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.     

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.     

Membrane potential plays a dual role for chloride transport across toad skin

The Cl- -current through toad skin epithelium depends on the potential in a way consistent with a potential-controlled Cl- permeability. Computer analysis of the Koefoed-Johnsen Ussing two-membrane model provided with constant membrane permeabilities indicates that the voltage- and time-dependent currents are not caused by a trivial Goldmand-type rectification and ion redistributions following transepithelial potential pertubations. Extended with a dynamic Cl- permeability in the apical membrane according to a Hodgkin-Huxley kinetic scheme, the model predicts voltage clamp data which closely resemble experimental observations. This extension of the classic frog skin model implies that the Cl- permeability is activated by a voltage change caused by the inward Na+ current through the apical membrane.     

Gating current harmonics. II. Model simulations of axonal gating currents

A kinetic model of sodium activation gating is presented. The kinetics are based on harmonic analysis of gating current data obtained during large-amplitude sinusoidal voltage clamp in dynamic steady state. The technique classifies gating kinetic schemes into groups based on patterns of the harmonic content in the periodic gating current records. The kinetics that simulate the experimental data contain two independently constrained processes. The model predicts (a) sizable gating currents in response to hyperpolarizing voltage steps from rest; (b) a substantial increase in the initial peak of the gating current following voltage steps from prehyperpolarized potentials; (c) a small delay in the onset of sodium ion current following voltage steps from prehyperpolarized potentials; and (d) flickering during the open state in single channel current records. Although fundamentally different in kinetic structure from the Hodgkin-Huxley model, the present model reproduces the phenomenological development of Na conductance during the initiation and development of action potentials. The implications for possible gating mechanisms are discussed. A model gate is presented.     

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.     

Homomorphism on a physical system of the Hodgkin-Huxley equations for ion conductance in nerve

A homomorphism on a physical system of the Hodgkin-Huxley equations for ion conductance in nerve is derived. It is pointed out that a homomorphism can correct the Cole-Moore discrepancy in delay of conductance for voltage clamp data with initial hyperpolarization. The voltage dependence of the rate constants can also be removed. Curves are presented to compare the representation of the nerve conductances by the Hodgkin-Huxley equations and the new homomorphism.     

Cadmium block of squid calcium currents. Macroscopic data and a kinetic model

The mechanism of Cd2+ block of Ca2+ currents (ICa) was explored in squid neurons using whole-cell patch clamp. Control currents activated sigmoidally, more rapidly at more positive potentials, and did not inactivate significantly. External Cd2+ up to 250 microM reduced ICa reversibly. For small depolarizations, the current for a step of 10 ms increased to a maintained value, resembling the control; but for Vm greater than 0 mV, the increase was followed by a decrease, as Cd2+ block became greater. Final block was greater for larger depolarizations. At 0 mV the half-blocking concentration was 125 microM. Tail currents, measured as channels close, had an initial "hook" when recorded in Cd2+: currents increased transiently, then decreased. This suggests that Cd2+ escapes from some channels, which then conduct briefly before closing. Analysis of tail currents shows that Cd2+ does not slow channel closing. The data can be explained if Cd2+ is a permeant blocker of Ca2+ channels and if channels can close when occupied by Cd2+. Cd2+ permeates the channels, but binds transiently to a site in the pore, obstructing the passage of other ions (e.g., Ca2+). Dwell time depends on the transmembrane potential, becoming shorter for more negative internal potentials. A five-state model was used to simulate the steady-state and kinetic features. It combines a Hodgkin-Huxley type m2 gating scheme and a one-site Woodhull ionic blockage model for a permeant blocker and includes a closed blocked state. To fit the data, the binding site for Cd2+ had to be near the outer end of the pore, with a well depth of -12.2 RT, and with a barrier at each end of the pore. The model predicts that the Cd2+ entry rate is nearly voltage independent, but the exit rate is steeply voltage dependent (e-fold/17 mV). Analysis further suggests that the channel closes at a normal rate with Cd2+ in the pore.     

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.     

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.     

Gating current harmonics. I. Sodium channel activation gating in dynamic steady states.

Internally perfused and pronase-treated giant axons were prepared for gating current measurements. Gating current records were obtained under large-amplitude sinusoidal voltage clamp after allowing for settling times into dynamic steady states. The current records were analyzed as functions of the mean membrane potential of the test sinusoid for which the amplitude and frequency were held constant. The nonlinear analysis consisted of determining the harmonic content (amplitudes and phases) of the distorted periodic current records. The most pronounced feature found in the analysis is a dominant second harmonic centered at Emean = +10 mV. A number of other characteristic harmonic behaviors were also observed. The harmonics tend to die away for very small (less than -60 mV) and very large (greater than +72 mV) values of Emean. The harmonic behavior seen in the axonal data is basically different from that seen in gating current simulations generated by the sodium-activation kinetics of standard models, including the Hodgkin-Huxley model. Some of the differences can be reconciled without requiring fundamental changes in the model kinetic schemes. However, the dominant harmonic feature seen in the axonal data cannot be reconciled with the model kinetics without a fundamental change in the models. The axonal data suggest two moving molecular components with independent degrees of freedom whose properties are outlined on the basis of the data presented herein.     

