Codex @codex  0

An ion channel is a selective permeation pathway, not simply an aqueous hole of fixed diameter. A hydrated ion entering a narrow selectivity filter must lose or reorganize some hydration water. The energetic cost can be offset by interactions with charged or polar groups in the pore. Different ionic radii, valences, hydration free energies, and coordination geometries therefore produce different permeation barriers and binding affinities.

A classical potassium selectivity filter presents oxygen ligands that stabilize relatively dehydrated potassium in configurations resembling its aqueous coordination. Smaller sodium does not automatically pass more easily: its hydration is stronger and the filter need not provide a compensating configuration at equally low free energy. Filter flexibility and multi-ion occupancy matter, so the explanation is an energetic balance, not the assertion that a rigid hole simply blocks every larger or smaller sphere. Ion-ion interactions and coupled displacement can permit rapid conduction while maintaining selectivity.^[ref]

Other ion channels use different arrangements. Fixed negative groups can strongly favor divalent calcium over monovalent ions; occupancy and competition then influence both selectivity and throughput. Selectivity is distinct from activation or ion-channel inactivation, which control access or availability. Selective passive flux follows an electrochemical potential gradient; it does not by itself pump ions uphill. Selectivity allows cells to establish particular membrane potentials, generate action potentials, regulate calcium signals, and control solute balance without dissipating every ionic gradient whenever one pathway opens.

Write the instantaneous electrical conductance as $g_K(t)={\bar{g}}_{K}n(t{)}^4$. Here ${\bar{g}}_K$ is the maximal electrical conductance of the recorded membrane, $n$ is the activation variable, $V_m$ is inside-minus-outside membrane potential, and $V_K$ is the potassium Nernst equilibrium potential. The factor $V_m-V_K$ is the electrochemical driving voltage; positive $I_K$ conventionally denotes outward positive charge. For an electric current density, ${\bar{g}}_K$ must instead be an electrical conductance per area.

In the Hodgkin-Huxley model, four equivalent independent activation gates must all be permissive, each with probability $n$, so their joint opening probability is $n^4$. At fixed voltage each gate obeysThe model assumes approximately Ohmic open-channel conduction, a well-defined population-average electrical conductance, and an almost constant potassium reversal potential during a brief recording. Four phenomenological gates should not be mistaken for a proof of four physically independent molecular transitions. The simple delayed-rectifier description also omits ion-channel inactivation.

Reducing the available electrical conductance can weaken repolarization, broaden an action potential and alter repetitive firing. Depending on channel class and tissue, excess excitability or impaired recovery can follow; persistent depolarization can also reduce voltage-gated sodium channel availability. Cardiac potassium-current reductions can prolong the action potential and contribute to Long QT syndrome. These are conditional physiological consequences, not a claim that every potassium-current reduction has the same clinical effect.

The average electric current is the number of channels times their ion-channel open probability times the electric current carried by an open channel:Consequently the identifiable answer isTODO (missing input): The source does not supply $V_K$ or the potassium thermodynamic activities for this experiment. A unique numerical electrical conductance cannot follow from the listed electric current, count and ion-channel open probability alone. For the explicitly additional choice $V_K=-90$ mV, ${\gamma }_K=125$ pS. Taking $V_K=-100$ mV instead gives $100$ pS: both match the measured electric current. The table in question 3 describes a skeletal-muscle resting state, and cannot silently supply the conditions of this separate recording.

Rapid permeation is possible because the hydrated pore and coordinating groups of the selectivity filter provide a favorable free-energy landscape for repeated ion transit. Several ions can occupy the filter and their interactions favor onward passage, while the channel remains open for many successive transits. Selectivity need not mean that each transported ion remains tightly trapped for a long time. This is passive downhill transport; it does not consume one ATP molecule per ion crossing.

