Fast Pseudo-Periodic Oscillation in the Rat Brain Voltage-gated Sodium Channel α Subunit

In the work reported here, we have investigated the changes in the activation and fast inactivation properties of the rat brain voltage-gated sodium channel (rNa_v 1.2a) α subunit, expressed heterologously in the Chinese Hamster Ovary (CHO) cells, by short depolarizing prepulses (10 – 1000 ms). The time constant of recovery from fast inactivation (τ_fast) and steady-state parameters for activation and inactivation varied in a pseudo-oscillatory fashion with the duration and amplitude of a sustained prepulse. A consistent oscillation was observed in most of the steady-state and non-inactivating current parameters with a time period close to 225 ms, although a faster oscillation of time period 125 ms was observed in the τ_fast. The studies on the non-inactivating current and steady-state activation indicate that the phase of oscillation varies from cell to cell. Co-expression of the β1 subunit with the α subunit channel suppressed the oscillation in the charge movement per single channel and free energy of steady-state inactivation, although the oscillation in the half steady-state inactivation potential remained unaltered. Incidentally, the frequencies of oscillation in the sodium channel parameters (4–8 Hz) correspond to the theta component of network oscillation. This fast pseudo-oscillatory mechanism, together with the slow pseudo-oscillatory mechanism found in these channels earlier, may contribute to the oscillations in the firing properties observed in various neuronal subtypes and many pathological conditions.     

A transient excited state model for sodium permeability changes in excitable membranes.

In this paper we explore the properties of a mathematical model for the passive sodium permeability system of excitable membranes. This model is distinguished by the explicit inclusion of a rate constant which depends not on instantaneous voltage, but on rate of voltage change. Actually, the model is a rather modest modification of the Hodgkin-Huxley model, but displays some behaviors which the H-H model does not. Among these behaviors are a pronounced inactivation shift (for certain parameter values), a difference between inactivation time constant as measured by turning off a sodium current under sustained depolarization and as measured by double pulse experiments, skip runs under sustained current stimulation, and accommodation to slowing rising currents.     

Analysis of models for crustacean stretch receptors

 The mechanisms underlying the diverse responses to step current stimuli of models [Edman et al. (1987) J Physiol (Lond) 384: 649–669] of lobster slowly adapting stretch receptor organs (SAO) and fast-adapting stretch receptor organs (FAO) are analyzed. In response to a step current, the models display three distinct types of firing reflecting the level of adaptation to the stimulation. Low-amplitude currents evoke transient firing containing one to several action potentials before the system stabilizes to a resting state. Conversely, high-amplitude stimulations induce a high frequency transient burst that can last several seconds before the model returns to its quiescent state. In the SAO model, the transition between the two regimes is characterized by a sustained pacemaker firing at an intermediate stimulation amplitude. The FAO model does not exhibit such a maintained firing; rather, the duration of the transient firing increases at first with the stimulus intensity, goes through a maximum and then decreases at larger intensities. Both models comprise seven variables representing the membrane potential, the sodium fast activation, fast inactivation, slow inactivation, the potassium fast activation, slow inactivation gating variables, and the intra cellular sodium concentration. To elucidate the mechanisms of the firing adaptations, the seven-variable model for the lobster stretch receptor neuron is first reduced to a three-dimensional system by regrouping variables with similar time scales. More precisely, we substituted the membrane potential V for the sodium fast activation equivalent potential V _m , the potassium fast inactivation V _n for the sodium fast inactivation V _h , and the sodium slow inactivation V _l for the potassium slow inactivation V _r . Comparison of the responses of the reduced models to those of the original models revealed that the main behaviors of the system were preserved in the reduction process. We classified the different types of responses of the reduced SAO and FAO models to constant current stimulation. We analyzed the transient and stationary responses of the reduced models by constructing bifurcation diagrams representing the qualitatively distinct dynamics of the models and the transitions between them. These revealed that (1) the transient firings prior to reaching the stationary state can be accounted for by the sodium slow inactivation evolving more slowly than the other two variables, so that the changes during the transient firings reflect the bifurcations that the two-dimensional system undergoes when the sodium slow inactivation, considered as a parameter, is varied; and (2) the stationary behaviors of the models are captured by the standard bifurcations of a two-dimensional system formed by the membrane potential and the potassium fast inactivation. We found that each type of firing and the transitions between them is due to the interplay between essentially three variables: two fast ones accounting for the action potential generation and the post-discharge refractoriness, and a third slow one representing the adaptation. Received: 28 February 2000 / Accepted in revised form: 4 October 2000     

