Spontaneous secondary spiking in excitable cells

Kepler & Marder (1993, Biol. Cybern.68, 209-214) proposed a model describing the electrical activity of a crab neuron in which a train of directly induced action potentials is sometimes followed by one or more spontaneous action potentials, referred to as spontaneous secondary spikes. We reduce their five-dimensional model to three dimensions in two different ways in order to gain insight into the mechanism underlying the spontaneous spikes. We then treat a slowly varying current as a parameter in order to give a qualitative explanation of the phenomenon using phase-plane and bifurcation analysis. We demonstrate that a three-dimensional model, consisting of a two-dimensional excitable system plus a slow inward current, is sufficient to produce the behaviour observed in the original model. The exact dynamics of the excitable system are not important, but the relative time constant and amplitude of the slow inward current are crucial. Using the numerical bifurcation analysis package AUTO (Doedel & Kernevez, 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology), we compute bifurcation diagrams using the maximum amplitude of the slow inward current as the bifurcation parameter. The full and reduced models have a stable resting potential for all values of the bifurcation parameter. At a critical value of the bifurcation parameter, a stable tonic firing mode arises via a saddle-node of periodics bifurcation. Whether or not the models can exhibit transient or continuous spontaneous spiking depends on their position in parameter space relative to this saddle-node of periodics.     

Intercellular conduction velocity variability as the basis for re-entrant arrhythmias in the ischemic myocardium.

Re-entrant arrhythmias are the major cause of death from cardiovascular disease. A number of models or mechanisms have been proposed to explain the generation of re-entrant arrhythmias in the ischemic or damaged heart. However, none of these models can qualitatively predict the formation of re-entry movements with no modifications of the basic electrophysiologic characteristics of the myocardium. In this presentation we evaluate the concept that the generation of re-entrant arrhythmias is due to increased variance in the conduction characteristics of the cardiac tissue, rather than to modification of these properties. Using a model of a homogeneous two-dimensional matrix of excitable conducting cells, we derived the relationship between the relative standard deviation (RSD) and the probability of occurrence of a local ordered dispersion of velocities shown to have the potential to result in circular propagation. This probability was found to be insignificant when the RSD was lower than 10%, but increased dramatically for RSDs greater than 10%. On the basis of experimental RSD, the calculated probability for circus movement formation is one in 10,000 normal heart beats and one in two ischemic heart beats. The model provides new insight into the mechanism of re-entrant arrhythmias as well as new tools for diagnosis.     

A model study of stability and oscillations in the myocardial cell membrane

As a step towards an improved understanding of cardiac arrhythmias caused by abnormal automaticity, we perform a stability analysis of a Hodgkin-Huxley model of the myocardial cell membrane (modified Beeler-Reuter, MBR). The bifurcation structure of the model is obtained as a function of three parameters: the intensity of an applied constant current; the potassium equilibrium potential representing the accumulation of K+ ions in the external medium; and the maximum conductance of the slow inward current mimicking the local application of catecholamines on the membrane. For a range of parameter values, the model exhibits either stable automaticity or bistability between two quiescent states or between a quiescent state and an oscillatory state. These transformations of the bifurcation structure are shown to depend on the interrelationship between three elements: the activation of the slow inward current, the region of high slope conductance of the time-independent potassium current functions, and the slow variables controlling the activation of the potassium current and the inactivation of the slow inward current. Reduced two- and three-dimensional models are shown to reproduce the main stability properties of the full MBR model and to facilitate the understanding of its dynamic behavior. The onset of instability and the oscillatory features of the MBR model are in good agreement with relevant experimental results, and possible sources of disagreement on certain points are discussed.     