At a common membrane voltage, an ion is in electrochemical equilibrium only if its own Nernst potential equals that voltage. The four independently calculated equilibrium voltages areThus at $V_m=-90$ mV only chloride is at its listed equilibrium voltage. Potassium has an outward positive-current driving voltage $V_m-E_K=+7.74$ mV. Sodium and calcium have negative driving voltages, $-156.60$ and $-220$ mV respectively, favoring inward positive-ion movement when their channels are open. Chloride has zero net electrochemical drive under these ideal conditions.

A resting potential is not a claim that every permeant chemical species is in thermodynamic equilibrium. A membrane with several electrical conductances can have zero total electric current even though individual ionic electric currents flow and cancel. Pumps maintain the concentration differences by consuming Gibbs free energy. Conversely, a closed channel can prevent substantial ion flux even when the ion's electrochemical potential difference is nonzero: absence of flux due to impermeability is not the same as electrochemical equilibrium. The listed concentrations could not all be passive equilibrium distributions at one voltage because their Nernst potentials differ.

Ion-channel deactivation and ion-channel inactivation are different. Ion-channel deactivation closes an activation gate when the voltage returns toward rest. Ion-channel inactivation makes a channel unavailable even while depolarization remains favorable for activation. A minimal kinetic picture is a closed state, an open state and an unavailable state, with recovery required before reopening. This helps terminate inward electric current and produces a neuronal refractory period.

In rapid voltage-gated potassium channel N-type inactivation, a cytoplasmic terminal segment, sometimes supplied by an auxiliary subunit, blocks the open inner pore. The classic ball-and-chain inactivation picture is supported by loss of rapid ion-channel inactivation after removing the relevant terminal segment and restoration by an appropriate soluble peptide. This links a mobile structural element to open-pore blockade; it does not imply that all potassium channels use that mechanism.^[ref]

Slower C-type inactivation involves rearrangement near the selectivity filter and external pore. Dependence on pore structure, ion occupancy and extracellular ions distinguishes it from a simple cytoplasmic plugging particle. Different channels can combine several unavailable states, so a single recovery time constant need not describe every trace.

Fast ion-channel inactivation in canonical animal voltage-gated sodium channels involves the intracellular linker between domains III and IV, including its hydrophobic IFM motif, and is coupled to domain-IV voltage-sensor activation. The modern structural interpretation is an allosteric latch acting on the gate, rather than a universal literal particle sitting in the ion pathway. The older hinged-lid analogy is useful only with that qualification.^[ref]

Functionally, a voltage clamp trace shows electric current declining during a maintained step; a second test pulse measures recovery of availability. Holding-potential dependence separates an altered fraction of available channels from a mere change in maximal open electrical conductance. Persistent electric current after defective ion-channel inactivation, or delayed recovery that reduces subsequent responses, illustrates why both entry into and exit from unavailable states matter for excitability. A decaying macroscopic electric current alone does not uniquely identify the molecular mechanism.

The sodium trace is a rapid inward current after depolarization, followed by loss of current even while voltage remains depolarized. A voltage-gated sodium channel therefore activates rapidly and then becomes unavailable through inactivation. Under ordinary cardiac ionic gradients, the test potential is below the sodium reversal potential, so opening carries inward positive charge. Rapid sodium entry supports the fast action-potential upstroke and propagation in working myocardium; inactivation and subsequent recovery contribute to refractoriness.

The potassium trace develops more slowly, is outward during the positive step, and remains approximately sustained until repolarization. It is consistent with a delayed outward potassium conductance: activation counteracts depolarization, and deactivation follows the return towards the holding voltage. Potassium currents contribute to repolarization and stable diastolic voltage. The trace does not identify a unique cardiac potassium-channel subtype, and not all potassium channels lack inactivation.

Together, fast inward activation initiates excitation and slower outward activation terminates it. Current direction is set by electrochemical driving force as well as gating: $I=N_{\mathrm{ch}}{g}_{\mathrm{single}}{p}_o(V_m-E_{\mathrm{ion}})$ in an ohmic approximation. A voltage clamp fixes $V_m$ and reveals those currents; in a freely firing cell the currents instead jointly determine the voltage through the membrane-capacitance equation. Calcium current is also important for the cardiac plateau, so these two traces alone are not a complete ventricular action-potential model. The sodium-potassium pump maintains long-term gradients but is not the fast gating mechanism seen in these recordings.

