A mathematical analysis of the action potential plateau duration of a human ventricular myocyte

The plateau phase of a human ventricular myocyte is analysed. The plateau duration is a function of the time required for a myocyte's transmembrane voltage to decrease by a certain voltage, DeltaV. The timing of the plateau is shown to be controlled by two slowly changing gate variables, the inactivation gate that controls the inward/depolarizing L-type calcium current and the inactivation gate that controls the outward/repolarizing slow rectifier potassium current. The amount of current controlled by these variables is a function of the net conductivity of the corresponding sodium and potassium channels. An equation is derived that relates action potential duration to these net conductivities and the time dependence of the slowly moving variables. This equation is used to estimate plateau duration for a given value of DeltaV. The initial conditions of the slowly moving inactivation variables are shown to affect plateau duration. These initial conditions depend on the amount of time that has elapsed between a previous repolarization and a current depolarization (diastolic interval). The analysis thus helps to quantify the characteristics of action potential duration restitution.     

Developmental change in calcium-activated chloride current during the differentiation of Xenopus spinal neurons in culture.

The duration and ionic dependence of action potentials change during the differentiation of embryonic amphibian spinal neurons both in vivo and in culture. The development of sodium, calcium, and potassium currents has been characterized in these cells and the shortening of the action potential has been shown to depend to a great extent on developmental changes of potassium currents. Previous evidence suggests that a chloride current may also be present in these embryonic neurons. Chloride currents were investigated with intracellular current-clamp and single-electrode and whole-cell voltage-clamp techniques. Most neurons exhibited a calcium-activated chloride current (ICl(Ca] that contributed to the postdepolarization following the action potential recorded in the absence of sodium and potassium currents. This current appeared to decrease in density and its deactivation rate increased during the first day in culture. Its incidence also declined during this period. A much larger Ca(2+)-dependent Cl- current was also observed in a subset of neurons after 24 hr, but was absent at earlier stages of development. The results suggest the presence of two Cl- currents with different developmental fates. The early current probably contributes to the repolarization of long calcium-dependent action potentials at initial stages of neuronal development, when potassium currents are small, and may serve to reduce the extent of repetitive firing.     

Slow Changes of Potassium Permeability in the Squid Giant Axon

A slow potassium inactivation i.e. decrease of conductance when the inside of the membrane is made more positive with respect to the outside, has been observed for the squid axon. The conductance-potential curve is sigmoid shaped, and the ratio between maximum and minimum potassium conductance is at least 3. The time constant for the change of potassium conductance with potential is independent of the concentration of potassium in the external solution, but dependent upon potential and temperature. At 9 degrees C and at the normal sea water resting potential, the time constant is 11 sec. For lower temperature or more depolarizing potentials, the time constant is greater. The inactivation can be described by modifying the Hodgkin-Huxley equation for potassium current, using one additional parameter. The modified equation is similar in form to the Hodgkin-Huxley equation for sodium current, suggesting that the mechanism for the passive transport of potassium through the axon membrane is similar to that for sodium.     

Extracellular potassium inhibits Kv7.1 potassium channels by stabilizing an inactivated state

Kv7.1 (KCNQ1) channels are regulators of several physiological processes including vasodilatation, repolarization of cardiomyocytes, and control of secretory processes. A number of Kv7.1 pore mutants are sensitive to extracellular potassium. We hypothesized that extracellular potassium also modulates wild-type Kv7.1 channels. The Kv7.1 currents were measured in Xenopus laevis oocytes at different concentrations of extracellular potassium (1–50 mM). As extracellular potassium was elevated, Kv7.1 currents were reduced significantly more than expected from theoretical calculations based on the Goldman-Hodgkin-Katz flux equation. Potassium inhibited the steady-state current with an IC₅₀ of 6.0 ± 0.2 mM. Analysis of tail-currents showed that potassium increased the fraction of channels in the inactivated state. Similarly, the recovery from inactivation was slowed by potassium, suggesting that extracellular potassium stabilizes an inactivated state in Kv7.1 channels. The effect of extracellular potassium was absent in noninactivating Kv7.1/KCNE1 and Kv7.1/KCNE3 channels, further supporting a stabilized inactivated state as the underlying mechanism. Interestingly, coexpression of Kv7.1 with KCNE2 did not attenuate the inhibition by potassium. In a number of other Kv channels, including Kv1.5, Kv4.3, and Kv7.2–5 channels, currents were only minimally reduced by an increase in extracellular potassium as expected. These results show that extracellular potassium modulates Kv7.1 channels and suggests that physiological changes in potassium concentrations may directly control the function of Kv7.1 channels. This may represent a novel regulatory mechanism of excitability and of potassium transport in tissues expressing Kv7.1 channels.     

