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An action potential is a brief regenerative excursion of membrane potential, produced by changing electrical resistance and conductance through voltage-gated ion channels. In a typical fast neuronal action potential, sodium entry raises membrane potential, activating further sodium channels. This feedback produces the rapid rising phase. Inactivation of sodium channels and increased conductance to potassium then lower membrane potential again; persisting potassium conductance can produce an undershoot below the resting level. See Purves and colleagues: the ionic basis of action potentials.

For an illustrative electrical calculation, write inward sodium electric current as $I_{\mathrm{Na}}=g_{\mathrm{Na}}(E_{\mathrm{Na}}-V)$. At $V=-50\mathrm{mV}$, taking $E_{\mathrm{Na}}=60\mathrm{mV}$ and $g_{\mathrm{Na}}=10\mathrm{nS}$ gives $I_{\mathrm{Na}}=1.1\mathrm{nA}$ inward. A potassium current $I_{\mathrm{K}}=g_{\mathrm{K}}(E_{\mathrm{K}}-V)$ with $E_{\mathrm{K}}=-80\mathrm{mV}$ and $g_{\mathrm{K}}=20\mathrm{nS}$ gives $-0.6\mathrm{nA}$, outward. These illustrative values expose opposing driving forces; a waveform requires the conductances to evolve with membrane potential and time, as in the Hodgkin-Huxley model.

An action potential propagates along an axon because local electric currents bring neighboring regions of cell membrane into the regime where their voltage-gated ion channels regenerate the signal. It is a traveling pattern of excitation, not a packet of sodium ions traversing the entire axon. Regeneration allows an action potential to travel without the passive attenuation of a small local voltage change. See Ion Channels and the Electrical Properties of Membranes.

The neuronal refractory period constrains how soon another action potential can follow. Under comparable conditions, successive action potentials have roughly stereotyped shapes, making event count and timing useful for neural coding; waveform details can nevertheless change with channel state and conditions. The leaky integrate-and-fire model abstracts each action potential to a threshold event and reset. Its reset jump is a mathematical rule, not the actual regenerative waveform produced by sodium and potassium currents.

Sustained membrane potential depolarization can prevent repeated action potentials because voltage-dependent ion-channel inactivation leaves too few available sodium channels. This differs from a reduced response to a passive current step: active Hodgkin-Huxley model dynamics and recovery from inactivation are essential to the mechanism.

A conductance-based Hodgkin-Huxley model couples membrane potential to voltage-dependent channel-gating variables. It explains generation of action potentials through regenerative and opposing ionic electric currents, while its parameters and channel assumptions determine its biological domain of validity.

In the classical model, $m$ describes sodium activation, $h$ describes availability against sodium inactivation, and $n$ describes potassium activation. Each follows two-state ion-channel gating, with its own voltage-dependent rates. The available electrical conductances areThe barred quantities are maximum electrical conductances. The factors have the interpretation of independent permissive gates in an ion channel: three activation factors and a distinct inactivation-availability factor for sodium, and four identical activation factors for potassium. Large $h$ means availability; small $h$ means inactivation. These are model assumptions, not a literal structural description of every ion channel. See Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics (2014), §2.2: model definition and stochastic opening.

With inward-positive electric current, a single-compartment form isHere $C$ is membrane capacitance, $I_{\mathrm{ext}}$ is imposed inward electric current, $g_L$ is leak electrical conductance, and each $E$ is a reversal potential. Use consistent units, and either whole-compartment quantities throughout or quantities per membrane area throughout. Opening changes electrical conductance; the reversal potential and present membrane potential determine the associated electric current. Coupling this equation to the voltage-dependent gates makes a generally nonlinear system of ordinary differential equations; a gate's fixed-voltage limit alone does not specify its evolving state during an action potential.

The original quantitative account is ^[ref]. The contrast with the leaky integrate-and-fire model is useful: here an action potential emerges from evolving ionic electric currents, whereas the simpler model records a threshold event and imposes a reset.

