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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.

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.

A sodium-channel gain-of-function mechanism in Long QT syndrome can reduce the inactivation rate $k_i$ or increase recovery/reopening from inactivated states, leaving a small persistent open population. The resulting late inward sodium current opposes repolarizing outward currents while $V<E_{\mathrm{Na}}$, prolonging the cardiac action potential. Slower entry into $I$ would also extend the typical conducting episode relative to its normal duration. The altered effect is prolonged sodium entry, not reversal of sodium's charge or a pump-driven uphill flux through the channel.

The precise rate change depends on the mutation: the clinical label alone does not uniquely specify one transition constant. Long QT syndromes also include other channel mechanisms; the question here concerns its stated sodium-channel variant.^[ref]

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.

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.

An ion channel supplies a water-accessible pathway through the otherwise strongly insulating, hydrophobic lipid bilayer. Moving a hydrated ion directly into that low-permittivity region is energetically expensive. The channel's polar pore and coordinating groups reduce the barrier while retaining selectivity. Transport is passive: the electrochemical potential difference drives net movement, and the channel does not itself pump an ion uphill. For an ideal species of charge number $z$,The net electric current of a pathway permeable only to that species reverses at its Nernst equilibrium potential. An open channel can carry zero net electric current at that voltage even while individual ions cross both ways.

The selectivity filter balances loss of the hydration shell against favorable coordination within the pore. In a typical potassium channel, backbone carbonyl oxygen atoms create appropriately arranged binding sites for a substantially dehydrated potassium ion. The smaller sodium ion is not automatically preferred simply because it fits geometrically: its hydration is stronger and the filter does not compensate its dehydration in the same way. Geometry, flexibility, coordination chemistry and ion occupancy all matter. Sodium-selective and calcium-selective channels use different arrangements, often involving charged side chains; a single rigid-sieve explanation is inadequate.

Rapid passage coexists with transient binding because several nearby sites have comparable free energies and relatively low barriers between them. Multiple ions occupying a pore influence one another through electrostatic interactions, enabling correlated exchange and displacement rather than irreversible trapping at one deep site. High throughput therefore requires favorable passage kinetics as well as strong discrimination against unwanted species. A more tightly bound species can even block a pathway instead of being its fastest permeant.

Gating and selectivity are distinct. A voltage-sensing domain couples membrane potential to activation-gate conformation; ligand-gated ion channels couple binding to opening; mechanosensitive ion channels couple membrane stress to opening. Ion-channel inactivation introduces additional unavailable states. None of these mechanisms alone identifies which ion the open pore prefers.

For $N$ similar independent channels, a local Ohmic approximation with outward-positive electric current isHere open probability controls availability, single-channel electrical conductance controls current per open pore, and $V-E_{\mathrm{rev}}$ is the electrical driving factor in this approximation. Nonlinear current-voltage relations and mixed-ion permeation may require a fuller model. Single-channel recording resolves discrete openings; reversal shifts when ionic concentrations change test selectivity; gating electric currents test sensor motion; structural evidence identifies the pore and its coordinating groups. Together these distinguish rapid selective permeation from the separate process of opening the gate.

Write $L_i$ for the lithium $2s$ orbital at site $i$ and $N_i$ for the sodium $3s$ orbital. Reading the printed axes, the lithium coordinates have signs $1=(-,+,+)$, $2=(+,-,+)$, $3=(+,+,-)$ and $4=(-,-,-)$; the sodium sites are $5=(+,+,+)$, $6=(-,-,+)$, $7=(-,+,-)$ and $8=(+,-,-)$. The normalized tetrahedral s-orbital symmetry combinations areEach $T_2$ row takes its signs from the corresponding coordinate, so the three functions transform together like $(x,y,z)$. These coefficient vectors are orthonormal when intersite orbital overlap is neglected. A common sign reversal of any row is physically immaterial.

Write $t=\mathrm{C}(\mathrm{C}{\mathrm{H}}_3{)}_3$. D is the sodium enolate $\mathrm{t}-\mathrm{C}({\mathrm{O}}^{-})=\mathrm{C}{\mathrm{H}}_2\mathrm{N}{\mathrm{a}}^{+}$. Its two resonances in the alkene-proton region belong to the two hydrogens of the terminal methylene group. One is cis to oxygen and the other trans to oxygen, so they are not related by a symmetry operation even though both belong to the same carbon. Replacing one or the other gives different alkene geometries, so they are diastereotopic. Their weak or unresolved geminal coupling is consistent with the reported singlet descriptions. Without an additional spatial assignment, it is not justified to decide uniquely which numerical shift belongs to which of those two orientations. The nine-proton singlet is the tert-butyl group. The formula ${\mathrm{C}}_6{\mathrm{H}}_{11}\mathrm{ONa}$ follows from removing one alpha hydrogen from pinacolone.

E is the enol-dominated 1,3-diketone $\mathrm{tCO}-\mathrm{C}{\mathrm{H}}_2-\mathrm{COt}$, observed as $\mathrm{tC}(\mathrm{OH})=\mathrm{CH}-\mathrm{COt}$. The very downfield broad signal belongs to the enol OH in a six-membered intramolecular O–H···O chelate. The one-proton signal in the vinyl region is its central enol CH; a predominantly diketo structure would instead require a two-proton central CH2 signal and no enol OH. The eighteen-proton singlet accounts for both tert-butyl groups. Rapid exchange between the two degenerate enol orientations can average their different local environments; accidental unresolved equality cannot be excluded by the supplied spectrum alone. Thus one should not infer a permanently symmetric single enol Lewis structure from that singlet. The total integration is $1+1+18=20$, agreeing with ${\mathrm{C}}_{11}{\mathrm{H}}_{20}{\mathrm{O}}_2$.

