A note on the asymptotic reduction of the Hodgkin-Huxley equations for nerve impulses

The (standard) FitzHugh reduction of the Hodgkin-Huxley equations for the propagation of nerve impulses ignores the dynamics of the activation gates. This assumption is invalid and leads to an over-estimation of the wave speed by a factor of 5 and the wrong dependence of wave speed on sodium channel conductance. The error occurs because a non-dimensional parameter, which is assumed to be small in the FitzHugh reduction, is in fact large (≈18). We analyse the Hodgkin-Huxley equations for propagating nerve impulses in the limit that this non-dimensional parameter is large, and show that the analytical results are consistent with numerical simulations of the Hodgkin-Huxley equations.     

Isomorphism on a physical system of the Hodgkin-Huxley equations for sodium conductance: toxin allosteric effects and gating currents

An isomorphism on a physical system of the Hodgkin-Huxley equations for sodium ion conductance in the nerve membrane is derived. The physical system consists of 8 states. It shows that the voltage dependence of the sodium conductance arises from a change in ionization of the molecule that provides the ion-selective conductance channels. It associates reversibly with singly charged (H+?) and doubly charged (Ca++?) ions. The inactivation process is the result of the associating of an ionized particle by half of the states. The effect of toxins and narcotics in blocking or inactivating sodium conductance can be understood as an enzyme or allosteric change of the standard free energy difference of the molecule that provides the sodium channels. The effect of changing pH and Ca++ substrate concentration on the sodium conductance is predicted. The gating charge current is predicted. The time constant predicted is in agreement with experiment.     

Some Models of Neuronal Variability

The pattern of nerve action potentials produced by unit permeability changes (quantal inputs) occurring at random is considered analytically and by computer simulation methods. The important parameters of a quantal input are size and duration. Varying both the mean and the probability density function of these parameters has calculable effects on the distribution of interspike intervals. Particular attention is paid to the relation between the mean rate of excitatory inputs and the mean frequency of nerve action potentials (input-output curve) and the relation between the coefficient of variation for the interval distribution and the mean interval (variability curve). In the absence of action potentials one can determine the parameters of the voltage distribution including the autocorrelation function and the power spectrum. These parameters can sometimes be used to approximate the variability of interspike intervals as a function of the threshold voltage. Different neuronal models are considered including one containing the Hodgkin-Huxley membrane equations. The negative feedback inherent in the Hodgkin-Huxley equations tends to produce a small negative serial correlation between successive intervals. The results are discussed in relation to the interpretation of experimental results.     

Corrections for space-clamp errors in measured parameters of voltage-dependent conductances in a cylindrical neurite

The ability to correct parameters of voltage-gated conductances measured under poor spatial control by point voltage clamp could rescue much flawed experimental data. We explore a strategy for correcting errors in experiments that employs a full-trace approach to parameter determination. Simulated soma voltage-clamp runs are made on a model neuron with a single voltage-gated, Hodgkin-Huxley channel type distributed uniformly along an elongate process. Estimates for both kinetic and I(V) parameters are obtained by fitting a form of the Hodgkin-Huxley equations to the complete time course of leak-subtracted current curves. The fitted parameters are used to determine how much correction in each parameter is needed to regenerate the set actually belonging to the channel. Corrections are generated for a range of neurite lengths, conductance densities, and channel characteristics.     

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.     

Modeling repetitive firing and bursting in a small unmyelinated nerve fiber.

The Hodgkin-Huxley equations, originally developed to describe the electrical events in the squid giant axon, have been modified to simulate the ionic and electrical events in a small unmyelinated nerve fiber. The modified equations incorporate an electrogenic sodium-potassium pump, finite intra-axonal volume, a periaxonal space, a calcium current, and calcium-dependent potassium conductance (GKCa). The model shows that adaptation can occur in two ways: increased Na-K pump activity because of periaxonal K accumulation or intra-axonal Na accumulation; or from an increase in (GKCa) caused by calcium accumulating within the axon. Bursting is an extension of adaptation and occurs when the sensitivity of the Na-K pump or (GKCa) to changes in ionic concentration is increased.     

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.     

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.     

An assessment of a coupled three-state kinetic model for sodium conductance changes.

The behavior of a coupled three-state kinetic scheme is examined to see if it might be a viable model for the conductance changes of sodium channels. It is found that for simulations of experiments which determine the properties of the Hodgkin-Huxley m and h gates, the three-state scheme performs approximately equivalently to the Hodgkin-Huxley model. In particular, the three-state scheme successfully simulates those experiments which the Hodgkin-Huxley model successfully simulates, but fails to simulate those newer voltage clamp experiments which give results anomalous to the H-H model. It is concluded that the three-state scheme is probably as good as the H-H model, but is not a viable successor to it.     

Voltage-noise-induced transitions in electrically excitable membranes.

A quantitative study of the steady-state behavior of the sodium and potassium conductance for the Hodgkin-Huxley axon under the influence of an externally driven voltage noise is reported. The dichotomous Markov noise (random telegraph signal) considered allows for an exact evaluation of the stationary probability density of the conductances. Phase diagrams are constructed to represent the response of the system as a function of the amplitude and the correlation time of the noise. The results obtained for the Hodgkin-Huxley axon are compared with some molecular models used in the literature.     

Sinusoidal voltage clamp of the Hodgkin-Huxley model.

A voltage clamp consisting of a sinusoidal voltage of amplitude V1 and frequency f, superimposed on a steady voltage level V0, is applied to the Hodgkin-Huxley model of the squid giant axon membrane. The steady-state response is a current composed of sinusoidal components of frequencies O, f, 2f, 3f,... The frequencies greater than f arise from the nonlinearity of the membrane. The total current is described by a power series in V1; each coefficient of this series is composed of current components for one or more frequencies. For different frequencies one can derive higher-order generalized admittances characterizing the nonlinear as well as the linear properties of the membrane. Formulas for the generalized admittances are derived from the Hodgkin-Huxley equations for frequencies up to 3f, using a perturbation technique. Some of the resulting theoretical curves are compared with experimental results, with good qualitative agreement.     