High-Frequency Stimulation of Excitable Cells and Networks

High-frequency (HF) stimulation has been shown to block conduction in excitable cells including neurons and cardiac myocytes. However, the precise mechanisms underlying conduction block are unclear. Using a multi-scale method, the influence of HF stimulation is investigated in the simplified FitzhHugh-Nagumo and biophysically-detailed Hodgkin-Huxley models. In both models, HF stimulation alters the amplitude and frequency of repetitive firing in response to a constant applied current and increases the threshold to evoke a single action potential in response to a brief applied current pulse. Further, the excitable cells cannot evoke a single action potential or fire repetitively above critical values for the HF stimulation amplitude. Analytical expressions for the critical values and thresholds are determined in the FitzHugh-Nagumo model. In the Hodgkin-Huxley model, it is shown that HF stimulation alters the dynamics of ionic current gating, shifting the steady-state activation, inactivation, and time constant curves, suggesting several possible mechanisms for conduction block. Finally, we demonstrate that HF stimulation of a network of neurons reduces the electrical activity firing rate, increases network synchronization, and for a sufficiently large HF stimulation, leads to complete electrical quiescence. In this study, we demonstrate a novel approach to investigate HF stimulation in biophysically-detailed ionic models of excitable cells, demonstrate possible mechanisms for HF stimulation conduction block in neurons, and provide insight into the influence of HF stimulation on neural networks.     

Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction–diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction–diffusion system is Baer and Rinzel’s continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon “complete accommodation.”     

Exploring neuronal bistability at the depolarization block

Many neurons display bistability-coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na(+) current. Our study suggests a connection between the amount of the Na(+) window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.     

Bifurcation analysis of a Morris–Lecar neuron model

In this paper, we investigate the dynamical behaviors of a Morris–Lecar neuron model. By using bifurcation methods and numerical simulations, we examine the global structure of bifurcations of the model. Results are summarized in various two-parameter bifurcation diagrams with the stimulating current as the abscissa and the other parameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay much attention to the emergence of periodic solutions and bistability. Different membrane excitability is obtained by bifurcation analysis and frequency-current curves. The alteration of the membrane properties of the Morris–Lecar neurons is discussed.     

Simulation of ciliary beating by an excitable dynein model: oscillations, quiescence and mechano-sensitivity

Murase et al. (1989, J. theor. Biol. 139, 413) showed that an excitable dynein model can generate flagellar-like bending waves of low amplitude along an axoneme suspended in a viscous fluid. Either regular base-to-tip and irregular tip-to-base propagating waves can be produced. The present study shows that if the force-vs.-distance functions (or the potential energy functions as their integral form) that represent the functional properties of dyneins differ in the basal region, as compared with the rest of the active length of a short axoneme, and also differ between the opposing doublets, ciliary-like repetitive beats can be simulated. Depending on the parameter values, a cilium beats once and then becomes resting or quiescent, at the end of either its recovery or effective stroke. Interestingly, a quiescent cilium exhibits repetitive beats when a steady flow of water is applied to a part of the cilium in a suitable direction and at an appropriate speed. This kind of responsiveness to external stimuli, called directional mechano-sensitivity, may account for metachronal waves over a layer of cilia. As in the previous model for flagellar movement, the present model requires a passive region at the tip, but does not need a curvature feedback control, to generate ciliary-like beating patterns.     

The phases of small networks of chemical reactors and neurons

We present an experimental study of the phase relationships observed in small reactor networks consisting of two and three continuous flow stirred tank reactors. In the three-reactor network one chemical oscillator is coupled to two other reactors in parallel in analogy to a small neural net. Each reactor contains an identical reaction mixture of the excitable Belousov-Zhabotinsky reaction which is characterized by its bifurcation diagram, where the electrical current is the bifurcation parameter. Coupling between the reactors is electrical via Pt-working electrodes and it can be either repulsive (inhibitory) or attractive (excitatory). An external electrical stimulus is applied to all three reactors in the form of an asymmetric electrical current pulse which sweeps across the bifurcation diagram. As a consequence, all three reactors oscillate with characteristic oscillation patterns or remain silent in analogy to the firing of neurons. The observed phase behavior depends on the type of coupling in a complex way. This situation is analogous to the in vivo measurements on single neurons (local neurons and projection neurons) performed by G. Laurent and co-workers on the olfactory system of the locust. We propose a simple neural network similar to the reactor network using the Hodgkin-Huxley model to simulate the action potentials of the coupled single neurons. Analogies between the reactor network and the neural network are discussed.     

Mother rotor anchoring in branching tissue with heterogeneous membrane properties.

