Effects of extracellular calcium on electrical bursting and intracellular and luminal calcium oscillations in insulin secreting pancreatic beta-cells.

The extracellular calcium concentration has interesting effects on bursting of pancreatic beta-cells. The mechanism underlying the extracellular Ca2+ effect is not well understood. By incorporating a low-threshold transient inward current to the store-operated bursting model of Chay, this paper elucidates the role of the extracellular Ca2+ concentration in influencing electrical activity, intracellular Ca2+ concentration, and the luminal Ca2+ concentration in the intracellular Ca2+ store. The possibility that this inward current is a carbachol-sensitive and TTX-insensitive Na+ current discovered by others is discussed. In addition, this paper explains how these three variables respond when various pharmacological agents are applied to the store-operated model.     

Sensing and refilling calcium stores in an excitable cell.

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ mobilization leads to depletion of the endoplasmic reticulum (ER) and an increase in Ca2+ entry. We show here for the gonadotroph, an excitable endocrine cell, that sensing of ER Ca2+ content can occur without the Ca2+ release-activated Ca2+ current (Icrac), but rather through the coupling of IP3-induced Ca2+ oscillations to plasma membrane voltage spikes that gate Ca2+ entry. Thus we demonstrate that capacitative Ca2+ entry is accomplished through Ca(2+)-controlled Ca2+ entry. We develop a comprehensive model, with parameter values constrained by available experimental data, to simulate the spatiotemporal behavior of agonist-induced Ca2+ signals in both the cytosol and ER lumen of gonadotrophs. The model combines two previously developed models, one for ER-mediated Ca2+ oscillations and another for plasma membrane potential-driven Ca2+ oscillations. Simulations show agreement with existing experimental records of store content, cytosolic Ca2+ concentration ([Ca2+]i), and electrical activity, and make a variety of new, experimentally testable predictions. In particular, computations with the model suggest that [Ca2+]i in the vicinity of the plasma membrane acts as a messenger for ER content via Ca(2+)-activated K+ channels and Ca2+ pumps in the plasma membrane. We conclude that, in excitable cells that do not express Icrac, [Ca2+]i profiles provide a sensitive mechanism for regulating net calcium flux through the plasma membrane during both store depletion and refilling.     

On the mechanism of spiking and bursting in excitable cells

A mathematical model previously developed to explain beta-cell membrane potential oscillations has been modified to accommodate the external variation of K+, Na+ and Ca2+ concentrations. Our model, which is applicable to excitable cells, incorporates the barrier kinetics. Hodgkin-Huxley-type gating mechanism, and an electrogenic Na+-K+ pump. Numerical solutions of our model are in agreement with many of the experimental results reported in the literature on excitable cells.     

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh–Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.     

Noise-induced transition in excitable neuron models

 We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model – as has been done previously – our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles. Received: 19 September 2000 / Accepted in revised form: 4 March 2001     

Control of ionic currents in guard cell vacuoles by cytosolic and luminal calcium

Activity of vacuolar ion channels can be regulated by the cytosolic free Ca²⁺ concentration ([Ca²⁺]_cyt). Using the whole-vacuole mode of patch-clamp with Vicia faba guard cell vacuoles, three distinct cation currents were apparent that were differentially regulated by [Ca²⁺]_cyt. At ‘zero’ to 100 nM [Ca²⁺]_cyt, instantaneous currents typical of Fast Vacuolar (FV) channels were activated. A 10 fold KCl gradient directed out of the vacuole increased FV currents (up to fivefold) at negative potentials compared with the currents in symmetrical KCl. At [Ca²⁺]_cyt higher than 100 nM, instantaneous currents became smaller and voltage-independent (non-rectifying) and were typical of Vacuolar K,+-selective (VK) channels. These currents were less sensitive to a KCl gradient than were the FV currents, being stimulated less than twofold at negative potentials. Reversal potentials measured in the presence of a KCl gradient indicated a high K⁺ permeability of both FV and VK currents. At [Ca²⁺]_cyt higher than 600 nM time-dependent currents elicited by positive potentials were typical of Slow Vacuolar (SV) channel activation. When the Ca²⁺ mole fraction in the cytosolic or luminal solution was varied the reversal potential of SV currents (determined by tail current analysis) passed through maximum or minimum values. The resultant calculated apparent permeability ratios varied with ionic conditions but indicated high Ca²⁺ and K⁺ permeabilities. If a Cl^? permeability was assumed then the apparent P_Ca was lower. However, substitution of Cl^? by the larger (impermeant) anion gluconate had no effect on the reversal potential of SV tail currents in the presence of Ca²⁺ and a K⁺ gradient, demonstrating that the assumption of Cl^? permeability of the SV channel is invalid. Single-channel SV currents also decreased with increasing cytosolic Ca²⁺ mole fraction. These data indicate that the SV channel is highly cation selective, shows characteristics typical of a multi-ion pore and derives ion selectivity by Ca²⁺ binding. The SV channel currents could also be Mg²⁺-activated and were demonstrated to be Mg²⁺-permeable in the absence of Ca²⁺. The apparent permeability ratio (P_Mg:P_K) also varied under different ionic conditions. The results indicate not only that FV, VK and SV channels are all present in a single cell type, but also that each is differentially regulated by [Ca²⁺]_cyt. The respective roles of these channels in vacuolar ion release are discussed, and possible conditions are presented in which these channels could be activated by disparate signalling pathways during stomatal closure.     

