Model for synchronization of pancreatic beta-cells by gap junction coupling

Finite volume and asymptotic methods for stochastic neuron models with correlated inputs

We consider a pair of stochastic integrate and fire neurons receiving correlated stochastic inputs. The evolution of this system can be described by the corresponding Fokker-Planck equation with non-trivial boundary conditions resulting from the refractory period and firing threshold. We propose a finite volume method that is orders of magnitude faster than the Monte Carlo methods traditionally used to model such systems. The resulting numerical approximations are proved to be accurate, nonnegative and integrate to 1. We also approximate the transient evolution of the system using an Ornstein-Uhlenbeck process, and use the result to examine the properties of the joint output of cell pairs. The results suggests that the joint output of a cell pair is most sensitive to changes in input variance, and less sensitive to changes in input mean and correlation.

Conflicting effects of excitatory synaptic and electric coupling on the dynamics of square-wave bursters

Using two-cell and 50-cell networks of square-wave bursters, we studied how excitatory coupling of individual neurons affects the bursting output of the network. Our results show that the effects of synaptic excitation vs. electrical coupling are distinct. Increasing excitatory synaptic coupling generally increases burst duration. Electrical coupling also increases burst duration for low to moderate values, but at sufficiently strong values promotes a switch to highly synchronous bursts where further increases in electrical or synaptic coupling have a minimal effect on burst duration. These effects are largely mediated by spike synchrony, which is determined by the stability of the in-phase spiking solution during the burst. Even when both coupling mechanisms are strong, one form (in-phase or anti-phase) of spike synchrony will determine the burst dynamics, resulting in a sharp boundary in the space of the coupling parameters. This boundary exists in both two cell and network simulations. We use these results to interpret the effects of gap-junction blockers on the neuronal circuitry that underlies respiration.

Calcium wave signaling in cancer cells

Ca(2+) functions as an important signaling messenger right from beginning of life to the final moments of the end of life. Ca(2+) is needed at several steps of the cell cycle such as early G(1), at the G(1)/S, and G(2)/M transitions. The Ca(2+) signals in the form of time-dependent changes in intracellular Ca(2+) concentrations, [Ca(2+)](i), are presented as brief spikes organized into regenerative Ca(2+) waves. Ca(2+)-mediated signaling pathways have also been shown to play important roles in carcinogenesis such as transformation of normal cells to cancerous cells, tumor formation and growth, invasion, angiogenesis and metastasis. Since the global Ca(2+) oscillations arise from Ca(2+) waves initiated locally, it results in stochastic oscillations because although each cell has many IP(3)Rs and Ca(2+) ions, the law of large numbers does not apply to the initiating event which is restricted to very few IP(3)Rs due to steep Ca(2+) concentration gradients. The specific Ca(2+) signaling information is likely to be encoded in a calcium code as the amplitude, duration, frequency, waveform or timing of Ca(2+) oscillations and decoded again at a later stage. Since Ca(2+) channels or pumps involved in regulating Ca(2+) signaling pathways show altered expression in cancer, one can target these Ca(2+) channels and pumps as therapeutic options to decrease proliferation of cancer cells and to promote their apoptosis. These studies can provide novel insights into alterations in Ca(2+) wave patterns in carcinogenesis and lead to the development of newer technologies based on Ca(2+) waves for the diagnosis and therapy of cancer.

Distant synchronization through a passive medium

This paper deals with the phenomenon of synchronization of oscillatory ensembles interacting distantly through the passive medium. Main characteristics of such a kind of synchronization are studied. The results of this work can be applied to describe the synchronization of cardiac oscillatory cells separated by the passive fibroblasts. In this work the phenomenological models (Bonhoeffer-Van der Pol) of cardiac cells as well as biologically relevant (Luo-Rudy, Sachse) models are used. We also propose equivalent model of distant synchronization and derive on its basis an analytical scaling of the frequency of synchronous oscillations.

Evidence that low-grade systemic inflammation can induce islet dysfunction as measured by impaired calcium handling

In obesity and the early stages of type 2 diabetes (T2D), proinflammatory cytokines are mildly elevated in the systemic circulation. This low-grade systemic inflammation exposes pancreatic islets to these circulating cytokines at much lower levels than seen within the islet during insulitis. These low-dose effects have not been well described. We examined mouse islets treated overnight with a low-dose cytokine combination commonly associated with inflammation (TNF-alpha, IL-1 beta, and IFN-gamma). We then examined islet function primarily using intracellular calcium ([Ca(2+)](i)), a key component of insulin secretion and cytokine signaling. Cytokine-treated islets demonstrated several features that suggested dysfunction including excess [Ca(2+)](i) in low physiological glucose (3mM), reduced responses to glucose stimulation, and disrupted [Ca(2+)](i) oscillations. Interestingly, islets taken from young db/db mice showed similar disruptions in [Ca(2+)](i) dynamics as cytokine-treated islets. Additional studies of control islets showed that the cytokine-induced elevation in basal [Ca(2+)](i) was due to both greater calcium influx through L-type-calcium-channels and reduced endoplasmic reticulum (ER) calcium storage. Many of these cytokine-induced disruptions could be reproduced by SERCA blockade. Our data suggest that chronic low-grade inflammation produces circulating cytokine levels that are sufficient to induce beta-cell dysfunction and may play a contributing role in beta-cell failure in early T2D.

Pancreatic islet cells: a model for calcium-dependent peptide release

In mammals the concentration of blood glucose is kept close to 5 mmol∕l. Different cell types in the islet of Langerhans participate in the control of glucose homeostasis. β-cells, the most frequent type in pancreatic islets, are responsible for the synthesis, storage, and release of insulin. Insulin, released with increases in blood glucose promotes glucose uptake into the cells. In response to glucose changes, pancreatic α-, β-, and δ-cells regulate their electrical activity and Ca(2+) signals to release glucagon, insulin, and somatostatin, respectively. While all these signaling steps are stimulated in hypoglycemic conditions in α-cells, the activation of these events require higher glucose concentrations in β and also in δ-cells. The stimulus-secretion coupling process and intracellular Ca(2+) ([Ca(2+)](i)) dynamics that allow β-cells to secrete is well-accepted. Conversely, the mechanisms that regulate α- and δ-cell secretion are still under study. Here, we will consider the glucose-induced signaling mechanisms in each cell type and the mathematical models that explain Ca(2+) dynamics.