Slow and incomplete inactivations of voltage-gated channels dominate encoding in synthetic neurons.

Electrically excitable channels were expressed in Chinese hamster ovary cells using a vaccinia virus vector system. In cells expressing rat brain IIA Na+ channels only, brief pulses (< 1 ms) of depolarizing current resulted in action potentials with a prolonged (0.5-3 s) depolarizing plateau; this plateau was caused by slow and incomplete Na+ channel inactivation. In cells expressing both Na+ and Drosophila Shaker H4 transient K+ channels, there were neuron-like action potentials. In cells with appropriate Na+/K+ current ratios, maintaining stimulation produced repetitive firing over a 10-fold range of frequencies but eventually led to "lock-up" of the potential at a positive value after several seconds of stimulation. The latter effect was due primarily to slow inactivation of the K+ currents. Numerical simulations of modified Hodgkin-Huxley equations describing these currents, using parameters from voltage-clamp kinetics studied in the same cells, accounted for most features of the voltage trajectories. The present study shows that insights into the mechanisms for generating action potentials and trains of action potentials in real excitable cells can be obtained from the analysis of synthetic excitable cells that express a controlled repertoire of ion channels.     

Sodium Channel NaV1.5 Expression is Enhanced in Cultured Adult Rat Skeletal Muscle Fibers

This study analyzes changes in the distribution, electrophysiological properties, and proteic composition of voltage-gated sodium channels (Na_V) in cultured adult rat skeletal muscle fibers. Patch clamp and molecular biology techniques were carried out in flexor digitorum brevis (FDB) adult rat skeletal muscle fibers maintained in vitro after cell dissociation with collagenase. After 4 days of culture, an increase of the Na_V1.5 channel type was observed. This was confirmed by an increase in TTX-resistant channels and by Western blot test. These channels exhibited increased activation time constant (τ_m) and reduced conductance, similar to what has been observed in denervated muscles in vivo, where the density of Na_V1.5 was increasing progressively after denervation. By real-time polymerase chain reaction, we found that the expression of β subunits was also modified, but only after 7 days of culture: increase in β₁ without β₄ modifications. β₁ subunit is known to induce a negative shift of the inactivation curve, thus reducing current amplitude and duration. At day 7, τ_h was back to normal and τ_m still increased, in agreement with a decrease in sodium current and conductance at day 4 and normalization at day 7. Our model is a useful tool to study the effects of denervation in adult muscle fibers in vitro and the expression of sodium channels. Our data evidenced an increase in Na_V1.5 channels and the involvement of β subunits in the regulation of sodium current and fiber excitability.     

Sea Anemone Toxin (ATX II) Modulation of Heart and Skeletal Muscle Sodium Channel α-Subunits Expressed in tsA201 Cells