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.     

Effects of transient receptor potential-like current on the firing pattern of action potentials in the Hodgkin-Huxley neuron during exposure to sinusoidal external voltage

Transient receptor potential vanilloid-1 (TRPV1) channels play a role in several inflammatory and nociceptive processes. Previous work showed that magnetic electrical field-induced antinociceptive [corrected] action is mediated by activation of capsaicin-sensitive sensory afferents. In this study, a modified Hodgkin-Huxley model, in which TRP-like current (ITRP) was incorporated, was implemented to predict the firing behavior of action potentials (APs), as the model neuron was exposed to sinusoidal changes in externally-applied voltage. When model neuron is exposed to low-frequency sinusoidal voltage, increased maximal conductance of ITRP can enhance repetitive bursts of APs accompanied by a shortening of inter-spike interval (ISI) in AP firing. The change in ISIs with number of interval is periodic with the phase-locking. In addition, increased maximal conductance of ITRP can abolish chaotic pattern of AP firing in model neuron during exposure to high-frequency voltage. The ISI pattern is converted from irregular to constant, as maximal conductance of ITRP is increased under such high-frequency voltage. Our simulation results suggest that modulation of TRP-like channels functionally expressed in small-diameter peripheral sensory neurons should be an important mechanism through which it can contribute to the firing pattern of APs.     

The Effect of Membrane Parameters on the Properties of the Nerve Impulse

The effect of varying membrane capacitance, conductance, and rate constants on the properties of the nerve impulse is considered in terms of the degree of regeneration in the Hodgkin-Huxley model for the squid giant axon. It is shown through computer simulation that reducing regeneration generally increases the duration of the action potential and decreases its amplitude, rate of rise, and conduction velocity. The threshold becomes much less sharp and the amplitude of the response of a patch of membrane grades with stimulus strength. A second stimulus, applied shortly after a first stimulus, considerably perturbs the membrane potential from its original time-course. Under certain conditions, the nerve signal can propagate with a small decrement.     

A model for conductance changes in the squid giant axon. II. Channel kinetics

The kinetics of the sodium and potassium channels in voltage clamped squid giant axon following a relaxation of the membrane subunits are examined and compared with the Hodgkin-Huxley equations. Mechanisms are suggested for the turn-off of the sodium conductance and a set of kinetic states are proposed for the potassium channel which are consistent with the experimental observations. Determination of the rate constants for relaxation of the surface subunits which triggers the subsequent changes within the independent channels provide information on the equilibrium constant and free energy for this process. The free energy is observed to approach zero as the depolarizing voltage of the clamp approaches $\text{E}{}_{\mathrm{<mtext>Na</mtext>}}$, the voltage for zero sodium current in voltage clamped axons. Analysis of the final rate constants in the kinetic sequence for potassium indicates a symmetry of the channel when it is in its steady-state configuration during clamp in the absence of external gradients.     

Exponentiated exponential model (Gompertz kinetics) of Na+ and K+ conductance changes in squid giant axon

The conductance changes, gK(t) and gNa(t), of squid giant axon under voltage clamp (Hodgkin and Huxley, 1952) may be modeled by exponentiated exponential functions (Gompertz kinetics) from any holding potential VO to any membrane clamp potential V. The equation constants are set by the membrane potential V, and include, for any voltage step in the case of gK, the initial conductance, gO, the asymptote conductance g, and rate constant k: gK = g exp(-be-kt) where b = 1n g/gO. Equations of similar form relate g and k to the voltage V, and govern the corresponding parameters of the gNa system. For the gNa, the fast phase y = y exp (-be-kt) is cut down in proportion to a slow process p = (1 - p)e-k't + p, and thus gNa = py. The expo-exponential functions involve fewer constants than the Hodgkin-Huxley model. In particular, the role of the n, m, h parameters appears to be filled largely by 1n (g/gO) in the case of gK and by 1n (y/yO) in the case of gNa. Membrane action potentials during current clamp may be computed from the conductances generated by use of the appropriate differential forms of the equations; diverse other membrane behaviors may be predicted.     

Modeling of time disparity detection by the Hodgkin-Huxley equations

Phase-sensitive neurons in the electrosensory lateral line lobe in the electrosensory pathway of the wave-type electric fish, Gymnarchus niloticus, are specialized for sensing the time disparity between sensory inputs at different parts of the body surface that is necessary for an electrical behavior, jamming avoidance response. These neurons are sensitive to time disparity in the microsecond range between synaptic inputs that represent occurrence times of electrosensory signals at different areas on the body surface. We showed that an ideal Hodgkin-Huxley equation may serve as a time disparity detector that fits physiological precision, and the precision for the time disparity detection is largely regulated by the maximal g(K) conductance in the Hodgkin-Huxley equations.     

Towards a molecular theory of the nerve membrane: the significance of the quasithreshold behaviour

The threshold properties of a model of the nerve membrane are examined. A graded quasithreshold is found, and it can be characterized by an amplification factor μ. Upon introducing an activation enthalpy for one relaxation process, the temperature dependence of μ can be matched over nine decades to that previously found from the Hodgkin-Huxley equations. The reason for the unusual form of the temperature dependence is clearly indicated in the model. The central mechanism postulated in the model is shown to provide an integrative rationalization of the basic phenomena of nerve excitation.     

A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance that allows calculation of both the speed and shape of the traveling pulses, in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses that is valid with good accuracy in a wide range of the parameters.