How synchronization protects from noise

The functional role of synchronization has attracted much interest and debate: in particular, synchronization may allow distant sites in the brain to communicate and cooperate with each other, and therefore may play a role in temporal binding, in attention or in sensory-motor integration mechanisms. In this article, we study another role for synchronization: the so-called “collective enhancement of precision”. We argue, in a full nonlinear dynamical context, that synchronization may help protect interconnected neurons from the influence of random perturbations—intrinsic neuronal noise—which affect all neurons in the nervous system. More precisely, our main contribution is a mathematical proof that, under specific, quantified conditions, the impact of noise on individual interconnected systems and on their spatial mean can essentially be cancelled through synchronization. This property then allows reliable computations to be carried out even in the presence of significant noise (as experimentally found e.g., in retinal ganglion cells in primates). This in turn is key to obtaining meaningful downstream signals, whether in terms of precisely-timed interaction (temporal coding), population coding, or frequency coding. Similar concepts may be applicable to questions of noise and variability in systems biology.

Synchronization phenomena are pervasive in biology, creating collective behavior out of local interactions between neurons, cells, or animals. On the other hand, many of these systems function in the presence of large amounts of noise or disturbances, making one wonder how meaningful behavior can arise in these highly perturbed conditions. In this paper we show mathematically, in a general context, that synchronization is actually a means to protect interconnected systems from effects of noise and disturbances. One possible mechanism for synchronization is that the systems jointly create and then share a common signal, such as a mean electrical field or a global chemical concentration, which in turn makes each system directly connected to all others. Conversely, extracting meaningful information from average measurements over populations of cells (as commonly used for instance in electro-encephalography, or more recently in brain-machine interfaces) may require the presence of synchronization mechanisms similar to those we describe.

Glucose metabolism, islet architecture, and genetic homogeneity in imprinting of [Ca2+](i) and insulin rhythms in mouse islets

We reported previously that islets isolated from individual, outbred Swiss-Webster mice displayed oscillations in intracellular calcium ([Ca²⁺]_i) that varied little between islets of a single mouse but considerably between mice, a phenomenon we termed “islet imprinting.” We have now confirmed and extended these findings in several respects. First, imprinting occurs in both inbred (C57BL/6J) as well as outbred mouse strains (Swiss-Webster; CD1). Second, imprinting was observed in NAD(P)H oscillations, indicating a metabolic component. Further, short-term exposure to a glucose-free solution, which transiently silenced [Ca²⁺]_i oscillations, reset the oscillatory patterns to a higher frequency. This suggests a key role for glucose metabolism in maintaining imprinting, as transiently suppressing the oscillations with diazoxide, a K_ATP-channel opener that blocks [Ca²⁺]_i influx downstream of glucose metabolism, did not change the imprinted patterns. Third, imprinting was not as readily observed at the level of single beta cells, as the [Ca²⁺]_i oscillations of single cells isolated from imprinted islets exhibited highly variable, and typically slower [Ca²⁺]_i oscillations. Lastly, to test whether the imprinted [Ca²⁺]_i patterns were of functional significance, a novel microchip platform was used to monitor insulin release from multiple islets in real time. Insulin release patterns correlated closely with [Ca²⁺]_i oscillations and showed significant mouse-to-mouse differences, indicating imprinting. These results indicate that islet imprinting is a general feature of islets and is likely to be of physiological significance. While islet imprinting did not depend on the genetic background of the mice, glucose metabolism and intact islet architecture may be important for the imprinting phenomenon.

Electrical coupling promotes fidelity of responses in the networks of model neurons

We consider an integrate-and-fire element subject to randomly perturbed synaptic input and an electrically coupled ensemble of such elements. The latter is interpreted as either a model of electrically coupled population of neurons or a multicompartment model of a dendrite. Random fluctuations blur the input signal and cause false responses in the system dynamics. For instance, under the influence of noise, the system may respond with an action potential to a subthreshold stimulus. We show that the responses of the elements within the network are more reliable than the responses of the same elements in isolation. Specifically, we show that the variances of the stochastic processes generated by the coupled model can be made arbitrarily small (i.e., the network responses can be made arbitrarily accurate) by increasing the number of elements in the network and the strength of electrical coupling. Our results suggest that the organization of cells in electrically coupled groups on the network level, or the dendritic morphology on the cellular level, may be involved in the filtering noise and therefore may play an important role in the information processing mechanisms operating on the network or cellular level respectively.

Firing patterns and synchronization in nonsynaptic epileptiform activity: the effect of gap junctions modulated by potassium accumulation

Several lines of evidence point to the modification of firing patterns and of synchronization due to gap junctions (GJs) as having a role in the establishment of epileptiform activity (EA). However, previous studies consider GJs as ohmic resistors, ignoring the effects of intense variations in ionic concentration known to occur during seizures. In addition to GJs, extracellular potassium is regarded as a further important factor involved in seizure initiation and sustainment. To analyze how these two mechanisms act together to shape firing and synchronization, we use a detailed computational model for in vitro high-K(+) and low-Ca(2+) nonsynaptic EA. The model permits us to explore the modulation of electrotonic interactions under ionic concentration changes caused by electrodiffusion in the extracellular space, altered by tortuosity. In addition, we investigate the special case of null GJ current. Increased electrotonic interaction alters bursts and action potential frequencies, favoring synchronization. The particularities of pattern changes depend on the tortuosity and array size. Extracellular potassium accumulation alone modifies firing and synchronization when the GJ coupling is null.

Existence of hysteresis in the Kuramoto model with bimodal frequency distributions

We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one due to the shallowness of the central dip. Here we show that proximity to the unimodal-bimodal border does not necessarily imply hysteresis when the width, but not the depth, of the central dip tends to zero. We draw this conclusion from a detailed study of the Kuramoto model with a suitable family of bimodal distributions.

Synchronization of chaotic early afterdepolarizations in the genesis of cardiac arrhythmias

The synchronization of coupled oscillators plays an important role in many biological systems, including the heart. In heart diseases, cardiac myocytes can exhibit abnormal electrical oscillations, such as early afterdepolarizations (EADs), which are associated with lethal arrhythmias. A key unanswered question is how cellular EADs partially synchronize in tissue, as is required for them to propagate. Here, we present evidence, from computational simulations and experiments in isolated myocytes, that irregular EAD behavior is dynamical chaos. We then show in electrically homogeneous tissue models that chaotic EADs synchronize globally when the tissue is smaller than a critical size. However, when the tissue exceeds the critical size, electrotonic coupling can no longer globally synchronize EADs, resulting in regions of partial synchronization that shift in time and space. These regional partially synchronized EADs then form premature ventricular complexes that propagate into recovered tissue without EADs. This process creates multiple premature ventricular complexes that propagate as [corrected] "shifting" foci resembling polymorphic ventricular tachycardia. Shifting foci encountering shifting repolarization gradients can also develop localized wave breaks leading to reentry and fibrillation. As predicted by the theory, rabbit hearts exposed to oxidative stress (H(2)O(2)) exhibited multiple shifting foci causing polymorphic tachycardia and fibrillation. This mechanism explains how collective cellular behavior integrates at the tissue scale to generate lethal cardiac arrhythmias over a wide range of heart rates.