We have expressed recombinant α-subunits of hH1 (human heart subtype 1), rSkM1 (rat skeletal muscle subtype 1) and hSkM1 (human skeletal muscle) sodium channels in human embryonic kidney cell line, namely the tsA201 cells and compared the effects of ATX II on these sodium channel subtypes. ATX II slows the inactivation phase of hH1 with little or no effect on activation. At intermediate concentrations of ATX II the time course of inactivation is biexponential due to the mixture of free (fast component, τ^fast _h ) and toxin-bound (slow component, τ^slow _h ) channels. The relative amplitude of τ^slow _h allows an estimate of the IC₅₀ values ∼11 nm. The slowing of inactivation in the presence of ATX II is consistent with destabilization of the inactivated state by toxin binding. Further evidence for this conclusion is: (i) The voltage-dependence of the current decay time constants (τ _h ) is lost or possibly reversed (time constants plateau or increase at more positive voltages in contrast to these of untreated channels). (ii) The single channel mean open times are increased by a factor of two in the presence of ATX II. (iii) The recovery from inactivation is faster in the presence of ATX II. Similar effects of ATX II on rSkM1 channel behavior occur, but only at higher concentrations of toxin (IC₅₀= 51 nm). The slowing of inactivation on hSkM1 is comparable to the one seen with rSkM1. A residual or window current appears in the presence of ATX II that is similar to that observed in channels containing mutations associated with some of the familial periodic paralyses. Received: 5 December 1995/Revised: 1 March 1996     

Knockout of the BK β2 subunit abolishes inactivation of BK currents in mouse adrenal chromaffin cells and results in slow-wave burst activity

Rat and mouse adrenal medullary chromaffin cells (CCs) express an inactivating BK current. This inactivation is thought to arise from the assembly of up to four β2 auxiliary subunits (encoded by the kcnmb2 gene) with a tetramer of pore-forming Slo1 α subunits. Although the physiological consequences of inactivation remain unclear, differences in depolarization-evoked firing among CCs have been proposed to arise from the ability of β2 subunits to shift the range of BK channel activation. To investigate the role of BK channels containing β2 subunits, we generated mice in which the gene encoding β2 was deleted (β2 knockout [KO]). Comparison of proteins from wild-type (WT) and β2 KO mice allowed unambiguous demonstration of the presence of β2 subunit in various tissues and its coassembly with the Slo1 α subunit. We compared current properties and cell firing properties of WT and β2 KO CCs in slices and found that β2 KO abolished inactivation, slowed action potential (AP) repolarization, and, during constant current injection, decreased AP firing. These results support the idea that the β2-mediated shift of the BK channel activation range affects repetitive firing and AP properties. Unexpectedly, CCs from β2 KO mice show an increased tendency toward spontaneous burst firing, suggesting that the particular properties of BK channels in the absence of β2 subunits may predispose to burst firing.     

Effect of Valproate,Lamotrigine and Levetiracetam on Excitability and Firing Properties of CA1 Neurons in Rat Brain Slices

The purpose of this study was to analyze the rapid effects of the antiepileptic drugs valproate, lamotrigine, and levetiracetam on excitability and firing properties of hippocampal neurons. The drug effects on resting potential, action potential, and repetitive firing properties were studied in whole-cell current-clamp recordings of CA1 neurons in rat brain slices. Lamotrigine changed action potential rising slope by −24 ± 38 V/s (mean ± SD), peak amplitude by −6.8 ± 5.0 mV, and maximum firing frequency by −60 ± 13%. Lamotrigine thereto increased the voltage threshold by 4.3 ± 4.2 mV and augmented the action potential attenuation during repetitive firing. All effects were significant (P < 0.01 to P < 0.0002) compared to control cells. Valproate and levetiracetam showed no significant effects on these parameters. None of the tested drugs had a significant effect on the resting potential. The lamotrigine effects are consistent with sodium channel blocking which may explain or contribute to the antiepileptic mode of action. Valproate and levetiracetam did not show these effects and the mechanism of their antiepileptic action need to be different. These findings (valproate) differ in some respects from findings reported in cultured or dissociated neurons. In a slice where the neurons have largely preserved connections, drug effects are likely to be more similar to the therapeutic action in the brain.     

