Hodgkin-Huxley Axon: Increased Modulation and Linearity of Response to Constant Current Stimulus

Repetitive response patterns resembling those of tonic receptors were obtained by increasing the potassium system time constant in the Hodgkin-Huxley (H-H) equations. The increase in time constant varied with membrane potential. Calculated spike frequencies varied linearly with the magnitude of the constant current stimulus; in addition, minimum frequencies were greatly reduced, and the frequency range increased. Modification of the maximum ionic conductances, membrane capacitance, and rate constant voltage dependence was found to vary the minimum frequency, current at that frequency, slope, and over-all modulation of the modified responses.     

Frequency entrainment of squid axon membrane

Summary Sinusoidally varying stimulating currents were applied to space-clamped squid giant axon membranes in a double sucrose gap apparatus. Stimulus parameters varied were peak-to-peak current amplitude, frequency, and DC offset bias. In response to these stimuli, the membranes produced action potentials in varying patterns, according to variation of input stimulus parameters. For some stimulus parameters the output patterns were stable and obviously periodic with the periods being simple multiples of the input period; for other stimulus parameters no obvious periodicity was manifest in the output. The experimental results were compared with simulations using a computer model which was modified in several ways from the Hodgkin-Huxley model to make it more representative of our preparation. The model takes into account K⁺ accumulation in the periaxonal space, features of Na⁺ inactivation which are anomalous to the Hodgkin-Huxley model, sucrose gap hyperpolarization current, and membrane current noise. Many aspects of the experiments are successfully simulated but some are not, possibly because some very slow process present in the preparation is not included in the model.     

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.     

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.     

Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.     

Oscillation and Repetitive Firing in Squid Axons : Comparison of experiments with computations

Space-clamped squid axons treated with low calcium and computed Hodgkin-Huxley (HH) axons were stimulated by steps of superthreshold current from 101 to 400% of the rheobasic value over a temperature range of 5–27°C. The natural frequency of sustained repetitive firing of real and computed axons depended weakly upon stimulus intensity and strongly upon temperature, with a Q₁₀ of 2.7 (experimental) and 2.6 (computed). For real axons, but not the computed axon, the intervals between the first two spikes were shorter than between subsequent spikes. Constant spike frequencies from 75 Hz at low intensities and temperatures to 330 Hz at high intensities and temperatures were soon achieved. Subthreshold and superthreshold responses were sometimes intermixed in a train of responses from a real axon responding to a constant step of current, but not predicted by HH. The time interval following a spike was always longer than that following a subthreshold oscillation in slightly decalcified real axons, as Huxley and FitzHugh also found for computed axons. There was a bias toward spikes at the beginning of the train and toward subthreshold responses later on. Some repeated patterns were found, every second, third, or fourth response being a spike. Neither the HH equations nor the computed or experimental threshold behaviors show a critical temperature to support a membrane phase transition.     

The frequency response function and sinusoidal threshold properties of the Hodgkin-Huxley model of action potential encoding

The behaviour of the space-clamped Hodgkin-Huxley model has been studied using bandlimited white noise (0–50 Hz) as the input membrane current and taking the output as a point process in time given by the peaks of the action potentials. The frequency response and coherence functions were measured by use of the Fourier transform and digital filtering of the spike train. The results obtained are in good agreement with those already published for the simple integrator and leaky integrator models of neuronal encoding, as well as the earlier studies on the response of the Hodgkin-Huxley model to steady currents. In addition, the threshold of the model to sinusoidal membrane currents has been measured as a function of frequency over the range of 0.1–100 Hz. This shows a relatively constant level up to 2 Hz and then a clear minimum at 60 Hz, in agreement with measured thresholds of squid axons. These results are discussed in terms of the possible contributions of action potential encoding mechanisms to the frequency responses and sinusoidal thresholds which have been measured for rapidly adapting receptors.     

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.     

Excitation dynamics: insights from simplified membrane models

Excitable nerve membranes and models for their electrical activity exhibit a broad repertoire of dynamic behavior. To reveal these behaviors the theoretician seeks a model that is simple enough to analyze yet one that retains adequate biophysical realism. Here we strike such a balance by describing a two-variable simplification of the Hodgkin-Huxley (HH) model, which exhibits many membrane phenomena and reproduces, with good agreement, many HH responses. Comparisons and illustrations are presented for the single spike response, repetitive firing (and its cessation by a brief current pulse), and bistable behavior for increased extracellular K+ concentrations.     

Nonlinear cable equations for axons. I. Computations and experiments with internal current injection

Steady-state potential and current distributions resulting from internal injection of current in the squid giant axon have been measured experimentally and also computed from nonlinear membrane cable equation models by numerical methods, using the Hodgkin-Huxley equations to give the membrane current density. The solutions obtained by this method satisfactorily reproduce experimental measurements of the steady-state distribution of membrane potential. Computations of the input current-voltage characteristic for a nonlinear cable were in excellent agreement with measurements on axons. Our results demonstrate the power of Cole's equation to extract the nonlinear membrane characteristics simply from measurement of the input resistance.     

