Rate of excitation propagation in a reduced Hodgkins-Huxley model. II. Slow relaxation of the sodium current]

The influence of sodium current activation on the value of nerve excitation conduction velocity is investigated on the basis of Hodgkin-Huxley model. The potassium activation and sodium inactivation are considered as slow processes which do not develop to an appreciable extent in the region of conduction velocity formation. The system of equations was derived and solved analytically after neglecting the dependency of sodium relaxation time on potential; the approximation of steady-state sodium activation was also used with the help of Hevyside function. The algebraic equation for conduction velocity was obtained; its solution has a simple analytical form in two limits of rapid and slow sodium current relaxation. The comparison with the experimental data has shown that at not very high temperatures the slow (compared to the potential dynamics) sodium current relaxation approximation is more appropriate. The dependency of impulse velocity on capacitance and conductance of the fiber was analyzed.     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. I. Rapid relaxation of the sodium current]

The question of calculating excitation propagation velocity is analyzed on the basis of the Hodgkin-Huxley model. The activation of the sodium current is assumed to be rapid as compared to the rate of potential variation. Because of slow variation of potassium activation and sodium inactivation the dynamics of these processes is assumed to be of negligible effect in the region of impulse velocity formation. By means of pieace-wise linear approximation of thus obtained voltage-current characteristics the characteristics the analytical solution of the problem was found. In two limiting cases this solution coincides with the solutions of Kolmogorov and Scott. The dependence of impulse velocity on parameters is analyzed and illustrated graphically.     

Interaction of deoxycholate with the sodium channel of squid axon membranes

Deoxycholate can react with sodium channels with a high potency. The apparent dissociation constant for the saturable binding reaction is 2 microM at 8 degrees C, and the heat of reaction is approximately -7 kcal/mol. Four independent test with Na-free media, K-free media, tetrodotoxin, and pancuronium unequivocally indicate that it is the sodium channel that is affected by deoxycholate. Upon depolarization of the membrane, the drug modified channel exhibits a slowly activating and noninactivating sodium conductance. The kinetic pattern of the modified channel was studied by increasing deoxycholate concentration, lowering the temperature, chemical elimination of sodium inactivation, or conditioning depolarization. The slow activation of the modified channel can be represented by a single exponential function with the time constant of 1--5 ms. The modified channel is inactivated only partially with a time constant of 1 S. The reversal potential is unchanged by the drug. Observations in tail currents and the voltage dependence of activation suggest that the activation gate is actually unaffected. The apparently slow activation may reflect an interaction betweem deoxycholate and the sodium channel in resting state.     

Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons.

The oscillatory properties of single thalamocortical neurons were investigated by using a Hodgkin-Huxley-like model that included Ca2+ diffusion, the low-threshold Ca2+ current (lT) and the hyperpolarization-activated inward current (lh). lh was modeled by double activation kinetics regulated by intracellular Ca2+. The model exhibited waxing and waning oscillations consisting of 1-25-s bursts of slow oscillations (3.5-4 Hz) separated by long silent periods (4-20 s). During the oscillatory phase, the entry of Ca2+ progressively shifted the activation function of lh, terminating the oscillations. A similar type of waxing and waning oscillation was also observed, in the absence of Ca2+ regulation of lh, from the combination of lT, lh, and a slow K+ current. Singular approximation showed that for both models, the activation variables of lh controlled the dynamics of thalamocortical cells. Dynamical analysis of the system in a phase plane diagram showed that waxing and waning oscillations arose when lh entrained the system alternately between stationary and oscillating branches.     

A Monte Carlo study of the dynamics of G-protein activation.