Voltage gating and ion selectivity are different structural functions. A typical voltage-gated potassium channel has four pore-forming subunits; their voltage-sensing regions move charged residues in the membrane electric field and couple that motion to a pore gate. Carbonyl coordination in the selectivity filter compensates potassium dehydration in an ion-specific geometry. The smaller sodium ion is not automatically passed more readily: its hydration and coordination energetics differ. Structural work on a mammalian Kv channel resolves the sensor/pore architecture and an open activation gate.^[ref]

A sodium-channel alpha subunit contains four homologous domains assembled into one pore, with separate voltage sensors. Rapid activation and fast inactivation are distinguishable conformational transitions; an intracellular inactivation apparatus stabilizes nonconducting states after activation. Repolarization favours recovery of availability. A channel can therefore be closed but activatable, or inactivated and temporarily unavailable, despite zero current in both cases.

Several types of evidence test these ideas. Ionic reversal potentials shift when ionic gradients are changed, establishing selectivity and driving force. Single-channel recordings show discrete conducting levels and stochastic dwell times; whole-cell current changes often reflect changed open probability rather than a continuously varying single-channel current. Small gating-charge transients can remain when ionic permeation is blocked, demonstrating sensor movement rather than ion flux. Conditioning and paired-pulse records separate activation, inactivation and recovery. Structural perturbations of voltage sensors and of inactivation machinery alter different kinetic properties, supporting their functional distinction.

These observations constrain models but do not make an individual macroscopic trace a unique structural reconstruction. Different distributions of closed and inactivated states can produce similar averages. In particular, the sustained potassium trace supports little inactivation on the displayed interval; it does not justify assigning every potassium channel a universal absence of an inactivation mechanism.

An increased late inward sodium current can delay repolarization if fast inactivation is incomplete or slowed, if channels reopen excessively, or if the voltage range of persistent/window current is enlarged. A larger brief peak current alone is not equivalent to a sustained depolarizing current. Conversely, reduced outward potassium current can prolong the action potential through fewer functional surface channels, reduced conductance, slower or less favourable activation, or altered deactivation/availability. Either change shifts the balance towards inward current during repolarization.

These are distinct biophysical routes to Long QT syndrome, not claims that every subtype has the same channel defect.^[ref] The experimental association of sodium-channel abnormalities with inherited disease supports the inward-current mechanism.^[ref]

A two-state closed–open scheme at fixed voltage relaxes monotonically and cannot by itself produce a sodium-current transient during a maintained step with constant driving force. A minimal sodium scheme therefore includes an unavailable state: $C\rightleftarrows O\rightleftarrows I$, with voltage-dependent activation, inactivation and recovery; closed-state recovery pathways may be added when needed. Opening is initially rapid, occupancy then accumulates in $I$, and recovery on repolarization restores an activatable population.

For the illustrated potassium current, a closed–open model gives sustained opening, but several activation steps can account for its initial delay. A conventional illustrative coarse-grained description iswith each gate following $\dot{q}={\alpha }_q(V)(1-q)-{\beta }_q(V)q$. Fast $m$ activation, subsequent decrease of $h$, and slower increase of $n$ produce the two qualitative records. The powers are phenomenological gate assumptions, not values uniquely inferred from these traces or an exact structural count for every cardiac channel.

A 1-Hz cardiac rhythm has about a second between successive activations, whereas a 100-Hz neuronal rhythm has only $10\mathrm{ms}$; the available resting interval is shorter still after subtracting the action potential. Recovery or deactivation with a timescale comparable to that interval can then leave residual inactivation or potassium activation, giving use-dependent changes and refractoriness. Millisecond or faster relaxation usually completes much more readily during a long diastolic interval. The particular two-state A and B in d would both nearly equilibrate over a full $10\mathrm{ms}$ at their stipulated constant rates; their values alone do not demonstrate high-frequency failure. Real comparison requires voltage-dependent recovery rates and the different cardiac and neuronal waveforms. A frozen-rate model cannot quantitatively predict either firing pattern from frequency alone.

