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.     

Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons.

The Hodgkin-Huxley equations for space-clamped squid axon (18 degrees C) have been modified to approximate voltage clamp data from repetitive-firing crustacean walking leg axons and activity in response to constant current stimulation has been computed. The m infinity and h infinity parameters of the sodium conductance system were shifted along the voltage axis in opposite directions so that their relative overlap was increased approximately 7 mV. Time constants tau m and tau h, were moved in a similar manner. Voltage-dependent parameters of delayed potassium conductance, n infinity and tau n, were shifted 4.3 mV in the positive direction and tau n was uniformly increased by a factor of 2. Leakage conductance and capacitance were unchanged. Repetitive activity of this modified circuit was qualitatively similar to that of the standard model. A fifth branch was added to the circuit representing a transient potassium conductance system present in the repetitive walking leg axons and in other repetitive neurons. This model, with various parameter choices, fired repetitively down to approximately 2 spikes/s and up to 350/s. The frequency vs. stimulus current plot could be fit well by a straight line over a decade of the low frequency range and the general appearance of the spike trains was similar to that of other repetitive neurons. Stimulus intensities were of the same order as those which produce repetitive activity in the standard Hodgkin-Huxley axon. The repetitive firing rate and first spike latency (utilization time) were found to be most strongly influenced by the inactivation time constant of the transient potassium conductance (tau b), the delayed potassium conductance (tau n), and the value of leakage conductance (gL). The model presents a mechanism by which stable low frequency discharge can be generated by millisecond-order membrane conductance changes.     

Sodium currents in the giant axon of the crab Carcinus maenas

Measurements were made of the kinetics and steady-state properties of the sodium conductance changes in the giant axon of the crab Carcinus maenas. The conductance measurements were made in the presence of small concentrations of tetrodotoxin and as much electrical compensation as possible in order to minimize errors caused by the series resistance. After an initial delay of 10-150 microsec, the conductance increase during depolarizing voltage clamp pulses followed the Hodgkin-Huxley kinetics. Values of the time constant for the activation of the sodium conductance lay on a bell-shaped curve with a maximum under 180 microsec at -40 mV (at 18 degrees C). Values of the time constant for the inactivation of the sodium conductance were also fitted using a bell-shaped curve with a maximum under 7 msec at -70 mV. The effects of membrane potential on the fraction of Na channels available for activation studied using double pulse protocols suggest that hyperpolarizing potentials more negative than -100 mV lock a fraction of the Na channels in a closed conformation.     

The initial inward current in spherical clusters of chick embryonic heart cells

The rapid inward sodium current in spherical clusters of 11-d-old embryonic chick heart cells, ranging in size between 65 and 90 micron diameter, was studied using the two-microelectrode voltage-clamp technique. Using these preparations, it was possible to resolve the activation phase of the rapid inward current for potentials negative to -25 mV at 37 degrees C. The rapid inward current exhibited a voltage and time dependence similar to that observed in other excitable tissues. It was initiated at potential steps more positive than -45 mV. The magnitude of the current reached its maximum value at a potential of approximately -20 mV. The measured reversal potential was that predicted by the Nernst equation for sodium ions. The falling phase of the current followed a single exponential time-course with a time constant of inactivation, tau h, ranging between 2.14 ms at -40 mV and 0.18 ms at -5 mV. The time constant of inactivation, tau h, determined by a single voltage-step protocol was compared to the constant, tau c, determined by a double voltage-step protocol and no significant different between the two constants of inactivation was found. Furthermore, the time constants of inactivation and reactivation at the same potential in the same preparation were similar. The results of this study demonstrate that the sodium current of heart cells recorded at 37 degrees C can be described by Hodgkin-Huxley kinetics with speeds approximately four times faster than the squid giant axon at 15 degrees C.     

On the voltage-dependent action of tetrodotoxin.

The use of the maximum rate-of-rise of the action potential (Vmax) as a measure of the sodium conductance in excitable membranes is invalid. In the case of membrane action potentials, Vmax depends on the total ionic current across the membrane; drugs or conditions that alter the potassium or leak conductances will also affect Vmax. Likewise, long-term depolarization of the membrane lessens the fraction of total ionic current that passes through the sodium channels by increasing potassium conductance and inactivating the sodium conductance, and thereby reduces the effect of Vmax of drugs that specifically block sodium channels. The resultant artifact, an apparent voltage-dependent potency of such drugs, is theoretically simulated for the effects of tetrodotoxin on the Hodgkin-Huxley squid axon.     

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.     

