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.     

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.     

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.     

Computation of axon gating currents from dipole moment changes in channel subunits.

The gating polarizational currents were computed on the basis of the dipole moment changes occurring in nerve membrane ionic channel subunits. Membrane thickness and surface density of channels were the only parameters used in addition to the Hodgkin-Huxley model. The gating currents computed for membrane potentials where the Hodgkin-Huxley empirical formulae are reliable were found to be in good agreement with the available experimental data. It is demonstrated that the gating currents of the n and h subunits are responsible for the late slowly decaying gating currents.     

A new look at the mechanism of activation and inactivation of voltage-gated ion channels.

Studies on the kinetics of activation and inactivation of the sodium channels of the squid giant axon, on the sodium gating current, and on the properties of the non-inactivating steady-state current, are briefly reviewed. Taken in conjunction with recent evidence on the structure of voltage-gated ion channels, they have led to the development of a series-parallel model of the sodium channel that can be regarded as a modernized version of the Hodgkin-Huxley model, with some novel features. It is suggested that activation results from conformational changes brought about by the four S4 voltage sensors operating in parallel, each of which makes two discrete steps to reach the fully activated state of the channel. There follows a voltage-independent hydration step, and the channel is ready to open. Inactivation is a potential-dependent process involving a third transition of voltage sensor S4d alone, which, rather than bringing a ball and chain blocking group into position to close the channels, serves to switch the system so that it passes from an initial activated mode, in which there is a high probability of arriving at an open state with a brief latency, to a second steady-state mode, in which the probability of opening is very much lower.     

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.     

Real-time kinetic modeling of voltage-gated ion channels using dynamic clamp

We propose what to our knowledge is a new technique for modeling the kinetics of voltage-gated ion channels in a functional context, in neurons or other excitable cells. The principle is to pharmacologically block the studied channel type, and to functionally replace it with dynamic clamp, on the basis of a computational model. Then, the parameters of the model are modified in real time (manually or automatically), with the objective of matching the dynamical behavior of the cell (e.g., action potential shape and spiking frequency), but also the transient and steady-state properties of the model (e.g., those derived from voltage-clamp recordings). Through this approach, one may find a model and parameter values that explain both the observed cellular dynamics and the biophysical properties of the channel. We extensively tested the method, focusing on Na_v models. Complex Markov models (10-12 states or more) could be accurately integrated in real time at >50 kHz using the transition probability matrix, but not the explicit Euler method. The practicality of the technique was tested with experiments in raphe pacemaker neurons. Through automated real-time fitting, a Hodgkin-Huxley model could be found that reproduced well the action potential shape and the spiking frequency. Adding a virtual axonal compartment with a high density of Na_v channels further improved the action potential shape. The computational procedure was implemented in the free QuB software, running under Microsoft Windows and featuring a friendly graphical user interface.     

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.     

Further evidence that membrane thickness influences voltage-gated sodium channels.

The short-chain phospholipid, diheptanoyl phosphatidylcholine, at 520 microM, reduced the maximum inward sodium current in voltage-clamped squid giant axons by greater than 50%. Analysis of these currents by means of the Hodgkin-Huxley equations showed this reduction to be mainly the result of a large depolarizing shift in the voltage dependence of the steady state activation parameter, m infinity. The voltage dependence of the steady state inactivation parameter, h infinity, was also moved in the depolarizing direction and the axonal membrane capacitance per unit area measured at 100 kHz was increased. A longer chain length derivative, didecanoyl phosphatidylcholine, had no significant effect on the axonal sodium current at concentrations of 3.7 and 18.5 microM. Dioctanoyl phosphatidylcholine was intermediate in its effects, 200 microM producing approximately the same current suppression as 520 microM diheptanoyl phosphatidylcholine, together with depolarizing shifts in m infinity and h infinity. These effects may be contrasted with those of the normal and cyclic alkanes (1-3), which tend to move both m infinity and h infinity in the hyperpolarizing direction and to reduce the capacitance per unit area at 100 kHz. The above results are all consistent with the hypothesis that small hydrocarbons thicken, while short-chain phospholipids thin, the axonal membrane. Thus membrane thickness changes may be of considerable importance in determining the behavior of the voltage-gated sodium channel.     

