Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

A marriage of convenience: beta-subunits and voltage-dependent K+ channels

The movement of ions across cell membranes is essential for a wide variety of fundamental physiological processes, including secretion, muscle contraction, and neuronal excitation. This movement is possible because of the presence in the cell membrane of a class of integral membrane proteins dubbed ion channels. Ion channels, thanks to the presence of aqueous pores in their structure, catalyze the passage of ions across the otherwise ion-impermeable lipid bilayer. Ion conduction across ion channels is highly regulated, and in the case of voltage-dependent K(+) channels, the molecular foundations of the voltage-dependent conformational changes leading to the their open (conducting) configuration have provided most of the driving force for research in ion channel biophysics since the pioneering work of Hodgkin and Huxley (Hodgkin, A. L., and Huxley, A. F. (1952) J. Physiol. 117, 500-544). The voltage-dependent K(+) channels are the prototypical voltage-gated channels and govern the resting membrane potential. They are responsible for returning the membrane potential to its resting state at the termination of each action potential in excitable membranes. The pore-forming subunits (alpha) of many voltage-dependent K(+) channels and modulatory beta-subunits exist in the membrane as one component of macromolecular complexes, able to integrate a myriad of cellular signals that regulate ion channel behavior. In this review, we have focused on the modulatory effects of beta-subunits on the voltage-dependent K(+) (Kv) channel and on the large conductance Ca(2+)- and voltage-dependent (BK(Ca)) channel.     

Axonal excitability revisited

The original papers of Hodgkin and Huxley (J. Physiol. 116 (1952a) 449, J. Physiol. 116 (1952b) 473, J. Physiol. 116 (1952c) 497, J. Physiol. 117 (1952d) 500) have provided a benchmark in our understanding of cellular excitability. Not surprisingly, their model of the membrane action potential (AP) requires revisions even for the squid giant axon, the preparation for which it was originally formulated. The mechanisms they proposed for the voltage-gated potassium and sodium ion currents, IK, and INa, respectively, have been superceded by more recent formulations that more accurately describe voltage-clamp measurements of these components. Moreover, the current-voltage relation for IK has a non-linear dependence upon driving force that is well described by the Goldman-Hodgkin-Katz (GHK) relation, rather than the linear dependence on driving force found by Hodgkin and Huxley. Furthermore, accumulation of potassium ions in the extracellular space adjacent to the axolemma appears to be significant even during a single AP. This paper describes the influence of these various modifications in their model on the mathematically reconstructed AP. The GHK and K+ accumulation results alter the shape of the AP, whereas the modifications in IK and INa gating have surprisingly little effect. Perhaps the most significant change in their model concerns the amplitude of INa, which they appear to have overestimated by a factor of two. This modification together with the GHK and the K+ accumulation results largely remove the discrepancies between membrane excitability of the squid giant axon and the Hodgkin and Huxley (J. Physiol. 117 (1952d) 500) model previously described (Clay, J. Neurophysiol. 80 (1998) 903).     

Bifurcation of the Hodgkin and Huxley equations: A new twist

The Hodgkin and Huxley equations model action potentials in squid giant axons. Variants of these equations are used in most models for electrial activity of excitable membranes. Computational tools based upon the theory of nonlinear dynamical systems are used here to illustrate how the dynamical behavior of the Hodgkin Huxley model changes as functions of two of the system parameters.     

Computer simulations of the effect of non-inactivating sodium channels on the electric behavior of excitable cells

