charge fluctuation on its fusion. We find that, at concentrations of millimolar range, multivalent cations such as calcium in solution induce a strong correlated-charge fluctuation on each membrane, leading to inversion and overcondensation of surface charges. When the charge fluctuation is cooperatively coupled to undulation, two apposing membranes undergo a dynamic instability to spontaneous growth of in-phase undulation with submicron wavelengths, thereby greatly reducing fusion barrier.     

Spatiotemporal multiple coherence resonances and calcium waves in a coupled hepatocyte system

Spatiotemporal multiple coherence resonances for calcium activities induced by weak Gaussian white noise in coupled hepatocytes are studied. It is shown that bi-resonances in hepatocytes are induced by the interplay and competition between noise and coupling of cells, in other words, the cell in network can be excited either by noise or by its neighbour via gap junction which can transfer calcium ions between cells. Furthermore, the intercellular annular calcium waves induced by noise are observed, in which the wave length decreases with noise intensity augmenting but increases monotonically with coupling strength increasing. And for a fixed noise level, there is an optimal coupling strength that makes the coherence resonance reach maximum.     

Chaotic patterns in a coupled oscillator-excitator biochemical cell system

In this paper we examine dynamical modes resulting from diffusion-like interaction of two model biochemical cells. Kinetics in each of the cells is given by the ICC model of calcium ions in the cytosol. Constraints for one of the cells are set so that it is excitable. One of the constraints in the other cell - a fraction of activated cell surface receptors-is varied so that the dynamics in the cell is either excitable or oscillatory or a stable focus. The cells are interacting via mass transfer and dynamics of the coupled system are studied as two parameters are varied-the fraction of activated receptors and the coupling strength. We find that (i) the excitator-excitator interaction does not lead to oscillatory patterns, (ii) the oscillator-excitator interaction leads to alternating phase-locked periodic and quasiperiodic regimes, well known from oscillator-oscillator interactions; torus breaking bifurcation generates chaos when the coupling strength is in an intermediate range, (iii) the focus-excitator interaction generates compound oscillations arranged as period adding sequences alternating with chaotic windows; the transition to chaos is accompanied by period doublings and folding of branches of periodic orbits and is associated with a Shilnikov homoclinic orbit. The nature of spontaneous self-organized oscillations in the focus-excitator range is discussed. (c) 1999 American Institute of Physics.     

Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling

We propose a novel mechanism leading to spatiotemporal oscillations in extended systems that does not rely on local bulk instabilities. Instead, oscillations arise from the interaction of two subsystems of different spatial dimensionality. Specifically, we show that coupling a passive diffusive bulk of dimension d with an excitable membrane of dimension d-1 produces a self-sustained oscillatory behavior. An analytical explanation of the phenomenon is provided for d=1. Moreover, in-phase and antiphase synchronization of oscillations are found numerically in one and two dimensions. This novel dynamic instability could be used by biological systems such as cells, where the dynamics on the cellular membrane is necessarily different from that of the cytoplasmic bulk.     

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.     

Remodelling of cellular excitation (reaction) and intercellular coupling (diffusion) by chronic atrial fibrillation represented by a reaction-diffusion system

Human atrial tissue is an excitable system, in which myocytes are excitable elements, and cell-to-cell electrotonic interactions are via diffusive interactions of cell membrane potentials. We developed a family of excitable system models for human atrium at cellular, tissue and anatomical levels for both normal and chronic atrial fibrillation (AF) conditions. The effects of AF-induced remodelling of cell membrane ionic channels (reaction kinetics) and intercellular gap junctional coupling (diffusion) on atrial excitability, conduction of excitation waves and dynamics of re-entrant excitation waves are quantified. Both ionic channel and gap junctional coupling remodelling have rate dependent effects on atrial propagation. Membrane channel conductance remodelling allows the propagation of activity at higher rates than those sustained in normal tissue or in tissue with gap junctional remodelling alone. Membrane channel conductance remodelling is essential for the propagation of activity at rates higher than 300/min as seen in AF. Spatially heterogeneous gap junction coupling remodelling increased the risk of conduction block, an essential factor for the genesis of re-entry. In 2D and 3D anatomical models, the dynamical behaviours of re-entrant excitation waves are also altered by membrane channel modelling. This study provides insights to understand the pro-arrhythmic effects of AF-induced reaction and diffusion remodelling in atrial tissue.     