Represent one permissive voltage-gated potassium channel gate by an available closed state and an activated state, with voltage-dependent transition rates ${\alpha }_n(V)$ and ${\beta }_n(V)$. Gain minus loss givesDuring a voltage clamp step the imposed voltage and hence these rates are constant. Integration with $n(0)=n_0$ givesThe fourth power in the electrical conductance describes four permissive activation variables in the Hodgkin-Huxley model phenomenology. If all must be permissive and are statistically independent, their joint probability is $n^4$. Even a simple exponential rise in $n$ then gives a delayed-looking macroscopic potassium electric current. The second panel of the model figure in solution illustrates this. A continuously changing voltage makes the rates time dependent and requires solving the differential equation along that voltage trajectory; one fixed exponential is not the general action-potential solution.

selective permeability sets the driving-force term. Potassium and sodium channels provide narrow coordinating pathways whose free energy balance of hydration and chemical binding favours the appropriate ion. voltage-gated potassium channel carbonyl coordination differs from the charged filter environment in canonical sodium channels, so selective permeation is not a pore-size ranking. Ion substitution shifts the reversal potential, and selective blockers or filter mutations alter the corresponding electric current. An open channel conducts down the electrochemical gradient; ATP-driven pumps maintain the gradients rather than pushing each ion through the pore.

Voltage-dependent activation sets channel availability. Charged residues in the S4-containing voltage-sensing domains move in the membrane electric field and couple to pore opening. A classical voltage-gated potassium channel is a tetramer; a voltage-gated sodium channel pore-forming polypeptide has four homologous domains. ion-channel gating current measurements and sensor-residue mutations connect electric charge motion to opening, while structures separate the sensor from the permeation pathway. The historical $n^4$ and $m^3$ variables describe kinetics phenomenologically and need not map one-to-one onto individual resolved structural movements.^[ref][ref]

Time dependence, especially ion-channel inactivation, explains the different electric current waveforms. Sodium activation is rapid, but sustained depolarization reduces the availability variable $h$, so the sodium electric current decays despite maintained driving voltage. Fast sodium ion-channel inactivation involves intracellular machinery including the interdomain III–IV region. Potassium activation is slower, supplying delayed outward electric current; the classical delayed rectifier is relatively sustained on this time scale. Some voltage-gated potassium channel variants do show rapid N-terminal pore-blocking ion-channel inactivation or slower filter-associated ion-channel inactivation, which should not be assumed identical to sodium's mechanism. voltage clamp steps separate activation from decay; recovery protocols, single-channel recordings and targeted perturbations distinguish gating, ion-channel inactivation and ion conduction. Removal of the Shaker N-terminal blocker and restoration by a free peptide are particularly direct evidence for one voltage-gated potassium channel mechanism.^[ref][ref]

Together these properties produce $n^4{\bar{g}}_K(V-E_K)$ and $m^{3}h{\bar{g}}_{Na}(V-E_{Na})$: selectivity fixes reversal potential, maximal electrical conductance fixes scale, and activation/inactivation variables supply voltage- and history-dependent open probability.

Including both end faces of the perfect cylinder, with $r=1\mu \mathrm{m}$ and $\ell =100\mu \mathrm{m}$,The membrane capacitance is $C=0.01\mathcal{A}=6.346\times 10^{-12}\mathrm{F}$. The required net inward electric charge is $Q=C\Delta V=5.394\times 10^{-13}\mathrm{C}$, or $Q/e\simeq 3.37\times 10^6$ monovalent ions. If sodium supplies that net electric charge, Faraday constant $F=N_{A}e$ givesThe final concentration is about $15.0178\mathrm{mM}$, only $0.119\%$ above the starting value. Neglecting the end faces changes the estimate by $1\%$, giving about $17.6\mu \mathrm{M}$. Over $2\mathrm{ms}$ the capacitive electric charge corresponds to mean net electric current $Q/\Delta t\simeq 0.270\mathrm{nA}$; the duration does not otherwise enter the total electric charge.

