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.     

Impact of fast sodium channel inactivation on spike threshold dynamics and synaptic integration

Neurons spike when their membrane potential exceeds a threshold value. In central neurons, the spike threshold is not constant but depends on the stimulation. Thus, input-output properties of neurons depend both on the effect of presynaptic spikes on the membrane potential and on the dynamics of the spike threshold. Among the possible mechanisms that may modulate the threshold, one strong candidate is Na channel inactivation, because it specifically impacts spike initiation without affecting the membrane potential. We collected voltage-clamp data from the literature and we found, based on a theoretical criterion, that the properties of Na inactivation could indeed cause substantial threshold variability by itself. By analyzing simple neuron models with fast Na inactivation (one channel subtype), we found that the spike threshold is correlated with the mean membrane potential and negatively correlated with the preceding depolarization slope, consistent with experiments. We then analyzed the impact of threshold dynamics on synaptic integration. The difference between the postsynaptic potential (PSP) and the dynamic threshold in response to a presynaptic spike defines an effective PSP. When the neuron is sufficiently depolarized, this effective PSP is briefer than the PSP. This mechanism regulates the temporal window of synaptic integration in an adaptive way. Finally, we discuss the role of other potential mechanisms. Distal spike initiation, channel noise and Na activation dynamics cannot account for the observed negative slope-threshold relationship, while adaptive conductances (e.g. K+) and Na inactivation can. We conclude that Na inactivation is a metabolically efficient mechanism to control the temporal resolution of synaptic integration.     

Use-dependent potentiation of the Nav1.6 sodium channel

Nav1.2 and Nav1.6 are two voltage-gated sodium channel isoforms that are abundant in the adult central nervous system. These channels are expressed in different cells and localized in different neuronal regions, which may reflect functional specialization. To examine this possibility, we compared the properties of Nav1.2 and Nav1.6 in response to a rapid series of repetitive depolarizations. Currents through Nav1.6 coexpressed with beta1 demonstrated use-dependent potentiation during a rapid train of depolarizations. This potentiation was in contrast to the use-dependent decrease in current for Nav1.2 with beta1. The voltage dependence of potentiation correlated with the voltage dependence of activation, and it still occurred when fast inactivation was removed by mutation. Rapid stimulation accelerated a slow phase of activation in the Nav1.6 channel that had fast inactivation removed, resulting in faster channel activation. Although the Nav1.2 channel with fast inactivation removed also demonstrated slightly faster activation, that channel showed very pronounced slow inactivation compared to Nav1.6. These results indicate that potentiation of Nav1.6 sodium currents results from faster channel activation, and that this effect is masked by slow inactivation in Nav1.2. The data suggest that Nav1.6 might be more resistant to inactivation, which might be helpful for high-frequency firing at nodes of Ranvier compared to Nav1.2.     

Interaction of scorpion alpha-toxins with cardiac sodium channels: binding properties and enhancement of slow inactivation

The effects of the scorpion alpha-toxins Lqh II, Lqh III, and LqhalphaIT on human cardiac sodium channels (hH1), which were expressed in human embryonic kidney (HEK) 293 cells, were investigated. The toxins removed fast inactivation with EC(50) values of <2.5 nM (Lqh III), 12 nM (Lqh II), and 33 nM (LqhalphaIT). Association and dissociation rates of Lqh III were much slower than those of Lqh II and LqhalphaIT, such that Lqh III would not dissociate from the channel during a cardiac activation potential. The voltage dependence of toxin dissociation from hH1 channels was nearly the same for all toxins tested, but it was different from that found for skeletal muscle sodium channels (muI; Chen et al. 2000). These results indicate that the voltage dependence of toxin binding is a property of the channel protein. Toxin dissociation remained voltage dependent even at high voltages where activation and fast inactivation is saturated, indicating that the voltage dependence originates from other sources. Slow inactivation of hH1 and muI channels was significantly enhanced by Lqh II and Lqh III. The half-maximal voltage of steady-state slow inactivation was shifted to negative values, the voltage dependence was increased, and, in particular for hH1, slow inactivation at high voltages became more complete. This effect exceeded an expected augmentation of slow inactivation owing to the loss of fast inactivation and, therefore, shows that slow sodium channel inactivation may be directly modulated by scorpion alpha-toxins.     

