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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]

At fixed voltage let $a$ and $b$ be opening and closing rates. The open probability obeysThis illustrates equal open probability with unequal ion-channel relaxation times: both channels have $p_{\infty }=1/2$, but channel A has ${\tau }_A=1\mathrm{ms}$ and channel B has ${\tau }_B=0.1\mathrm{ms}$. Starting closed, $p_A(t)=\frac{1}{2}(1-e^{-t/(1\mathrm{ms})})$ and $p_B(t)=\frac{1}{2}(1-e^{-t/(0.1\mathrm{ms})})$. If channel number, unitary conductance and driving force agree, both have the same steady macroscopic current; B approaches it ten times faster.

An individual channel remains either open or closed. Its mean closed and open dwell times are $1/a$ and $1/b$, respectively: both are $2\mathrm{ms}$ in A and $0.2\mathrm{ms}$ in B. Thus identical equilibrium open probability does not imply identical kinetics or noise correlation time. Ensemble probability relaxation depends on the sum of rates, while the mean dwell in a given state depends on its exit rate.

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.

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.

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.

For two-state ion-channel gating, write $C{\rightleftarrows }_{k_c}^{k_o}O$. Gain minus loss of open probability gives the ordinary differential equationThus the ratio of the transition rates determines the final open probability, while their sum determines the response speed. Both given channels have $p_{\infty }=1/2$. For an initially closed population, with $t$ measured in milliseconds,The mean individual open and closed lifetimes are $1/k_c$ and $1/k_o$: both are $2\mathrm{ms}$ for A and $0.2\mathrm{ms}$ for B. With the same channel number, single-channel electrical conductance and driving force, the steady macroscopic electric currents are equal, but B reaches that value ten times faster. If both populations were already at $p_O=1/2$, neither would show a mean relaxation under these fixed rates: a response also requires an initial condition or a change in rates.

The stationary variance of the number open is $N/4$ for independent channels. Its autocovariance is $(N/4)e^{-|t|/\tau }$, so B's fluctuations decorrelate ten times faster even though their instantaneous variance is unchanged. Faster gating allows an ion channel population to follow a brief stimulus and contribute to rapid action potential timing. Slower gating integrates over brief changes and can provide a delayed opposing electric current, useful in setting action potential duration or frequency selectivity. Those examples describe consequences of kinetics; the stipulated voltage-independent rates alone do not define a complete excitable neuron.

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.

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.

In a classical action potential, delayed activation of voltage-gated potassium channels supplies outward electric current that drives repolarization. In the Hodgkin-Huxley model, for example, $I_K={\bar{g}}_K{n}^4(V-E_K)$ and $\dot{n}={\alpha }_n(V)(1-n)-{\beta }_n(V)n$. Increasing the opening rate at depolarized membrane potential, other properties fixed, recruits outward electric current earlier: the action potential generally becomes narrower and may have a smaller peak. Slower opening delays repolarization and broadens the action potential.

Slower closing after repolarization retains excess potassium electrical conductance, lengthening the after-hyperpolarization and often the interval of reduced excitability. Faster closing shortens this tail and permits earlier recovery. If activation or the available electrical conductance is insufficient, normal repolarization may fail; changing rates can also change the steady open fraction. Duration is therefore determined by the whole balance of inward and outward electric currents, not by a closing rate in isolation.

An ion channel can disturb cellular electrical activity by changing unitary electrical conductance, open probability, gating kinetics or the number delivered to the cell membrane. A useful organizing relation isHere outward electric current is positive. Altering one channel's biophysical behavior changes the balance governing membrane potential and hence the timing or recurrence of action potentials.

Long QT syndrome, type 3. Certain disease-associated SCN5A variants impair fast ion-channel inactivation of the cardiac voltage-gated sodium channel NaV1.5. Instead of vanishing promptly after the upstroke, a small persistent inward sodium electric current remains during the plateau and repolarization. Because $V<E_{\mathrm{Na}}$ in this phase, the residual electric current is inward and opposes the outward electric currents that would terminate the cardiac action potential. The ventricular action potential lasts longer, extending the QT interval on the electrocardiogram. Excessively prolonged or uneven repolarization permits early afterdepolarizations and dangerous ventricular rhythm disturbances.