Bifurcation Analysis and Transition Mechanism in a Modified Model of Ca~(2+) Oscillations

Some new elements are introduced into a mathematical model of intracellular calcium oscillations, which make it particularly suitable for the study of bifurcation. In addition to generating regular oscillations, such a modified model can be used to reproduce the burst discharges similar to those recorded in experiments and to describe two new types of oscillatory phenomena. By means of a fast/slow dynamical analysis, we explore the bifurcation and transition mechanisms associated with two types of bursters due to changes in the interaction of two slow variables with different timescales.     

Model for Influences of Magnetic Fields on Intracellular Calcium Oscillations

Based on the model of the two calcium stores developed by Goldbeter, the influence of external magnetic field on the calcium concentration has been discussed. We believe that the cell membrane is a major site of interaction for extremely-low-frequency (ELF) electromagnetic fields, and the permeability of ions can be changed with the changing electromagnetic fields. It is shown that magnetic field initiatesintracellular calcium oscillation with a threshold in flux density, and that the calcium oscillations do not occur if the density of magnetic field is below the threshold. The results of theoretical calculation are consistent with that of the experiment. The intracellular free calcium concentrations of different cells exposed to the same magnetic fields are different from each other. It is indicated that the different behaviors such as oscillation, rise and invariability of calcium concentration are associated with the sort of cells.     

Reading the patterns in living cells —the physics of ca2+ signaling

Ca²⁺ is one of the most important messengers. It transmits signals inside living cells and takes part in intercellular coordination. The dynamics of the Ca²⁺ concentration shows a transition from elemental, stochastic events to global events like waves and oscillations. This transition renders it an ideal tool for studying basic concepts of pattern formation, especially since access to the most important experimental parameters is given. Ca²⁺ dynamics in living cells has been a major topic of biophysical modelling in the last 15 years. Modelling has reached the level of predictive power. The theoretical analysis of waves provided new insight into the mechanisms of Ca²⁺ signaling and led to new concepts of analysis of wave equations with concentration dependent diffusion and novel wave bifurcations. Modelling of oscillations provided understanding especially of complex oscillations and allowed to extract information about the underlying cellular parameters and mechanisms. The investigation of the stochastic aspects of intracellular Ca²⁺ dynamics demonstrated the fundamental role of fluctuations arising from the control of the release channel by Ca²⁺ and IP₃. This review presents an overview of current theoretical research on Ca²⁺ dynamics in living cells driven by the inositol 1,4,5-trisphosphate receptor channel.     

Spatiotemporal dynamics of Ca2+ signaling and its physiological roles

Changes in the intracellular Ca²⁺ concentration regulate numerous cell functions and display diverse spatiotemporal dynamics, which underlie the versatility of Ca²⁺ in cell signaling. In many cell types, an increase in the intracellular Ca²⁺ concentration starts locally, propagates within the cell (Ca²⁺ wave) and makes oscillatory changes (Ca²⁺ oscillation). Studies of the intracellular Ca²⁺ release mechanism from the endoplasmic reticulum (ER) showed that the Ca²⁺ release mechanism has inherent regenerative properties, which is essential for the generation of Ca²⁺ waves and oscillations. Ca²⁺ may shuttle between the ER and mitochondria, and this appears to be important for pacemaking of Ca²⁺ oscillations. Importantly, Ca²⁺ oscillations are an efficient mechanism in regulating cell functions, having effects supra-proportional to the sum of duration of Ca²⁺ increase. Furthermore, Ca²⁺ signaling mechanism studies have led to the development of a method for specific inhibition of Ca²⁺ signaling, which has been used to identify hitherto unrecognized functions of Ca²⁺ signals.     