Bursting excitable cell models by a slow Ca2+ current

Bursting in excitable cells is a phenomenon that has attracted the interest of many electrophysiologists and non-linear dynamicists. In this paper, we present two models that give rise to bursting in action potentials. The membrane of the first model contains a voltage-activated Ca2+ channel that inactivates very slowly upon depolarization and a delayed K+ channel that is activated by voltage. This model consists of three dynamic variables--the gating variable of K+ channel (n), inactivation gating variable of the Ca2+ channel (f), and membrane potential (V). The membrane of the second model contains a voltage-activated Na+ channel that inactivates rather fast upon depolarization. This model contains altogether five dynamic variables--the Na+ inactivation gating variable (h) and Ca2+ activation variable (d), in addition to the three dynamic variables in the first model. With the first model, we show how various interesting bursting patterns may arise from such a simple three dynamic variable model. We also demonstrate that a slowly inactivating voltage-dependent Ca2+ channel may play the key role in the genesis of bursting. With the second model, we show how the participation of a quickly inactivating fast inward current may lead to a neuronal type of bursting, multi-peaked oscillations, and chaos, as the rates of the gating variables change.     

Electric oscillation in an excitable model membrane impregnated with lipid analogues

A model membrane constructed from a Millipore filter, whose pores were impregnated with dioleyl phosphate, exhibited an electric self-oscillation under nonequilibrium conditions. The membrane interposed between two solutions with the same KCl concentrations showed no temporal change in membrane potential. However, the potential became oscillatory on application of an electric current to the membrane. The frequency was proportional to the magnitude of the electric current. When both KCl solutions were replaced by NaCl solutions, a similar trend was observed, although the oscillation was not as regular as in the case of KCl. A membrane placed between equimolar solutions of KCl and NaCl, on the other hand, gave rise to an oscillation even without current application. When a membrane was placed between 5 mM KCl and 100 mM KCl, it was found that NaCl added to the 5 mM KCl side had a pronounced effect on the membrane with respect to the frequency response of the oscillation. These results indicate that the dioleyl phosphate membrane discriminates Na+ from K+.     

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.     

Total calcium ultrastructure: advances in excitable cells

This is a review which is written on the basis of a cell calcium lecture delivered on 22 July 2000 at the European Research Meeting 'Calcium as a molecule of cellular integration'.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Mathematical models for excitable systems in biology and medicine.

The excitable systems play a very important role in Biology and Medicine. Phenomena such as the transmission of impulses between neurons, the cardiac arrhythmia, the aggregation of amoebas, the appearance of organized structures in the cortex of egg cells, all derive from the activity of excitable media. In the first part of this work a general definition of excitable system is given; we then analyze some cases of excitability, distinguishing between electrical and chemical excitability and comparing experimental observations with simulations carried out by appropriate mathematical models. Such models are almost always formulated by partial differential equations of "reaction-diffusion" type and they have the characteristic to describe propagations of electrical waves (neurons, pacemaker cardiac cells, pancreatic b-cells) or chemical and mechanical waves (propagation of Ca++ waves or mechanical waves in the endoplasmic reticulum). The aim is to put in evidence that the biological systems can show not only excitability of electrical type, but also excitability of chemical nature, which can be observed in the first steps of development of egg cells or, for example, in the formation of pigments in vertebrate skin or in clam shells.     

Retardation currents in excitable membranes and models of flicker noise

We have presented (Holden and Rubio, 1976) a model for flicker noise in nerve membranes in which there are interactions between adjacent ionic channels. These interactions are proposed to be mediated by order-disorder transitions in the membrane matrix. In this paper we explore the relaxation behaviour of our model, and, using transition state theory, predict a new class of membrane ionic currents which we call retardation currents. Such retardation currents have slow (hundreds of ms) kinetics, a low temperature dependence and appear as inactivation processes. We consider some candidate retardation currents.     

Electrical characteristics in an excitable element of lipid membrane.