Non-inactivating sodium channels have been discovered in various cell types. Additionally, normal voltage-gated sodium channels can be induced to lose their ability to inactivate by treatment with proteolytic enzymes, with certain chemical reagents, or with toxins. The presence of non-inactivating sodium channels in the outer membrane of a cell is expected to profoundly modify the electrical properties of the cell, because the electrical depolarization of the cell and the opening of these channels reciprocally reinforce each other without intrinsic control. The normal resting state may thus be destabilized and a new resting state at depolarized resting potentials may become possible. In this study, computer simulations were carried out to systematically explore the patterns of behavior of excitable cells which have non-inactivating sodium channels in their plasma membrane. The cells were assumed to be space clamped and the relevant Hodgkin and Huxley equations were assumed to describe the electrical behavior of the cells, except that some or all of the sodium channels could not inactivate. The sodium currents were thus represented by the sum of two terms: FI.gNa.m3.h.(V-ENa) + (1-FI).gNa.m3(V-ENa), where FI(0 less than or equal to FI less than or equal to 1) is the fraction of sodium channels which inactivate normally, and the other symbols have their usual significance. The behavior of non-inactivating sodium channels created by pronase treatment or reaction with chemical reagents was found to conform with that predicted by the second term in this expression. The simulations thus quantitatively apply to excitable cells thus treated, but may serve additionally to qualitatively illustrate patterns of electrical activity induced by non-inactivating sodium channels also in other cases. A variety of possible types of electrical behavior was obtained: Normal behavior, including capability of firing action potentials, requires values of FI which are not far from unity, the permissible range depending on the fully activated potassium ion conductance, gK. Bistability, at which the cell may exist in one of two stable states of different resting potential, occurs when the value of FI is lowered. Transitions from the polarized to the depolarized resting states, and vice versa, may be brought about by depolarizing and hyperpolarizing triggers, respectively. Such behavior is like that of memory storage devices. Monostability at depolarized potentials is favored by low FI values and can occur if gK is less than the Hodgkin and Huxley value.(ABSTRACT TRUNCATED AT 400 WORDS)     

Time course of TEA(+)-induced anomalous rectification in squid giant axons

Changes in the voltage clamp currents of squid giant axons wrought by low axoplasmic TEA+ (tetraethylammonium chloride) concentrations (0.3 mM and above) are described. They are: (a) For positive steps from the resting potential in sea water, the K+ current increases, decreases, then increases, instead of increasing monotonically. (b) For positive steps from the resting potential in 440 mM external K+, the current has an exponentially decaying component, whose decay rate increases with axoplasmic [TEA+]. The control currents increase monotonically. (c) For negative steps from the resting potential in 440 mM external K+, the current record has a peak followed by a decay that is slow relative to the control. The control record decreases monotonically. Qualitatively these findings can be described by a simple kinetic model, from which, with one assumption, it is possible to calculate the rate at which K+ ions move through the K+ channels. An interesting conclusion from (c) is that the channels cannot be closed by the normal voltage-sensitive mechanism (described by Hodgkin and Huxley) until they are free of TEA+.     

Das Membranpotenzial. Biophysik im Experiment

The membrane potential Every living cell is surrounded by a cell membrane separating the cell's contents from the environment. It enables the generation of an electrical potential difference, the membrane potential or membrane voltage. Simple experiments using model membranes (ion exchanger membranes) help to understand the characteristics and the generation of the membrane potential, as well as analysis of the data by the theoretical equations derived by Nernst, Goldman, Hodgkin and Huxley.     

Membrane potentials and resistances of giant mitochondria. Metabolic dependence and the effects of valinomycin

The membrane potentials and resistances of giant mitochondria from mice fed cuprizone have been studied. They were found to correspond approx. 10-20 mV, positive inside, and 2 M omega, respectively. These properties were found to be independent of the metabolic state. The microelectrodes were in the inner mitochondrial space since (a) the potentials in the presence of valinomycin depended on the K+ concentration of the medium and magnitude of the K+ diffusion potentials was consistent with the presence of a high internal concentration of K+, (b) almost identical results were obtained with mitochondria from which the external membrane had been removed and the cristae were evaginated, and (c) punch-through experiments, in which the microelectrodes were advanced until they emerged through the other side of the mitochondria, showed an identical membrane potential both in the presence and in the absence of valinomycin. The potentials were stable under a variety of conditions and showed no sign of decay of membrane leakiness. Detailed evidence that the impaled mitochondria are metabolically viable will be presented in a separate publication.     

Relationship between thermodynamic driving force and one-way fluxes in reversible processes

Chemical reaction systems operating in nonequilibrium open-system states arise in a great number of contexts, including the study of living organisms, in which chemical reactions, in general, are far from equilibrium. Here we introduce a theorem that relates forward and reverse fluxes and free energy for any chemical process operating in a steady state. This relationship, which is a generalization of equilibrium conditions to the case of a chemical process occurring in a nonequilibrium steady state in dilute solution, provides a novel equivalent definition for chemical reaction free energy. In addition, it is shown that previously unrelated theories introduced by Ussing and Hodgkin and Huxley for transport of ions across membranes, Hill for catalytic cycle fluxes, and Crooks for entropy production in microscopically reversible systems, are united in a common framework based on this relationship.     