The alkene attacks electrophilic mercury(II), with acetate departure as appropriate, to give a bridged mercurinium ion between the ring carbon and the exocyclic methylene. That bridge is polarized: the more substituted ring carbon can better stabilize developing positive charge than the terminal CH2 carbon. water attacks the ring carbon, opening the mercury bridge there, and deprotonation gives the tertiary alcohol with an exocyclic CH2–HgOAc substituent.

Subsequent sodium-borohydride demercuration replaces the terminal C–Hg bond by C–H. Thus the original exocyclic CH2 becomes Me, and the new OH remains on the substituted ring carbon. The result is Markovnikov hydration of the exocyclic alkene. There is no need for a freely equilibrating tertiary carbocation, and no rearrangement of the carbohydrate carbon skeleton or cleavage of the methyl acetal is required. The question does not require the mercury-stage facial selection or a detailed demercuration mechanism.

Both permitted alternatives follow the same condensation–activation–migration sequence. sodium acetate liberates a useful concentration of hydroxylamine from its hydrochloride while providing a buffered medium. nitrogen of NH2OH attacks the ketone carbonyl; proton transfer gives a carbinolamine. protonation of its carbon-derived OH and loss of water, with nitrogen lone-pair donation, then gives the oxime after deprotonation. The two C=N geometries can be separated.

For the B-to-D route, oxime O-tosylation begins when oxime oxygen attacks sulfur of tosyl chloride and chloride leaves; triethylamine accepts the released proton. The resulting O-tosyl oxime has an excellent leaving group. The $K-R_a$ bond, anti to N–OTs, migrates to nitrogen while N–O cleaves, giving a ring-expanded nitrilium ion. Its local connectivity is $R_s-K\equiv N^{+}-R_a$, with the existing scaffold still joining the two paths. water attacks the nitrilium carbon $K$. proton transfers and imidic-acid/amide tautomerization produce D, with nitrogen inserted on the aromatic side of the old carbonyl.

For the C-to-E route, the same O-tosylation activates the other oxime geometry. Now the anti bond is $K-R_s$; its migration gives $R_a-K\equiv N^{+}-R_s$, and aqueous trapping produces E, with nitrogen inserted on the spiro-junction side. Each Beckmann rearrangement expands the ketone's six-membered ring to a seven-membered lactam. The original carbonyl carbon remains the amide carbonyl carbon; nitrogen is the inserted atom. In this mechanism the ketone oxygen is lost during oxime condensation, the oxime oxygen leaves with the sulfonate group, and water supplies the new amide oxygen.

In the Hodgkin-Huxley model, $V$ is the membrane potential, the voltage inside the axon relative to outside. The space clamp assumption makes it spatially uniform. The gating variables lie between zero and one: $m$ describes sodium activation, $h$ describes availability against sodium inactivation, and $n$ describes potassium activation. Thus large $h$ means channels remain available, not that inactivation is strong.

The left side $C_m\dot{V}$ is capacitive electric current per unit membrane area. The imposed $I$ is inward-positive applied electric current. The other terms are outward-positive ionic electric currents: sodium with electrical conductance $g_{Na}{m}^{3}h$, potassium with conductance $g_K{n}^4$, and a passive leak with conductance $g_L$. Each current is its conductance times the difference between $V$ and the corresponding reversal potential $V_{Na}$, $V_K$ or $V_L$. The coefficients $g_{Na},g_K$ are maximal conductances; the gate factors represent independent permissive gates in an ion channel.

For each gate, $k_{\infty }(V)$ is the equilibrium opening/availability fraction at fixed voltage and ${\tau }_k(V)>0$ its exponential relaxation time. The equation follows from two-state ion-channel gating: if $\dot{k}={\alpha }_k(V)(1-k)-{\beta }_k(V)k$, thenVoltage feedback couples these otherwise elementary gating equations to the membrane potential and produces action potentials.

Write $M(v)=m^3(v)$. In the physically relevant strip $-v_K<v<1$, $0<n<H$, the zero-current nullcline satisfiesThe left side increases strictly from zero to infinity, so there is exactly one nullcline height $n=N(v)$ for each interior voltage. If $M$ is positive at the endpoints, its curve joins $(-v_K,H)$ to $(1,0)$; when low-voltage activation is idealized as exactly zero, the limiting left branch can lie on $v=-v_K$.

The signs of the derivatives explain its folds:At low voltage, potassium's approximately linear contribution dominates and $g_v>0$. During rapid sodium activation the last negative term can dominate, giving $g_v<0$. Once activation saturates, $M^{\prime }$ becomes small and $g_v>0$ again. This folded conductance-model nullcline has the familiar shape: the curve has descending outer branches and an ascending middle branch in the $(v,n)$ plane. Equivalently, an intermediate fixed $n$ can give three zeros of $g$ as voltage rises, with positive, negative and positive $g_v$ at those crossings.

The rapid-activation description permits this folded shape; exact fold positions depend on $M,{\gamma }_K,H,v_K$. The figure supplies an explicit admissible sigmoid example. Its right panel shows $g=0$ as well as the recovery nullcline and an oscillating trajectory.