Genetic association identifies the channel, while voltage clamp distinguishes defective ion-channel inactivation from a mere increase in channel abundance. Recordings of disease-associated channels show persistent or bursting openings; coupling measured mutant sodium electric current into an otherwise modeled cardiac membrane through dynamic clamp tests whether that kinetic defect prolongs the action potential. The phenotype is variant-specific: not every SCN5A variant increases late electric current or causes long QT syndrome.^[ref][ref]

Myotonia congenita. CLCN1 encodes the skeletal-muscle ClC-1 chloride channel, which normally supplies a large resting stabilizing electrical conductance. Because the chloride reversal potential is near the resting membrane potential, that pathway resists small depolarizing disturbances. Loss of functional channels, reduced electrical conductance, or a shift in opening toward more positive membrane potential reduces the resting electrical shunt. Potassium accumulation in the transverse-tubule space during activity can then produce larger depolarizing afterpotentials and trigger further muscle action potentials after the intended stimulus has ceased. Continued electrical firing sustains contraction and delays relaxation: this is myotonia, rather than a primary inability of the contractile apparatus to relax.

Patient-family linkage and variant analysis connect the disease to ClC-1. Functional recordings of disease-associated channels demonstrate shifted voltage dependence and reduced availability near resting membrane potential; single-channel and noise measurements also resolve two conductive protopores and their gating behavior. Different variants can act through different mechanisms, so a gating shift is an example rather than an explanation of every case.^{[ref][ref][ref]}

These examples connect contrasting current defects to their consequences: persistent inward sodium electric current prolongs cardiac action potentials, whereas loss of stabilizing resting chloride electrical conductance permits repetitive skeletal-muscle action potentials. Both causal chains require functional channel measurements in addition to a disease-associated sequence change.

For one spin-degenerate parabolic two-dimensional band, $k_F=\sqrt{2\pi n}=1.3729\times 10^8$ m${}^{-1}$. The free-electron model with $m=m_e$ givesThe given $10^3\Omega$ is a sheet resistivity, not a bulk resistivity in ohm-metres. The two-dimensional Drude sheet conductance is $n{e}^2\tau /m_e$, soThus $v_F=\hslash k_F/m_e=1.589\times 10^4$ m/s andThe sample temperature is not given. This is specifically the Fermi-surface mean free path, appropriate in the degenerate regime $T\ll 8.3$ K for the bare-mass model. At higher temperatures the characteristic thermal speed should replace $v_F$ for a thermal transport distance. If a real two-dimensional band has mass $m^{\ast }$, the stated results scale as $E_F,T_F\propto 1/m^{\ast }$, $\tau \propto m^{\ast }$ and $v_F\propto 1/m^{\ast }$; the inferred ${\ell }_F=\hslash k_F/(n{e}^2{\rho }_{\square })$ is mass independent under these same Drude assumptions. No semiconductor effective mass or valley degeneracy is silently introduced.

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.

In the Hodgkin-Huxley model, $V$ is the intracellular-minus-extracellular membrane potential, $C_m$ is the membrane capacitance per unit area, and $I_{\mathrm{app}}$ is the applied inward electric current per unit area. A space clamp makes $V$ spatially uniform, leaving temporal ordinary differential equations. With outward ionic electric current positive, the capacitance balance is $C_m\dot{V}=I_{\mathrm{app}}-I_i$.

The variables $m,n,h$ lie between zero and one. They describe sodium activation, potassium activation and availability against sodium inactivation, respectively. Thus larger $h$ means less inactivation. Their voltage-dependent limiting values are $m_{\infty },n_{\infty },h_{\infty }$, and ${\tau }_m,{\tau }_n,{\tau }_h$ are their positive relaxation time constants.

Each ionic term has the Ohm's law form conductance times driving potential. The sodium term has instantaneous electrical conductance $g_{\mathrm{Na}}{m}^{3}h$, the potassium term has instantaneous electrical conductance $g_{\mathrm{K}}{n}^4$, and the leak term has constant electrical conductance $g_{\mathrm{L}}$. Here the $g$ constants are maximal electrical conductances per area, and $V_{\mathrm{Na}},V_{\mathrm{K}},V_{\mathrm{L}}$ are the corresponding reversal potentials. The factors $m^{3}h$ and $n^4$ arise from independent permissive gates in an ion channel. A conducting channel carries zero net ionic electric current at its reversal potential.

