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.     

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.     

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.     

A study of the crustacean axon repetitive response. 3. A comparison of the effect of veratrine sulfate solution and potassium-rich solutions

The crustacean single nerve fiber gives rise to trains of impulses during a prolonged depolarizing stimulus. It is well known that the alkaloid veratrine itself causes a prolonged depolarization; and consequently it was of interest to investigate the effect of this chemically produced depolarization on repetitive firing in the single axon and compare it with the effect of depolarization by an applied stimulating current or by a potassium-rich solution. It was found that veratrine depolarization, though similar in some respects to a potassium-rich depolarization of depolarizing current effect, was in many respects quite different. (1) At low veratrine concentration, less than 1 Mg%, the negative after potential following a spike action potential was prolonged and augmented. At higher concentrations or after a long period of time, veratrine caused a prolonged steady state depolarization of the membrane, the “veratrine response”. The prolonged plateau depolarization response could be elicited with or without an action potential spike by a short or long duration stimulating pulse, but only if the veratrine depolarization was prevented or offset by an applied conditioning hyperpolarizing inward current. (2) The “veratrine response” resembled the potassium-rich solution response in the plateau-like contour of the depolarization and the very low membrane resistance during this plateau phase. Like the potassium response, it was possible to obtain a typical hyperpolarizing response with an inwardly directed current pulse if applied during the plateau phase. During the negative after potential augmented with veratrine, however, this hyperpolarizing response was not observed. (3) In contrast to the potassium response, however, the “veratrine response” is intimately associated with the sodium concentration in the external medium. The depolarization in millivolts is linearly related to the log of the concentration of external sodium. Moreover, during veratrine action there is a continuous and progressive inactivation of the sodium mechanism which ultimately terminates repetitive firing and abolishes the spike action potential. Then even with conditioning hyperpolarization only the slow response may be elicited in veratrine, occasionally with a spike superimposed if sodium is present, but without repetitive firing. (4) It is concluded that veratrine action is the result of a chemical or metabolic reaction by the alkaloid in the membrane. It is suggested that veratrine may inhibit the sodium extrusion mechanism, or may itself compete for sites in the membrane with calcium and/or sodium. This explains the inhibiting effect of high calcium, the abolition of the “veratrine response” with low temperature and high calcium combined and the progressive inactivation of the sodium system.     

Variability of bursting patterns in a neuron model in the presence of noise

Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.     

Functional Consequences of a Decreased Potassium Affinity in a Potassium Channel Pore : Ion Interactions and C-Type Inactivation

Ions bound near the external mouth of the potassium channel pore impede the C-type inactivation conformational change (Lopez-Barneo, J., T. Hoshi, S. Heinemann, and R. Aldrich. 1993. Receptors Channels. 1:61– 71; Baukrowitz, T., and G. Yellen. 1995. Neuron. 15:951–960). In this study, we present evidence that the occupancy of the C-type inactivation modulatory site by permeant ions is not solely dependent on its intrinsic affinity, but is also a function of the relative affinities of the neighboring sites in the potassium channel pore. The A463C mutation in the S6 region of Shaker decreases the affinity of an internal ion binding site in the pore (Ogielska, E.M., and R.W. Aldrich, 1998). However, we have found that this mutation also decreases the C-type inactivation rate of the channel. Our studies indicate that the C-type inactivation effects observed with substitutions at position A463 most likely result from changes in the pore occupancy of the channel, rather than a change in the C-type inactivation conformational change. We have found that a decrease in the potassium affinity of the internal ion binding site in the pore results in lowered (electrostatic) interactions among ions in the pore and as a result prolongs the time an ion remains bound at the external C-type inactivation site. We also present evidence that the C-type inactivation constriction is quite local and does not involve a general collapse of the selectivity filter. Our data indicate that in A463C potassium can bind within the selectivity filter without interfering with the process of C-type inactivation.     