Potassium flux ratio in voltage-clamped squid giant axons

The potassium flux ratio across the axolemma of internally perfused, voltage-clamped giant axons of Loligo pealei has been evaluated at various membrane potentials and internal potassium concentrations ([K]i). Four different methods were used: (a) independent measurement of one-way influx and efflux of 42K; (b) simultaneous measurement of net K current (IK) and 42K influx; (c) simultaneous measurement of IK and 42K efflux; and (d) measurement of potassium conductance and 42K influx at the potassium equilibrium potential. The reliability of each of these methods is discussed. The average value of the exponent n' in the Hodgkin-Keynes equation ranged from 1.5 at -4mV and 200 mM [K]i to 3.3 at -38 mV and 350 mM [K]i and appeared to be a function of membrane potential and possibly of [K]i. It is concluded that the potassium channel of squid giant axon is a multi-ion, single-file pore with three or more sites.     

Transient potassium currents in slow muscle fibers.

Transient changes in potassium conductance in chronically depolarized slow muscle fibers have been studied using a voltage clamp method. The transient behavior included current decays from initial to steady state for hyperpolarizing and depolarizing voltage clamp steps. A two-pulse voltage clamp sequence (conditioning step followed by test step) showed the initial potassium test current to depend sigmoidally on conditioning potential implicating the involvement of a membrane-bound charged group in regulating potassium current.     

The Ventral Photoreceptor Cells of Limulus : III. A voltage-clamp study

In the dark, the ventral photoreceptor of Limulus exhibits time-variant currents under voltage-clamp conditions; that is, if the membrane potential of the cell is clamped to a depolarized value there is an initial large outward current which slowly declines to a steady level. The current-voltage relation of the cell in the dark is nonlinear. The only ion tested which has any effect on the current-voltage relation is potassium; high potassium shifts the reversal potential towards zero and introduces a negative slope-conductance region. When the cell is illuminated under voltage-clamp conditions, an additional current, the light-induced current, flows across the cell membrane. The time course of this current mimics the time course of the light response (receptor potential) in the unclamped cell; namely, an initial transient phase is followed by a steady-state phase. The amplitude of the peak transient current can be as large as 60 times the amplitude of the steady-state current, while in the unclamped cell the amplitude of the peak transient voltage never exceeds 4 times the amplitude of the steady-state voltage. The current-voltage relations of the additional light-induced current obtained for different instants of time are also nonlinear, but differ from the current-voltage relations of the dark current. The ions tested which have the greatest effect on the light-induced current are sodium and calcium; low sodium decreases the current, while low calcium increases the current. The data strongly support the hypothesis that two systems of electric current exist in the membrane. Thus the total ionic current which flows in the membrane is accounted for as the sum of a dark current and a light-induced current.     

Potassium Ion Current in the Squid Giant Axon: Dynamic Characteristic

Measurements of the potassium current in the squid axon membrane have been made, after changes of the membrane potential to the sodium potential of Hodgkin and Huxley (HH), from near the resting potential, from depolarizations of various durations and amplitudes, and from hyperpolarizations of up to 150 mv. The potassium currents I given by I = I_∞ {1 - exp [- (t + t₀)/τ]}²⁵, where t₀ is determined by the initial conditions, represent the new data and approximate the HH functions in the regions for which they are adequate. A corresponding modification for the sodium current does not appear necessary. The results support the HH assumptions of the independence of the potassium and sodium currents, the dependence of the potassium current upon a single parameter determined by the membrane potential, and the expression of this parameter by a first order differential equation, and, although the results drastically modify the analytical expressions, they very considerably extend the range of apparent validity of these assumptions. The delay in the potassium current after severe hyperpolarization is used to estimate a potassium ion mobility in the membrane as 10^-5 of its value in aqueous solutions.     