Evidence of diminished glucose stimulation and endoplasmic reticulum function in nonoscillatory pancreatic islets

Pulsatility is a fundamental feature of pancreatic islets and a hallmark of hormone secretion. Isolated pancreatic islets endogenously generate rhythms in secretion, metabolic activity, and intracellular calcium ([Ca(2+)](i)) that are important to normal physiological function. Few studies have directly compared oscillatory and nonoscillatory islets to identify possible differences in function. We investigated the hypothesis that the loss of these oscillations is a leading indicator of islet dysfunction by comparing oscillatory and nonoscillatory mouse islets for multiple parameters of function. Nonoscillatory islets displayed elevated basal [Ca(2+)](i) and diminished [Ca(2+)](i) response and insulin secretory response to 3-28 mm glucose stimulation compared with oscillatory islets, suggesting diminished glucose sensitivity. We investigated several possible mechanisms to explain these differences. No differences were observed in mitochondrial membrane potential, estimated ATP-sensitive potassium channel and L-type calcium channel activity, or cell death rates. Nonoscillatory islets, however, showed a reduced response to the sarco(endo)plasmic reticulum calcium ATPase inhibitor thapsigargin, suggesting a disruption in calcium homeostasis in the endoplasmic reticulum (ER) compared with oscillatory islets. The diminished ER calcium homeostasis among nonoscillatory islets was also consistent with the higher cytosolic calcium levels observed in 3 mm glucose. Inducing mild damage with low-dose proinflammatory cytokines reduced islet oscillatory capacity and produced similar effects on glucose-stimulated [Ca(2+)](i), basal [Ca(2+)](i), and thapsigargin response observed among untreated nonoscillatory islets. Our data suggest the loss of oscillatory capacity may be an early indicator of diminished islet glucose sensitivity and ER dysfunction, suggesting targets to improve islet assessment.

The Cardiac Physiome: perspectives for the future

The Physiome Project, exemplified by the Cardiac Physiome, is now 10 years old. In this article, we review past progress and future challenges in developing a quantitative framework for understanding human physiology that incorporates both genetic inheritance and environmental influence. Despite the enormity of the challenge, which is certainly greater than that facing the pioneers of the human genome project 20 years ago, there is reason for optimism that real and accelerating progress is being made.

Beneficial effects of intercellular interactions between pancreatic islet cells in blood glucose regulation

Glucose homeostasis is controlled by the islets of Langerhans which are equipped with alpha-cells increasing the blood glucose level, beta-cells decreasing it, and delta-cells the precise role of which still needs identifying. Although intercellular communications between these endocrine cells have recently been observed, their roles in glucose homeostasis have not been clearly understood. In this study, we construct a mathematical model for an islet consisting of two-state alpha-, beta-, and delta-cells, and analyze effects of known chemical interactions between them with emphasis on the combined effects of those interactions. In particular, such features as paracrine signals of neighboring cells and cell-to-cell variations in response to external glucose concentrations as well as glucose dynamics, depending on insulin and glucagon hormone, are considered explicitly. Our model predicts three possible benefits of the cell-to-cell interactions: First, the asymmetric interaction between alpha- and beta-cells contributes to the dynamic stability while the perturbed glucose level recovers to the normal level. Second, the inhibitory interactions of delta-cells for glucagon and insulin secretion prevent the wasteful co-secretion of them at the normal glucose level. Finally, the glucose dose-responses of insulin secretion is modified to become more pronounced at high glucose levels due to the inhibition by delta-cells. It is thus concluded that the intercellular communications in islets of Langerhans should contribute to the effective control of glucose homeostasis.

Variety of synchronous regimes in neuronal ensembles

We consider a Hodgkin-Huxley-type model of oscillatory activity in neurons of the snail Helix pomatia. This model has a distinctive feature: It demonstrates multistability in oscillatory and silent modes that is typical for the thalamocortical neurons. A single neuron cell can demonstrate a variety of oscillatory activity: Regular and chaotic spiking and bursting behavior. We study collective phenomena in small and large arrays of nonidentical cells coupled by models of electrical and chemical synapses. Two single elements coupled by electrical coupling show different types of synchronous behavior, in particular in-phase and antiphase synchronous regimes. In an ensemble of three inhibitory synaptically coupled elements, the phenomenon of sequential synchronous dynamics is observed. We study the synchronization phenomena in the chain of nonidentical neurons at different oscillatory behavior coupled with electrical and chemical synapses. Various regimes of phase synchronization are observed: (i) Synchronous regular and chaotic spiking; (ii) synchronous regular and chaotic bursting; and (iii) synchronous regular and chaotic bursting with different numbers of spikes inside the bursts. We detect and study the effect of collective synchronous burst generation due to the cluster formation and the oscillatory death.

Synchronization of genetic oscillators

Synchronization of genetic or cellular oscillators is a central topic in understanding the rhythmicity of living organisms at both molecular and cellular levels. Here, we show how a collective rhythm across a population of genetic oscillators through synchronization-induced intercellular communication is achieved, and how an ensemble of independent genetic oscillators is synchronized by a common noisy signaling molecule. Our main purpose is to elucidate various synchronization mechanisms from the viewpoint of dynamics, by investigating the effects of various biologically plausible couplings, several kinds of noise, and external stimuli. To have a comprehensive understanding on the synchronization of genetic oscillators, we consider three classes of genetic oscillators: smooth oscillators (exhibiting sine-like oscillations), relaxation oscillators (displaying jump dynamics), and stochastic oscillators (noise-induced oscillation). For every class, we further study two cases: with intercellular communication (including phase-attractive and repulsive coupling) and without communication between cells. We find that an ensemble of smooth oscillators has different synchronization phenomena from those in the case of relaxation oscillators, where noise plays a different but key role in synchronization. To show differences in synchronization between them, we make comparisons in many aspects. We also show that a population of genetic stochastic oscillators have their own synchronization mechanisms. In addition, we present interesting phenomena, e.g., for relaxation-type stochastic oscillators coupled to a quorum-sensing mechanism, different noise intensities can induce different periodic motions (i.e., inhomogeneous limit cycles).