Distributed circuit model. A transmission line must be treated as a distributed system when its propagation time is appreciable compared with the signal timescale. Divide a uniform two-conductor line into elements of length $dx$. Each has series resistance $R^{\prime }dx$ and inductance $L^{\prime }dx$, and shunt electrical conductance $G^{\prime }dx$ and capacitance $C^{\prime }dx$. The primes denote quantities per unit length, with units $\mathrm{\Omega }{\mathrm{m}}^{-1}$, $\mathrm{H}{\mathrm{m}}^{-1}$, $\mathrm{S}{\mathrm{m}}^{-1}$ and $\mathrm{F}{\mathrm{m}}^{-1}$, respectively.

Kirchhoff's voltage and current laws give the Telegrapher's equations:For a lossless line, $R^{\prime }=G^{\prime }=0$, differentiation gives $V_{xx}=L^{\prime }{C}^{\prime }{V}_{tt}$ and the same wave equation for $I$. The propagation speed and characteristic impedance areA general lossless signal has $V=V^{+}(t-x/u)+V^{-}(t+x/u)$ and $I=[V^{+}(t-x/u)-V^{-}(t+x/u)]/Z_0$. Thus characteristic impedance is the voltage/current ratio of one travelling wave, not simply the resistance of the wires. It can be real even when the line dissipates no energy, because energy is carried away by propagation.

For sinusoidal quantities with convention $e^{i\omega t}$, the general lossy-line formulas areChoose the square-root branches for forward attenuation and positive forward power. The real part of ${\gamma }_{\mathrm{prop}}$ is attenuation per unit length; its imaginary part is the phase wavenumber. For small losses, attenuation is approximately $R^{\prime }/(2Z_0)+G^{\prime }{Z}_0/2$, using the lossless real $Z_0$ in this estimate.

Fields and geometry. Two separated conductors support a transverse electromagnetic mode, with electric and magnetic fields transverse to the propagation axis. For a homogeneous dielectric and negligible conductor penetration, both examples below have $L^{\prime }{C}^{\prime }=\mu ϵ$ and hence $u=1/\sqrt{\mu ϵ}$. The distributed circuit model follows from the field energy and charges in each slice.

For a coaxial cable with inner radius $a$ and inner surface of the outer conductor at $b$, Gauss's law and Ampère's law giveThe electric field is radial and the magnetic field circles the axis. Integration of voltage and field energy givesThe equal and opposite return current on the outer conductor confines the ideal external field to the annulus.

For wide parallel plates of width $w$ separated by $d\ll w$, neglect fringing. The electric field crosses the gap, $E\simeq V/d$, while the magnetic field lies in the gap parallel to the plates and perpendicular to propagation, $H\simeq I/w$. ThereforeFor a forward wave, $\mathbf{E}\times \mathbf{H}$ points along the line. Integrating this Poynting vector over either cross-section gives instantaneous power $VI$. For complex peak amplitudes, average power is $\mathrm{Re}(V{I}^{\ast })/2$, or $|V^{+}{|}^2/(2Z_0)$ for a single forward wave on a lossless line. The conductors guide an energy flux predominantly in the dielectric between them.

Terminations and reflection. At a load $Z_L$, require $V_L=Z_L{I}_L$. If $V^{+}$ and $V^{-}$ are the incident and reflected voltage amplitudes there, the total current is $(V^{+}-V^{-})/Z_0$. ThusA matched load, $Z_L=Z_0$, absorbs the incident power with no reflection. An open circuit has ${\Gamma }_V=+1$, a current node and a doubled voltage at the end. A short circuit has ${\Gamma }_V=-1$, a voltage node and a doubled current there. A purely reactive load on a lossless line has $|{\Gamma }_V|=1$ and accepts no average power. For real $Z_0$, the reflected power fraction is $|{\Gamma }_V{|}^2$ and the accepted fraction is $1-|{\Gamma }_V{|}^2$.