Conditioning prepulses and kinetics of potassium conductance in the frog node

Summary The kinetics of potassium conductance were analyzed in response to voltage-clamp steps with holding potential (–75 mV) as initial condition and after a positive prepulse to-wards +45 mV of 10-msec duration. As the potassium reversal potentialE _K altered during potassium current flow, a method to obtain the conductance independent ofE _K was used. Conductance kinetics at 15°C were analyzed according to the Hodgkin-Huxley (HH) model. The time constant of potassium activation, with holding potential as initial condition, is a monotonous decreasing function of membrane potential. Its value ofca. 9 msec at –50 mV decreases to 1 msec at +30 mV. Changes inE _K did not affect the voltage dependency of this time constant. The time constant of potassium deactivation, i.e. the off-response following a 10-msec prepulse towards +45 mV, shows a completely different voltage dependency. At a membrane potential of –90 mV it is approximately 2 msec and gradually increases for more positive voltages towards a maximum value of about 6 msec, that is reached between –5 and 0 mV. At still larger values of membrane voltage this time constant starts to fall again. It is concluded that a HH-model, as applied for a single population of potassium channels, has to be rejected. Computer simulations indicate that an extension to two populations of independent potassium channels, each with HH-kinetics, is also inconsistent with the observed results.     

Leak Current Rectification in Myxicola Giant Axons : Constant field and constant conductance components

Early leak current, i.e. for times similar to the time to peak of the transient current was measured in Myxicola giant axons in the presence of tetrodotoxin. The leak current-voltage relation rectifies, showing more current for strong depolarizing pulses than expected from symmetry around the holding potential. A satisfactory practical approximation for most leak corrections is constant resting conductance. The leak current-voltage curve rectifies less than expected from the constant field equation. These curves cannot be reconstructed by summing the constant field currents for sodium and potassium using a P_Na/P_K ratio obtained in the usual way, from zero current constant field fits to resting membrane potential data. Nor can they be reconstructed by summing the constant field current for potassium with that for any other single ion. They can be reconstructed, however, by summing the constant field current for potassium with a constant conductance component. It is concluded that the leak current and the resting membrane potential, therefore, are determined by multiple ionic components, at least three and possibly many. Arguments are presented suggesting that ion permeability ratios obtained in the usual way, by fitting the constant field equation to resting membrane potential data should be viewed with skepticism.     

Computer simulation for studying calcium dependent abnormalities in firing mechanism of molluscan neurones

Computer modelling technique is proposed to assist in physiological research on invertebrate neuronal membranes. The firing mechanism of a single patch of invertebrate neuronal membrane has been studied in dependence on maximum Ca++ conductance. The calculations are based on modification of Hodgkin-Huxley's data completed by a straight line approximation between experimental points of the kinetic parameters of Ca++ current and early transient potassium current. The time course of conductance changes is assumed to be proportional to m2h for Ca++ current. Three distinct potassium currents are involved into the model, viz. transient potassium current, delayed potassium current and Ca++-dependent potassium current. The modified Euler method run on a digital computer has been used for numerical integration of kinetic equations. Significant effects of Ca++ conductance on spike broadening, plateau development and spike afterhyperpolarization are represented. In the range of small Ca++ conductance an infinite spontaneous activity can be triggered by a short (suprathreshold) current pulse which may be considered a model of pacemaker activity. Plateau development resulting from potassium blocking or decreasing potassium equilibrium is facilitated by Ca++ conductance in the range of greater Ca++ conductance. The effects of voltage sensitivity of the coupling coefficient describing the current of Ca++-dependent K+ channels were studied and compared to the voltage independent case. The coupling coefficient seems to be a crucial factor in broadening the range of Ca++ conductance responsible for pacemaker activity. For greater values of Ca++ conductance, a decrease of the coupling coefficient leads to a transition from prolonged bursting to interruption of burst activity by burst-afterhyperpolarization. The blocking effect of 4-aminopyridine on fast outward current has been studied by the model which has a practical significance considering that aminopyridine is known as a convulsive agent. We suppose that it is reasonable to study the convulsive effects of aminopyridine by the model based on the kinetics of the isolated neuronal membrane. The model may help in understanding the ionic background underlying abnormal network activity during epileptic discharges of mammalian neurones.     