Bursting Ca2+ Oscillations and Synchronization in Coupled Cells

A mathematical model proposed by Grubelnk et al. [Biophys. Chem. 94 （2001） 59] is employed to study the physiological role of mitochondria and the cytosolic proteins in generating complex Ca^2＋ oscillations, lntracellulax bursting calcium oscillations of point-point, point-cycle and two-folded limit cycle types are observed and explanations are given based on the fast/slow dynamical analysis, especially for point-cycle and two-folded limit cycle types, which have not been reported before. Furthermore, synchronization of coupled bursters of Ca^2＋ oscillations via gap junctions and the effect of bursting types on synchronization of coupled cells are studied. It is argued that bursting oscillations of point-point type may be superior to achieve synchronization than that of point cycle type.     

Selective effects of noise by stochastic multi-resonance in coupled cells system

By investigating a stochastic model for intracellular calcium oscillations proposed by Höfer, we have analyzed the transmission behavior of calcium signaling in a one-dimensional two-way coupled hepatocytes system. It is shown that when the first cell is subjected to the external noise, the output signal-to-noise ratio (SNR) in the cell exhibits two maxima as a function of external noise intensity, indicating the occurrence of stochastic bi-resonance (SBR). It is more important that when cells are coupled together, the resonant behavior in the 1st cell propagates along the chain with different features through the coupling effect. The cells whose locations are comparatively close to or far from the 1st cell can show SBR, while the cells located in the middle position can display stochastic multi-resonance (SMR). Furthermore, the number of cells that can show SMR increases with coupling strength enhancing. These results indicate that the cells system may make an effective choice in response to external signaling induced by noise, through the mechanism of SMR by adjusting coupling strength.     

Synchronization in multicell systems exhibiting dynamic plasticity

Collective behaviour in multicell systems arises from exchange of chemicals/ signals between cells and may be different from their intrinsic behaviour. These chemicals are products of regulated networks of biochemical pathways that underlie cellular functions, and can exhibit a variety of dynamics arising from the non-linearity of the reaction processes. We have addressed the emergent synchronization properties of a ring of cells, diffusively coupled by the end product of an intracellular model biochemical pathway exhibiting non-robust birhythmic behaviour. The aim is to examine the role of intercellular interaction in stabilizing the non-robust dynamics in the emergent collective behaviour in the ring of cells. We show that, irrespective of the inherent frequencies of individual cells, depending on the coupling strength, the collective behaviour does synchronize to only one type of oscillations above a threshold number of cells. Using two perturbation analyses, we also show that this emergent synchronized dynamical state is fairly robust under external perturbations. Thus, the inherent plasticity in the oscillatory phenotypes in these model cells may get suppressed to exhibit collective dynamics of a single type in a multicell system, but environmental influences can sometimes expose this underlying plasticity in its collective dynamics.      

Enhanced Synchronization of Intercellular Calcium Oscillations by Noise Contaminated Signals

We consider the dynamics of locally coupled calcium oscillation systems, each cell is subjected to extracellular contaminated signal, which contains common sub-threshold signal and independent Gaussian noise. It is found that intermediate noise can enhance synchronized oscillations of calcium ions, where the frequency of noise-induced oscillations is matched with the one of sub-threshold external signal. We show that synchronization is enhanced as a result of the entrainment of external signal. Furthermore, the effect of coupling strength is considered. We find above-mentioned phenomenon exists only when coupling strength is very small. Our findings may exhibit that noise can enhance the detection of feeble external signal through the mechanism of synchronization of intercellular calcium ions.     

Symmetry breaking,bifurcations, quasiperiodicity,and chaos due to electric fields in a coupled cell model

A model for the asymmetric coupling of two oscillatory cells is considered. The coupling between the cells is both through diffusional exchange (symmetric) and through the electromigration of ionic reactant species from one cell to the other (asymmetric) in applied electric fields. The kinetics in each cell are the same and based on the Gray-Scott scheme. Without the electric field, only simple, stable dynamics are seen. The effect of the asymmetry (applying electric fields) is to create a wide variety of stable dynamics, multistability, multiperiodic oscillations, quasiperiodicity and chaos being observed, this complexity in response being more prevalent at weaker coupling rates and at weaker field strengths. The results are obtained using a standard dynamical systems continuation program, though asymptotic results are obtained for strong coupling rates and strong electric fields. These are seen to agree well with the numerically determined values in the appropriate parameter regimes. (c) 2002 American Institute of Physics.