Noise effects on spike propagation in the stochastic Hodgkin-Huxley models

Effects of membrane current noise on spike propagation along a nerve fiber are studied. Additive current noise and channel noise are considered by using stochastic versions of the Hodgkin-Huxley model. The results of computer simulation show that the membrane noise causes considerable variation of the propagation time of a spike (thus changes in interspike intervals) for a small unmyelinated fiber of radius 0.1 approximately 1 micron.     

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.     

The excitable membrane-biophysical theory and experiment

Very different biophysical theories can lead to similar or even identical predictions for a wide range of experiments. This was true for the original Rashevsky theory of membrane excitation, and the later Hill formulation. Similarly, the Hodgkin-Huxley equations for membrane current are based on two postulates: current proportional to total thermodynamic potential difference across the membrane, and the independence principle. Experiments used to confirm these postulates can be predicted as well by a model in which neither postulate applies.     

On the complex dynamics of intracellular ganglion cell light responses in the cat retina

We recorded intracellular responses from cat retinal ganglion cells to sinusoidal flickering lights, and compared the response dynamics with a theoretical model based on coupled nonlinear oscillators. Flicker responses for several different spot sizes were separated in a smooth generator (G) potential and corresponding spike trains. We have previously shown that the G-potential reveals complex, stimulus-dependent, oscillatory behavior in response to sinusoidally flickering lights. Such behavior could be simulated by a modified van der Pol oscillator. In this paper, we extend the model to account for spike generation as well, by including extended Hodgkin-Huxley equations describing local membrane properties. We quantified spike responses by several parameters describing the mean and standard deviation of spike burst duration, timing (phase shift) of bursts, and the number of spikes in a burst. The dependence of these response parameters on stimulus frequency and spot size could be reproduced in great detail by coupling the van der Pol oscillator and Hodgkin-Huxley equations. The model mimics many experimentally observed response patterns, including non-phase-locked irregular oscillations. Our findings suggest that the information in the ganglion cell spike train reflects both intraretinal processing, simulated by the van der Pol oscillator, and local membrane properties described by Hodgkin-Huxley equations. The interplay between these complex processes can be simulated by changing the coupling coefficients between the two oscillators. Our simulations therefore show that irregularities in spike trains, which normally are considered to be noise, may be interpreted as complex oscillations that might carry information.To the memory of Prof. Otto-Joachim Grusser     

Discrete-state theory in nerve impulse modelling.

Hodgkin-Huxley models have been the standard for describing ionic current kinetics. However, many single channel behaviors cannot be described using traditional Hodgkin-Huxley models; they can be described by expanding the Hodgkin-Huxley models to have multiple resting and inactivated states. The model, based on charge translocation between a finite number of discrete Markovian states, is a biophysical kinetic model, according to current generalizations of channel structure, capable of reproducing channel behavior. The elaboration of the model is based on the Markov process. This type of model assumes that each channel has a discrete number of states that are connected by a kinetic diagram that defines the allowable transitions between these states and the rates at which these transitions occur. The application of the model presented here leads to results in accordance with the experimental data regarding the shape and characteristics of the nerve impulse registered along the nerve fibre. Unlike the traditional Hodgkin-Huxley models, the model based on the Markov processes has the advantage of removing the empirical equations, simplifying the computation of the membrane potential and revealing the single-channel variables. The average behavior is obtained by the repetition in one channel of the same stimulus, a number of times equal to the number of channels, which means that the macroscopic variables are predictable by the repetition, a certain number of times, of the same observations in a single channel.     

Null space in the Hodgkin-Huxley Equations. A critical test.

Voltage perturbation methods based upon topological concepts are used to elicit responses from the Hodgkin-Huxley (HH) nonlinear differential equations. These responses present a critical check upon the validity of the HH model for electrical activity across squid axon membrane. It is shown that when a constant current is applied such that a stable equilibrium and rhythmic firing are present, the following predictions are inherent in the HH system of equations: (a) Small instantaneous voltage perturbations to the axon given at points along its firing spike result in phase resetting curves (when new phase versus old phase is plotted) with an average slope of 1. (b) A larger voltage perturbation (from certain points along the firing spike) results in the permanent cessation of periodic firing, with membrane voltage rapidly approaching the equilibrium value. (c) A still larger perturbation yields phase resetting curves with an average slope equal to 0. These predictions, coupled with Tasaki's experimental demonstration that squid axons in excellent condition do give repetitive firing under constant current, provide a critical test of the validity of the HH model.     

The Voltage Dependence of the Cardiac Membrane Conductance

Solutions have been computed for the point polarization of a sheet-like membrane obeying the equations used previously (Noble, 1960, 1962) to reproduce the Purkinje fiber action potential. It was found that, in spite of the gross non-linearity of the membrane current-voltage relations, the relations between total polarizing current and displacement of membrane potential at various distances from the polarizing electrode are remarkably linear. It is therefore concluded that Johnson and Tille's (1960, 1961) results showing linear polarizing current-voltage relations obtained by passing current through the membrane from a microelectrode during the plateau of the rabbit ventricular action potential do not conflict with the Hodgkin-Huxley theory of electrical activity.     

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.