Temporal relationship between neurotransmitter release and ion flux during spreading depression and anoxia

Brain ion homeostasis is severely perturbed during spreading depression of Leao and during anoxia. The ionic composition of the extracellular space changes abruptly and approaches the intracellular concentrations owing to an increase in cell permeability. In spreading depression, synchronous transmitter efflux caused by a depolarization of the presynaptic terminals has been implicated as a possible mechanism that would explain the concomitant movement of ions. Anoxia, having many features in common with spreading depression, may follow the same mechanism. We have measured the concentrations of extracellular potassium with ion-selective microelectrodes and dopamine by in vivo voltammetry with carbon fiber microelectrodes during spreading depression and anoxia to compare the temporal relationship between the release of dopamine and ion movements in the striatum. There is a pronounced release of dopamine during both spreading depression and anoxia. In spreading depression, the sharp increase of potassium concentration that follows an initial smaller and slower increase of potassium is accompanied by the release of dopamine. In anoxia, the dopamine release clearly precedes the fast rise of extracellular potassium concentration. We conclude that in striatum, there is a pronounced dopamine release during spreading depression and anoxia, but that the relationships between ionic changes and transmitter release for these two phenomena are different and probably reflect different mechanisms.     

Potassium removal accompanied by enhanced biological phosphate removal

The effects of potassium on excess uptake of phosphate in an aerobic-anaerobic activated sludge process were examined by the fill-and-draw procedure. The presence of sufficient potassium was necessary for excess uptake to occur. The contents of potassium and phosphate in the sludge at the end of each cycle of the process were correlated with each other by a non-linear equation, with some scattering. However, the sum of the molar ion valences of magnesium (Mg) and potassium (K), 2 Mg + K, was well correlated by a linear equation with the moles of P expressed as mmol/g-VSS, with a correlation coefficient of 0.993. When the potassium concentration was insufficient for the enhanced uptake of phosphate, its concentration in treated water was of the order of 0.1 mg/l. In the first anaerobic period phosphate was released into the liquid phase, but potassium was released after initial instantaneous removal, and in the successive aerobic period they were both taken up again.     

Depletion and accumulation of potassium in the extracellular clefts of cardiac Purkinje fibers during voltage clamp hyperpolarization and depolarization: experiments in sodium-free bathing media

Voltage clamp hyperpolarization and depolarization result in currents consistent with depletion and accumulation of potassium in the extracellular clefts o cardiac Purkinje fibers exposed to sodium-free solutions. Upon hyperpolarization, an inward current that decreased with time (id) was observed. The time course of tail currents could not be explained by a conductance exhibiting voltage-dependent kinetics. The effect of exposure to cesium, changes in bathing media potassium concentration and osmolarity, and the behavior of membrane potential after hyperpolarizing pulses are all consistent with depletion of potassium upon hyperpolarization. A declining outward current was observed upon depolarization. Increasing the bathing media potassium concentration reduced the magnitude of this current. After voltage clamp depolarizations, membrane potential transiently became more positive. These findings suggest that accumulation of potassium occurs upon depolarization. The results indicate that changes in ionic driving force may be easily and rapidly induced. Consequently, conclusions based on the assumption that driving force remains constant during the course of a voltage step may be in error.     

Neurohumoral control of ionic pumping in molluscan muscle—II. evidence for an electrogenic sodium pump in radular protractor muscle of Busycon canaliculatum

1. A sucrose gap technique was used to study the effects of brief periods of superfusion with solutions in which the potassium content of artificial sea water was reduced or omitted.2. Stepwise reduction in bath potassium had a complex effect, culminating in the response to potassium-free solution. This was composed of a rapid initial hyperpolarizing phase, overtaken by a slower depolarizing phase, which was accompanied by force.3. Readmission of bath potassium induced a transient after-hyperpolarization.4. There was a high degree of individual variability in RPM preparations from different animals. This was particularly evident in cases in which either the hyperpolarizing phase or the depolarizing phase predominated, in the response to zero-potassium, but the muscles from any one animal showed reproducible responses.5. The RPM behaved as predicted on Nernst equation grounds, to the extent that initial hyperpolarization showed stepwise increases with stepwise reduction in [K⁺]₀, but as the steps approached zero-potassium there was a stepwise increase in the slower depolarizing response, suggesting reduction in electrogenic Na-K exchange.6. In Na-free solution the depolarizing phase of the response to zero-K was abolished, leaving only an enhanced hyperpolarizing phase.7. Abrupt chilling had a depolarizing effect.8. There was only a slight increase in resistance during the action of zero K.     