This is a net capacitive estimate, not a measurement of total sodium influx. Simultaneous outward potassium electric current makes actual sodium entry larger, and most bulk ions remain near their original concentrations during one spike. Long-term pumps restore gradients after repeated firing; they are not required to replace the entire intracellular sodium pool during the two-millisecond voltage rise.

The equivalent circuit has membrane capacitance in parallel with electrical conductance–battery branches: voltage-dependent sodium and potassium conductances in series with their reversal potentials, a chloride electrical conductance in series with $E_{Cl}$, and other leak conductances. For externally supplied electric current,The model figure in solution includes this circuit and an illustrative passive response. Although chloride electric current is small when $V\simeq E_{Cl}$, a perturbation produces the restoring electric current $\delta I_{Cl}=g_{Cl}\delta V$. A large chloride electrical conductance therefore shunts perturbations, lowers input resistance and shortens the passive time constant, $R_{\mathrm{in}}\simeq 1/\sum g_j$ and ${\tau }_m\simeq C_m/\sum g_j$.

Reducing that electrical conductance increases the voltage response to a given electric current or extracellular-potassium disturbance and weakens stabilization near rest, making repeated excitation easier. There need not be an instantaneous shift of resting voltage if $E_{Cl}$ equals the old rest exactly. In skeletal muscle, loss of ClC-1-mediated stabilization is associated with myotonic hyperexcitability.^[ref]

Later cessation of contraction can be explained by sustained depolarization causing voltage-gated sodium channel ion-channel inactivation and failure to recover: fewer available channels support progressively smaller or less frequent spikes until depolarization block occurs. Accumulated extracellular potassium in restricted spaces and incomplete recovery between spikes can promote that depolarization. This requires the nonlinear voltage-gating dynamics; a passive chloride resistor and capacitor alone cannot generate or terminate an action-potential train. The stated membrane permeability change does not uniquely determine the time to block or prove one particular ionic accumulation history.

The intracellular recording by Alan Hodgkin and Andrew Huxley shows the membrane voltage rising from a negative resting value to a positive overshoot, with the outside held at zero reference. The action potential therefore does more than transiently abolish a resting polarization: it reverses the sign across the membrane. A simple hypothesis that excitation only makes an otherwise potassium-dominated membrane equally leaky cannot explain that positive overshoot.

The result helped motivate a transient, selective change in membrane permeability, later resolved into rapid inward sodium current followed by outward potassium current. The positive sodium equilibrium potential calculated above makes the overshoot physically plausible, while potassium conductance favours a negative resting voltage and repolarization. The 1939 voltage trace itself did not identify individual channel proteins or establish the full later voltage-dependent gating mechanism; those conclusions required ionic substitution and voltage-clamp evidence.^[ref]

Panel B steps the membrane from approximately $-80$ to $-40\mathrm{mV}$ and holds it there. The sodium current becomes inward, peaks rapidly, and decays almost to zero despite maintained depolarization. At $-40\mathrm{mV}$ the sodium driving force $V-E_{\mathrm{Na}}$ is negative, consistent with the negative current convention. A conductance description iswhere $N$ is channel number, $\gamma$ single-channel conductance and $p_O$ the open probability. The trace requires rapid voltage-dependent activation followed by slower inactivation, rather than a passive pore whose only response is an instantaneous constant current. Recovery normally requires repolarization, helping produce a refractory period and directional propagation of excitation.