Gating molecule interactions in K+ channels. A consistent explanation for cole-moore shifts and potential ramp experiments

The enhanced induction period of potassium channel currents in squid giant axon induced by hyperpolarizing prepulses (the Cole-Moore shift) is observed and analyzed for a range of depolarizing step potentials. The induction periods produced when the axon is voltage clamped with ascending potential ramps are also analyzed since both sets of experiments are incompatible with fourth-power dependence proposed by Hodgkin and Huxley for the squid K⁺ channels. When the Hodgkin-Huxley equations are modified to include the effects of interactions between gating molecules within individual channels, both the Cole-Moore and ascending potential ramp data are described with a fourth-power dependence. The constant interaction parameters which provide a consistent fit for all the data are based on the geometrical arrangement of gating molecules within the channel and the total interaction energy which stabilizes the four gating molecules in their closed configuration. A tetrahedral gating molecule geometry and an interaction energy of only 471 cal/mole provide optimal fits of all the data; the modified equations retain the ability to describe data presently described by the Hodgkin-Huxley equations in the depolarizing regime.     

A fully coupled transient excited state model for the sodium channel

Summary The behavior under voltage clamp conditions of a coupled kinetic scheme for the sodium channel is examined. The scheme is given diagrammatically by: tano Numerical simulations are presented which show that this model fits the voltage clamp data which are well described by the Hodgkin-Huxley equations, but also gives the sorts of behavior anomalous to the Hodgkin-Huxley model which have been seen experimentally. Further, straightforward changes in parameter values are shown to be capable of mimicking the ways in which some axonal preparations differ from others. Detailed, but admittedly heuristic, arguments are presented for the propositions that: 1) the model is minimal; i.e. no simpler kinetic model will fit the array of data simulated, and: 2) the transient excited state is necessary; i.e. no model of comparable simplicity with pure voltage dependent kinetics will fit the array of data simulated.     

The role of voltage-gated sodium channels in neuropathic pain

Use-dependent inhibitors of voltage-gated sodium channels (VGSC) are important therapeutic tools for chronic pain management, but are limited by possible severe side effects. Recent studies have provided much new information on the function of several voltage-gated sodium channels that are predominantly expressed in peripheral sensory neurons, and on their possible link to pathological pain states arising from injuries to the sensory nerve. The use of antisense oligonucleotides to target specific channel subtypes shows that the functional localization of the channel subtype Na(V)1.8 after nerve injury is essential for persistent pain states. The putative roles of Na(V)1.3 and Na(V)1.9 in neuropathic pain are also discussed. These studies may form a basis for developing inhibitors to target specific channel subtype(s) for use in chronic pain treatment.     

The periodic axon membrane skeleton leads to Na nanodomains but does not impact action potentials

Recent work has established that axons have a periodic skeleton structure comprising of azimuthal actin rings connected via longitudinal spectrin tetramer filaments. This structure endows the axon with structural integrity and mechanical stability. Additionally, voltage-gated sodium channels follow the periodicity of the active-spectrin arrangement, spaced ～190 nm segments apart. The impact of this periodic arrangement of sodium channels on the generation and propagation of action potentials is unknown. To address this question, we simulated an action potential using the Hodgkin-Huxley formalism in a cylindrical compartment, but instead of using a homogeneous distribution of voltage-gated sodium channels in the membrane, we applied the experimentally determined periodic arrangement. We found that the periodic distribution of voltage-gated sodium channels does not significantly affect the generation or propagation of action potentials but instead leads to large, localized sodium action currents caused by high-density sodium nanodomains. Additionally, our simulations show that the distance between periodic sodium channel strips could control axonal excitability, suggesting a previously underappreciated mechanism to regulate neuronal firing properties. Together, this work provides a critical new insight into the role of the periodic arrangement of sodium channels in axons, providing a foundation for future experimental studies.     