Quantitative description of the slowly inactivated barium current through the somatic membrane of molluscan neurons

The potential-dependent slowly inactivated barium current through the somatic membrane of identified neurons of the molluskHelix pomatia was investigated under voltage clamp conditions. The experimental data were analyzed by the equations of Hodgkin and Huxley. Steady-state values of the inactivation variable of this current were shown to be a function, described approximately by an analytical equation, of membrane potential. The slowly inactivated barium current was shown to be proportional to the third power of the activation variable. The dependence of the time constant of activation of this current on membrane potentials was investigated.     

Parameter estimation in Hodgkin-Huxley-Type equations for membrane action potentials in nerve and heart muscle

A method is presented which renders parameter estimation possible in systems of non-linear differential equations where normally no solution exists in terms of analytic functions and which have to be solved numerically. The method uses the concept of sensitivity equations. Two examples are given, taking mathematical models for membrane action potentials in nerve and heart muscle by Hodgkin and Huxley and by Beeler and Reuter. The model equations together with the corresponding system of sensitivity equations are given, which are necessary to estimate maximum conductivity coefficients defining the interactions of different ionic current components. A computer program is described and results of action potential numerical analysis are presented using simulated data. It can be seen, that even with superimposed simulated noise the real parameter values are estimated in an excellent manner. The method can be used to interpret observed changes in action potential time courses under physiological and pharmacological conditions.     

A minimal biophysical model for an excitable and oscillatory neuron

Presented here is a minimal biophysical cell model, based on work by Hodgkin and Huxley and by Rinzel, that can exhibit both excitable and oscillatory behavior. Two versions of the model are studied, which conform to data for squid and lobster giant axons.     

Na channel kinetics during the spontaneous heart beat in embryonic chick ventricle cells.

Using cell-attached and whole-cell recording techniques simultaneously allows the measurement of Na currents during action potentials in beating heart cells. In 7-d chick ventricle, the dynamic reversal potential for Na is 30 mV, which is 20 mV less than the reversal potential in nonbeating cells. This result implies that the spontaneous activity of heart cells raises the Na concentration at the internal face of the membrane to nearly 40 mM. Fitting the Na action currents to the Hodgkin and Huxley equations shows that patches may contain from 50 to 700 Na channels, with an average density of 23 +/- 21 per micron2. Only approximately 2% of the available Na channels are open at the peak of the Na action current. This low probability is a consequence of the channels' continual inactivation during the diastolic depolarization phase.     

Built for speed

Many of us were taught in high school biology that the action potential waveform in nerves and other excitable tissues was generated by an initial rapid influx of external Na⁺ ions across the plasma membrane, followed by an outward movement of intracellular K⁺ ions. The former event, mediated by voltage-gated Na⁺ channels, is responsible for the fast depolarizing upstroke of the action potential, while voltage-gated K+ channels are responsible for the subsequent repolarizing phase, which largely controls action potential duration. Although Hodgkin and Huxley described the fundamental importance of this sequential activation process more than 60 y ago, the molecular and structural details underlying the faster activation of voltage-gated Na⁺ (Nav) vs. K⁺ (Kv) channels have yet to be fully resolved.     

Contributions of secondary active transport processes to membrane potentials

Summary Equations are developed to examine the effects of secondary active transport processes on the steady-state membrane potential of symmetrical cells. It is shown that, with suitable modifications, equations of the type developed by Goldman, Hodgkin and Katz may be derived to accommodate the contributions to the membrane potential of both electroneutral and electrogenic transporters. Where the membrane potential is function of the dominant medium ions (Na, K, and Cl), other contributions can come only from an electrogenic Na pump and from neutral co- and counter-transporters if, and only if, these involve the dominant ions. Experimental approaches to measure the parameters necessary to solve the equations developed here are discussed.     

电压依赖性离子通道门控的分子机制

５０年代Ｈｏｄｇｋｉｎ和Ｈｕｘｌｅｙ双通道模型及其激活与失活学说,正逐步被８０年代以来的分子生物学和电生理学研究所证实。Ｎａ＾＋、Ｋ＾＋离子通道的激活主要决定于高度保守的带正电荷氨基酸残基密集的Ｓ４段,由膜内向膜外方向的拧改锥样旋转。Ｎａ＾＋通道的失活主要与其Ⅲ－Ⅳ功能区之间的胞内连结襻的“铰链盖”样运动有关;Ｋ＾＋通的失活分Ｎ－、Ｃ－、Ｐ－三型,分别发生在Ｎ－、Ｃ－末端和Ｐ区,其Ｎ型失活与Ｎ－末     

Effects of rectification on synaptic efficacy.