Incident and reflected waves produce a standing-wave pattern with voltage standing wave ratio $(1+|{\Gamma }_V|)/(1-|{\Gamma }_V|)$. The input impedance of a lossless length $\ell$ isThis gives $i{Z}_0\tan (k\ell )$ for a short and $-i{Z}_0\cot (k\ell )$ for an open. A quarter-wave section gives $Z_{\mathrm{in}}=Z_0^2/Z_L$, enabling impedance transformation. After a source launches a pulse, a distant termination affects it only after the reflected pulse returns; the return time is $2\ell /u$. Real conductor resistance, dielectric loss and departures from a TEM field profile limit the ideal model at high frequency.

For the spin separation at a smooth quantum Hall edge, linearize the confinement energy at a crossing. Compensating a spin energy difference by a potential-energy change givesHere $V^{\prime }$ is interpreted as the specified slope of electron potential energy. If it denotes electrostatic voltage per length instead, the denominator is $e|V^{\prime }|$. The two edges of a neck and the two sides of an antidot have oppositely directed slopes; the formula is a positive local separation on any one edge. For local filling two, only the $n=0$ spin pair is occupied at the neck minimum: $E_{0,+}+U_{\min }<E_F<E_{1,-}+U_{\min }$. Raising both branches with the confinement produces two crossings on each side, as shown in the original neck sketch. This fixed-slope estimate precedes self-consistent electrostatic screening.

An antidot is a depleted potential hill. guiding centres circulate around its closed equipotentials, with chirality opposite to the outer sample edge. A phase-coherent round trip contributes magnetic phase $2\pi BS/{\Phi }_0$, with ${\Phi }_0=h/e$. A localized orbit resonant with the incident energy can transfer an electron between extended channels. If the outgoing channel returns toward the injecting reservoir, the process is resonant reflection and reduces electrical conductance. If coupling instead connects an otherwise blocked incident channel to the far reservoir, it is resonant transmission and adds electrical conductance. Transmission peaks require that forward connectivity; a closed orbit by itself does not guarantee them. Interchannel scattering can change which orbit participates.^[ref]

For the antidot resonance conductance bounds with one active spin-resolved resonance, writeHere ${\Gamma }_L,{\Gamma }_R$ are energy broadenings; the corresponding escape rates are ${\Gamma }_L/\hslash ,{\Gamma }_R/\hslash$. In a reflection-dominated configuration with two otherwise transmitted channels, $G=(e^2/h)(2-\mathcal{L})$: a $2e^2/h$ background can dip to $e^2/h$. In a transmission-dominated configuration with one spectator channel and the second otherwise blocked, $G=(e^2/h)(1+\mathcal{L})$: peaks rise from $e^2/h$ toward $2e^2/h$. These are the two ideal channel-count examples plotted below. More generally, $G=(e^2/h)(N_{\mathrm{open}}-R)$ or $(e^2/h)(N_{\mathrm{dir}}+T)$; a different spectator count changes the integer offset. The constriction filling alone does not specify every tunnel coupling or resonance height. Equal couplings permit unit resonant transfer in the isolated-level model; asymmetry, temperature and dephasing reduce the contrast. Holding the effective orbit area approximately fixed gives $\Delta B\simeq h/(eS)$ for one branch, not a universal charging-corrected period.

In quantum-dot versus antidot resonance topology, a quantum dot is a potential well holding a finite electron island. Its closed guiding-centre circulation is reversed relative to an antidot at the same field. With two barriers in series, current follows source $\to$ dot $\to$ drain; off resonance there need not be a direct conducting path. A gate shifts a level or an addition electrochemical potential into alignment with the reservoirs and produces a electrical conductance peak. For an isolated coherent spin-resolved level, the zero-temperature peak is at most $e^2/h$ and reaches it only for equal couplings; unresolved noninteracting spin degeneracy doubles this bound. In weak sequential tunnelling, peak height depends on rates and temperature rather than being quantized. Charging produces Coulomb blockade peaks with spacing set by addition energy and gate lever arm. An antidot also supports discrete orbit/charging resonances, but can be side-coupled to already open channels and produce dips. Gate voltage changes both orbit area and confinement, whereas a magnetic sweep directly changes enclosed flux.