Selective modification of sodium channel gating in lobster axons by 2, 4, 6-trinitrophenol: Evidence for two inactivation mechanisms

Trinitrophernol (TNP) selectively alters the sodium conductance system of lobster giant axons as measured in current clamp and voltage clamp experiments using the double sucrose gap technique. TNP has no measurable effect on potassium currents but reversibly prolongs the time-course of sodium currents during maintained depolarizations over the full voltage range of observable currents. Action potential durations are increased also. Tm of the Hodgkin-Huxley model is not markedly altered during activation of the sodium conductance but is prolonged during removal of activation by repolarization, as observed in sodium tail experiments. The sodium inactivation versus voltage curve is shifted in the hyperpolarizing direction as is the inactivation time constant curve, measured with conditioning voltage steps. This shift speeds the kinetics of inactivation over part of the same voltage range in which sodium currents are prolonged, a contradiction incompatible with the Hodgkin-Huxley model. These results are interpreted as support for a hypothesis of two inactivation processes, one proceeding directly from the resting state and the other coupled to the active state of sodium conductance.     

Transfer properties of neuronal dendrites with tonically activated conductances

The somatopetal current transfer was studied in the mathematical models of a reconstructed brainstem motoneuron with tonically activated excitatory synaptic inputs uniformly distributed over dendritic arborization. The soma and axon provided a constant passive leak. The extrasynaptic dendritic membrane was either passive or active (of a Hodgkin-Huxley type). The longitudinal membrane current density (per unit path length) was used as an estimate of the current transfer effectiveness of different dendritic paths. Introduction of a steady uniform voltage-independent conductance per unit membrane area simulated such a synaptic activation. This actions always produced a spatially inhomogeneous membrane depolarization decaying from the distal dendritic tips toward the soma. The reason for such an inhomogeneity was the preponderance of somatopetal over somatofugal input conductance at every site in the dendrites with sealed distal ends and a leaky somatic end. In active dendrites, partial voltage-dependent extrasynaptic conductances followed this depolarization according to their activation-inactivation kinetics. The greater the local depolarization, the greater the contribution of the non-inactivating potassium conductance to the total membrane conductance. The contribution of the inactivated sodium conductance was one order of magnitude smaller. Correspondingly, the effective equilibrium potential of the total transmembrane current became spatially inhomogeneous and shifted to the potassium equilibrium potential. In the passive dendrites, the equilibrium potential remained spatially homogeneous. Inhomogeneities of the dendritic geometry (abrupt change in the diameter and, especially, asymmetrical branching) caused characteristic perturbations in the voltage gradient, so that the path profiles of the voltage, conductances, and currents diverged. This indicated a geometry-induced separation of the dendritic paths in their transfer effectiveness. Active dendrites of the same geometry were less effective than passive ones due to the effect of the potassium conductance associated with the hyperpolarizing equilibrium potential.     

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.     

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.     

Differential blockage of two types of potassium channels in the crab giant axon

Summary Measurements were made of the kinetic and steady-state characteristics of the potassium conductance in the giant axon of the crabsCarcinus maenas andCancer pagirus. The conductance increase during depolarizing voltage-clamp pulses was analyzed assuming that two separate types of potassium channels exist in these axons (M. E. Quinta-Ferreira, E. Rojas and N. Arispe,J. Membrane Biol. 66:171–181, 1982). It is shown here that, with small concentrations of conventional K⁺-channel blockers, it is possible to differentially inhibit these channels. The potassium channels with activation and fast inactivation gating (m³h, Hodgkin-Huxley kinetics) were blocked by external application of 4 amino-pyridine (4-AP). The potassium channels with standard gating (n⁴, Hodgkin-Huxley kinetics) were preferentially inhibited by externally applied tetraethylammonium (TEA). The differential blockage of the two types of potassium conductance changes suggests that they represent two different populations of potassium channels.It is further shown here that blocking the early transient conductance increase leads to the inhibition of the repetitive electrical activity induced by constant depolarizing current injection in fibers fromCardisoma guanhumi.     

Some Relations between Action Potential and Resting Potential of the Lobster Giant Axon

Experiments were performed to determine the quantitative relation existing between action potential and resting potential of the lobster giant axon. Resting potential changes were induced by either increasing the external potassium concentration or by reducing the external calcium concentration. For either treatment the action potential amplitude is proportional to the logarithm of the resting potential minus a constant. This constant is equivalent to the minimum resting potential at which a propagated spike is possible, and is larger for depolarization in low calcium than in high potassium. Thus the change in action potential per unit change in resting potential is greater in low external calcium than in high external potassium. Analog computer solutions to the Hodgkin-Huxley equations for squid axon membrane potentials show that, if the initial conditions are properly specified, the action potential is proportional to the logarithm of the potassium potential minus a constant. The experimental results and the analog computations suggest that reducing external calcium produces changes in the invertebrate axon that cannot be accounted for solely on the basis of alterations in the potassium potential.     