The hyperbolic effect of density and strength of inter beta-cell coupling on islet bursting: a theoretical investigation

BACKGROUND

Insulin, the principal regulating hormone of blood glucose, is released through the bursting of the pancreatic islets. Increasing evidence indicates the importance of islet morphostructure in its function, and the need of a quantitative investigation. Recently we have studied this problem from the perspective of islet bursting of insulin, utilizing a new 3D hexagonal closest packing (HCP) model of islet structure that we have developed. Quantitative non-linear dependence of islet function on its structure was found. In this study, we further investigate two key structural measures: the number of neighboring cells that each beta-cell is coupled to, nc, and the coupling strength, gc.

RESULTS

BETA-cell clusters of different sizes with number of beta-cells nbeta ranging from 1-343, nc from 0-12, and gc from 0-1000 pS, were simulated. Three functional measures of islet bursting characteristics--fraction of bursting beta-cells fb, synchronization index lambda, and bursting period Tb, were quantified. The results revealed a hyperbolic dependence on the combined effect of nc and gc. From this we propose to define a dimensionless cluster coupling index or CCI, as a composite measure for islet morphostructural integrity. We show that the robustness of islet oscillatory bursting depends on CCI, with all three functional measures fb, lambda and Tb increasing monotonically with CCI when it is small, and plateau around CCI = 1.

CONCLUSION

CCI is a good islet function predictor. It has the potential of linking islet structure and function, and providing insight to identify therapeutic targets for the preservation and restoration of islet beta-cell mass and function.

Optimization of synchronization in complex clustered networks

There has been mounting evidence that many types of biological or technological networks possess a clustered structure. As many system functions depend on synchronization, it is important to investigate the synchronizability of complex clustered networks. Here we focus on one fundamental question: Under what condition can the network synchronizability be optimized? In particular, since the two basic parameters characterizing a complex clustered network are the probabilities of intercluster and intracluster connections, we investigate, in the corresponding two-dimensional parameter plane, regions where the network can be best synchronized. Our study yields a quite surprising finding: a complex clustered network is most synchronizable when the two probabilities match each other approximately. Mismatch, for instance caused by an overwhelming increase in the number of intracluster links, can counterintuitively suppress or even destroy synchronization, even though such an increase tends to reduce the average network distance. This phenomenon provides possible principles for optimal synchronization on complex clustered networks. We provide extensive numerical evidence and an analytic theory to establish the generality of this phenomenon.

Investigating the role of islet cytoarchitecture in its oscillation using a new beta-cell cluster model

The oscillatory insulin release is fundamental to normal glycemic control. The basis of the oscillation is the intercellular coupling and bursting synchronization of β cells in each islet. The functional role of islet β cell mass organization with respect to its oscillatory bursting is not well understood. This is of special interest in view of the recent finding of islet cytoarchitectural differences between human and animal models. In this study we developed a new hexagonal closest packing (HCP) cell cluster model. The model captures more accurately the real islet cell organization than the simple cubic packing (SCP) cluster that is conventionally used. Using our new model we investigated the functional characteristics of β-cell clusters, including the fraction of cells able to burst f_b, the synchronization index λ of the bursting β cells, the bursting period T_b, the plateau fraction p_f, and the amplitude of intracellular calcium oscillation [Ca]. We determined their dependence on cluster architectural parameters including number of cells n_β, number of inter-β cell couplings of each β cell n_c, and the coupling strength g_c. We found that at low values of n_β, n_c and g_c, the oscillation regularity improves with their increasing values. This functional gain plateaus around their physiological values in real islets, at n_β∼100, n_c∼6 and g_c∼200 pS. In addition, normal β-cell clusters are robust against significant perturbation to their architecture, including the presence of non-β cells or dead β cells. In clusters with n_β>∼100, coordinated β-cell bursting can be maintained at up to 70% of β-cell loss, which is consistent with laboratory and clinical findings of islets. Our results suggest that the bursting characteristics of a β-cell cluster depend quantitatively on its architecture in a non-linear fashion. These findings are important to understand the islet bursting phenomenon and the regulation of insulin secretion, under both physiological and pathological conditions.

Size distribution of mouse Langerhans islets

Pancreatic beta-cells are clustered in islets of Langerhans, which are typically a few hundred micrometers in a variety of mammals. In this study, we propose a theoretical model for the growth of pancreatic islets and derive the islet size distribution, based on two recent observations: First, the neogenesis of new islets becomes negligible after some developmental stage. Second, islets grow via a random process, where any cell in an islet proliferates with the same rate regardless of the present size of the islet. Our model predicts either log-normal or Weibull distributions of the islet sizes, depending on whether cells in an islet proliferate coherently or independently. To confirm this, we also measure the islet size by selectively staining islets, which are exposed from exocrine tissues in mice after enzymatic treatment. Indeed revealed are skewed distributions with the peak size of approximately 100 cells, which fit well to the theoretically derived ones. Interestingly, most islets turned out to be bigger than the expected minimal size (approximately 10 or so cells) necessary for stable synchronization of beta-cells through electrical gap-junction coupling. The collaborative behavior among cells is known to facilitate synchronized insulin secretion and tends to saturate beyond the critical (saturation) size of approximately 100 cells. We further probe how the islets change as normal mice grow from young (6 weeks) to adult (5 months) stages. It is found that islets may not grow too large to maintain appropriate ratios between cells of different types. Our results implicate that growing of mouse islets may be regulated by several physical constraints such as the minimal size required for stable cell-to-cell coupling and the upper limit to keep the ratios between cell types. Within the lower and upper limits the observed size distributions of islets can be faithfully regenerated by assuming random and uncoordinated proliferation of each beta-cell at appropriate rates.

Neural synchronization at tonic-to-bursting transitions

We studied the synchronous behavior of two electrically-coupled model neurons as a function of the coupling strength when the individual neurons are tuned to different activity patterns that ranged from tonic firing via chaotic activity to burst discharges. We observe asynchronous and various synchronous states such as out-of-phase, in-phase and almost in-phase chaotic synchronization. The highest variety of synchronous states occurs at the transition from tonic firing to chaos where the highest coupling strength is also needed for in-phase synchronization which is, essentially, facilitated towards the bursting range. This demonstrates that tuning of the neuron's internal dynamics can have significant impact on the synchronous states especially at the physiologically relevant tonic-to-bursting transitions.

