Analysis of the noise-induced bursting-spiking transition in a pancreatic beta-cell model

A chaotic bursting-spiking transition in a pancreatic beta-cells system: Observation of an interior glucose-induced crisis

Nonlinear systems are commonly able to display abrupt qualitative changes (or transitions) in the dynamics. A particular type of these transitions occurs when the size of a chaotic attractor suddenly changes. In this article, we present such a transition through the observation of a chaotic interior crisis in the Deng bursting-spiking model for the glucose-induced electrical activity of pancreatic β-cells. To this chaos-chaos transition corresponds precisely the change between the bursting and spiking dynamics, which are central and key dynamical regimes that the Deng model is able to perform. We provide a description of the crisis mechanism at the bursting-spiking transition point in terms of time series variations and based on certain amplitudes of invariant intervals associated with return maps. Using symbolic dynamics, we are able to accurately compute the points of a curve representing the transition between the bursting and spiking regimes in a biophysical meaningfully parameter space. The analysis of the chaotic interior crisis is complemented by means of topological invariants with the computation of the topological entropy and the maximum Lyapunov exponent. Considering very recent developments in the literature, we construct analytical solutions triggering the bursting-spiking transition in the Deng model. This study provides an illustration of how an integrated approach, involving numerical evidences and theoretical reasoning within the theory of dynamical systems, can directly enhance our understanding of biophysically motivated models.

Coupling and noise induced spiking-bursting transition in a parabolic bursting model

The transition from tonic spiking to bursting is an important dynamic process that carry physiologically relevant information. In this work, coupling and noise induced spiking-bursting transition is investigated in a parabolic bursting model with specific discussion on their cooperation effects. Fast/slow analysis shows that weak coupling may help to induce the bursting by changing the geometric property of the fast subsystem so that the original unstable periodical solution are stabilized. It turned out that noise can play the similar stabilization role and induce bursting at appropriate moderate intensity. However, their cooperation may either strengthen or weaken the overall effect depending on the choice of noise level.

Mathematical models for insulin secretion in pancreatic β-cells

Insulin secretion is one of the most characteristic features of β-cell physiology. As it plays a central role in glucose regulation, a number of experimental and theoretical studies have been performed since the discovery of the pancreatic β-cell. This review article aims to give an overview of the mathematical approaches to insulin secretion. Beginning with the bursting electrical activity in pancreatic β-cells, we describe effects of the gap-junction coupling between β-cells on the dynamics of insulin secretion. Then, implications of paracrine interactions among such islet cells as α-, β-, and δ-cells are discussed. Finally, we present mathematical models which incorporate effects of glycolysis and mitochondrial glucose metabolism on the control of insulin secretion.

Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing

Spontaneous spatiotemporal wave activity occurs in groups of excitable particles for groups larger than a critical size. Experiments are carried out with particles loaded with the catalyst of the Belousov-Zhabotinsky reaction that are immersed in catalyst-free reaction mixture. The particles diffusively exchange activator and inhibitor species with the surrounding solution. All particles are nonoscillatory when separated from the other particles; however, target and spiral waves are exhibited in sufficiently large groups. A cellular particle model of the system also exhibits transitions from excitable steady state behavior to spatiotemporal wave activity with increasing group size.

Bursting regimes in map-based neuron models coupled through fast threshold modulation

A system consisting of two map-based neurons coupled through reciprocal excitatory or inhibitory chemical synapses is discussed. After a brief explanation of the basic mechanism behind generation and synchronization of bursts, parameter space is explored to determine less obvious but biologically meaningful regimes and effects. Among them, we show how excitatory synapses without any delays may induce antiphase synchronization; that a synapse may change its character from excitatory to inhibitory and vice versa by changing its conductance, without any change in reversal potential; and that small variations in the synaptic threshold may result in drastic changes in the synchronization of spikes within bursts. Finally we show how the synchronization effects found in the two-neuron system carry over to larger networks.

Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos

The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying the external current I. For increasing current values, the model exhibits a peculiar cascade of nonchaotic and chaotic period-adding bifurcations leading the system from the silent regime to a chaotic state dominated by bursting events. At higher I-values, this phase is substituted by a regime of continuous chaotic spiking and finally via an inverse period doubling cascade the system returns to silence. The analysis is focused on the transition between the two chaotic phases displayed by the model: one dominated by spiking dynamics and the other by bursts. At the transition an abrupt shrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent is observable. However, the transition appears to be continuous and smoothed out over a finite current interval, where bursts and spikes coexist. The beginning of the transition (from the bursting side) is signaled from a structural modification in the interspike interval return map. This change in the map shape is associated with the disappearance of the family of solutions responsible for the onset of the bursting chaos. The successive passage from bursting to spiking chaos is associated with a progressive pruning of unstable long-lasting bursts.

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.

Patterns in inhibitory networks of simple map neurons

We study the dynamics of networks of inhibitory map-based bursting neurons. Linear analysis allows us to understand how the patterns of bursting are determined by network topology and how they depend on the strength of synaptic connections, when inhibition is balanced. Two kinds of patterns are found depending on the symmetry of the network: slow cyclic patterns riding on subthreshold oscillations where almost all neurons contribute bursts in a sparse manner and fast patterns of bursts in which only one of two mutually exclusive groups of neurons take part. We also discuss the properties of the neuron model that underlie the described phenomena, comment on the limitations of the technique of analysis, and point to some possible ways to overcome them.

Rigid rod anchored to infinite membrane

We investigate the shape deformation of an infinite membrane anchored by a rigid rod. The density profile of the rod is calculated by the self-consistent-field theory and the shape of the membrane is predicted by the Helfrich membrane elasticity theory [W. Helfrich, Z. Naturforsch. 28c, 693 (1973)]. It is found that the membrane bends away from the rigid rod when the interaction between the rod and the membrane is repulsive or weakly attractive (adsorption). However, the pulled height of the membrane at first increases and then decreases with the increase of the adsorption strength. Compared to a Gaussian chain with the same length, the rigid rod covers much larger area of the membrane, whereas exerts less local entropic pressure on the membrane. An evident gap is found between the membrane and the rigid rod because the membrane's curvature has to be continuous. These behaviors are compared with that of the flexible-polymer-anchored membranes studied by previous Monte Carlo simulations and theoretical analysis. It is straightforward to extend this method to more complicated and real biological systems, such as infinite membrane/multiple chains, protein inclusion, or systems with phase separation.

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.