A slow component of gating current in crayfish giant axons resembles inactivation charge movement.

Recent experimental evidence from a number of preparations indicates that sodium channel inactivation may be intrinsically voltage sensitive. Intrinsically voltage sensitive inactivation should produce a charge movement. Crayfish giant axons provide a unique opportunity to reexamine the slower components of gating currents (Ig) for a contribution from inactivation (Igh). In reference to other axon preparations, this preparation has relatively rapid inactivation, and steady-state inactivation has a comparatively steep voltage dependence. As predicted by a two-state scheme for voltage-sensitive sodium channel inactivation, Ig in crayfish axons includes a slow component with time constant comparable to the time constant of decay of the sodium current. Allowing for some delay in its onset (60 microseconds), inactivation as described by this slow component of Ig carries roughly the amount of charge predicted by the voltage dependence of inactivation.     

Model of adaptation of the nerve fiber ranvier node membrane to the action of a steady current as the result of sodium channel inactivation

The effect of the second-order kinetics of the sodium channel inactivation system on spike activity was studied on a modified Hodgkin-Huxley model of the amphibian Ranvier node (Dodge model). The results of calculations based on the experimental data of Chiu, Kniffki, et al. suggest that the cause of adaptation of the amphibian nerve fiber membrane may be the second-order kinetics of the sodium channel inactivation system. Unlike the potassium mechanism of adaptation, the sodium mechanism is associated with constancy or a progressive decline of response amplitudes.     

G-protein activation modulates pseudo-periodic oscillation of Na channel

We have shown before that the duration and amplitude of both prolonged (1-160 s) and short (100-1000 ms) depolarizing prepulse altered all the steady-state and kinetic parameters of rNav1.2a voltage-gated sodium channel in a pseudo-oscillatory fashion with variable time period and amplitude, often superimposed on a linear trend. In this study, we have examined the effect of G-protein activation on pseudo-oscillatory properties of the rNav1.2a sodium channel alpha subunit, heterologously expressed in Chinese hamster ovary cells. G-protein modification caused insignificant changes in the slow pseudo-periodic oscillation of the activation properties of sodium channel; only the time period of the oscillation was altered from approximately 30 to 21s. In contrast, G-protein activation abolished the faster component of pseudo-periodic oscillation in steady-state inactivation properties of sodium channel; the conditioning duration dependence of steady-state inactivation becomes monotonic in nature.     

Biophysical mechanism of spike threshold dependence on the rate of rise of the membrane potential by sodium channel inactivation or subthreshold axonal potassium current

Spike threshold filters incoming inputs and thus gates activity flow through neuronal networks. Threshold is variable, and in many types of neurons there is a relationship between the threshold voltage and the rate of rise of the membrane potential (dVm/dt) leading to the spike. In primary sensory cortex this relationship enhances the sensitivity of neurons to a particular stimulus feature. While Na⁺ channel inactivation may contribute to this relationship, recent evidence indicates that K⁺ currents located in the spike initiation zone are crucial. Here we used a simple Hodgkin-Huxley biophysical model to systematically investigate the role of K⁺ and Na⁺ current parameters (activation voltages and kinetics) in regulating spike threshold as a function of dVm/dt. Threshold was determined empirically and not estimated from the shape of the Vm prior to a spike. This allowed us to investigate intrinsic currents and values of gating variables at the precise voltage threshold. We found that Na⁺ inactivation is sufficient to produce the relationship provided it occurs at hyperpolarized voltages combined with slow kinetics. Alternatively, hyperpolarization of the K⁺ current activation voltage, even in the absence of Na⁺ inactivation, is also sufficient to produce the relationship. This hyperpolarized shift of K⁺ activation allows an outward current prior to spike initiation to antagonize the Na⁺ inward current such that it becomes self-sustaining at a more depolarized voltage. Our simulations demonstrate parameter constraints on Na⁺ inactivation and the biophysical mechanism by which an outward current regulates spike threshold as a function of dVm/dt.     