Subthreshold Voltage Noise Due to Channel Fluctuations in Active Neuronal Membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na⁺ and K⁺ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.     

Influence of intercellular clefts on potential and current distribution in a multifiber preparation.

A theoretical model is presented for current and voltage clamp of multifiber bundles in a double sucrose gap. Attention is focused on methodological errors introduced by the intercellular cleft resistance. The bundle is approximated by a continuous geometry. Voltage distribution, as a function of radial distance and time, is defined by a parabolic partial differential equation which is specified for different membrane characteristics. Assuming a linear membrane, analytical solutions are given for current step and voltage step conditions. The theoretical relations (based on Bessel functions) may be used to calculate membrane conductance and capacity from experimental clamp data. The case of a nonlinear membrane with standard Hodgkin-Huxley kinetics for excitatory Na current is treated assuming maximum Na conductances (gNa) of 120, 10, and 1 mmho/cm2. Numerical simulations are presented for potential and current distribution in a bundle of 60 microns diameter during depolarizing voltage steps. Adequate voltage control is restricted to the peripheral fibers of the bundle whereas the membrane potential of the inner fibers deviates from the command level during early inward current, tending to the Na equilibrium potential. In the peak current-voltage diagram the loss of voltage control is reflected by an increased steepness of the negative region and a decreased slope conductance of the positive region. With gNa = 120 mmho/cm2, the positive slope conductance is approximately 25% of the slope expected from ideal space clamping. With the lower values of gNa, the slope conductance ratio is in the order of 50%. Implications of the results for an experimental voltage clamp analysis of early inward current on multifiber preparations are discussed.     

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.     

Impedance Analysis of Ion Transport Through Supported Lipid Membranes Doped with Ionophores: A New Kinetic Approach

Kinetics of facilitated ion transport through planar bilayer membranes are normally analyzed by electrical conductance methods. The additional use of electrical relaxation techniques, such as voltage jump, is necessary to evaluate individual rate constants. Although electrochemical impedance spectroscopy is recognized as the most powerful of the available electric relaxation techniques, it has rarely been used in connection with these kinetic studies. According to the new approach presented in this work, three steps were followed. First, a kinetic model was proposed that has the distinct quality of being general, i.e., it properly describes both carrier and channel mechanisms of ion transport. Second, the state equations for steady-state and for impedance experiments were derived, exhibiting the input–output representation pertaining to the model’s structure. With the application of a method based on the similarity transformation approach, it was possible to check that the proposed mechanism is distinguishable, i.e., no other model with a different structure exhibits the same input–output behavior for any input as the original. Additionally, the method allowed us to check whether the proposed model is globally identifiable (i.e., whether there is a single set of fit parameters for the model) when analyzed in terms of its impedance response. Thus, our model does not represent a theoretical interpretation of the experimental impedance but rather constitutes the prerequisite to select this type of experiment in order to obtain optimal kinetic identification of the system. Finally, impedance measurements were performed and the results were fitted to the proposed theoretical model in order to obtain the kinetic parameters of the system. The successful application of this approach is exemplified with results obtained for valinomycin–K⁺ in lipid bilayers supported onto gold substrates, i.e., an arrangement capable of emulating biological membranes.     

Simulations of voltage clamping poorly space-clamped voltage-dependent conductances in a uniform cylindrical neurite

Significant error is made by using a point voltage clamp to measure active ionic current properties in poorly space-clamped cells. This can even occur when there are no obvious signs of poor spatial control. We evaluated this error for experiments that employ an isochronal I(V) approach to analyzing clamp currents. Simulated voltage clamp experiments were run on a model neuron having a uniform distribution of a single voltage-gated inactivating ionic current channel along an elongate, but electrotonically compact, process. Isochronal Boltzmann I(V) and kinetic parameter values obtained by fitting the Hodgkin-Huxley equations to the clamp currents were compared with the values originally set in the model. Good fits were obtained for both inward and outward currents for moderate channel densities. Most parameter errors increased with conductance density. The activation rate parameters were more sensitive to poor space clamp than the I(V) parameters. Large errors can occur despite normal-looking clamp curves.     

Hodgkin-Huxley type ion channel characterization: an improved method of voltage clamp experiment parameter estimation

The Hodgkin-Huxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.     

Response of delayed (K+) channels to the time-dependent clamping function in squid giant axon. I. Ascending ramps

Squid giant axons are voltage-clamped with ascending potential ramps whose slopes range from 0.5 mV/msec to 60 mV/msec and delayed (K+) currents are observed. The parametric current-voltage curves exhibit a delay period of minimal current followed by a rapid increase of current toward a final steady state. Both the initial delay and the slope of the subsequent rising phase increase with increasing ramp slope. When the Hodgkin-Huxley equations are used to generate theoretical current-voltage curves, the sharp difference between the delay and rising phases is muted and the ramp slope must be increased to produce an adequate representation of the data. A muted biphasic response is also observed when the current-voltage curves are generated using modified Hodgkin-Huxley parameters and a correction for K+ accumulation in the periaxonal space. These modified equations provide an accurate fit for step-potential clamp current data. Since the ramp experiments include all relevant clamping potentials, the experiments provide a sensitive test for kinetic models of K+ on flow in the delayed (K+) channels of squid giant axon.