A Threshold Equation for Action Potential Initiation

In central neurons, the threshold for spike initiation can depend on the stimulus and varies between cells and between recording sites in a given cell, but it is unclear what mechanisms underlie this variability. Properties of ionic channels are likely to play a role in threshold modulation. We examined in models the influence of Na channel activation, inactivation, slow voltage-gated channels and synaptic conductances on spike threshold. We propose a threshold equation which quantifies the contribution of all these mechanisms. It provides an instantaneous time-varying value of the threshold, which applies to neurons with fluctuating inputs. We deduce a differential equation for the threshold, similar to the equations of gating variables in the Hodgkin-Huxley formalism, which describes how the spike threshold varies with the membrane potential, depending on channel properties. We find that spike threshold depends logarithmically on Na channel density, and that Na channel inactivation and K channels can dynamically modulate it in an adaptive way: the threshold increases with membrane potential and after every action potential. Our equation was validated with simulations of a previously published multicompartemental model of spike initiation. Finally, we observed that threshold variability in models depends crucially on the shape of the Na activation function near spike initiation (about −55 mV), while its parameters are adjusted near half-activation voltage (about −30 mV), which might explain why many models exhibit little threshold variability, contrary to experimental observations. We conclude that ionic channels can account for large variations in spike threshold.     

Trafficking and regulation of neuronal voltage-gated calcium channels

The importance of voltage-gated calcium channels is underscored by the multitude of intracellular processes that depend on calcium, notably gene regulation and neurotransmission. Given their pivotal roles in calcium (and hence, cellular) homeostasis, voltage-gated calcium channels have been the subject of intense research, much of which has focused on channel regulation. While ongoing research continues to delineate the myriad of interactions that govern calcium channel regulation, an increasing amount of work has focused on the trafficking of voltage-gated calcium channels. This includes the mechanisms by which calcium channels are targeted to the plasma membrane, and, more specifically, to their appropriate loci within a given cell. In addition, we are beginning to gain some insights into the mechanisms by which calcium channels can be removed from the plasma membrane for recycling and/or degradation. Here we highlight recent advances in our understanding of these fundamentally important mechanisms.     

Conformations of 3-carboxylic esters essential for neurotoxicity in veratrum alkaloids are loosely restricted and fluctuate

The lipid-soluble veratrum alkaloids, veratridine and cevadine, are plant neurotoxins that are agonists of voltage-gated sodium channel. Their conformations in a hydrophobic environment were analyzed by NMR spectroscopy in solution phase chloroform at low temperatures. The conformations around the 3-carboxylic esters which is essential for their neurotoxicity, was completely different from the previously reported X-ray crystallographic structure. The carbonyl oxygen atom (O28) of the carboxylic ester forms a weak intramolecular hydrogen bond with the OH proton at C4 (4-OH) that loosely restricts the conformation of the 3-veratroyl ester in veratridine and the 3-angeloyl ester in cevadine. Methylation at C4 hydroxyl group of veratridine had much reduced its neurotoxic activity relating to voltage-gated sodium channel. The results suggested that the loose conformational restrictions of the carboxylic esters are important for neurotoxicity of the veratrum alkaloids.     