Effects of ionizing radiation on artificial (planar) lipid membranes. I. Radiation inactivation of the ion channel gramicidin A

The ion channel formed by the pentadecapeptide gramicidin A in planar lipid membranes is extremely sensitive to ionizing radiation. The membrane conductance may drop by several orders of magnitude under appropriate experimental conditions (low pH and presence of oxygen). The radiation sensitivity is strongly reduced for gramicidin M-. This analogue has the four tryptophan residues replaced by phenylalanines. Experiments performed in the presence of various radical scavengers suggest that the inactivation of the channel is due to a combined action of OH and of HO2 radicals at the tryptophan residues. The shape of the inactivation curves following continuous radiolysis or pulse radiolysis were found to be in fair agreement with a simple model which assumes that the damage of a single tryptophan residue is sufficient for channel inactivation. The conductance of inactivated channels could not be resolved within the experimental accuracy. This is contrary to photolysis of gramicidin channels found by Busath and Waldbilling (1983), where a broad distribution of low conductance states was observed. The inactivation by radiolysis seems to represent an 'all-or-none-process' of the channel conductance.     

Modulation of potassium channel gating by external divalent cations

We have examined the actions of Zn2+ ions on Shaker K channels. We found that low (100 microM) concentrations of Zn2+ produced a substantial (approximately three-fold) slowing of the kinetics of macroscopic activation and inactivation. Channel deactivation was much less affected. These results were obtained in the presence of 5 mM Mg2+ and 4 mM Ca2+ in the external solution and so are unlikely to be due to modification of membrane surface charges. Furthermore, the action of 100 microM Zn2+ on activation was equivalent to a 70-mV reduction of a negative surface potential whereas the effects on deactivation would require a 15-mV increase in surface potential. External H+ ions reduced the Zn-induced slowing of macroscopic activation with an apparent pK of 7.3. Treatment of Shaker K channels with the amino group reagent, trinitrobenzene sulfonic acid (TNBS), substantially reduced the effects of Zn2+. All these results are qualitatively similar to the actions of Zn2+ on squid K channels, indicating that the binding site may be a common motif in potassium channels. Studies of single Shaker channel properties showed that Zn2+ ions had little or no effect on the open channel current level or on the open channel lifetime. Rather, Zn2+ substantially delayed the time to first channel opening. Thus, K channels appear to contain a site to which divalent cations bind and in so doing act to slow one or more of the rate constants controlling transitions among closed conformational states of the channel.     

Morphological and electrophysiological aspects of dorsal median paired neurons generating plateau action potentials in the terminal abdominal ganglion of the insect central nervous system

Among the three clusters of dorsal unpaired median neurons of the Periplaneta americana terminal abdominal ganglion, another type of neuron has been characterized by anterograde cobalt stainings and microelectrode technique. These neurons are bilaterally distributed in the ganglion. Their axons ipsilaterally exit the ganglion via the anterior proctodeal nerves, to innervate the proctodeum. They are characterized by a long-duration overshooting action potentials and a low firing frequency. Most often the depolarizing phase is composed of two peaks: a fast spike followed by a slow phase. Tetrodotoxin suppressed the fast peak and blocked the spontaneous activity suggesting that sodium channels are involved in the depolarizing phase as well as in the initiation of the action potential. Calcium channel blockers induced a disappearing of the slow depolarizing phase indicating the participation of calcium ions and a reduction of the afterhyperpolarization reflecting the participation of calcium-activated potassium channels. Furthermore, cadmium, as lanthanum or barium, induced a long-lasting plateau potential, which would be due to a persistent sodium conductance. Tetraethylammonium increased the duration of the action potential indicating that potassium channels are implicated in the falling phase. The results demonstrate that these neurons are different from other cells, especially dorsal unpaired median neurons, of the central nervous system of the cockroach.Abbreviations DUM dorsal unpaired median - SDP slow depolarizing phase - AP action potential - PAP plateau action potential - TAG terminal abdominal ganglion - CNS central nervous system     

Cardiac Na currents and the inactivating, reopening, and waiting properties of single cardiac Na channels