Repetitive activity of a branched Hodgkin-Huxley axon with multiple encoding sites

Afferent activity in a receptor afferent fiber with several encoding sites is generally believed to represent the activity of the fastest pacemaker that resets all more slowly encoding sites. Alternatively, some impulse mixing as well as some nonlinear summation of receptor current to a single encoder have been considered. In this article the repetitive firing activity of a Hodgkin-Huxley axon consisting of two branches that join into a single stem axon was investigated. The model axon was stimulated by constant-current injection into either the right or the left or both branches. It was found that the model axon generated an (infinite) train of action potentials if the input current was large enough. The discharge frequency found was constant, and on combined stimulation of both branches with different current, the site of impulse initiation was always in the branch receiving the higher input current, excluding a simple impulse mixing. On the other hand, the combined stimulation of both branches evoked repetitive firing with a higher frequency than expected by the pacemaker-resetting hypothesis. Moreover, a stimulus that is subthreshold for repetitive firing if injected into one branch yields repetitive firing when it is injected into both branches, a behavior inconsistent with impulse mixing and pacemaker resetting. On the other hand, current injection into one branch allowed repetitive activity only within a rather limited range of firing frequencies. Using distributed current injection into both branches, however, allowed many more different firing frequencies. Such behavior is inconsistent with both pacemaker resetting and (nonlinear) input current summation. Consequently, the repetitive firing behavior of a branched Hodgkin-Huxley axon with multiple encoding sites appears to be more complex than postulated in the simple hypotheses.     

Theoretical reconstruction of myotonia and paralysis caused by incomplete inactivation of sodium channels.

Muscle fibers from individuals with hyperkalemic periodic paralysis generate repetitive trains of action potentials (myotonia) or large depolarizations and block of spike production (paralysis) when the extracellular K+ is elevated. These pathologic features are thought to arise from mutations of the sodium channel alpha subunit which cause a partial loss of inactivation (steady-state Popen approximately 0.02, compared to < 0.001 in normal channels). We present a model that provides a possible mechanism for how this small persistent sodium current leads to repetitive firing, why the integrity of the T-tubule system is required to produce myotonia, and why paralysis will occur when a slightly larger proportion of channels fails to inactivate. The model consists of a two-compartment system to simulate the surface and T-tubule membranes. When the steady-state sodium channel open probability exceeds 0.0075, trains of repetitive discharges occur in response to constant current injection. At the end of the current injection, the membrane potential may either return to the normal resting value, continue to discharge repetitive spikes, or settle to a new depolarized equilibrium potential. This after-response depends on both the proportion of noninactivating sodium channels and the magnitude of the activity-driven K+ accumulation in the T-tubular space. A reduced form of model is presented in which a two-dimensional phase-plane analysis shows graphically how this diversity of after-responses arises as extracellular [K+] and the proportion of noninactivating sodium channels are varied.     

A gating charge interaction required for late slow inactivation of the bacterial sodium channel NavAb

Voltage-gated sodium channels undergo slow inactivation during repetitive depolarizations, which controls the frequency and duration of bursts of action potentials and prevents excitotoxic cell death. Although homotetrameric bacterial sodium channels lack the intracellular linker-connecting homologous domains III and IV that causes fast inactivation of eukaryotic sodium channels, they retain the molecular mechanism for slow inactivation. Here, we examine the functional properties and slow inactivation of the bacterial sodium channel NavAb expressed in insect cells under conditions used for structural studies. NavAb activates at very negative membrane potentials (V_1/2 of approximately −98 mV), and it has both an early phase of slow inactivation that arises during single depolarizations and reverses rapidly, and a late use-dependent phase of slow inactivation that reverses very slowly. Mutation of Asn49 to Lys in the S2 segment in the extracellular negative cluster of the voltage sensor shifts the activation curve ∼75 mV to more positive potentials and abolishes the late phase of slow inactivation. The gating charge R3 interacts with Asn49 in the crystal structure of NavAb, and mutation of this residue to Cys causes a similar positive shift in the voltage dependence of activation and block of the late phase of slow inactivation as mutation N49K. Prolonged depolarizations that induce slow inactivation also cause hysteresis of gating charge movement, which results in a requirement for very negative membrane potentials to return gating charges to their resting state. Unexpectedly, the mutation N49K does not alter hysteresis of gating charge movement, even though it prevents the late phase of slow inactivation. Our results reveal an important molecular interaction between R3 in S4 and Asn49 in S2 that is crucial for voltage-dependent activation and for late slow inactivation of NavAb, and they introduce a NavAb mutant that enables detailed functional studies in parallel with structural analysis.     

