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.     

Evidence for functional sodium and calcium ion channels in the membrane of cultured cardiomyocytes of the adult rat

Characteristics are reported for electrical activity of adult rat cardiomyocytes in long-term primary culture. Cells in vitro for 12 to 28 days have mean membrane potential of -53 mV, are electrically excitable, and some are spontaneously contractile. The action potential of these cells has a slow rate of depolarization and is abolished by methoxyverapamil (D-600) but not by tetrodotoxin (TTX). When cells are hyperpolarized by passage of an inward current, spontaneous action potentials cease and action potentials evoked by depolarizing pulses are then TTX sensitive. Fetal bovine serum is a constituent of the culture medium. Its temporary removal causes spontaneous contractility to cease but the cells remain electrically excitable.     

Intermittency in the Hodgkin-Huxley model

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.     

Augmentation and suppression of action potentials by estradiol in the myometrium of pregnant rat.

The purpose of this study was to investigate the actions of estradiol on spontaneous and evoked action potentials in the isolated longitudinal smooth muscle cells of the pregnant rat. Single cells were obtained by enzymatic digestion from pregnant rat longitudinal myometrium. Action potentials and currents were recorded by whole-cell current-clamp and voltage-clamp methods, respectively. The acute effects of 17beta-estradiol on action potentials and inward and outward currents were investigated. The following results were obtained. The average resting membrane potential of single myometrial cells was -54 mV (n = 40). In many cells, an electrical stimulation evoked a membrane depolarization, and action potentials were superimposed on the depolarization. In some cells, spontaneous action potentials were observed. Estradiol (30 microM) slightly depolarized the membrane (ca. 5 mV) and attenuated the generation of action potentials by reducing the frequency and amplitude of the spikes. Afterhyperpolarization was also attenuated by estradiol (30 microM). On the other hand, in 5 of 35 cells, estradiol increased the first spike amplitude and action potential duration, while frequency of the spike generation and afterhyperpolarization were inhibited. In voltage-clamped muscle cells, estradiol inhibited both inward and outward currents. Acute inhibition or augmentation of spike generation by estradiol is due to the balance of inhibition of inward and outward currents. Inhibition of both currents also prevented afterhyperpolarization, causing potential-dependent block of Ca spikes.     

The assembly of ionic currents in a thalamic neuron. II. The stability and state diagrams

In the previous model of a thalamic neuron (R.M. Rose & J.L. Hindmarsh, Proc. R. Soc. Lond. B237, 267-288 (1989], which we referred to as the z-model, the burst response was terminated by the slow activation of a subthreshold outward current. In this paper we show that similar results can be obtained if the burst response is terminated by slow inactivation of the subthreshold inward current, Isa. We illustrate the use of this new model, which we refer to as the ha-model, by using it to explain the response of a thalamic neuron to a double ramp current. The main aim of the paper is to show how the stability and state diagrams introduced previously can be used to explain various types of firing pattern of thalamic and other neurons. We show that increasing the threshold for the fast action potentials leads to low threshold spikes of increased amplitude. Also, addition of a second subthreshold inward current adds a new stability region, which enables us to explain the origin of plateau potentials. In addition, various types of subthreshold oscillation are produced by relocating a previously stable equilibrium point in an unstable region. Finally, we predict a sequence of responses to current steps from different levels of background current that extends the burst, rest, tonic sequence of thalamic neurons. The stability and state diagrams therefore provide us with a useful way of explaining further properties of thalamic neurons and appear to have further applications to other mammalian neurons.     

Studies on re-entrant arrhythmias and ectopic beats in excitable tissues by bifurcation analyses.