Tetrodotoxin (TTX)-sensitive Na currents were examined in single dissociated ventricular myocytes from neonatal rats. Single channel and whole cell currents were measured using the patch-clamp method. The channel density was calculated as 2/micron 2, which agreed with our usual finding of four channels per membrane patch. At 20 degrees C, the single channel conductance was 20 pS. The open time distributions were fit by a single-exponential function with a mean open time of approximately 1.0 ms at membrane potentials from -60 to -40 mV. Averaged single channel and whole cell currents were similar when scaled and showed both fast and slow rates of inactivation. The inactivation and activation gating shifted quickly to hyperpolarized potentials for channels in cell-attached as well as excised patches, whereas a much slower shift occurred in whole cells. Slowly inactivating currents were present in both whole cell and single channel current measurements at potentials as positive as -40 mV. In whole cell measurements, the potential range could be extended, and slow inactivation was present at potentials as positive as -10 mV. The curves relating steady state activation and inactivation to membrane potential had very little overlap, and slow inactivation occurred at potentials that were positive to the overlap. Slow inactivation is in this way distinguishable from the overlap or window current, and the slowly inactivating current may contribute to the plateau of the rat cardiac action potential. On rare occasions, a second set of Na channels having a smaller unit conductance and briefer duration was observed. However, a separate set of threshold channels, as described by Gilly and Armstrong (1984. Nature [Lond.]. 309:448), was not found. For the commonly observed Na channels, the number of openings in some samples far exceeded the number of channels per patch and the latencies to first opening or waiting times were not sufficiently dispersed to account for the slowly inactivating currents: the slow inactivation was produced by channel reopening. A general model was developed to predict the number of openings in each sample. Models in which the number of openings per sample was due to a dispersion of waiting times combined with a rapid transition from an open to an absorbing inactivated state were unsatisfactory and a model that was more consistent with the results was identified.     

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.     

Differences in Na and Ca Spikes As Examined by Application of Tetrodotoxin, Procaine, and Manganese Ions

The effects of tetrodotoxin, procaine, and manganese ions were examined on the Ca spike of the barnacle muscle fiber injected with Ca-binding agent as well as on the action potential of the ventricular muscle fiber of the frog heart. Although tetrodotoxin and procaine very effectively suppress the "Na spike" of other tissues, no suppressing effects are found on "Ca spike" of the barnacle fiber, while the initiation of the Ca spike is competitively inhibited by manganese ions. The initial rate of rise of the ventricular action potential is suppressed by tetrodotoxin and procaine but the plateau phase of the action potential is little affected. In contrast the suppressing effect of manganese ions is mainly on the plateau phase. The results suggest that the plateau phase of the ventricular action potential is related to the conductance increase in the membrane to Ca ions even though Na conductance change may also contribute to the plateau.     

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.     

Activation-inactivation of potassium channels and development of the potassium-channel spike in internally perfused squid giant axons

A spike that is the result of calcium permeability through potassium channels was separated from the action potential is squid giant axons internally perfused with a 30 mM NaF solution and bathed in a 100 mM CaCl2 solution by blocking sodium channels with tetrodotoxin. Currents through potassium channels were studied under voltage clamp. The records showed a clear voltage-dependent inactivation of the currents. The inactivation was composed of at least two components; one relatively fast, having a time constant of 20--30 ms, and the other very slow, having a time constant of 5--10 s. Voltage clamp was carried out with a variety of salt compositions in both the internal and external solutions. A similar voltage-dependent inactivation, also composed of the two components, was recognized in all the current through potassium channels. Although the direction and intensity of current strongly depended on the salt composition of the solutions, the time-courses of these currents at corresponding voltages were very similar. These results strongly suggest that the inactivation of the currents in attributable to an essential, dynamic property of potassium channels themselves. Thus, the generation of a potassium-channel spike can be understood as an event that occurs when the equilibrium potential across the potassium channel becomes positive.     