Abstract Current understanding of atrial fibrillation is based on the co-existence of multiple re-entrant waves propagating randomly throughout the tissue. However, recent experimental results indicate that in many cases one or a small number of periodic, high-frequency re-entrant sources (mother rotors) can drive the arrhythmia. Owing to the high activation rate, mother rotors seem to be located in regions of shortened action potential duration. In this study a computer model of cardiac propagation was applied to investigate mechanisms leading to the formation and maintenance of such mother rotors. For this purpose, a region of short action potential duration was generated by varying the acetylcholine concentration across the tissue. A mother rotor initiated in the center of this region drifts away, and the activation terminates. If an additional heterogeneity such as a bundle is included into the model, a further drift mechanism directed to the bundle is observed and the rotor can be stabilized. Therefore, bundle insertions may play an important role in the maintenance of mother rotors. The influence of the driving rotor on the activation pattern was studied in a three-dimensional model of rectangular shape and a monolayer model of anatomically correct atrial geometry.     

Elimination of proliferating cells unmasks the shift from senescence to quiescence caused by rapamycin

Background

Depending on cellular context, p53-inducing agents (such as nutlin-3a) cause different outcomes including reversible quiescence and irreversible senescence. Inhibition of mTOR shifts the balance from senescence to quiescence. In cell lines with incomplete responses to p53, this shift may be difficult to document because of a high proportion of proliferating cells contaminating arrested (quiescent and senescent) cells. This problem also complicates the study of senescence caused by minimal levels of p21 that are capable to arrest a few cells.

Methodology

During induction of senescence by low levels of endogenous p53 and ectopic p21, cells were co-treated with nocodazole, which eliminated proliferating cells. As a result, only senescent and quiescent cells remained.

Results and Discussion

This approach revealed that rapamycin efficiently converted nutlin-induced-senescence into quiescence. In the presence of rapamycin, nutlin-arrested MCF-7 cells retained the proliferative potential and small/lean morphology. Using this approach, we also unmasked senescence in cells arrested by low levels of ectopic p21, capable to arrest only a small proportion of HT1080-p21-9 cells. When p21 did cause arrest, mTOR caused senescent phenotype. Rapamycin and high concentrations of nutlin-3a, which inhibit the mTOR pathway in these particular cells, suppressed senescence, ensuring quiescence instead. Thus, p21 causes senescence passively, just by causing arrest, while still active mTOR drives senescent phenotype.     

Relationship between burst properties and sensitivity to input: A theoretical analysis

This paper examines the sensitivity of endogenous bursters to a brief input pulse. The interneurons of the lobster cardiac ganglion were selected as a case study.Using a mathematical model specifically developed for the neurons in the cardiac ganglion of the lobster (Av-Ron et al., 1993), we show a tight link between burst characteristics and certain other parameters. We show that cells with different burst properties differ in their sensitivity to an input of a brief pulse.Irrespective of these differences, all cells display a bimodal response to a brief pulse applied during the quiescent period. During the first three-quarters of the quiescent period, they respond by producing a single spike at most. During the remaining one-quarter, the brief pulse can initiate the cells' intrinsic burst. Our predictions fit experimental results obtained by Tazaki and Cooke (1979).The results obtained herein are discussed with respect to fault tolerance considerations.     

Continuation and bifurcation analysis of a periodically forced excitable system

The response of an excitable cell to periodic electrical stimulation is modeled using the FitzHugh-Nagumo (FHN) system submitted to a gaussian-shaped pacing, the width of which is small compared with the action potential duration. The influence of the amplitude and the period of the stimulation is studied using numerical continuation and bifurcation techniques (AUTO97 software). Results are discussed in the light of prior experimental and theoretical findings. In particular, agreement with the documented behavior of periodically stimulated cardiac cells and squid axons is discussed. As previously reported, we find many different "M:N" periodic solutions, period-doubling sequences leading to seemingly chaotic regimes, and bistability phenomena. In addition, the use of continuation techniques has allowed us to track unstable solutions of the system and thus to determine how the different stable rhythms are connected with each other in a bifurcation diagram. Depending on the stimulus amplitude, the aspect of the bifurcation diagram with the stimulus period as main varying parameter can vary from very simple to very complex. In its most developed structure, this bifurcation diagram consists of a main "tree" of period-2(P) branches, where the 1:1, 1:0, 2:2, 2:1,... rhythms are located, and of several closed loops made up of period-{N x 2(P)} branches (N>2), isolated from each other and from the main tree. It is mainly on such loops that N:1 rhythms (N>2) on one hand, and N:N-1 or Wenckebach rhythms (N>2) on the other hand, are located. Stable M:N and M:N-1 rhythms (M>or=N) can be found on the same branch of solutions. They are separated by a region of unstable solutions at small stimulus amplitudes, but this region shrinks gradually as the stimulus amplitude is raised, until it finally disappears. We believe that this property is related to the excitability characteristics of the FHN system. It would be interesting to know if it has any correspondence in the behavior of real excitable cells.     