Structural determinants of slow inactivation in human cardiac and skeletal muscle sodium channels.

Skeletal and heart muscle excitability is based upon the pool of available sodium channels as determined by both fast and slow inactivation. Slow inactivation in hH1 sodium channels significantly differs from slow inactivation in hSkM1. The beta(1)-subunit modulates fast inactivation in human skeletal sodium channels (hSkM1) but has little effect on fast inactivation in human cardiac sodium channels (hH1). The role of the beta(1)-subunit in sodium channel slow inactivation is still unknown. We used the macropatch technique on Xenopus oocytes to study hSkM1 and hH1 slow inactivation with and without beta(1)-subunit coexpression. Our results indicate that the beta(1)-subunit is partly responsible for differences in steady-state slow inactivation between hSkM1 and hH1 channels. We also studied a sodium channel chimera, in which P-loops from each domain in hSkM1 sodium channels were replaced with corresponding regions from hH1. Our results show that these chimeras exhibit hH1-like properties of steady-state slow inactivation. These data suggest that P-loops are structural determinants of sodium channel slow inactivation, and that the beta(1)-subunit modulates slow inactivation in hSkM1 but not hH1. Changes in slow inactivation time constants in sodium channels coexpressed with the beta(1)-subunit indicate possible interactions among the beta(1)-subunit, P-loops, and the slow inactivation gate in sodium channels.     

Cardiac sodium channel Markov model with temperature dependence and recovery from inactivation

A Markov model of the cardiac sodium channel is presented. The model is similar to the CA1 hippocampal neuron sodium channel model developed by Kuo and Bean (1994. Neuron. 12:819-829) with the following modifications: 1) an additional open state is added; 2) open-inactivated transitions are made voltage-dependent; and 3) channel rate constants are exponential functions of enthalpy, entropy, and voltage and have explicit temperature dependence. Model parameters are determined using a simulated annealing algorithm to minimize the error between model responses and various experimental data sets. The model reproduces a wide range of experimental data including ionic currents, gating currents, tail currents, steady-state inactivation, recovery from inactivation, and open time distributions over a temperature range of 10 degrees C to 25 degrees C. The model also predicts measures of single channel activity such as first latency, probability of a null sweep, and probability of reopening.     

Physico-chemical model of a voltage-gated sodium channel

A permeant ion is known to create in the channel pore a local electrical field, the intensity of which exceeds the intensity of an electrical field produced by the membrane potential. In our study, we consider a sodium channel model, in which the effects of a permeant ion, an inactivating particle, and pharmacological agents on mobile charged groups of the channel are semi-phenomenologically taken into account by using motion equations for a generalized structural variable. Stationary solutions for the equation correspond to “open,” “closed,” and “inactivated” channel states. Because of this, the channel free energy profile, as a function of the structural variable, has three local minima. The three energy values of these states depend both on the electrical field applied externally and on the near-membrane concentrations of permeant ions and acting pharmacological agents. Sodium channel activation and inactivation kinetics are considered resulting from relative changes of the free energy typical of the above three states of the channel. The results we obtained in the course of channel activation and inactivation modeling and their voltage dependence are qualitatively consistent with the commonly known experimental data. The proposed model allows one to qualitatively predict the dependence of the sodium channel kinetic characteristics on the concentrations of permeant ions and pharmacological agents.     