Ion mechanisms of hyperpolarizing afterpotentials accompanying epileptic discharges in neocortical neurons

Hyperpolarizing afterpotentials of penicillin-induced (local application) paroxysmal depolarizing shifts (PDS) in neurons of the sensorimotor cortex of the cat were studied. The pattern of membrane conductance changes within different segments of hyperpolarization and the data on the role of various ion currents in its generation allow us to conclude that hyperpolarizing afterpotentials accompanying PDS are of a composite nature and include the following components: (i) the initial component provided by an increased membrane permeability to chloride ions (presumably a synaptic GABA_A response); (ii) the second component resulting predominantly from a potassium current and representing presumably a GABA_B response; and (iii) the final component comprising mainly a calcium-activated potassium current. These components are present in all neurons, are not clearly demarcated as separate waves, and partially overlap with each other, thus forming a prolonged hyperpolarizing deflection of the potential.     

Extracellular [K+] fluctuations in voltage-clamped canine cardiac Purkinje fibers.

Membrane currents and extracellular [K+] were measured in canine Purkinje strands during voltage-clamp steps to plateau or diastolic potentials. Extracellular [K+] increased during step depolarizations and decreased during step hyperpolarizations. On hyperpolarization, the largest fraction of the K+ depletion occurred during the initial 500 ms of the voltage-clamp step and was correlated with a potassium depletion current, the id. A slower component of the depletion also occurred on hyperpolarization and had a time constant consistent with cylindrical diffusion of potassium within the Purkinje strands. On depolarization, there is an accumulation of K+ that is correlated with the plateau current ix. On termination of depolarizing test pulses, the K+ accumulation decays with a time course similar to the ix tail current. Surprisingly, no accumulation of K+ occurred during the arrhythmogenic transient inward current, TI, suggesting that the selectivity of this current should be reevaluated.     

Aminopyridine block of transient potassium current

The blocking action of 4-aminopyridine (4-AP) and 3, 4-diaminopyridine (Di-AP) on transient potassium current (IA) in molluscan central neurons was studied in internal perfusion voltage-clamp experiments. Identical blocking effects were seen when the drugs were applied either externally or internally. It was found that aminopyridines have two kinds of effects on IA channels. The first involves block of open channels during depolarizing pulses and results in a shortening of the time to peak current and an increase in the initial rate of decay of current. This effect of the drug is similar to the block of delayed potassium current by tetraethylammonium (TEA). The other effect is a steady block that increases in strength during hyperpolarization, is removed by depolarization, and is dependent on the frequency of stimulation. The voltage dependence of steady state block approximates the voltage dependence of inactivation gating a changes e-fold in approximately 10 mV. These data suggest that the strength of block may depend on the state of IA gating such that the resting state of the channel with open inactivation gate is more susceptible to block than are the open or inactivated states. A multistate sequential model for IA gating and voltage-dependent AP block is developed.     

Effects of N-alcohols on potassium conductance in squid giant axons

The effect of bath application of several short chain N-alcohols on voltage-dependent potassium conductance has been studied in intact giant axons of Loligo forbesi under voltage-clamp conditions. All tested alcohols (methanol, ethanol, propanol, butanol, heptanol and octanol) were found to depress potassium conductance only at concentrations much larger than those necessary to reduce sodium conductance. The efficacy of the different molecules was correlated with the carbon-chain length. In all cases the effects were found to be at least partly reversible. Low concentrations of propanol (100 mM) or heptanol (1 mM) were found to increase potassium conductance whereas higher concentrations had the usual depressing effect. The two alcohols were found to induce a slow inactivation of the potassium conductance. A detailed analysis of the time course of the turning-on of the potassium current for various pulse potentials in the presence of TTX revealed that, for membrane potential values more positive than -20 mV, the time constant of activation was reduced in the presence of propanol or heptanol. The delay which separates the change in potential and the turning-on of the potassium current, which was systematically analysed for different pulse and prepulse potential values, was increased by the two alcohols, the curve relating this delay to prepulse potential being shifted towards larger (positive) delays. This high degree of complexity in the effects on potassium conductance suggests that the alcohol molecules modify several more or less independent mechanisms associated with the turning-on of the potassium current.