A phase-plane bifurcation analysis is a useful way to theoretically understand how various types of arrhythmias may arise from excitable tissues. In this paper, we have performed phase-plane bifurcation analysis to characterize arrhythmogenic states in excitable tissues. To achieve this, we have first formulated a model which is simple enough to be mathematically tractable, yet captures the non-linear features of cardiac excitation and conduction. In this model, single cells are connected in a circular fashion by gap conductances. Each cell carries the following two types of currents: a passive outward current and an inward "excitable" current which contains an activation and an inactivation gate. The activation gate is responsible for the upstroke of action potential and inactivation gate is responsible for the termination of the plateau potential. With this model, we have constructed bifurcation diagrams as a function of a bifurcation parameter. The parameter chosen as the bifurcation parameter has the property of raising maximum diastolic potential while shorting the refractory period. Our analysis revealed the existence of three distinct multi-stable phases in certain ranges of the bifurcation parameter: (1) bistability between a rotor and a quiescent state, (2) bistability between rotor and ectopic beats, and (3) three stable states co-existing among quiescent state, rotor, and ectopic beats. In these three regions, external impulses exert very distinct effects: In region 1, a brief current pulse can annihilate a re-entrant arrhythmia to quiescence. To initiate re-entry from a quiescent tissue, however, it takes two pulses (a primary pulse followed by a premature pulse at a site different from the "primary" site). In region 2, a brief pulse can convert a re-entrant arrhythmia to ectopic beats. To convert the ectopic beats back to circus movement, these beats have to be suppressed by a few brief current pulses to initiate one-way propagation. Depending on the frequency and strength of impulses in region 3, the tissue can switch back and forth among quiescence, circus movement, and ectopic beats. For comparison, we have also included a more complete Beeler-Reuter cardiac cell model in our analysis and obtained essentially the same results. From the behavioral similarities of these models, we conclude that re-entrant and ectopic arrhythmias must be intrinsic properties of excitable tissues and external stimuli can convert one mode of arrhythmia to another in the multistability regions.(ABSTRACT TRUNCATED AT 400 WORDS)     

Axonal contribution to subthreshold currents inaplysia bursting pacemaker neurons

The contribution of axonal activity to the ionic currents which generate bursting pacemaker activity was studied by using the two-electrode voltage-clamp technique in Aplysia bursting neuron somata in conjunction with intraaxonal voltage recordings. Depolarizing voltage-clamp pulses applied to bursting cell somata triggered axonal action potentials. The voltage-clamp current recording exhibited transient inward current "notches" corresponding to each of the axonal spikes. The addition of 50 microM tetrodotoxin (TTX) to the bathing medium blocked the fast axonal spikes and current notches, revealing a slower axonal spike which was blocked by the replacement of external Ca2+ with Co2+. The inward current evoked by applying a depolarizing voltage-clamp pulse in the soma is distorted by the occurrence of the axonal Ca2+ spike. Elimination of the axonal spike, by injecting hyperpolarizing current into the axon, changes both the time course and the magnitude of the inward current. The axonal Ca2+ spikes are followed by a series of Ca2+-dependent afterpotentials: a rapid postspike hyperpolarization, a depolarizing afterpotential (DAP) and, finally, a long-lasting postburst hyperpolarization. The long-lasting hyperpolarization is not blocked by 50 mM external tetraethyl ammonium, an effective blocker of Ca2+-activated K+ current [IK(Ca)], and does not appear to reverse at EK. Hence, the axonal long-lasting hyperpolarization may not be due to IK(Ca). Somatic voltage-clamp pulses in bursting neurons are followed by a slow inward tail current, which is sometimes coincident with a DAP in the axon. In some cells, the amplitude of the slow inward tail current is greatly reduced if axonal spikes and DAPs are prevented by hyperpolarization of the axon, while, in other cells, elimination of axonal activity has little effect. Therefore, the slow inward tail current is not necessarily an artifact of poor voltage-clamp control over the axonal membrane potential but probably results from the activation of an ionic conductance mechanism located partly in the axon and partly in the soma.     

Ghostbursting: A Novel Neuronal Burst Mechanism

Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observed to produce high-frequency burst discharge with constant depolarizing current (Turner et al., 1994). We present a two-compartment model of an ELL pyramidal cell that produces burst discharges similar to those seen in experiments. The burst mechanism involves a slowly changing interaction between the somatic and dendritic action potentials. Burst termination occurs when the trajectory of the system is reinjected in phase space near the ghost of a saddle-node bifurcation of fixed points. The burst trajectory reinjection is studied using quasi-static bifurcation theory, that shows a period doubling transition in the fast subsystem as the cause of burst termination. As the applied depolarization is increased, the model exhibits first resting, then tonic firing, and finally chaotic bursting behavior, in contrast with many other burst models. The transition between tonic firing and burst firing is due to a saddle-node bifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type I intermittency, and we present experimental analysis of ELL pyramidal cell burst trains that support this model prediction. By varying parameters in a way that changes the positions of both saddle-node bifurcations in parameter space, we produce a wide gallery of burst patterns, which span a significant range of burst time scales.     