Sodium Channel Inactivation Is Altered by Substitution of Voltage Sensor Positive Charges

The role of the voltage sensor positive charges in fast and slow inactivation of the rat brain IIA sodium channel was investigated by mutating the second and fourth conserved positive charges in the S4 segments of all four homologous domains. Both charge-neutralizing mutations (by glutamine substitution) and charge-conserving mutations were constructed in a cDNA encoding the sodium channel α subunit. To determine if fast inactivation altered the effects of the mutations on slow inactivation, the mutations were also constructed in a channel that had fast inactivation removed by the incorporation of the IFMQ3 mutation in the III–IV linker (West, J.W., D.E. Patton, T. Scheuer, Y. Wang, A.L. Goldin, and W.A. Catterall. 1992. Proc. Natl. Acad. Sci. USA. 89:10910– 10914). Most of the mutations shifted the v_1/2 of fast inactivation in the negative direction, with the largest effects resulting from mutations in domains I and II. These shifts were in the opposite direction compared with those observed for activation. The effects of the mutations on slow inactivation depended on whether fast inactivation was intact or not. When fast inactivation was eliminated, most of the mutations resulted in positive shifts in the v_1/2 of slow inactivation. The largest effects again resulted from mutations in domains I and II. When fast inactivation was intact, the mutations in domains II and III resulted in negative shifts in the v_1/2 of slow inactivation. Neutralization of the fourth charge in domain I or II resulted in the appearance of a second component in the voltage dependence of slow inactivation that was only observable when fast inactivation was intact. These results suggest the S4 regions of all four domains of the sodium channel are involved in the voltage dependence of inactivation, but to varying extents. Fast inactivation is not strictly coupled to activation, but it derives some independent voltage sensitivity from the charges in the S4 domains. Finally, there is an interaction between the fast and slow inactivation processes.     

A Model of sodium channel-inactivation based upon the modulated blocker

An attempt is made to model sodium channel inactivation based upon real physical processes. The principle involved, which is supported by calculation and by direct appeal to experimental results, is that the gating dipole reversal or gating charge transfer that occurs when the channel is activated, markedly modulates the electrical properties of charged groups at the channel ends. Four examples of possible mechanisms that lead to channel inactivation are described. The simple four-state model that results is able to predict: (a) the steep voltage dependence of the equilibrium inactivation characteristic without the presence of any appreciable displacement current associated with inactivation; (b) the negative shift in membrane voltage of the equilibrium inactivation characteristic relative to the activation characteristic; (c) the bell-shaped dependence of inactivation time constant on membrane voltage; (d) the similarity of the membrane voltage dependence of the time constant of recovery from inactivation, to that of inactivation itself. A brief discussion of a model for sodium channel activation based upon the same physical principle is included.     

Fast and slow voltage sensor movements in HERG potassium channels

HERG encodes an inwardly-rectifying potassium channel that plays an important role in repolarization of the cardiac action potential. Inward rectification of HERG channels results from rapid and voltage-dependent inactivation gating, combined with very slow activation gating. We asked whether the voltage sensor is implicated in the unusual properties of HERG gating: does the voltage sensor move slowly to account for slow activation and deactivation, or could the voltage sensor move rapidly to account for the rapid kinetics and intrinsic voltage dependence of inactivation? To probe voltage sensor movement, we used a fluorescence technique to examine conformational changes near the positively charged S4 region. Fluorescent probes attached to three different residues on the NH₂-terminal end of the S4 region (E518C, E519C, and L520C) reported both fast and slow voltage-dependent changes in fluorescence. The slow changes in fluorescence correlated strongly with activation gating, suggesting that the slow activation gating of HERG results from slow voltage sensor movement. The fast changes in fluorescence showed voltage dependence and kinetics similar to inactivation gating, though these fluorescence signals were not affected by external tetraethylammonium blockade or mutations that alter inactivation. A working model with two types of voltage sensor movement is proposed as a framework for understanding HERG channel gating and the fluorescence signals.     