Gap junctional communication in morphogenesis

Gap junctions permit the direct passage of small molecules from the cytosol of one cell to that of its neighbor, and thus form a system of cell-cell communication that exists alongside familiar secretion/receptor signaling. Because of the rich potential for regulation of junctional conductance, and directional and molecular gating (specificity), gap junctional communication (GJC) plays a crucial role in many aspects of normal tissue physiology. However, the most exciting role for GJC is in the regulation of information flow that takes place during embryonic development, regeneration, and tumor progression. The molecular mechanisms by which GJC establishes local and long-range instructive morphogenetic cues are just beginning to be understood. This review summarizes the current knowledge of the involvement of GJC in the patterning of both vertebrate and invertebrate systems and discusses in detail several morphogenetic systems in which the properties of this signaling have been molecularly characterized. One model consistent with existing data in the fields of vertebrate left-right patterning and anterior-posterior polarity in flatworm regeneration postulates electrophoretically guided movement of small molecule morphogens through long-range GJC paths. The discovery of mechanisms controlling embryonic and regenerative GJC-mediated signaling, and identification of the downstream targets of GJC-permeable molecules, represent exciting next areas of research in this fascinating field.

Stable concurrent synchronization in dynamic system networks

In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple "rhythms" interacting and functional assemblies combining neural oscillators of many different types. Mathematically, stable concurrent synchronization corresponds to convergence to a flow-invariant linear subspace of the global state space. We derive a general condition for such convergence to occur globally and exponentially. We also show that, under mild conditions, global convergence to a concurrently synchronized regime is preserved under basic system combinations such as negative feedback or hierarchies, so that stable concurrently synchronized aggregates of arbitrary size can be constructed. Robustnesss of stable concurrent synchronization to variations in individual dynamics is also quantified. Simple applications of these results to classical questions in systems neuroscience and robotics are discussed.

Mathematical model of morphogen electrophoresis through gap junctions

Gap junctional communication is important for embryonic morphogenesis. However, the factors regulating the spatial properties of small molecule signal flows through gap junctions remain poorly understood. Recent data on gap junctions, ion transporters, and serotonin during left-right patterning suggest a specific model: the net unidirectional transfer of small molecules through long-range gap junctional paths driven by an electrophoretic mechanism. However, this concept has only been discussed qualitatively, and it is not known whether such a mechanism can actually establish a gradient within physiological constraints. We review the existing functional data and develop a mathematical model of the flow of serotonin through the early Xenopus embryo under an electrophoretic force generated by ion pumps. Through computer simulation of this process using realistic parameters, we explored quantitatively the dynamics of morphogen movement through gap junctions, confirming the plausibility of the proposed electrophoretic mechanism, which generates a considerable gradient in the available time frame. The model made several testable predictions and revealed properties of robustness, cellular gradients of serotonin, and the dependence of the gradient on several developmental constants. This work quantitatively supports the plausibility of electrophoretic control of morphogen movement through gap junctions during early left-right patterning. This conceptual framework for modeling gap junctional signaling -- an epigenetic patterning mechanism of wide relevance in biological regulation -- suggests numerous experimental approaches in other patterning systems.

Signal transmission between gap-junctionally coupled passive cables is most effective at an optimal diameter

We analyze simple morphological configurations that represent gap-junctional coupling between neuronal processes or between muscle fibers. Specifically, we use cable theory and simulations to examine the consequences of current flow from one cable to other gap-junctionally coupled passive cables. When the proximal end of the first cable is voltage clamped, the amplitude of the electrical signal in distal portions of the second cable depends on the cable diameter. However, this amplitude does not simply increase if cable diameter is increased, as expected from the larger length constant; instead, an optimal diameter exists. The optimal diameter arises because the dependency of voltage attenuation along the second cable on cable diameter follows two opposing rules. As cable diameter increases, the attenuation decreases because of a larger length constant yet increases because of a reduction in current density arising from the limiting effect of the gap junction on current flow into the second cable. The optimal diameter depends on the gap junction resistance and cable parameters. In branched cables, dependency on diameter is local and thus may serve to functionally compartmentalize branches that are coupled to other cells. Such compartmentalization may be important when periodic signals or action potentials cause the current flow across gap junctions.

Synchronization and propagation of bursts in networks of coupled map neurons

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.

Neuronal electrical rhythms described by composite mapped clock oscillators

The Mapped Clock Oscillator (MCO) model is a representation of omnipresent transmembrane voltage oscillations in excitable cells. We present a generalized version of the MCO that can model neuronal electrical oscillations, both labile and omnipresent, entirely within the framework of a system of ordinary differential equations. The previously described MCO was a second-order system, whereas the model presented here, which we call the composite MCO (cMCO) is a fourth-order system. Furthermore, we show how this cMCO can also be adapted to describe a pair of cells that forms a functional unit, as illustrated here by a model of the CA3 pyramidal cell and its basket cell interneuron feedback loop. The model was able to reproduce the high frequencies (super gamma) and possibly chaotic dynamics observed in the biological system.

Effect of intracellular delivery of energy metabolites on intracellular Ca2+ in mouse islets of Langerhans

Regulation of glucose-induced oscillations in intracellular Ca2+ concentration ([Ca2+]i) was investigated by using a novel technique, electroporation from an electrolyte-filled capillary, to deliver energy metabolites to the intracellular compartment of mouse islets. Intracellular application of ATP resulted in a nifedipine-sensitive increase in [Ca2+]i, consistent with a KATP-channel dependent mechanism of Ca2+ influx. [Ca2+]i in islets exposed to 10 mM glucose oscillated with a period of approximately 3 min, often superimposed with faster oscillations. Electroporation of ATP blocked all types of oscillations and elevated [Ca2+]i while delivery of ADP had no effect on oscillations. Intracellular delivery of glucose-6-phosphate or fructose-1,6-bisphosphate tended to transform slow oscillations to fast oscillations. These results demonstrate that modulation of ATP concentrations and glycolytic flux are important in development of slow oscillations.

How noise and coupling induce bursting action potentials in pancreatic {beta}-cells

Unlike isolated beta-cells, which usually produce continuous spikes or fast and irregular bursts, electrically coupled beta-cells are apt to exhibit robust bursting action potentials. We consider the noise induced by thermal fluctuations as well as that by channel-gating stochasticity and examine its effects on the action potential behavior of the beta-cell model. It is observed numerically that such noise in general helps single cells to produce a variety of electrical activities. In addition, we also probe coupling via gap junctions between neighboring cells, with heterogeneity induced by noise, to find that it enhances regular bursts.