We have investigated the effects of postsynaptic membrane properties on the shape of synaptic potentials generated by time-varying synaptic conductances. We used numerical simulation techniques to model cells of several different geometrical forms, from an isopotential sphere to a neuron with a soma and a dendritic tree. A variety of postsynaptic membrane properties were tested: (a) a passive resistance-capacitance membrane, (b) a membrane represented by the Hodgkin and Huxley (HH) equations, and (c) a membrane that was passive except for a delayed rectification represented by a voltage- and time-dependent increase in GK. In all cases we investigated the effects of these postsynaptic membrane properties on synaptic potentials produced by synaptic conductances that were fast or slow compared with the membrane time constant. In all cases the effects of postsynaptic rectification occurred on postsynaptic potentials of amplitudes as low as 1 mV. The HH model (compared with the passive model) produced an increased peak amplitude (from the increase in GNa) but a decreased half-width and a decreased time integral (from the increase in GK). These effects of the HH GK change were duplicated by a simple analytical rectifier model.     

Mathematical modelling of intra- and extracellular potentials generated by active structures: effects of a step change in structure diameter

A mathematical model developed in our laboratory is used to estimate and analyse extracellular potentials generated in a volume conductor by a geometrically inhomogeneous structure with a step increase or a step decrease in its diameter. The transmembrane potentials were calculated using the model of Hodgkin and Huxley (1952) and the method of Joyner et al. (1978). Variations in waveforms of the transmembrane and extracellular potentials were described and discussed. Differences in waveforms of the extracellular potentials and in declines of their components are due to changes in the source which generates these potentials. In case of a propagation block the peak-to-peak amplitude of the extracellular potentials calculated over the area of the block may be higher than that over the area of propagation of action potentials. The possible applications of the results to the analysis of extracellular potentials recorded around actual motoneurons during their orthodromic or antidromic activation are discussed.     

Noise-induced spatiotemporal patterns in Hodgkin–Huxley neuronal network

The effect of noise on the pattern selection in a regular network of Hodgkin–Huxley neurons is investigated, and the transition of pattern in the network is measured from subexcitable to excitable media. Extensive numerical results confirm that kinds of travelling wave such as spiral wave, circle wave and target wave could be developed and kept alive in the subexcitable network due to the noise. In the case of excitable media under noise, the developed spiral wave and target wave could coexist and new target-like wave is induced near to the border of media. The averaged membrane potentials over all neurons in the network are calculated to detect the periodicity of the time series and the generated traveling wave. Furthermore, the firing probabilities of neurons in networks are also calculated to analyze the collective behavior of networks.     

Characterization of the inward-rectifying potassium current in cat ventricular myocytes

Whole-cell membrane currents were measured in isolated cat ventricular myocytes using a suction-electrode voltage-clamp technique. An inward-rectifying current was identified that exhibited a time-dependent activation. The peak current appeared to have a linear voltage dependence at membrane potentials negative to the reversal potential. Inward current was sensitive to K channel blockers. In addition, varying the extracellular K+ concentration caused changes in the reversal potential and slope conductance expected for a K+ current. The voltage dependence of the chord conductance exhibited a sigmoidal relationship, increasing at more negative membrane potentials. Increasing the extracellular K+ concentration increased the maximal level of conductance and caused a shift in the relationship that was directly proportional to the change in reversal potential. Activation of the current followed a monoexponential time course, and the time constant of activation exhibited a monoexponential dependence on membrane potential. Increasing the extracellular K+ concentration caused a shift of this relationship that was directly proportional to the change in reversal potential. Inactivation of inward current became evident at more negative potentials, resulting in a negative slope region of the steady state current-voltage relationship between -140 and -180 mV. Steady state inactivation exhibited a sigmoidal voltage dependence, and recovery from inactivation followed a monoexponential time course. Removing extracellular Na+ caused a decrease in the slope of the steady state current-voltage relationship at potentials negative to -140 mV, as well as a decrease of the conductance of inward current. It was concluded that this current was IK1, the inward-rectifying K+ current found in multicellular cardiac preparations. The K+ and voltage sensitivity of IK1 activation resembled that found for the inward-rectifying K+ currents in frog skeletal muscle and various egg cell preparations. Inactivation of IK1 in isolated ventricular myocytes was viewed as being the result of two processes: the first involves a voltage-dependent change in conductance; the second involves depletion of K+ from extracellular spaces. The voltage-dependent component of inactivation was associated with the presence of extracellular Na+.