Tuning of EAG K+ channel inactivation: Molecular determinants of amplification by mutations and a small molecule

Ether-à-go-go (EAG) and EAG-related gene (ERG) K(+) channels are close homologues but differ markedly in their gating properties. ERG1 channels are characterized by rapid and extensive C-type inactivation, whereas mammalian EAG1 channels were previously considered noninactivating. Here, we show that human EAG1 channels exhibit an intrinsic voltage-dependent slow inactivation that is markedly enhanced in rate and extent by 1-10 μM 3-nitro-N-(4-phenoxyphenyl) benzamide, or ICA105574 (ICA). This compound was previously reported to have the opposite effect on ERG1 channels, causing an increase in current magnitude by inhibition of C-type inactivation. The voltage dependence of 2 μM ICA-induced inhibition of EAG1 current was half-maximal at -73 mV, 62 mV negative to the half-point for channel activation. This finding suggests that current inhibition by the drug is mediated by enhanced inactivation and not open-channel block, where the voltage half-points for current inhibition and channel activation are predicted to overlap, as we demonstrate for clofilium and astemizole. The mutation Y464A in the S6 segment also induced inactivation of EAG1, with a time course and voltage dependence similar to that caused by 2 μM ICA. Several Markov models were investigated to describe gating effects induced by multiple concentrations of the drug and the Y464A mutation. Models with the smallest fit error required both closed- and open-state inactivation. Unlike typical C-type inactivation, the rate of Y464A- and ICA-induced inactivation was not decreased by external tetraethylammonium or elevated [K(+)](e). EAG1 channel inactivation introduced by Y464A was prevented by additional mutation of a nearby residue located in the S5 segment (F359A) or pore helix (L434A), suggesting a tripartite molecular model where interactions between single residues in S5, S6, and the pore helix modulate inactivation of EAG1 channels.     

Slow inactivation in human cardiac sodium channels.

The available pool of sodium channels, and thus cell excitability, is regulated by both fast and slow inactivation. In cardiac tissue, the requirement for sustained firing of long-duration action potentials suggests that slow inactivation in cardiac sodium channels may differ from slow inactivation in skeletal muscle sodium channels. To test this hypothesis, we used the macropatch technique to characterize slow inactivation in human cardiac sodium channels heterologously expressed in Xenopus oocytes. Slow inactivation was isolated from fast inactivation kinetically (by selectively recovering channels from fast inactivation before measurement of slow inactivation) and structurally (by modification of fast inactivation by mutation of IFM1488QQQ). Time constants of slow inactivation in cardiac sodium channels were larger than previously reported for skeletal muscle sodium channels. In addition, steady-state slow inactivation was only 40% complete in cardiac sodium channels, compared to 80% in skeletal muscle channels. These results suggest that cardiac sodium channel slow inactivation is adapted for the sustained depolarizations found in normally functioning cardiac tissue. Complete slow inactivation in the fast inactivation modified IFM1488QQQ cardiac channel mutant suggests that this impairment of slow inactivation may result from an interaction between fast and slow inactivation.     

Neuromodulation of Na+ channel slow inactivation via cAMP-dependent protein kinase and protein kinase C

Neurotransmitters modulate sodium channel availability through activation of G protein-coupled receptors, cAMP-dependent protein kinase (PKA), and protein kinase C (PKC). Voltage-dependent slow inactivation also controls sodium channel availability, synaptic integration, and neuronal firing. Here we show by analysis of sodium channel mutants that neuromodulation via PKA and PKC enhances intrinsic slow inactivation of sodium channels, making them unavailable for activation. Mutations in the S6 segment in domain III (N1466A,D) either enhance or block slow inactivation, implicating S6 segments in the molecular pathway for slow inactivation. Modulation of N1466A channels by PKC or PKA is increased, whereas modulation of N1466D is nearly completely blocked. These results demonstrate that neuromodulation by PKA and PKC is caused by their enhancement of intrinsic slow inactivation gating. Modulation of slow inactivation by neurotransmitters acting through G protein-coupled receptors, PKA, and PKC is a flexible mechanism of cellular plasticity controlling the firing behavior of central neurons.     

Slow inactivation of a tetrodotoxin-sensitive current in canine cardiac Purkinje fibers.