Intra- and inter-islet synchronization of metabolically driven insulin secretion

Insulin secretion from pancreatic beta-cells is pulsatile with a period of 5-10 min and is believed to be responsible for plasma insulin oscillations with similar frequency. To observe an overall oscillatory insulin profile it is necessary that the insulin secretion from individual beta-cells is synchronized within islets, and that the population of islets is also synchronized. We have recently developed a model in which pulsatile insulin secretion is produced as a result of calcium-driven electrical oscillations in combination with oscillations in glycolysis. We use this model to investigate possible mechanisms for intra-islet and inter-islet synchronization. We show that electrical coupling is sufficient to synchronize both electrical bursting activity and metabolic oscillations. We also demonstrate that islets can synchronize by mutually entraining each other by their effects on a simple model "liver," which responds to the level of insulin secretion by adjusting the blood glucose concentration in an appropriate way. Since all islets are exposed to the blood, the distributed islet-liver system can synchronize the individual islet insulin oscillations. Thus, we demonstrate how intra-islet and inter-islet synchronization of insulin oscillations may be achieved.

Three roads to islet bursting: emergent oscillations in coupled phantom bursters

Glucose-induced membrane potential and Ca(2+) oscillations in isolated pancreatic beta-cells occur over a wide range of frequencies, from >6/min (fast) to <1/min (slow). However, cells within intact islets generally oscillate with periods of 10-60 s (medium). The phantom bursting concept addresses how beta-cells can generate such a wide range of frequencies. Here, we explore an updated phantom bursting model to determine how heterogeneity in a single parameter can explain both the broad frequency range observed in single cells and the rarity of medium oscillations. We then incorporate the single-cell model into an islet model with parameter heterogeneity. We show that strongly coupled islets must be composed of predominantly medium oscillating single cells or a mixture of fast and slow cells to robustly produce medium oscillations. Surprisingly, we find that this constraint does not hold for moderate coupling, and that robustly medium oscillating islets can arise from populations of single cells that are essentially all slow or all fast. Thus, with coupled phantom bursters, medium oscillating islets can be constructed out of cells that are either all fast, all slow, or a combination of the two.

Mitochondrial metabolism reveals a functional architecture in intact islets of Langerhans from normal and diabetic Psammomys obesus

The cells within the intact islet of Langerhans function as a metabolic syncytium, secreting insulin in a coordinated and oscillatory manner in response to external fuel. With increased glucose, the oscillatory amplitude is enhanced, leading to the hypothesis that cells within the islet are secreting with greater synchronization. Consequently, non-insulin-dependent diabetes mellitus (NIDDM; type 2 diabetes)-induced irregularities in insulin secretion oscillations may be attributed to decreased intercellular coordination. The purpose of the present study was to determine whether the degree of metabolic coordination within the intact islet was enhanced by increased glucose and compromised by NIDDM. Experiments were performed with isolated islets from normal and diabetic Psammomys obesus. Using confocal microscopy and the mitochondrial potentiometric dye rhodamine 123, we measured mitochondrial membrane potential oscillations in individual cells within intact islets. When mitochondrial membrane potential was averaged from all the cells in a single islet, the resultant waveform demonstrated clear sinusoidal oscillations. Cells within islets were heterogeneous in terms of cellular synchronicity (similarity in phase and period), sinusoidal regularity, and frequency of oscillation. Cells within normal islets oscillated with greater synchronicity compared with cells within diabetic islets. The range of oscillatory frequencies was unchanged by glucose or diabetes. Cells within diabetic (but not normal) islets increased oscillatory regularity in response to glucose. These data support the hypothesis that glucose enhances metabolic coupling in normal islets and that the dampening of oscillatory insulin secretion in NIDDM may result from disrupted metabolic coupling.

On-line analysis of gap junctions reveals more efficient electrical than dye coupling between islet cells

Pancreatic beta-cells constitute a well-communicating multicellular network that permits a coordinated and synchronized signal transmission within the islet of Langerhans that is necessary for proper insulin release. Gap junctions are the molecular keys that mediate functional cellular connections, which are responsible for electrical and metabolic coupling in the majority of cell types. Although the role of gap junctions in beta-cell electrical coupling is well documented, metabolic communication is still a matter of discussion. Here, we have addressed this issue by use of a fluorescence recovery after photobleaching (FRAP) approach. This technique has been validated as a reliable and noninvasive approach to monitor functional gap junctions in real time. We show that control pancreatic islet cells did not exchange a gap junction-permeant molecule in either clustered cells or intact islets of Langerhans under conditions that allowed cell-to-cell exchange of current-carrying ions. Conversely, we have detected that the same probe was extensively transferred between islet cells of transgenic mice expressing connexin 32 (Cx32) that have enhanced junctional coupling properties. The results indicate that the electrical coupling of native islet cells is more extensive than dye communication. Dye-coupling domains in islet cells appear more restricted than previously inferred with other methods.

Synchronization of time-continuous chaotic oscillators

Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded in the fully synchronized state lose their transverse stability, and the appearance of globally and locally riddled basins of attraction is discussed, respectively, in terms of the subcritical, supercritical nature of the riddling bifurcations. We show how the introduction of a small parameter mismatch between the interacting chaotic oscillators causes a shift of the synchronization manifold. The presence of a coupling asymmetry is found to lead to further modifications of the destabilization process. Finally, the paper considers the problem of partial synchronization in a system of four coupled Rössler oscillators.

A model for glucose-induced wave propagation in pancreatic islets of Langerhans

A reaction-diffusion type model is constructed, describing the spatio-temporal dynamics of the basic intracellular variables assumed to be involved in the initiation of the insulin secretion process by beta -cells in the pancreatic islets of Langerhans. The model includes equations for the electric membrane potential of the cells, with respective kinetics for ionic currents, for concentrations of both free and stored intracellular calcium, and for the intra- and extracellular concentrations of glucose. An empirical expression connecting the equation for the intracellular glucose concentration to the electrical equation is introduced. The model reproduces the events observed in experiments in vitro upon external glucose application to the islets of Langerhans, such as usual bursting oscillations of the membrane potential and corresponding oscillations of the intracellular calcium concentration. It also allows simulation of electric wave propagation through the islet, initiated by the spatial gradient of glucose concentration within the islet. The gradient emerges due to glucose diffusing into the islets from the external medium, being high at the edges. The latter results show that glucose diffusion presents a means for wave initiation in the islets, which supports our previous assumption (Aslanidi et al., 2001).