Coulomb interactions require a self-consistent confinement potential. A gradual decrease in electron density cannot leave arbitrary uncompensated donor charge at no cost. Partly occupied Landau levels have a large charge response: electrostatic screening flattens their effective energy near the Fermi energy over compressible quantum Hall edge strips. Between them, integer-filled incompressible quantum Hall edge strips have a gap, weak density response and most of the potential variation. In an idealized cross-section the effective branch thus alternates between flat, partially occupied portions and rising portions crossing a gap; its carrier density varies oppositely. The bare levels still have Landau level and spin splittings locally, but the common electrostatic potential is no longer an imposed straight slope. exchange interaction can additionally renormalize spin gaps. Widths must follow electrostatics and the density gradient, so the unscreened $E_Z/|V^{\prime }|$ separation is not a quantitative screened-strip width.^[ref]

A modulation-doped GaAs-AlGaAs electron gas is confined on the GaAs side of the interface. The larger-gap AlGaAs supplies a conduction-band barrier; charge transfer from remote donors and the resulting electric field bend the bands into an approximately triangular well. transverse electronic mode wavefunctions are confined normal to the interface but free in its plane. electrons settle into the well until their electrochemical potential agrees with the donor/contact environment. An undoped spacer separates mobile electrons from ionized donors, reducing impurity scattering. Equilibrium has one Fermi level across the structure; the conduction-band edge is not flat. Surface depletion and the cap also affect the profile. The original drawing illustrates one occupied confinement transverse electronic mode; a single spatial electron gas need not in general imply only one occupied vertical transverse electronic mode.

molecular beam epitaxy can grow a clean GaAs buffer/channel, an undoped AlGaAs spacer, a Si-doped AlGaAs supply region and a protective GaAs cap on a suitable substrate. Control composition, interface quality and background impurities during growth; placing dopants directly in the channel defeats the principal mobility advantage. Define a mesa or isolation region, fabricate low-resistance ohmic contacts to the buried gas, and pattern fine metal split gates by electron beam lithography and the lift-off process. Negative gate bias depletes the underlying gas, leaving a narrow channel between the gates. Check contact resistance, leakage, carrier density and mobility; operation with ballistic electron transport requires channel length shorter than the relevant scattering lengths and smooth confinement on the wavelength scale. The lithographic gap does not equal the electrical width because of lateral depletion.

For a long uniform channel supporting ballistic electron transport, transverse confinement creates thresholds $E_n$ with continuous longitudinal propagation above each threshold. One directed spin branch of a transverse electronic mode has states per length $dk/(2\pi )$ and group velocity $v={\hslash }^{-1}dE/dk$. Hence $vdk/(2\pi )=dE/h$: velocity cancels the one-dimensional density of states. Including two unresolved spin states,For low temperature and small bias, a fully open transverse electronic mode has $T_n(E_F)=1$, givingThe assumptions are ideal reservoirs/contacts, negligible reflection and interactions that would invalidate the independent-subband description, and unresolved spin splitting. A resolved spin branch contributes $e^2/h$. The finite two-terminal electrical resistance is a contact/mode-injection effect, despite the absence of dissipative scattering inside the ideal channel.

For voltage partition in a ballistic constriction, a short saddle has strong local electric-field and electrostatic screening variation near its entrances and barrier, rather than the uniform Ohmic voltage gradient of a diffusive wire. Right-moving incident states inherit the source distribution and left-moving states the drain distribution; there is no unique local equilibrium chemical potential inside an ideal ballistic saddle. The electrostatic potential must be determined self-consistently. In a symmetric rigid-barrier approximation, the barrier follows the mean lead potential and half the electrochemical bias lies on each side. This is a stated approximation, not a universal spatial voltage profile.

For a saddle $U=U_0-m^{\ast }{\omega }_x^2{x}^2/2+m^{\ast }{\omega }_y^2{y}^2/2$, transverse electronic mode thresholds are $E_n=U_0+\hslash {\omega }_y(n+1/2)$. The smooth-barrier transmission is $T_n(E)=[1+\exp (-2\pi (E-E_n)/(\hslash {\omega }_x)){]}^{-1}$, approaching a step for a sharp threshold on the bias/thermal scale.^[ref]

Let ${\mu }_{S,D}=E_F\pm eV/2$ and assume no additional bias dependence of $T_n$. At zero temperature,When the source and drain sample $N+1$ and $N$ thresholds, the differential electrical conductance is $(N+1/2)2e^2/h$. These half plateaus arise from the two ends of the bias window, not fractional carriers. The dc ratio $I/V$ need not share their plateau values. Heating, spin splitting and interaction-induced features must be distinguished experimentally.