We used the two-microelectrode voltage clamp technique and tetrodotoxin (TTX) to investigate the possible occurrence of slow inactivation of sodium channels in canine cardiac Purkinje fibers under physiologic conditions. The increase in net outward current during prolonged (5-20 s) step depolarizations (range -70 to +5 mV) following the application of TTX is time dependent, being maximal immediately following depolarization, and declining thereafter towards a steady value. To eliminate the possibility that this time-dependent current was due to inadequate voltage control of these multicellular preparations early during square clamp pulses, we also used slowly depolarizing voltage clamp ramps (range 5-100 mV/s) to ensure control of membrane potential. TTX-sensitive current also was observed with these voltage ramps; the time dependence of this current was demonstrated by the reduction of the peak current magnitude as the ramp speed was reduced. Reducing the holding potential within the voltage range of sodium channel inactivation also decreased the TTX-sensitive current observed with identical speed ramps. These results suggest that the TTX-sensitive time-dependent current is a direct measure of slow inactivation of canine cardiac sodium channels. This current may play an important role in modulating the action potential duration.     

Voltage-dependent regulation of modal gating in the rat SkM1 sodium channel expressed in Xenopus oocytes

The TTX-sensitive rat skeletal muscle sodium channel (rSkM1) exhibits two modes of inactivation (fast vs slow) when the alpha subunit is expressed alone in Xenopus oocytes. In this study, two components are found in the voltage dependence of normalized current inactivation, one having a V1/2 in the expected voltage range (approximately -50 mV, I(N)) and the other with a more hyperpolarized V1/2 (approximately -130 mV, IH) at a holding potential of -90 mV. The I(N) component is associated with the gating mode having rapid inactivation and recovery from inactivation of the macroscopic current (N-mode), while IH corresponds to the slow inactivation and recovery mode (H-mode). These two components are interconvertible and their relative contribution to the total current varies with the holding potential: I(N) is favored by hyperpolarization. The interconversion between the two modes is voltage dependent and is well fit to a first-order two-state model with a voltage dependence of e-fold/8.6 mV and a V1/2 of -62 mV. When the rat sodium channel beta 1-subunit is coinjected with rSkM1, IH is essentially eliminated and the inactivation kinetics of macroscopic current becomes rapid. These two current components and their associated gating modes may represent two conformations of the alpha subunit, one of which can be stabilized either by hyperpolarization or by binding of the beta 1 subunit.     

Slow Sodium Channel Inactivation and Use-dependent Block Modulated by the Same Domain IV S6 Residue

Voltage- and/or conformation-dependent association and dissociation of local anesthetic-class drugs from a putative receptor site in domain IV S6 of the sodium channel and slow conformation transitions of the drug-associated channel have been proposed as mechanisms of use- and frequency-dependent reduction in sodium current. To distinguish these possibilities, we have explored the reactivity to covalent modification by thiols and block of the mutations F1760C and F1760A at the putative receptor site of the cardiac sodium channel expressed as stable cell lines in HEK-293 cells. Both mutations decreased steady-state fast inactivation, shifting V1/2h from −86 ± 1.3 mV (WT) to −72.3 ± 1.4 mV (F1760C) and −67.7 ± 1 mV (F1760A). In the absence of drug, the F1760C mutant channel displayed use-dependent current reduction during pulse-train stimulation, and faster onset of slow inactivation. This mutant also retained some sensitivity to lidocaine. In contrast, the F1760A mutant showed no use-dependent current reduction or sensitivity to lidocaine. The covalent-modifying agent MTS-ET enhanced use-dependent current reduction of the F1760C mutant channel only. The use-dependent reduction in current of the covalently modified channel completely recovered with rest. Lidocaine produced no additional block during exposure to MTS-ET-treated cells (MTS-ET 43 ± 2.7%: MTS-ET lidocaine 47 ± 4.5%), implying interaction at a common binding site. The data suggest that use-dependent binding at the F1760 site results in enhanced slow inactivation rather than alteration of drug association and dissociation from that site and may be a general mechanism of action of sodium-channel blocking agents.