Electrical coupling between model midbrain dopamine neurons: effects on firing pattern and synchrony

The role of gap junctions between midbrain dopamine (DA) neurons in mechanisms of firing pattern generation and synchronization has not been well characterized experimentally. We modified a multi-compartment model of DA neuron by adding a spike-generating mechanism and electrically coupling the dendrites of two such neurons through gap junctions. The burst-generating mechanism in the model neuron results from the interaction of a N-methyl-D-aspartate (NMDA)-induced current and the sodium pump. The firing patterns exhibited by the two model neurons included low frequency (2-7 Hz) spiking, high-frequency (13-20 Hz) spiking, irregular spiking, regular bursting, irregular bursting, and leader/follower bursting, depending on the parameter values used for the permeability for NMDA-induced current and the conductance for electrical coupling. All of these firing patterns have been observed in physiological neurons, but a systematic dependence of the firing pattern on the covariation of these two parameters has not been established experimentally. Our simulations indicate that electrical coupling facilitates NMDA-induced burst firing via two mechanisms. The first can be observed in a pair of identical cells. At low frequencies (low NMDA), as coupling strength was increased, only a transition from asynchronous to synchronous single-spike firing was observed. At high frequencies (high NMDA), increasing the strength of the electrical coupling in an identical pair resulted in a transition from high-frequency single-spike firing to burst firing, and further increases led to synchronous high-frequency spiking. Weak electrical coupling destabilizes the synchronous solution of the fast spiking subsystems, and in the presence of a slowly varying sodium concentration, the desynchronized spiking solution leads to bursts that are approximately in phase with spikes that are not in phase. Thus this transitional mechanism depends critically on action potential dynamics. The second mechanism for the induction of burst firing requires a heterogeneous pair that is, respectively, too depolarized and too hyperpolarized to burst. The net effect of the coupling is to bias at least one cell into an endogenously burst firing regime. In this case, action potential dynamics are not critical to the transitional mechanism. If electrical coupling is indeed more prominent in vivo due to basal level of modulation of gap junctions in vivo, these results may indicate why NMDA-induced burst firing is easier to observe in vivo as compared in vitro.

Bifurcation structure of a model of bursting pancreatic cells

One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic beta-cells reveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one side of the arms and saddle-node bifurcations on the other. The transition from this structure to the so-called period-adding structure is found to involve a subcritical period-doubling bifurcation and the emergence of type-III intermittency. The period-adding transition itself is not smooth but consists of a saddle-node bifurcation in which (n+1)-spike bursting behavior is born, slightly overlapping with a subcritical period-doubling bifurcation in which n-spike bursting behavior loses its stability.

Loss of coherence in a system of globally coupled maps

We study the formation of symmetric (i.e., equally sized) or nearly symmetric clusters in an ensemble of globally coupled, identical chaotic maps. It is shown that the loss of synchronization for the coherent state and the emergence of subgroups of oscillators with synchronized behavior are two distinct processes, and that the type of behavior that arises after the loss of total synchronization depends sensitively on the dynamics of the individual map. For our system of globally coupled logistic maps, symmetric two-cluster formation is found to proceed through a periodic state associated with the stabilization either of an asynchronous period-2 cycle or of an asynchronous period-4 cycle. With further reduction of the coupling strength, each of the principal clustering states undergoes additional bifurcations leading to cycles of higher periodicity or to quasiperiodic and chaotic dynamics. If desynchronization of the coherent chaotic state occurs before the formation of stable clusters becomes possible, high-dimensional chaotic motion is observed in the intermediate parameter interval.

Excitation wave propagation as a possible mechanism for signal transmission in pancreatic islets of Langerhans

In response to glucose application, beta-cells forming pancreatic islets of Langerhans start bursting oscillations of the membrane potential and intracellular calcium concentration, inducing insulin secretion by the cells. Until recently, it has been assumed that the bursting activity of beta-cells in a single islet of Langerhans is synchronized across the whole islet due to coupling between the cells. However, time delays of several seconds in the activity of distant cells are usually observed in the islets of Langerhans, indicating that electrical/calcium wave propagation through the islets can occur. This work presents both experimental and theoretical evidence for wave propagation in the islets of Langerhans. Experiments with Fura-2 fluorescence monitoring of spatiotemporal calcium dynamics in the islets have clearly shown such wave propagation. Furthermore, numerical simulations of the model describing a cluster of electrically coupled beta-cells have supported our view that the experimentally observed calcium waves are due to electric pulses propagating through the cluster. This point of view is also supported by independent experimental results. Based on the model equations, an approximate analytical expression for the wave velocity is introduced, indicating which parameters can alter the velocity. We point to the possible role of the observed waves as signals controlling the insulin secretion inside the islets of Langerhans, in particular, in the regions that cannot be reached by any external stimuli such as high glucose concentration outside the islets.

Channel sharing in pancreatic beta -cells revisited: enhancement of emergent bursting by noise

Secretion of insulin by electrically coupled populations of pancreatic beta -cells is governed by bursting electrical activity. Isolated beta -cells, however, exhibit atypical bursting or continuous spike activity. We study bursting as an emergent property of the population, focussing on interactions among the subclass of spiking cells. These are modelled by equipping the fast subsystem with a saddle-node-loop bifurcation, which makes it monostable. Such cells can only spike tonically or remain silent when isolated, but can be induced to burst with weak diffusive coupling. With stronger coupling, the cells revert to tonic spiking. We demonstrate that the addition of noise dramatically increases, via a phenomenon like stochastic resonance, the coupling range over which bursting is seen.

Insulin release at the molecular level: metabolic-electrophysiological modeling of the pancreatic beta-cells

The role of pancreatic beta-cells is fundamental in the control endocrine system, maintaining the blood glucose homeostais in a physiological regime, via the glucose-induced release of insulin. An increasing amount of detailed experimental evidences at the cellular and molecular biology levels have been collected on the key factors determining the insulin release by the pancreatic beta-cells. The direct transposition of such experimental data into accurate mathematical descriptions might contribute to considerably clarify the impact of each cellular component on the global glucose metabolism. Under these perspectives, we model and computer-simulate the stimulus-secretion coupling in beta-cells by describing four interacting cellular subsystems, consisting in the glucose transport and metabolism, the excitable electrophysiological behavior, the dynamics of the intracellular calcium ions, and the exocytosis of granules containing insulin. We explicit the molecular nature of each subsystem, expressing the mutual relationships and the feedbacks that determine the metabolic-electrophysiological behavior of an isolated beta-cell. Finally, we discuss the simulation results of the behavior of isolated beta-cells as well as of population of electrically coupled beta-cells in Langerhans islets, under physiological and pathological conditions, including noninsulin-dependent diabetes mellitus (NIDDM) and hyperinsulinemic hypoglycaemia (PHHI).