To link quantitatively the cell surface binding of ligand to receptor with the production of cellular responses, it may be necessary to explore early events in signal transduction such as G-protein activation. Two different model frameworks relating receptor/ligand binding to G-protein activation are examined. In the first framework, a simple ordinary differential equation model is used to describe receptor/ligand binding and G-protein activation. In the second framework, the events leading to G-protein activation are simulated using a dynamic Monte Carlo model. In both models, reactions between ligand-bound receptors and G-proteins are assumed to be diffusion-limited. The Monte Carlo model predicts two regimes of G-protein activation, depending upon whether the lifetime of a receptor/ligand complex is long or short compared with the time needed for diffusional encounters of complexes and G-proteins. When the lifetime of a complex is relatively short compared with the diffusion time, the movement of ligand among free receptors by binding and unbinding ("switching") significantly enhances G-protein activation. Receptor antagonists dramatically reduce G-protein activation and, thus, signal transduction in this case, and significant clustering of active G-proteins near receptor/ligand complexes results. The simple ordinary differential equation model poorly predicts G-protein activation for this situation. In the alternative case, when diffusion is relatively fast, ligand movement among receptors is less important and the simple ordinary differential equation model and Monte Carlo model results are similar. In this case, there is little clustering of active G-proteins near receptor/ligand complexes. Results also indicate that as the GTPase activity of the alpha-subunit decreases, the steady-state level of alpha-GTP increases, although temporal sensitivity is compromised.     

Validity of the rapid buffering approximation near a point source of calcium ions.

In the presence of rapid buffers the full reaction-diffusion equations describing Ca2+ transport can be reduced using the rapid buffering approximation to a single transport equation for [Ca2+]. Here we simulate the full and reduced equations, exploring the conditions necessary for the validity of the rapid buffering approximation for an isolated Ca2+ channel or a cluster of channels. Using a point source and performing numerical simulations of different durations, we quantify the error of the rapid buffering approximation as a function of buffer and source parameters as well as the time and spatial scale set by the resolution of confocal microscopic measurements. We carry out simulations of Ca2+ "sparks" and "puffs," both with and without the indicator dye Ca2+ Green-1, and find that the rapid buffering approximation is excellent. These calculations also show that the traditional calculation of [Ca2+] from a fluorescence signal may grossly underestimate the true value of [Ca2+] near a source. Finally, we use the full model to simulate the transient Ca2+ domain near the pore of an open Ca2+ channel in a cell dialyzed with millimolar concentrations of 1,2-bis(2-aminophenoxy)ethane-N,N,N,N-tetraacetic acid or EGTA. In this regime, where the rapid buffering approximation is poor. Neher's equation for the steady-state Ca2+ profile is shown to be a reliable approximation adjacent to the pore.     

Etiology of the supernormal period.

In many excitable cells, there is a time after the action potential when the threshold for eliciting a second action potential is lower than it is in the steady state. The Hodgkin and Huxley (1952) equations predict such a supernormal period. Using their model, it is shown that the supernormal period results from the slow kinetics of the potassium current and does not depend on sodium current activation or inactivation or on the after-depolarization.     

Bistable nerve conduction

It has been demonstrated experimentally that slow and fast conduction waves with distinct conduction velocities can occur in the same nerve system depending on the strength or the form of the stimulus, which give rise to two modes of nerve functions. However, the mechanisms remain to be elucidated. In this study, we use computer simulations of the cable equation with modified Hodgkin-Huxley kinetics and analytical solutions of a simplified model to show that stimulus-dependent slow and fast waves recapitulating the experimental observations can occur in the cable, which are the two stable conduction states of a bistable conduction behavior. The bistable conduction is caused by a positive feedback loop of the wavefront upstroke speed, mediated by the sodium channel inactivation properties. Although the occurrence of bistable conduction only requires the presence of the sodium current, adding a calcium current to the model further promotes bistable conduction by potentiating the slow wave. We also show that the bistable conduction is robust, occurring for sodium and calcium activation thresholds well within the experimentally determined ones of the known sodium and calcium channel families. Since bistable conduction can occur in the cable equation of Hodgkin-Huxley kinetics with a single inward current, i.e., the sodium current, it can be a generic mechanism applicable to stimulus-dependent fast and slow conduction not only in the nerve systems but also in other electrically excitable systems, such as cardiac muscles.     

Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the squid giant axon

The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.     

