Diffusively coupled bursters: Effects of cell heterogeneity

The interaction of a pair of weakly coupled biological bursters is examined. Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the β-cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition, non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity. An erratum to this article is available at .     

Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans.

Electrical bursting oscillations of mammalian pancreatic beta-cells are synchronous among cells within an islet. While electrical coupling among cells via gap junctions has been demonstrated, its extent and topology are unclear. The beta-cells also share an extracellular compartment in which oscillations of K+ concentration have been measured (Perez-Armendariz and Atwater, 1985). These oscillations (1-2 mM) are synchronous with the burst pattern, and apparently are caused by the oscillating voltage-dependent membrane currents: Extracellular K+ concentration (Ke) rises during the depolarized active (spiking) phase and falls during the hyperpolarized silent phase. Because raising Ke depolarizes the cell membrane by increasing the potassium reversal potential (VK), any cell in the active phase should recruit nonspiking cells into the active phase. The opposite is predicted for the silent phase. This positive feedback system might couple the cells' electrical activity and synchronize bursting. We have explored this possibility using a theoretical model for bursting of beta-cells (Sherman et al., 1988) and K+ diffusion in the extracellular space of an islet. Computer simulations demonstrate that the bursts synchronize very quickly (within one burst) without gap junctional coupling among the cells. The shape and amplitude of computed Ke oscillations resemble those seen in experiments for certain parameter ranges. The model cells synchronize with exterior cells leading, though incorporating heterogeneous cell properties can allow interior cells to lead. The model islet can also be forced to oscillate at both faster and slower frequencies using periodic pulses of higher K+ in the medium surrounding the islet. Phase plane analysis was used to understand the synchronization mechanism. The results of our model suggest that diffusion of extracellular K+ may contribute to coupling and synchronization of electrical oscillations in beta-cells within an islet.     

Bifurcations of emergent bursting in a neuronal network

Complex neuronal networks are an important tool to help explain paradoxical phenomena observed in biological recordings. Here we present a general approach to mathematically tackle a complex neuronal network so that we can fully understand the underlying mechanisms. Using a previously developed network model of the milk-ejection reflex in oxytocin cells, we show how we can reduce a complex model with many variables and complex network topologies to a tractable model with two variables, while retaining all key qualitative features of the original model. The approach enables us to uncover how emergent synchronous bursting can arise from a neuronal network which embodies known biological features. Surprisingly, the bursting mechanisms are similar to those found in other systems reported in the literature, and illustrate a generic way to exhibit emergent and multiple time scale oscillations at the membrane potential level and the firing rate level.     

Slow variable dominance and phase resetting in phantom bursting

Bursting oscillations are common in neurons and endocrine cells. One type of bursting model with two slow variables has been called ‘phantom bursting’ since the burst period is a blend of the time constants of the slow variables. A phantom bursting model can produce bursting with a wide range of periods: fast (short period), medium, and slow (long period). We describe a measure, which we call the ‘dominance factor’, of the relative contributions of the two slow variables to the bursting produced by a simple phantom bursting model. Using this tool, we demonstrate how the control of different phases of the burst can be shifted from one slow variable to another by changing a model parameter. We then show that the dominance curves obtained as a parameter is varied can be useful in making predictions about the resetting properties of the model cells. Finally, we demonstrate two mechanisms by which phase-independent resetting of a burst can be achieved, as has been shown to occur in the electrical activity of pancreatic islets.     

Phantom bursting is highly sensitive to noise and unlikely to account for slow bursting in beta-cells: considerations in favor of metabolically driven oscillations

Pancreatic beta-cells show bursting electrical activity with a wide range of burst periods ranging from a few seconds, often seen in isolated cells, over tens of seconds (medium bursting), usually observed in intact islets, to several minutes. The phantom burster model [Bertram, R., Previte, J., Sherman, A., Kinard, T.A., Satin, L.S., 2000. The phantom burster model for pancreatic beta-cells. Biophys. J. 79, 2880-2892] provided a framework, which covered this span, and gave an explanation of how to obtain medium bursting combining two processes operating on different time scales. However, single cells are subjected to stochastic fluctuations in plasma membrane currents, which are likely to disturb the bursting mechanism and transform medium bursters into spikers or very fast bursters. We present a polynomial, minimal, phantom burster model and show that noise modifies the plateau fraction and lowers the burst period dramatically in phantom bursters. It is therefore unlikely that slow bursting in single cells is driven by the slow phantom bursting mechanism, but could instead be driven by oscillations in glycolysis, which we show are stable to random ion channel fluctuations. Moreover, so-called compound bursting can be converted to apparent slow bursting by noise, which could explain why compound bursting and mixed Ca(2+) oscillations are seen mainly in intact islets.     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

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.     

From spikers to bursters via coupling: Help from heterogeneity

It has been shown previously that identical spiking cells, incapable of bursting by themselves, may burst under weak diffusive coupling conditions. With stronger coupling, the coupled system reverts to bursting can be recovered by introducing heterogeneity in the model parameters. For a two-cell system, we explain the phenomenon with bifurcation analysis. For larger clusters of coupled cells, we demonstrate by numerical simulation that the phenomenon carries over. In addition, we use a perturbation analysis to deduce the dependence of the heterogeneity parameter used in the bifurcation analysis on the original heterogeneity parameters and the coupling strength. Implications of the phenomenon of emergent bursting are discussed in the context of electrical activity in pancreatic beta cells.     