Neuronal Spike Initiation Modulated by Extracellular Electric Fields

Based on a reduced two-compartment model, the dynamical and biophysical mechanism underlying the spike initiation of the neuron to extracellular electric fields is investigated in this paper. With stability and phase plane analysis, we first investigate in detail the dynamical properties of neuronal spike initiation induced by geometric parameter and internal coupling conductance. The geometric parameter is the ratio between soma area and total membrane area, which describes the proportion of area occupied by somatic chamber. It is found that varying it could qualitatively alter the bifurcation structures of equilibrium as well as neuronal phase portraits, which remain unchanged when varying internal coupling conductance. By analyzing the activating properties of somatic membrane currents at subthreshold potentials, we explore the relevant biophysical basis of spike initiation dynamics induced by these two parameters. It is observed that increasing geometric parameter could greatly decrease the intensity of the internal current flowing from soma to dendrite, which switches spike initiation dynamics from Hopf bifurcation to SNIC bifurcation; increasing internal coupling conductance could lead to the increase of this outward internal current, whereas the increasing range is so small that it could not qualitatively alter the spike initiation dynamics. These results highlight that neuronal geometric parameter is a crucial factor in determining the spike initiation dynamics to electric fields. The finding is useful to interpret the functional significance of neuronal biophysical properties in their encoding dynamics, which could contribute to uncovering how neuron encodes electric field signals.     

Dual modulation of a potassium channel by the m1 muscarinic and beta2-adrenergic receptors

Neurotransmitter receptors alter membrane excitability and synaptic efficacy by generating intracellular signals that ultimately change the properties of ion channels. Given their critical role in controlling cell membrane potential, potassium channels are frequently the targets of modulatory signals from many different G protein-coupled receptors. However, due to the heterogeneity of potassium channel expression in vivo, it has been difficult to determine the molecular mechanisms governing the regulation of molecularly defined potassium channels. Through expression studies in Xenopus oocytes and mammalian cells, we found that the m1 muscarinic acetylcholine receptor (mAChR) potently suppresses a cloned delayed rectifier potassium channel, termed RAK, through a pathway involving phospholipase C activation and direct tyrosine phosphorylation of the RAK protein. In contrast, we found that RAK channel activity is strongly enhanced following agonist activation of beta2-adrenergic receptors; this effect requires a single PKA consensus phosphorylation site located near the amino terminus of the channel protein. These results demonstrate that a specific type of potassium channel that is widely expressed in the mammalian brain and heart is subject to both positive and negative regulation by G protein-dependent pathways.     

The action potential in the smooth muscle of the guinea pig taenia coli and ureter studied by the double sucrose-gap method

The configuration of the electrotonic potential and the action potential observed by the double sucrose-gap method was similar to that observed with a microelectrode inserted into a cell in the center pool between the gaps. In the taenia and the ureter, the evoked spike was larger in low Na or in Na-free (sucrose substitute) solution than in normal solution. However, the plateau component in the ureter was suppressed in the absence of Na. In Ca-free solution containing Mg (3–5 mM) and Na (137 mM), the membrane potential and membrane resistance were normal, but no spike could be elicited in both the taenia and ureter. Replacement of Ca with Sr did not affect the spike in the taenia, nor the spike component of the ureter but prolonged the plateau component. The prolonged plateau disappeared on removal of Na, while repetitive spikes could still be evoked. It was concluded that the spike activity in the taenia and in the ureter of the guinea pig is due to Ca entry, that the plateau component in the ureter is due to an increase in the Na conductance of the membrane, and that both mechanisms, for the spike and for the plateau, are separately controlled by Ca bound in the membrane.     

The effect of pentylenetetrazol on spike broadening and potassium inactivation

1. The effect of the convulsant drug, pentylenetetrazol (PTZ) on spike broadening and potassium current inactivation was studied.2. PTZ was found to decrease the time taken for a cell to reach maximal broadening as well as causing a decrease in the total amount of broadening.3. Voltage clamp studies showed that in the presence of PTZ potassium current inactivated less but exhibited a faster time constant of inactivation.4. By exerting an effect on potassium inactivation and thereby spike broadening, PTZ may alter synaptic efficacy.5. Such an effect on synaptic efficacy may partially underlie the drug's convulsive activity.     

Molecular basis for different rates of recovery from inactivation in the Shaker potassium channel family

The two alternative carboxyl-termini of Shaker K+ channels strongly influence the rates of inactivation and of recovery from channel inactivation. We show that this distinct inactivation behaviour is due to an alanine/valine amino acid replacement within the Shaker carboxyl-terminus at a site that occurs within the proposed membrane spanning segment S6.