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.     

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.     

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.”     

Biophysical basis for three distinct dynamical mechanisms of action potential initiation

Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.     

Dependence of pituitary hormone secretion on the pattern of spontaneous voltage-gated calcium influx. Cell type-specific action potential secretion coupling.

In excitable cells, voltage-gated calcium influx provides an effective mechanism for the activation of exocytosis. In this study, we demonstrate that although rat anterior pituitary lactotrophs, somatotrophs, and gonadotrophs exhibited spontaneous and extracellular calcium-dependent electrical activity, voltage-gated calcium influx triggered secretion only in lactotrophs and somatotrophs. The lack of action potential-driven secretion in gonadotrophs was not due to the proportion of spontaneously firing cells or spike frequency. Gonadotrophs exhibited calcium signals during prolonged depolarization comparable with signals observed in somatotrophs and lactotrophs. The secretory vesicles in all three cell types also had a similar sensitivity to voltage-gated calcium influx. However, the pattern of action potential calcium influx differed among three cell types. Spontaneous activity in gonadotrophs was characterized by high amplitude, sharp spikes that had a limited capacity to promote calcium influx, whereas lactotrophs and somatotrophs fired plateau-bursting action potentials that generated high amplitude calcium signals. Furthermore, a shift in the pattern of firing from sharp spikes to plateau-like spikes in gonadotrophs triggered luteinizing hormone secretion. These results indicate that the cell type-specific action potential secretion coupling in pituitary cells is determined by the capacity of their plasma membrane oscillator to generate threshold calcium signals.     

Local and global bifurcations at infinity in models of glycolytic oscillations

 We investigate two models of glycolytic oscillations. Each model consists of two coupled nonlinear ordinary differential equations. Both models are found to have a saddle point at infinity and to exhibit a saddle-node bifurcation at infinity, giving rise to a second saddle and a stable node at infinity. Depending on model parameters, a stable limit cycle may blow up to infinite period and amplitude and disappear in the bifurcation, and after the bifurcation, the stable node at infinity then attracts all trajectories. Alternatively, the stable node at infinity may coexist with either a stable sink (not at infinity) or a stable limit cycle. This limit cycle may then disappear in a heteroclinic bifurcation at infinity in which the unstable manifold from one saddle at infinity joins the stable manifold of the other saddle at infinity. These results explain prior reports for one of the models concerning parameter values for which the system does not admit any physical (bounded) behavior. Analytic results on the scaling of amplitude and period close to the bifurcations are obtained and confirmed by numerical computations. Finally, we consider more realistic modified models where all solutions are bounded and show that some of the features stemming from the bifurcations at infinity are still present. Received 4 September 1995; received in revised form 18 September 1996     

Effects of tetraethylammonium and 4-aminopyridine on the plateau potential of circular myometrium from the pregnant rat

In the pregnant rat, spontaneous electrical activity of circular muscle (CM) changes from single, plateau-type action potentials at early and mid-term to repetitive spike trains at term. To examine mechanisms underlying the plateau, we studied the effects of potassium channel blockers tetraethylammonium (TEA) and 4-aminopyridine (4-AP) on membrane potentials in CM from rats on gestation Days 14, 15, 16, 21 (term). Apparent membrane conductance was measured at rest and during the plateau in Day 14 muscles with and without TEA. 4-AP depolarized the resting membrane on all gestation days. Therefore, a direct action of 4-AP on plateau configuration could not be separated from an indirect effect of depolarization. TEA did not affect the resting potential but increased action potential size and depolarization rate on all gestation days. On Day 16, TEA reduced plateau amplitude, unmasking small, repetitive depolarizations. D-600 decreased plateau amplitude and duration and attenuated these effects of TEA. Plateau conductance increased initially then decreased before membrane repolarization. Membrane conductance and outward rectification during the plateau were reduced by TEA. The plateau potential may result from an outwardly rectifying TEA-sensitive current combined with a slow inward current, the plateau magnitude being determined by the relative intensity of each current.     