Effect of N-bromoacetamide on single sodium channel currents in excised membrane patches

We have studied the effect of N-bromoacetamide (NBA) on the behavior of single sodium channel currents in excised patches of rat myotube membrane at 10 degree C. Inward sodium currents were activated by voltage steps from holding potentials of about -100 mV to test potentials of -40 mV. The cytoplasmic-face solution was isotonic CsF. Application of NBA or pronase to the cytoplasmic face of the membrane irreversibly removed sodium channel inactivation, as determined by averaged single-channel records. Teh lifetime of the open channel at - 40 mV was increased about 10-fold by NBA treatment without affecting the amplitude of single-channel currents. A binomial analysis was used both before and after treatment to determine the number of channels within the excised patch. NBA was shown to have little effect on activation kinetics, as determined by an examination of both the rising phase of averaged currents and measurements f the delay between the start of the pulse and the first channel opening. Our data support a kinetic model of sodium channel activation in which the rate constant leading back from the open state to the last closed state is slower than expected from a strict Hodgkin-Huxley model. The data also suggest that the normal open-channel lifetime is primarily determined by the inactivation process in the voltage range we have examined.     

A model for axonal propagation incorporating both radial and axial ionic transport.

We present an axonal model that explicitly includes ionic diffusion in the intracellular, periaxonal, and extracellular spaces and that incorporates a Hodgkin-Huxley membrane, extended with potassium channel inactivation and active ion transport. Although ionic concentration changes may not be significant in the time course of one action potential, they are important when considering the long-term behavior (seconds to minutes) of an axon. We demonstrate this point with simulations of transected axons where ions are moving between the intra- and extracellular spaces through an opening that is sealing with time. The model predicts that sealing must occur within a critical time interval after the initial injury to prevent the entire axon from becoming permanently depolarized. This critical time interval becomes considerably shorter when active ion transport is disabled. Furthermore, the model can be used to study the effects of sodium and potassium channel inactivation; e.g., sodium inactivation must be almost complete (within 0.02%) to obtain simulation results that are realistic.     

The potassium A-current,low firing rates and rebound excitation in Hodgkin-Huxley models

It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I _A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI _A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI _A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI _A parameters given by Connor and Stevens for the crab axon. When theI _A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg _A decreases and as inactivation rate decreases. We also study howI _A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.     

High-Frequency Stimulation of Excitable Cells and Networks

High-frequency (HF) stimulation has been shown to block conduction in excitable cells including neurons and cardiac myocytes. However, the precise mechanisms underlying conduction block are unclear. Using a multi-scale method, the influence of HF stimulation is investigated in the simplified FitzhHugh-Nagumo and biophysically-detailed Hodgkin-Huxley models. In both models, HF stimulation alters the amplitude and frequency of repetitive firing in response to a constant applied current and increases the threshold to evoke a single action potential in response to a brief applied current pulse. Further, the excitable cells cannot evoke a single action potential or fire repetitively above critical values for the HF stimulation amplitude. Analytical expressions for the critical values and thresholds are determined in the FitzHugh-Nagumo model. In the Hodgkin-Huxley model, it is shown that HF stimulation alters the dynamics of ionic current gating, shifting the steady-state activation, inactivation, and time constant curves, suggesting several possible mechanisms for conduction block. Finally, we demonstrate that HF stimulation of a network of neurons reduces the electrical activity firing rate, increases network synchronization, and for a sufficiently large HF stimulation, leads to complete electrical quiescence. In this study, we demonstrate a novel approach to investigate HF stimulation in biophysically-detailed ionic models of excitable cells, demonstrate possible mechanisms for HF stimulation conduction block in neurons, and provide insight into the influence of HF stimulation on neural networks.     