Electrical characteristics in a membrane constructed from a porous filter adsorbed with a lipid analogue, dioleoyl phosphate (DOPH), were investigated in a situation interposed between 100 mM NaCl + 3 mM CaCl2 and 100 mM KCl. Calcium ions affected significantly the membrane characteristics. The membrane potential was negative on the KCl side, which implies the higher permeability to K+ than Na+; this tendency was increased by a tiny amount of Ca2+. While the membrane showed a low electrical resistance of several k omega . cm2 under K+/Na+ gradient, it showed several M omega . cm2 by Ca2+. The surface structure of the membrane exhibited many voids in the low-resistance state, but the surface was covered by oil droplets in the high-resistance state. Oscillations of the membrane potential appeared spontaneously with application of the electrical current from the KCl side to the NaCl + CaCl2 side. The frequency was increased with the electrical current. All these results were explained comprehensively using an electrochemical kinetic model taking account of the Ca2+ binding effect, where DOPH assemblies make a phase transition between oil droplets due to Ca2+ and multi-bilayers with excess K+. The oscillation arises from coupling of the phase transition to accumulation and release of K+ or Ca2+. This membrane can be used as an excitable element regulated by Ca2+ in neuro-computer devices.     

Electrical oscillation and contact inhibition of reproduction in cells

The assumption that electrical oscillatory phenomena must be present during the cellular reproductive cycle is examined and found to be supported by the presently available evidence, mainly that on mouse cells.As a corollary, it suggests an electrically oscillatory aspect to the contact inhibition of growth, and three mechanisms for the invasiveness of cancer cells: changes in either the power level, degree of insulation, or frequency of the oscillatory phenomena associated with reproduction. The available evidence is in support of this, but much more is needed. A number of critical questions are presented.     

Spontaneous oscillation and mechanically induced calcium waves in chondrocytes

The characteristics of spontaneous calcium (Ca(2+)) oscillation and mechanically induced Ca(2+) waves in articular chondrocytes were studied. In some, but not all, chondrocytes in sliced cartilage and primary cultures, we observed spontaneous oscillation of intracellular Ca(2+) that never spread to adjacent cells. In contrast, a mechanical stimulus to a single cell by touching with a glass rod induced an increase of intracellular Ca(2+) that spread to neighboring cells in a wave-like manner, even though there was no physical contact between the cells. This indicated the release of some paracrine factor from the mechanically stimulated cells. Application of ultrasonic vibration also induced an oscillation of intracellular Ca(2+). The application of a uridine 5'-triphosphate (UTP), UTP, induced a transient increase in intracellular Ca(2+) and the release of adenosine 5'-triphosphate (ATP) in cultured chondrocytes. A P2 receptor antagonist (suramin) and blockers of Cl(-) channels, niflumic acid and 4,4'-diisothiocyanostilbene-2,2'-disulfonic acid (DIDS), reduced the UTP-induced ATP release. The results indicated that Cl(-) channels were involved in the extracellular release of ATP following mechanical or P2Y receptor stimulation. Thus, ATP stimulation of P2Y receptors elicits an increase in intracellular Ca(2+), triggering further release of ATP from adjacent cells, thereby expanding the Ca(2+) wave in chondrocytes.     

Capacitative calcium entry deficits and elevated luminal calcium content in mutant presenilin-1 knockin mice

Dysregulation of calcium signaling has been causally implicated in brain aging and Alzheimer's disease. Mutations in the presenilin genes (PS1, PS2), the leading cause of autosomal dominant familial Alzheimer's disease (FAD), cause highly specific alterations in intracellular calcium signaling pathways that may contribute to the neurodegenerative and pathological lesions of the disease. To elucidate the cellular mechanisms underlying these disturbances, we studied calcium signaling in fibroblasts isolated from mutant PS1 knockin mice. Mutant PS1 knockin cells exhibited a marked potentiation in the amplitude of calcium transients evoked by agonist stimulation. These cells also showed significant impairments in capacitative calcium entry (CCE, also known as store-operated calcium entry), an important cellular signaling pathway wherein depletion of intracellular calcium stores triggers influx of extracellular calcium into the cytosol. Notably, deficits in CCE were evident after agonist stimulation, but not if intracellular calcium stores were completely depleted with thapsigargin. Treatment with ionomycin and thapsigargin revealed that calcium levels within the ER were significantly increased in mutant PS1 knockin cells. Collectively, our findings suggest that the overfilling of calcium stores represents the fundamental cellular defect underlying the alterations in calcium signaling conferred by presenilin mutations.     

Thrombin-induced calcium oscillation in human platelets and MEG-01, a megakaryoblastic leukemia cell line.

Digital imaging microscopy revealed that human platelets show periodic intracellular Ca++ elevation in response to 0.01 U/ml thrombin. MEG-01, a megakaryoblastic leukemia cell line, also responded with oscillatory intracellular Ca++ elevation (0.7-1 times/min) to thrombin (0.001-0.003U/ml). Ca++ transients appears to be fused with higher thrombin doses. With extracellular Ca++ concentrations of 0.1 mM or less, Ca++ oscillation could not be elicited, or even when present, it disappeared after a few spikings of [Ca++]i. Extracellular Ca++ concentrations of 0.3 mM or more were required to facilitate ongoing Ca++ oscillation, suggesting an important role of Ca++ influx for Ca++ oscillation.