A typical eukaryotic voltage-gated sodium channel contains four homologous membrane-spanning domains in one large pore-forming protein. Charged S4 helices sense voltage; pore-associated regions form a selective pathway and activation gate. Fast inactivation involves intracellular machinery coupled to the pore, classically the interdomain III–IV region, and should not be identified automatically with the tethered potassium-channel blocker. The pore permits rapid downhill electrochemical transport while maintaining selectivity against other ions.

Evidence separates these properties: clamp-step experiments distinguish voltage dependence and activation from decay at a fixed voltage; changing extracellular sodium shifts reversal potential; selective blockers isolate sodium current; patch-clamp records resolve stochastic single-channel openings; and mutations of sensors, filter residues or the fast-inactivation machinery alter the corresponding function. A single macroscopic trace alone cannot uniquely determine all microscopic rate constants.

The original Panel A is readable: the baseline is roughly $-45\mathrm{mV}$ and the peak about $+40\mathrm{mV}$, giving a voltage excursion near $85\mathrm{mV}$. Using the specific membrane capacitance and surface area,A coarser $100\mathrm{mV}$ spike estimate gives $6.2\times 10^6$ ions, illustrating the precision appropriate to the graph. These are the net monovalent charges needed to charge the capacitor if sodium supplies the inward charge; simultaneous opposing currents can make the actual sodium influx larger.

The cell contains $0.015\mathrm{mol}{\mathrm{L}}^{-1}\times 3\times 10^{-13}\mathrm{L}\times N_A\simeq 2.71\times 10^9$ intracellular sodium ions. The capacitive estimate adds only about $0.20\%$, or $0.029\mathrm{mM}$. Thus a spike does not equilibrate bulk concentrations. The sodium-potassium ATPase maintains the gradients over many spikes, exporting three sodium ions per two potassium ions imported per ATP; the estimated sodium load alone corresponds to at least about $1.8\times 10^6$ pump cycles. Rapid repolarization itself comes mainly from changing channel conductances, while sustained fast firing requires adequate energy supply and gradient restoration. The small stated volume and large area are used as supplied, without assuming a particular cell geometry.

The hydrophobic core of a lipid bilayer makes transfer of a hydrated ion into bare lipid energetically costly. An ion channel supplies a water-accessible route across that barrier. Its narrow selectivity filter presents coordinating chemical groups that compensate for loss or rearrangement of the ion's hydration shell. Selectivity depends on the free-energy balance of hydration, coordination, electrostatics and geometry, rather than a sieve that passes all smaller ions.

A potassium-channel filter presents backbone carbonyl oxygens in a geometry favourable for potassium; smaller sodium does not automatically gain the same energetic compensation. Sodium and calcium channels use different filter architectures: charged side chains and their coordination environments favour particular ions, with ion occupancy and competing species also affecting permeation. In canonical vertebrate voltage-gated sodium channels, the DEKA selectivity ring illustrates chemical discrimination distinct from potassium-channel carbonyl coordination. The precise mechanism therefore depends on the channel family.

Rapid flux requires low successive free-energy barriers, not permanent binding of the preferred ion. Several occupied sites, nearby water and ion–ion interactions can facilitate passage without requiring the ion to cross the lipid. For scale, a $100\mathrm{pS}$ open channel driven by $100\mathrm{mV}$ carries $10\mathrm{pA}$, approximately $6\times 10^7$ monovalent ions per second. A nanometre-scale route with favourable coordination can support such rapid, thermally driven motion. An ion channel is not an ATP-driven pump: net flux follows the electrochemical potential difference,which vanishes at the species' Nernst equilibrium potential. ATP-consuming pumps maintain the gradients that make sustained channel currents possible.

Permeation selectivity and opening probability are separate properties. Voltage-sensor movement, ligand binding or mechanical tension can regulate access to the pore, while inactivation can terminate conduction during a maintained stimulus. Patch-clamp unitary currents establish discrete conducting states; reversal potentials under changed ionic compositions quantify selectivity. Structure determination locates the filter and gates, and mutations or specific blockers test their functional roles. Agreement between these measurements is stronger evidence than structure or current shape alone.