Pathophysiology of impaired pulsatile insulin release

Plasma insulin displays 5-10 min oscillations. In Type 2 diabetes the regularity of the oscillations disappears, which may lead to insulin receptor down-regulation and glucose intolerance and explain why pulsatile delivery of the hormone has a greater hypoglycemic effect than continuous delivery. The rhythm is intrinsic to the islet. Variations in metabolism, cytoplasmic Ca(2+) concentration ([Ca(2+)](i)), other hormones, neuronal signaling and possibly beta-cell insulin receptor expression have been implicated in the regulation of plasma insulin oscillations. Most of these factors are important for amplitude-regulation of the insulin pulses. Although evidence exists supporting a role of both metabolism and [Ca(2+)](i) as pacemakers of the pulses, metabolic oscillations probably have a primary role and [Ca(2+)](i) oscillations a permissive role. Results from islets from animal models of diabetes suggest that altered plasma insulin pattern could be due to lowering of pulse amplitude of insulin oscillations rather than alterations in their frequency. Supporting a role of metabolism, altered plasma insulin oscillations were found in MODY2, MIDD and glycogenosis Type VII, which are linked to alterations in glucokinase, mitochondrial tRNALeu(UUR) and phosphofructokinase. Plasma insulin oscillations require coordination of islet secretory activities in the pancreas. The intrapancreatic ganglia have been suggested as coordinators. The diabetes-associated neuropathy may contribute to the deranged pattern as indicated by glucose intolerance in chagasic patients. Continued investigation of the role and regulation of pulsatile insulin release will lead to better understanding of the pathophysiology of impaired pulsatile insulin release, which could lead to new approaches to restore normal plasma insulin oscillations in diabetes and related diseases.

Gap junction effects on precision and frequency of a model pacemaker network

We investigated the precision of spike timing in a model of gap junction-coupled oscillatory neurons. The model incorporated the known physiology, morphology, and connectivity of the weakly electric fish's high-frequency and extremely precise pacemaker nucleus (Pn). Two neuron classes, pacemaker and relay cells, were each modeled with two compartments containing Hodgkin-Huxley sodium and potassium currents. Isolated pacemaker cells fired periodically, due to a constant current injection; relay cells were silent but slightly depolarized at rest. When coupled by gap junctions to other neurons, a model neuron, like its biological correlate, spiked at frequencies and amplitudes that were largely independent of current injections. The phase distribution in the network was labile to intracellular current injections and to gap junction conductance changes. The model predicts a biologically plausible gap junction conductance of 4-5 nS (200-250 MOmega). This results in a coupling coefficient of approximately 0.02, as observed in vitro. Network parameters were varied to test which could improve the temporal precision of oscillations. Increased gap junction conductances and larger numbers of cells (holding total junctional conductance per cell constant) both substantially reduced the coefficient of variation (CV = standard deviation/mean) of relay cell spike times by 74-85% and more, and did so with lower gap junction conductance when cells were contacted axonically compared with somatically. Pacemaker cell CV was only reduced when the probability of contact was increased, and then only moderately: a fivefold increase in the probability of contact reduced CV by 35%. We conclude that gap junctions facilitate synchronization, can reduce CV, are most effective between axons, and that pacemaker cells must have low intrinsic CV to account for the low CV of cells in the biological network.

Influence of cell number on the characteristics and synchrony of Ca2+ oscillations in clusters of mouse pancreatic islet cells

1. The cytoplasmic Ca2+ concentration ([Ca2+]i) was measured in single cells and cell clusters of different sizes prepared from mouse pancreatic islets. 2. During stimulation with 15 mM glucose, 20 % of isolated cells were inert, whereas 80 % showed [Ca2+]i oscillations of variable amplitude, duration and frequency. Spectral analysis identified a major frequency of 0.14 min-1 and a less prominent one of 0.27 min-1. 3. In contrast, practically all clusters (2-50 cells) responded to glucose, and no inert cells were identified within the clusters. As compared to single cells, mean [Ca2+]i was more elevated, [Ca2+]i oscillations were more regular and their major frequency was slightly higher (but reached a plateau at approximately 0.25 min-1). In some cells and clusters, faster oscillations occurred on top of the slow ones, between them or randomly. 4. Image analysis revealed that the regular [Ca2+]i oscillations were well synchronized between all cells of the clusters. Even when the Ca2+ response was irregular, slow and fast [Ca2+]i oscillations induced by glucose were also synchronous in all cells. 5. In contrast, [Ca2+]i oscillations resulting from mobilization of intracellular Ca2+ by acetylcholine were restricted to certain cells only and were not synchronized. 6. Heptanol and 18alpha-glycyrrhetinic acid, two agents widely used to block gap junctions, altered glucose-induced Ca2+ oscillations, but control experiments showed that they also exerted effects other than a selective uncoupling of the cells. 7. The results support theoretical models predicting an increased regularity of glucose-dependent oscillatory events in clusters as compared to isolated islet cells, but contradict the proposal that the frequency of the oscillations increases with the number of coupled cells. Islet cell clusters function better as electrical than biochemical syncytia. This may explain the co-ordination of [Ca2+]i oscillations driven by depolarization-dependent Ca2+ influx during glucose stimulation.

Bursting calcium rotors in cultured cardiac myocyte monolayers

Rotating waves (rotors) of cellular activity were observed in nonconfluent cultures of embryonic chick heart cells by using a macroscopic imaging system that detected fluorescence from intracellular Ca2+. Unlike previous observations of rotors or spiral waves in other systems, the rotors did not persist but exhibited a repetitive pattern of spontaneous onset and offset leading to a bursting rhythm. Similar dynamics were observed in a cellular automaton model of excitable media that incorporates spontaneous initiation of activity, and a decrease of excitability as a consequence of rapid activity (fatigue). These results provide a mechanism for bursting dynamics in normal and pathological biological processes.