A numerical approach to ion channel modelling using whole-cell voltage-clamp recordings and a genetic algorithm

The activity of trans-membrane proteins such as ion channels is the essence of neuronal transmission. The currently most accurate method for determining ion channel kinetic mechanisms is single-channel recording and analysis. Yet, the limitations and complexities in interpreting single-channel recordings discourage many physiologists from using them. Here we show that a genetic search algorithm in combination with a gradient descent algorithm can be used to fit whole-cell voltage-clamp data to kinetic models with a high degree of accuracy. Previously, ion channel stimulation traces were analyzed one at a time, the results of these analyses being combined to produce a picture of channel kinetics. Here the entire set of traces from all stimulation protocols are analysed simultaneously. The algorithm was initially tested on simulated current traces produced by several Hodgkin-Huxley–like and Markov chain models of voltage-gated potassium and sodium channels. Currents were also produced by simulating levels of noise expected from actual patch recordings. Finally, the algorithm was used for finding the kinetic parameters of several voltage-gated sodium and potassium channels models by matching its results to data recorded from layer 5 pyramidal neurons of the rat cortex in the nucleated outside-out patch configuration. The minimization scheme gives electrophysiologists a tool for reproducing and simulating voltage-gated ion channel kinetics at the cellular level.     

Differential sialylation modulates voltage-gated Na+ channel gating throughout the developing myocardium

Voltage-gated sodium channel function from neonatal and adult rat cardiomyocytes was measured and compared. Channels from neonatal ventricles required an approximately 10 mV greater depolarization for voltage-dependent gating events than did channels from neonatal atria and adult atria and ventricles. We questioned whether such gating shifts were due to developmental and/or chamber-dependent changes in channel-associated functional sialic acids. Thus, all gating characteristics for channels from neonatal atria and adult atria and ventricles shifted significantly to more depolarized potentials after removal of surface sialic acids. Desialylation of channels from neonatal ventricles did not affect channel gating. After removal of the complete surface N-glycosylation structures, gating of channels from neonatal atria and adult atria and ventricles shifted to depolarized potentials nearly identical to those measured for channels from neonatal ventricles. Gating of channels from neonatal ventricles were unaffected by such deglycosylation. Immunoblot gel shift analyses indicated that voltage-gated sodium channel alpha subunits from neonatal atria and adult atria and ventricles are more heavily sialylated than alpha subunits from neonatal ventricles. The data are consistent with approximately 15 more sialic acid residues attached to each alpha subunit from neonatal atria and adult atria and ventricles. The data indicate that differential sialylation of myocyte voltage-gated sodium channel alpha subunits is responsible for much of the developmental and chamber-specific remodeling of channel gating observed here. Further, cardiac excitability is likely impacted by these sialic acid-dependent gating effects, such as modulation of the rate of recovery from inactivation. A novel mechanism is described by which cardiac voltage-gated sodium channel gating and subsequently cardiac rhythms are modulated by changes in channel-associated sialic acids.     

Extracellular pH selectively modulates recovery from sodium inactivation in frog myelinated nerve.

The Hodgkin-Huxley kinetic parameters, alpha h and beta h, which govern the rate of recovery from and development of sodium channel inactivation, respectively, have been measured as a function of membrane potential and external pH using a three-pulse protocol. alpha h but not beta h is substantially accelerated by reducing external pH from 7.4 to 6.4. The alpha h vs. voltage curve appears to be selectively shifted in the depolarizing direction by approximately 12 mV for this pH change, giving an apparent, h infinity curve shift of approximately 6 mV in the same direction (less inactivation).     

Potassium currents in the giant axon of the crab Carcinus maenas

Measurements were made of the kinetic and steady-state characteristics of the potassium conductance in the giant axon of the crab Carcinus maenas. These measurements were made in the presence of tetrodotoxin, using the feedback amplifier concept introduced by Dodge and Frankenhaeuser (J. Physiol, (London) 143:76-90). The conductance increase during depolarizing voltage-clamp pulses was analyzed assuming that two separate potassium channels exist in these axons. The first potassium channel exhibited activation and fast inactivation gating which could be fitted using the m3h, Hodgkin-Huxley formalism. The second potassium channel exhibited the standard n4 Hodgkin-Huxley kinetics. These two postulated channels are blocked by internal application of caesium, tetraethylammonium and sodium ions. External application of 4 amino-pyridine also blocks these channels.