Inactivation mode of sodium channels defines the different maximal firing rates of conventional versus atypical midbrain dopamine neurons

Two subpopulations of midbrain dopamine (DA) neurons are known to have different dynamic firing ranges in vitro that correspond to distinct projection targets: the originally identified conventional DA neurons project to the dorsal striatum and the lateral shell of the nucleus accumbens, whereas an atypical DA population with higher maximum firing frequencies projects to prefrontal regions and other limbic regions including the medial shell of nucleus accumbens. Using a computational model, we show that previously identified differences in biophysical properties do not fully account for the larger dynamic range of the atypical population and predict that the major difference is that originally identified conventional cells have larger occupancy of voltage-gated sodium channels in a long-term inactivated state that recovers slowly; stronger sodium and potassium conductances during action potential firing are also predicted for the conventional compared to the atypical DA population. These differences in sodium channel gating imply that longer intervals between spikes are required in the conventional population for full recovery from long-term inactivation induced by the preceding spike, hence the lower maximum frequency. These same differences can also change the bifurcation structure to account for distinct modes of entry into depolarization block: abrupt versus gradual. The model predicted that in cells that have entered depolarization block, it is much more likely that an additional depolarization can evoke an action potential in conventional DA population. New experiments comparing lateral to medial shell projecting neurons confirmed this model prediction, with implications for differential synaptic integration in the two populations.     

A simple Markov model of sodium channels with a dynamic threshold

Characteristics of action potential generation are important to understanding brain functioning and, thus, must be understood and modeled. It is still an open question what model can describe concurrently the phenomena of sharp spike shape, the spike threshold variability, and the divisive effect of shunting on the gain of frequency-current dependence. We reproduced these three effects experimentally by patch-clamp recordings in cortical slices, but we failed to simulate them by any of 11 known neuron models, including one- and multi-compartment, with Hodgkin-Huxley and Markov equation-based sodium channel approximations, and those taking into account sodium channel subtype heterogeneity. Basing on our voltage-clamp data characterizing the dependence of sodium channel activation threshold on history of depolarization, we propose a 3-state Markov model with a closed-to-open state transition threshold dependent on slow inactivation. This model reproduces the all three phenomena. As a reduction of this model, a leaky integrate-and-fire model with a dynamic threshold also shows the effect of gain reduction by shunt. These results argue for the mechanism of gain reduction through threshold dynamics determined by the slow inactivation of sodium channels.     

Gain-of-function mutation of Nav1.5 in atrial fibrillation enhances cellular excitability and lowers the threshold for action potential firing

Genetic mutations of the cardiac sodium channel (SCN5A) specific only to the phenotype of atrial fibrillation have recently been described. However, data on the biophysical properties of SCN5A variants associated with atrial fibrillation are scarce. In a mother and son with lone atrial fibrillation, we identified a novel SCN5A coding variant, K1493R, which altered a highly conserved residue in the DIII-IV linker and was located six amino acids downstream from the fast inactivation motif of sodium channels. Biophysical studies of K1493R in tsA201 cells demonstrated a significant positive shift in voltage-dependence of inactivation and a large ramp current near resting membrane potential, indicating a gain-of-function. Enhanced cellular excitability was observed in transfected HL-1 atrial cardiomyocytes, including spontaneous action potential depolarizations and a lower threshold for action potential firing. These novel biophysical observations provide molecular evidence linking cellular “hyperexcitability” as a mechanism inducing vulnerability to this common arrhythmia.