The point of maximum cell water volume excursion in case of presence of an impermeable solute

A relativistic permeability model of cell osmotic response (Cryobiology 40:64-83; 41:366-367) is applied to a two-solute system with one impermeable solute. The use of the normalized water volume (w), and the amount of intracellular permeable solute (x), which is the product of the water volume and intracellular osmolality (y), as the main variables allowed us to obtain a homogeneous differential equation dx(Delta)/dw(Delta)=f(x(Delta)/w(Delta)), where w(Delta)=w-w(f), x(Delta)=x-x(f), and f refers to the final (equilibrium) values. The solution of this equation is an explicit function, w(Delta)=g(x(Delta)), which is given in the text. This approach allows us to obtain an analytical (exact) expression of the water volume at the moment of the maximum excursion (water extremum w(m)). Results are compared with numeration of basic osmotic equations and with approximation given in (Cryobiology 40:64-83). Assumption that, dw/dt approximately 0 gives good approximations of the kinetics of water and permeable CPA after the point of maximum volume excursion (the slow phase of osmotic response). Practical aspects of the relativistic permeability approach are also discussed.     

Fast and slow activation of voltage-dependent ion channels in radish vacuoles.

The molecular processes associated with voltage-dependent opening and closing (gating) of ion channels were investigated using a new preparation from plant cells, i.e., voltage and calcium-activated ion channels in radish root vacuoles. These channels display a main single channel conductance of approximately 90 pS and are characterized by long activation times lasting several hundreds of milliseconds. Here, we demonstrate that these channels have a second kinetically distinct activation mode which is characterized by even longer activation times. Different membrane potential protocols allowed to switch between the fast and the slow mode in a controlled and reversible manner. At transmembrane potentials of -100 mV, the ratio between the fast and slow activation time constant was around 1:5. Correspondingly, activation times lasting several seconds were observed in the slow mode. The molecular process controlling fast and slow activation may represent an effective modulator of voltage-dependent gating of ion channels in other plant and animal systems.     

Impaired slow inactivation in mutant sodium channels.

Hyperkalemic periodic paralysis (HyperPP) is a disorder in which current through Na+ channels causes a prolonged depolarization of skeletal muscle fibers, resulting in membrane inexcitability and muscle paralysis. Although HyperPP mutations can enhance persistent sodium currents, unaltered slow inactivation would effectively eliminate any sustained currents through the mutant channels. We now report that rat skeletal muscle channels containing the mutation T698M, which corresponds to the human T704M HyperPP mutation, recover very quickly from prolonged depolarizations. Even after holding at -20 mV for 20 min, approximately 25% of the maximal sodium current is available subsequent to a 10-ms hyperpolarization (-100 mV). Under the same conditions, recovery is less than 3% in wild-type channels and in the F1304Q mutant, which has impaired fast inactivation. This effect of the T698M mutation on slow inactivation, in combination with its effects on activation, is expected to result in persistent currents such as that seen in HyperPP muscle.     

Numerical Calculation of the Bridge Function for Soft-Sphere Supercooled Fluids via Molecular Dynamics Simulations

Abstract

Isokinetic molecular dynamics simulations have been performed for 13,500 soft-spheres interacting through the inverse-power potential, ε([sgrave]/r)ⁿ , near and below the freezing temperature. The bridge function for the integral equation of the theory of liquids is extracted from the pair distribution function (PDF) obtained by the computer simulations for n = 6 and 12. The result is compared with that of approximate theories, i.e., the Rogers-Young (RY) approximation and a modified hypernetted-chain approximation for supercooled soft-sphere fluids (MHNCS approximation). Below the freezing temperature, the bridge function obtained by the computer simulation begins to oscillate around zero at intermediate distances where the second peak of the PDF appears. Such oscillatory behavior of the bridge function is well reproduced by the MHNCS approximation which includes correlations given by the leading elementary diagram, in remarkable contrast to that of the RY approximation. The present result suggests that the split second peak of the PDF for highly supercooled liquids is essentially dominated by the intermediate-distance-range correlation of the leading elementary diagram.     

Ultra-slow inactivation of the ionic currents through the membrane of myelinated nerve.