In bias spectroscopy of one-dimensional subbands, plot differential electrical conductance or its gate derivative against gate bias and source-drain bias. Threshold branches occur where $E_n(V_g)={\mu }_S$ or ${\mu }_D$. When one bias-window edge meets $E_n$ and the other $E_{n+1}$,Thus the intersections of adjacent threshold branches calibrate the one-dimensional spacing directly in energy. A gate lever arm $\alpha =-e^{-1}\partial E_n/\partial V_g$ can then convert gate intervals to energy; it should be measured rather than guessed. Asymmetric bias partition, a moving barrier or a gate-dependent spacing modifies slopes and plateau weights, requiring the corresponding self-consistent bias model.

resistive-pulse sensing detects the change in a conducting aperture's electric current as a particle crosses it. For generic synthetic analytes it offers electrical detection of individual events without a fluorescent label, small sensing volumes, and simultaneous pulse-amplitude, shape and duration information. Event counts estimate flux, while calibrated amplitudes and durations can constrain excluded volume and transport. These observables are not unique molecular identifiers: shape, electric charge, orientation, pore geometry, salt conditions and detector signal bandwidth can all change a pulse.

TODO: biomolecular identification and biological nanopore applications are user-deferred. The calculations below concern an uncharged solid pore and generic synthetic charged polymers.

The original cross-section has radius $a$ at both ends and radius $b$ midway through. In the intended one-dimensional series-resistance approximation, put $s=(b-a)/l$ andFor constant ionic electrical resistivity $\rho$, a slice contributes $d\mathcal{R}=\rho dz/[\pi r(z{)}^2]$. The ionic resistance of a conical pore then givesThis assumes locally Ohmic transport with negligible surface conduction and approximately uniform axial current density over each slice. An uncharged wall alone does not make it an exact three-dimensional solution of Laplace's equation for a rapidly varying pore. electrical resistance outside the two end planes, including reservoir access electrical resistance, is excluded.

Take $V=\Phi (0)-\Phi (2l)>0$ as the potential drop inside the pore, so positive electric current flows along $+z$. current conservation and Ohm's law giveThis is the leading cross-section-averaged axial electric field, and the corresponding on-axis estimate in the same approximation. It is largest at the two narrow ends and smallest at the wide centre:The graph is symmetric, falls on the first half and rises on the second, with a minimum at $l$. The electric potential difference check is ${\int }_0^{2l}E(z)dz=V$. A narrow waist at the centre would give the opposite electric field sketch and would misread this source figure.

Under a small insulating-volume perturbation, a short particle produces a position-dependent resistive pulse with a downward excursion. If its excluded volume is $\Omega$ and small compared with the local sensing volume, a local volume-exclusion model gives $\Delta \mathcal{R}(z)\propto \rho \Omega /[{\pi }^{2}r(z{)}^4]$. Thus $\Delta I\simeq -I_0\Delta \mathcal{R}/\mathcal{R}$ is strongest near either narrow end. A sufficiently resolved event can show two deeper end features separated by a shallower middle depression, then return to baseline. Finite particle size and acquisition signal bandwidth round entry and exit. The original illustration below shows this conditional small-particle mechanism, not a fitted amplitude. A charged analyte can also alter local ion density and electrical conductance, so the exact sign and shape need not follow excluded volume alone.

For positive drift in this coordinate, require $\mu V>0$. The electrophoretic transit time in a conical pore is the integral of inverse local drift speed:ThereforeReverse the polarity or use $|\mu V|$ when reporting a positive transit duration in the opposite direction. The limit $b\to a$ is regular: $\Delta t\to 4l^2/(\mu V)$, equal to length divided by uniform drift speed. Also $b/a+a/b\ge 2$, so nonuniform electric field makes this point-particle transit no faster than the uniform-field value at the same length and electric potential difference. The wide, weak-field interior dominates residence time.