Nonlinear cable equations for axons. II. Computations and experiments with external current electrodes

We have investigated the steady-state potential and current distributions resulting from current injection into a close-fitting channel into which a squid axon is placed. Hybrid computer solutions of the cable equations, using the Hodgkin-Huxley equations to give the membrane current density, were in good agreement with experimental observations. A much better fit was obtained when the Hodgkin-Huxley leakage conductance was reduced fivefold.     

A model for axonal propagation incorporating both radial and axial ionic transport.

We present an axonal model that explicitly includes ionic diffusion in the intracellular, periaxonal, and extracellular spaces and that incorporates a Hodgkin-Huxley membrane, extended with potassium channel inactivation and active ion transport. Although ionic concentration changes may not be significant in the time course of one action potential, they are important when considering the long-term behavior (seconds to minutes) of an axon. We demonstrate this point with simulations of transected axons where ions are moving between the intra- and extracellular spaces through an opening that is sealing with time. The model predicts that sealing must occur within a critical time interval after the initial injury to prevent the entire axon from becoming permanently depolarized. This critical time interval becomes considerably shorter when active ion transport is disabled. Furthermore, the model can be used to study the effects of sodium and potassium channel inactivation; e.g., sodium inactivation must be almost complete (within 0.02%) to obtain simulation results that are realistic.     

The Effects of Several Alcohols on the Properties of the Squid Giant Axon

The effects of several alcohols on the resting potential, action potential, and voltage-clamp currents of the squid giant axon have been measured. All the alcohols employed are similar in that they depress maximum sodium conductance much more than maximum potassium conductance. Octyl alcohol differs from the others (C₂ through C₅) in that it has less tendency to depolarize the axon. Depolarization is always accompanied by a decrease of g_K near the resting potential, such that the ratio g_K/g_leak is decreased. Steady-state inactivation of the sodium ion current is unaffected by alcohols, as is membrane capacity. Resting membrane conductance is usually decreased by alcohols. The findings are discussed in relation to work on monomolecular films.     

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.     

A fully coupled transient excited state model for the sodium channel

Summary The axon membrane is simulated by standard Hodgkin-Huxley leakage and potassium channels plus a coupled transient excited state kinetic scheme for the sodium channel. This scheme for the sodium channel is as proposed previously by the author. Simultations are presented showing the form of the action potential, threshold behavior, accommodation, and repetitive firing. It is seen that the form of the individual action potential, its all-or-none nature, and its refractory period are well simulated by this model, as they are by the standard Hodgkin-Huxley model. However, the model differs markedly from the Hodgkin-Huxley model with respect to repetitive firing and accommodation to stimulating currents of slowly rising intensity, in ways that are anomn to be related to those features of the sodium inactivation which are anomalous to the H-H model. The tendency for repetitive firing is highly dependent on that parameter which primarily determintes the existence of the inactivation shift in voltage clamp experiments, in such a way that the more pronounced the inactivation shift, the less the tendency for repetitive firing,. The tendency for accommodation is highly dependent on that parameter which primarily determines the “τ_c − τ_h” separation, in such a way that the greater the separation the greater the tendency for the membrane to accommodate without firing action potentials to a slowly rising current.     

Effects of proteolytic enzymes on ionic conductances of squid axon membranes.

The effects of proteolytic enzymes on ionic conductances of squid axon membranes have been studied by means of the voltage clamp technique. When perfused internally alpha-chymotrypsin (1 mg/ml) increased and prolonged the depolarizing after-potential. Sodium inactivation was partially inhibited causing a prolonged sodium current, and peak sodium and steady-state potassium currents were suppressed. The time for sodium current to reach its peak was not affected. Leakage conductance increased later. On the other hand, carboxypeptidases A and B, both at 1mg/ml, suppressed the sodium and potassium conductance increases with little or no change in sodium inactivation. The mechanism that controls sodium inactivation appears to be associated with the structure of membrane proteins which is modified by alpha-chymotrypsin but not by carboxypeptidases and is located in a position accessible to alpha-chymotrypsin only from inside the membrane.     

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.