Plasmalemmal,voltage-dependent ionic currents from excitable pulvinar motor cells ofMimosa pudica

Summary Plasmalemmal ionic currents from excitable motor cells of the primary pulvinus ofMimosa pudica were investigated by patch-clamp techniques. In almost all of the enzymatically isolated protoplasts, a delayed rectifier potassium current was activated by depolarization, while no currents were detected upon hyperpolarization. This sustained outward current was reversibly blocked by Ba and TEA and serves to repolarize the membrane potential. Outward single channel currents that very likely underly the macroscopic outward potassium current had an elementary conductance of 20 pS. In addition, in a few protoplasts held at hyperpolarized potentials, depolarization-activated transient inward currents were observed, and under current clamp, action potential-like responses were triggered by depolarizing current injections or by mechanical perturbations. The activation characteristics of both inward currents and spikes showed striking similarities compared to those of action potentialsin situ.     

Dynamical mechanisms of pacemaker generation in IK1-downregulated human ventricular myocytes: insights from bifurcation analyses of a mathematical model

Dynamical mechanisms of the biological pacemaker (BP) generation in human ventricular myocytes were investigated by bifurcation analyses of a mathematical model. Equilibrium points (EPs), periodic orbits, stability of EPs, and bifurcation points were determined as functions of bifurcation parameters, such as the maximum conductance of inward-rectifier K+ current (I(K1)), for constructing bifurcation diagrams. Stable limit cycles (BP activity) abruptly appeared around an unstable EP via a saddle-node bifurcation when I(K1) was suppressed by 84.6%. After the bifurcation at which a stable EP disappears, the I(K1)-reduced system has an unstable EP only, which is essentially important for stable pacemaking. To elucidate how individual sarcolemmal currents contribute to EP instability and BP generation, we further explored the bifurcation structures of the system during changes in L-type Ca2+ channel current (I(Ca,L)), delayed-rectifier K+ currents (I(K)), or Na(+)/Ca2+ exchanger current (I(NaCa)). Our results suggest that 1), I(Ca,L) is, but I(K) or I(NaCa) is not, responsible for EP instability as a requisite to stable BP generation; 2), I(K) is indispensable for robust pacemaking with large amplitude, high upstroke velocity, and stable frequency; and 3), I(NaCa) is the dominant pacemaker current but is not necessarily required for the generation of spontaneous oscillations.     

Calcium dependence of electrical and mechanical activity in rat ileum examined by the sucrose-gap technique

1. Single sucrose gap recordings showed that spontaneous action potentials of rat ileal smooth muscle consisted of slow waves and superimposed spikes which generated rhythmic contractions. As external potassium was raised, the resting potential progressively depolarized.2. Calcium-free salines inhibited spontaneous mechanical activity and inhibited the plateau phase of the action potential, but spontaneous spike depolarizations persisted.3. Verapamil, nifedipine and diltiazem all inhibited spontaneous mechanical activity and the plateau phase of the action potential, while in addition diltiazem augmented spike amplitude.4. Mn ions also inhibited mechanical activity and the action potential plateau, without affecting spike activity while the calcium ionophore A23187 enhanced both mechanical and electrical activity with a pronounced effect on spike amplitude.5. These results are consistent with the view that the plateau phase of the ileal smooth muscle action potential is dependent upon an influx of extracellular calcium possibly through voltage dependent slow calcium channels.     

Various forms of potentials recorded from caudate nucleus neurons

Spontaneous and evoked activity of caudate nucleus neurons was recorded extracellularly in acute experiments on cats. Different forms of potentials were found by analysis of the results. The potentials recorded belong to three types: ordinary action potentials; prepotentials or incomplete spikes differing from ordinary action potentials in their lower amplitude and slower decline, and complex discharges in which a spike of somewhat reduced amplitude is followed by a slow positive-negative wave. In the spontaneous activity prepotentials were observed both in complete action potentials and in isolation. The frequency of the complex discharges was 0.5–1 per second. The slow wave of these discharges blocked prepotential and action potential formation. The origin of these forms of potentials in neurons of the caudate nucleus is discussed and they are compared with analogous forms of potentials described for the Purkinje cells of the cerebellum.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukranian SSR, Kiev. Translated from Neirofiziologiya, Vol. 9, No. 2, pp. 149–156, March–April, 1977.