The final length-dependent observations do not uniquely identify a microscopic mechanism. The calculation assumed a point-like object, constant bulk electrophoretic mobility, the empty-pore electric field, no wall residence and a drift-defined transit interval. An extended chain can sample entrance and reservoir fields simultaneously, alter its conformation and tension, and change the effective driving-to-friction ratio. These effects can make a bulk-mobility, local-field prediction underestimate the driving. For a small object, wall interactions, reversible trapping, or repeated entry excursions can add residence time that the deterministic formula omits. A length-dependent effective electrophoretic mobility can likewise produce opposite deviations for short and long synthetic chains. Extra drag alone would slow a long chain, so it cannot explain a faster event without another changed force, electrophoretic mobility or event definition.

Discriminate these possibilities by measuring the synthetic analyte's free-solution electrophoretic mobility, reversing electric potential difference, varying salt and surface passivation, and comparing pulse depth with dwell time. Resolved multiple occupancy or folds can alter amplitude as well as timing. Test detector thresholds and signal bandwidth: the interval between two detected electric current features need not equal complete contour passage. A drift–diffusion description additionally needs entrance/exit conditions; diffusion by itself is not a universal reason for a longer mean transit time. No unique anomalous-duration prediction follows from the provided parameters alone.

Choose the following original phenomenological mathematical model solely to calculate a paired-pulse readout. It is not experimental data, a fit to the teaching experiment, or a simulation of the Hodgkin-Huxley model. The conditioning action potential defines time zero. Test pulses have fixed width $0.2\mathrm{ms}$; $I$ denotes their inward electric current amplitude. Impose an absolute neuronal refractory period $t_{\mathrm{abs}}=2\mathrm{ms}$. No finite test amplitude generates an event for $0<\Delta <t_{\mathrm{abs}}$. After that hold, stipulate a required amplitudewith $I_{\infty }=0.2\mathrm{nA}$, dimensionless $A=2$ and recovery time constant ${\tau }_r=3\mathrm{ms}$. Emit a test event exactly when $I\ge I_{\mathrm{req}}(\Delta )$, including equality. For this ideal event readout, assign events to pulse onset and assume the conditioning event is successfully evoked. There is no calculated membrane potential, spike waveform, response latency or stochastic success probability.

The exponential decay represents a combined recovery deficit; it is neither the Hodgkin-Huxley model's $h$ nor a measured potassium electrical conductance. One such curve cannot separately identify sodium recovery and persistent potassium electrical conductance. A constant pulse-width amplitude criterion also does not establish an equivalent rule for other pulse widths or voltage trajectories.

At the hold boundary, $I_{\mathrm{req}}(2\mathrm{ms})=0.2(1+2)=0.6\mathrm{nA}$. At $\Delta =4\mathrm{ms}$,at $\Delta =6\mathrm{ms}$,Thus a $0.35\mathrm{nA}$ test fails at $1\mathrm{ms}$ because of the imposed hold, fails at $4\mathrm{ms}$ because recovery is insufficient for that amplitude, and succeeds at $6\mathrm{ms}$. At $4\mathrm{ms}$, a $0.5\mathrm{nA}$ pulse would instead succeed: failure of the weaker pulse was relative refractoriness.

For $I>I_{\infty }$, solve $I\ge I_{\mathrm{req}}(\Delta )$ using the logarithm. The earliest allowed delay isFor $I=0.35\mathrm{nA}$ this is $2+3\ln (8/3)\approx 4.942488\mathrm{ms}$; for $I=0.5\mathrm{nA}$ it is $2+3\ln (4/3)\approx 2.863046\mathrm{ms}$. If $I\ge 0.6\mathrm{nA}$, the imposed $2\mathrm{ms}$ hold is the only delay restriction. If $I\le I_{\infty }$, no finite delay passes the rule: the required amplitude stays strictly above its asymptotic baseline. A finite experimental “fully recovered” time therefore also needs a tolerance and a readout definition.

A quantum dot is an electron-confining potential well, whereas an antidot is a depleted potential hill. At the same magnetic field, their closed guiding-centre drift has opposite circulation. A series dot needs transmission through two barriers; a side-coupled antidot can connect already propagating quantum Hall edge states and induce backscattering. Both can have discrete energies and charging effects, but a electrical conductance peak or dip depends on lead connectivity rather than solely on the existence of a closed orbit.