Stability and oscillations in coupled systems.

In this paper are given criteria for stability and oscillations in coupled systems. The criteria are calculated by the determination of eigenvalues of a coupling matrix of an input-output relation which can be described by a set of differential equations of 1st order.     

Modeling observed chaotic oscillations in bursting neurons: the role of calcium dynamics and IP3

We study the dynamics and stability of legged locomotion in the horizontal plane. We discuss the relevance of idealized mechanical models, developed in a companion paper, to recent experiments and simulations on insect running and turning. Applying our results to rapidly running cockroaches, we show that the models' gait and force characteristics match observations reasonably well. Received: 6 September 1999 / Accepted in revised form: 8 May 2000     

A novel network of multipolar bursting interneurons generates theta frequency oscillations in neocortex

GABAergic interneurons can phase the output of principal cells, giving rise to oscillatory activity in different frequency bands. Here we describe a new subtype of GABAergic interneuron, the multipolar bursting (MB) cell in the mouse neocortex. MB cells are parvalbumin positive but differ from fast-spiking multipolar (FS) cells in their morphological, neurochemical, and physiological properties. MB cells are reciprocally connected with layer 2/3 pyramidal cells and are coupled with each other by chemical and electrical synapses. MB cells innervate FS cells but not vice versa. MB to MB cell as well as MB to pyramidal cell synapses exhibit paired-pulse facilitation. Carbachol selectively induced synchronized theta frequency oscillations in MB cells. Synchrony required both gap junction coupling and GABAergic chemical transmission, but not excitatory glutamatergic input. Hence, MB cells form a distinct inhibitory network, which upon cholinergic drive can generate rhythmic and synchronous theta frequency activity, providing temporal coordination of pyramidal cell output.     

Self-organized criticality and emergent oscillations in models of termite architecture with crowding

The termite architecture model of O'Toole et'al. (1999) is extended to incorporate arbitrary halting time-scales. It is shown that this also means that the assumption of synchronous building must be relaxed. Numerical simulations show that ordered nest architecture emerges under a wide range of time-scales but also that there is an optimal region of halting times. This optimal region is explained by the emergence of synchronized periods of termite activity. The correlation length of the building distribution is shown to diverge providing strong evidence that the model is self-organized critical.     

大鼠损伤神经的三种诱发簇放电节律

实验运用单纤维记录技术,观察了损伤神经起步点自发放电在改变[Ca^2 ]。和veratridine作用下放电节律的变化。结果表明：在每一标本上,记录到的相同背景的自发放电在低与高Ca^2 浓度和veratridine的作用下,转化为三种不同类型的簇放电。结果提示,神经元放电的节律形式与刺激的性质相关,不同的节律形式可能携带着不同的神经信息。     

Method for constructing the boundary of the bursting oscillations region in the neuron model

We examine the problem of constructing the boundary of bursting oscillations on a parameter plane for the system of equations describing the electrical behaviour of the membrane neuron arising from the interaction of fast oscillations of the cytoplasma membrane potential and slow oscillations of the intracellular calcium concentration. As the boundary point on the parameter plane we consider the values at which the limit cycle of the slow subsystem is tangent to the Hopf bifurcation curve of the fast subsystem. The method suggested for determining the boundary is based on the dissection of the system variables into slow and fast. The strong point of the method is that it requires the integration of the slow subsystem only. An example of the application of the method for the stomatogastric neuron model [Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Philos Trans R Soc Lond B 341: 345–359] is given. Received: 31 May 1999 / Accepted in revised form: 19 November 1999     

Epigenetic reprogramming: roads to pluripotency

Epigenetic reprogramming provides valuable resources for customized pluripotent stem cells generation, which are thought to be important bases of future regenerative medicine. Here we review the commonly used methods for epigenetic reprogramming: somatic cell nuclear transfer, cell fusion, cell extract treatment, inducing pluripotency by defined molecules, and briefly discuss their advantages and limitations. Finally we propose that mechanisms underlying epigenetic reprogramming and safety evaluation platform will be future research directions.     

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 β-cell is coupled to, n _c, and the coupling strength, g _c.     

Spontaneous oscillations in two 2-component cells coupled by diffusion

Mathematical examples are presented of oscillators with two variables which do not oscillate in isolation, but which do oscillate stably when coupled with a twin via difiusion. Two examples are presented, the LefeverPrigogine Brusselator and a system used to model glycolytic oscillations. The mathematical method is not the usual bifurcation theory, but rather a type of singular perturbation theory combined with bifurcation theory. For both examples, it is shown that all stationary solutions are unstable for appropriate parameter settings. In the case of the Brusselator, it is further shown that there exist limit cycles; i.e. stable oscillations, in this parameter range. A numerical example is presented.Partially supported by NSF     

Cdc42: new roads to travel

The roles of Cdc42 in stimulating changes in the actin cytoskeleton and in the establishment of cell polarity have been known for some time. However, evidence implicating Cdc42 in intracellular trafficking and in the regulation of cell growth have continued to surface. Recently, exciting developments have highlighted how Cdc42 might participate in these cellular functions by regulating the processing of the EGF receptor. Here, I discuss these findings within the broader context of how Cdc42 serves as a multifaceted signaling switch that links trafficking events to the control of cell growth.