Initiation and propagation of ectopic waves: insights from an in vitro model of ischemia-reperfusion injury

The objective of the present study was to directly visualize ectopic activity associated with ischemia-reperfusion and its progression to arrhythmia. To accomplish this goal, we employed a two-dimensional network of neonatal rat cardiomyocytes and a recently developed model of localized ischemia-reperfusion. Washout of the ischemia-like solution resulted in tachyarrhythmic episodes lasting 15-200 s. These episodes were preceded by the appearance of multiple ectopic sources and propagation of ectopic activity along the border of the former ischemic zone. The ectopic sources exhibited a slow rise in diastolic calcium, which disappeared upon return to the original pacing pattern. Border zone propagation of ectopic activity was followed by its escape into the surrounding control network, generating arrhythmias. Together, these observations suggest that upon reperfusion, a distinct layer, which consists of ectopically active, poorly coupled cells, is formed transiently over an injured area. Despite being neighbored by a conductive and excitable tissue, this transient functional layer is capable of sustaining autonomous waves and serving as a special conductive medium through which ectopic activity can propagate before spreading into the surrounding healthy tissue.     

Temperature Dependence of Bistability in Squid Giant Axons with Alkaline Intracellular pH

Raising the intracellular pH (pHi) above 7.7 in intracellularly perfused squid giant axons causes spontaneous firing of action potentials. The firing frequency ranged from 20 Hz at 0 degrees C to 200 Hz at 23 degrees C. Above 23 degrees C, the axons were quiescent. They were bistable for 13 相似文献   

Use of Oxprenolol in Cardiac Arrhythmias Associated with Acute Myocardial Ischaemia

Oxprenolol, a new beta-receptor blocking drug with intrinsic sympathomimetic activity, was used to treat 63 episodes of cardiac arrhythmia occurring in 43 patients with acute myocardial infarction or myocardial ischaemia. The drug was most effective in abolishing ventricular ectopic beats and supraventricular tachycardia. The best method of administration was by continuous intravenous infusion and the most satisfactory bolus does was 6 mg. The main side effect was hypotension, which occurred in 59% of episodes of arrhythmia that had responded previously to intravenous administration. Oxprenolol was often effective in lignocaine-resistant arrhythmia. The two main advantages of oxprenolol over propranolol are the reduced likelihood of adversely affecting myocardial function and the diminished tendency to produce bronchospasm.     

Subharmonic resonance and chaos in forced excitable systems

 Forced excitable systems arise in a number of biological and physiological applications and have been studied analytically and computationally by numerous authors. Existence and stability of harmonic and subharmonic solutions of a forced piecewise-linear Fitzhugh-Nagumo-like system were studied in Othmer ad Watanabe (1994) and in Xie et al. (1996). The results of those papers were for small and moderate amplitude forcing. In this paper we study the existence of subharmonic solutions of this system under large-amplitude forcing. As in the case of intermediate-amplitude forcing, bistability between 1 : 1 and 2 : 1 solutions is possible for some parameters. In the case of large-amplitude forcing, bistability between 2 : 2 and 2 : 1 solutions, which does not occur in the case of intermediate-amplitude forcing, is also possible for some parameters. We identify several new canonical return maps for a singular system, and we show that chaotic dynamics can occur in some regions of parameter space. We also prove that there is a direct transition from 2 : 2 phase-locking to chaos after the first period-doubling bifurcation, rather than via the infinite sequence of period doublings seen in a smooth quadratic interval map. Coexistence of chaotic dynamics and stable phase-locking can also occur. Received: 6 July 1998 / Revised version: 2 October 1998     

Ionic channel density of excitable membranes can act as a bifurcation parameter

As the maximal K⁺-conductance (or K⁺-channel density) of the Hodgkin-Huxley equations is reduced, the stable resting membrane potential bifurcates at a subcritical Hopf bifurcation into small amplitude unstable oscillations. These small amplitude solutions jump to large amplitude periodic solutions that correspond to a repetitive discharge of action potentials. Thus the specific channel density can act as a bifurcation parameter, and can control the excitability and autorhythmicity of excitable membranes.