(1) Voltage-clamp experiments were performed with myelinated fibres isolated from the sciatic nerve of the frog to study slow changes of the specific sodium and potassium currents as a function of membrane (holding) potential and time. (2) The level of the peak sodium current depends on holding potential VH. This dependence can be described by a sigmoidal function uinfinity(VH). The underlying process is called "ultra-slow sodium inactivation" and is different and separable from the short time steady-state inactivation, hinfinity(V), and from the slow inactivation depending on the extracellular potassium concentration (Adelman, Jr., W. J. and Palti, Y. (1969), J Gen. Physiol. 54, 589-606; Peganov, E. M., Khodorov, B.I. and Shishkova, L. D. (1973), Bull. Exp. Biol. Med. 25, 15-19; Khodorov, B. I. Shishkova, L. D. and Peganov, E. M. (1974), Bull. Exp. Biol. Med. 3, 10-14). (3) After a sudden change of the holding potential the sodium current reaches a new steady-state level (due to the transition of uinfinity(VH) to the corresponding value) within approx. 4 min. The kinetics of the transition cannot be described by a single exponential function. (4) A corresponding voltage- and time-dependent process of ultra-slow inactivation exists for the potassium current in the node of Ranvier. The kinetics are faster than those of the sodium system.     

Demonstration of subcomponents of PIII in the ERG of isolated rabbit retina using sodium asparaginate]

Experiments on the sum potential to dark flashes after eliminating the positive components by low temperature (25 degrees C) and low content of plasma (10% instead of 50%) in the perfusion fluid have shown that the cornea-negative component PIII is extensively reduced by application of 10 mM sodium aspartate. The existence of an aspartate sensitive PIII-subcomponent, which was first discovered by intraretinal records, is not in agreement with the opinion that the whole PIII represents receptor activity. Fast and slow PIII-subcomponents are also discernible in the cornea-negative PIII - recorded with cross-electrodes - by their time course without and with adding sodium aspartate. The fast subcomponents are demonstrable in a rough approximation by condenser-coupling the amplifier (t = 0.3 sec). Changing the temperature from 22 degrees C to 27 degrees C the temperature quotient amounts to 1,3 for the fast subcomponents, and to 2.1 for the whole PIII including the slow subcomponents.     

On the pseudo-steady-state approximation and Tikhonov theorem for general enzyme systems

The effectiveness of the Tikhonov theorem for justifying the so-called pseudo-steady-state approximation (PSSA) in general closed enzyme systems is proved under a widely satisfied sufficient condition on the ligand stoichiometry, which also allows us to state that all slow variables of the degenerate system are driven by a single scalar function (the pseudo- steady-state rate equation). Finally, remarks on the perturbative small parameter and open enzyme systems are made.     

Multiple mechanisms of spike-frequency adaptation in motoneurones.

Spike-frequency adaptation is the continuous decline in discharge rate in response to a constant stimulus. We have described three distinct phases of adaptation in rat hypoglossal motoneurones: initial, early and late. The initial phase of adaptation is over in one or two intervals, and is primarily due to summation of the calcium-activated potassium conductance underlying the medium duration afterhyperpolarization (mAHP). The biophysical mechanisms underlying the later phases of adaptation are not well understood. Two of the previously-proposed mechanisms for adaptation are an increase in outward current flowing through calcium-activated potassium channels and increasing outward current produced by the electrogenic sodium-potassium pump. We found that neither of these mechanisms are necessary for the expression of the early and late phases of adaptation. The magnitude of the initial phase of adaptation was reduced when the calcium in the external solution was replaced with manganese, but the magnitudes of the early and late phases were consistently increased under these conditions. Partial blockade of the sodium-potassium pump with ouabain had no significant effect on any of the three phases of adaptation. Our current working hypothesis is that the magnitude of late adaptation depends upon the interplay between slow inactivation of sodium currents, that tends to decrease discharge rate, and the slow activation or facilitation of a calcium current that tends to increase discharge rate. Adaptation is often associated with a progressive decrease in the peak amplitude and rate of rise of action potentials, and a computer model that incorporated slow inactivation of sodium channels reproduced this phenomenon. However, the time course of adaptation does not always parallel changes in spike shape, indicating that the progressive activation of another inward current might oppose the decline in frequency caused by slow sodium inactivation.