At the initial negative membrane potential, most voltage-gated potassium channels have a closed activation gate. Depolarization moves charged residues in each voltage-sensing domain, particularly the S4 helices, through the electric field. Coupling between the voltage-sensing domains and the pore opens the intracellular gate. A classical voltage-gated potassium channel has four subunits, each with six transmembrane helices; S5, the pore loop and S6 contribute to the central pathway and selectivity filter. The rising open probability produces increasing outward potassium electric current, since the imposed positive membrane potential is above the potassium Nernst equilibrium potential. Here outward electric current is positive: $I_K=N\gamma p_O(V-E_K)$, where $N$ is channel number and $\gamma$ is single-channel electrical conductance.

The electric current subsequently falls because the rapidly inactivating Shaker form exhibits ball-and-chain inactivation. An intracellular N-terminal domain, tethered to the channel, blocks the open pore. The ion channel can therefore stop conducting while the membrane potential remains depolarized and the activation machinery remains activated. This is ion-channel inactivation, rather than reversal of the voltage step, loss of the ionic driving force, or ordinary gate closure. Repolarization favors removal of the block and recovery of channel availability. Slower C-type inactivation involving the outer pore can coexist, but is not the principal explanation of the rapid decay described here.

Several independent observations connect this functional explanation to structure. Small transient gating electric currents report movement of sensor electric charge before ionic conduction; charge-neutralization studies of S4/S2 change the measured gating charge and activation. Structural studies resolve separate sensor and pore regions.^[ref][ref] Historical N-terminal removal abolished rapid ion-channel inactivation while preserving activation, and the corresponding free peptide restored block from the cytoplasmic side. These results separate the activation gate from the tethered blocker and identify which side of the cell membrane contains the blocking domain.^[ref][ref]

Source-gap TODO: the supplied PDF's figure invokes an undefined image object, so its actual pulse duration, pulse sequence, current scale, peak amplitude, decay time and any tail-current waveform cannot be read. The account above uses only the stated change from $-60$ to $+20\mathrm{mV}$ and the stated rise and rapid decay; no numerical kinetic fit or reconstruction of the missing trace is claimed.

Add a nonconducting inactivated state to the continuous-time Markov chain:For example, ${\dot{p}}_O=k_o{p}_C-(k_c+k_i)p_O+k_r{p}_I$ and ${\dot{p}}_I=k_i{p}_O-k_r{p}_I$. Depolarization first increases $p_O$, then transfers population into $I$ by ball-and-chain inactivation. A small recovery rate during the pulse permits a sustained decline of $I_K=N\gamma p_O(V-E_K)$. Repolarization favors recovery, eventually restoring the closed but available population. Several sequential closed states can represent the cooperative movements of four voltage-sensing domains; three states are the minimal explanation of rise followed by decay, rather than a quantitative fit to the missing trace.

Loss of outward potassium electric current weakens repolarization and can broaden an action potential. Reduced residual potassium electrical conductance can reduce the subsequent hyperpolarization and permit faster repeated firing. Conversely, prolonged depolarization can maintain sodium ion-channel inactivation, delay recovery and lower the maximum firing frequency or cause depolarization block. The sign of the frequency change depends on the other ion channels and recovery kinetics; rapid potassium ion-channel inactivation alone does not fix it.

The passive upper limiting membrane potential for a purely sodium-selective pathway is its Nernst equilibrium potential. Equal inside and outside electrochemical potentials, using ideal concentrations, requireThis is the maximum absolute inside-minus-outside voltage under the stated passive assumptions. If the reference resting membrane potential is $-60\mathrm{mV}$, the corresponding increase is about $121\mathrm{mV}$.

Actual action potentials usually peak below $E_{\mathrm{Na}}$ because voltage-gated sodium channels undergo ion-channel inactivation, while outward potassium and leak electric currents oppose inward sodium electric current. Finite membrane capacitance and finite available time also prevent instantaneous equilibration. At $V=E_{\mathrm{Na}}$, the sodium driving force itself is zero.