A permissive gate allows conduction when it is open. Suppose an ion channel conducts only when all $q$ gates are open, where $q$ is a positive integer. If the gate-open events are mutually independent events and each has the same open probability $p$ at the time considered, thenThis follows from the multiplication rule for independent events. Identical transition rates alone do not guarantee independence: the initial gate states must also be independently prepared, and the model must not couple their transitions. Gates sharing a fluctuating membrane potential can be correlated; the simple product requires an appropriate independence assumption, for example independent gates at an externally fixed voltage.

At $p=0.5$, channels requiring $q=1,2,4$ permissive gates have open probabilities $0.5$, $0.25$ and $0.0625$. For four independent gates there are $2^4=16$ possible open/closed configurations, equally probable at $p=0.5$; only the all-open configuration conducts. Thus $0.5$ is the gate-open probability, while $0.0625$ is the whole-channel open probability. Neither is a fractional state of one observed channel. Among $160$ independently prepared identical four-gate channels, the expected value of the open count is $160/16=10$, although an actual count fluctuates.

For unequal independent gate-open probabilities $p_j$, use $P_{\mathrm{open}}={\prod }_j{p}_j$ rather than $p^q$. This is the distinction behind the Hodgkin-Huxley model's $m^{3}h$: $m$ and $h$ are different activation and inactivation-availability variables, rather than one common $p$. Its $n^4$ uses four identical activation factors. Independence and the all-open requirement are model assumptions; correlated gates or alternative conducting configurations require a different calculation. The simple product is useful because a modest gate-open probability can produce sharply suppressed whole-channel availability without any additional gate interaction.

The leaky integrate-and-fire model combines a passive equation for membrane potential with a firing rule. Between events, inward-positive applied electric current $I(t)$ obeyswhere $C>0$ is membrane capacitance, $g_L>0$ is the leak conductance, $R$ is the corresponding electrical resistance, and $E_L$ is the leak reversal potential. Here the leak conductance is constant. The time constant $\tau$ describes passive relaxation. When $V$ first reaches a threshold $V_{\theta }$ from below, record an action potential event and reset $V$ to $V_r<V_{\theta }$. In the version used here, hold $V=V_r$ for a fixed neuronal refractory period $t_{\mathrm{ref}}$, then resume integration. See 1.3 Integrate-And-Fire Models.

For constant $I$, with $t=0$ at the start of an integration interval and $V(0)=V_{\mathrm{initial}}$, the exact subthreshold solution isThe equilibrium point of this passive equation is $V_{\infty }=E_L+RI$. Subtracting $V_{\infty }$ leaves $d(V-V_{\infty })/dt=-(V-V_{\infty })/\tau$, explaining the exponential function. This solution applies only until a threshold event or a change in $I$. Starting below threshold, constant input produces repeated events if $V_{\infty }>V_{\theta }$; if $V_{\infty }=V_{\theta }$, the threshold is approached asymptotically and never reached at finite time.

For a worked example, chooseThese imply $g_L=10\mathrm{nS}$ and $C=200\mathrm{pF}$. All these parameters are illustrative model choices, not measurements from a particular neuron. With $I=200\mathrm{pA}$, Ohm's law gives $RI=20\mathrm{mV}$ and hence $V_{\infty }=-45\mathrm{mV}$. ThusAt threshold, $e^{-t/\tau }=1/4$, so the first event occurs after $T=20\ln 4\mathrm{ms}\approx 27.7259\mathrm{ms}$. Each subsequent cycle includes the $2\mathrm{ms}$ neuronal refractory period, givingThis is the ideal sustained firing rate; the first-event latency is $T$, not $T+t_{\mathrm{ref}}$.

With $I=100\mathrm{pA}$ instead, $RI=10\mathrm{mV}$ and $V_{\infty }=-55\mathrm{mV}$. From the same initial state, $V(t)=-55\mathrm{mV}-10\mathrm{mV}{e}^{-t/\tau }$ stays below $-50\mathrm{mV}$: no events occur, however long this current is held. The critical sustained current is $(V_{\theta }-E_L)/R=150\mathrm{pA}$; equality gives asymptotic approach rather than finite-time firing.

The leak gives old inputs progressively less influence, while capacitance prevents an ordinary finite electric current step from changing $V$ instantly. This combination makes the leaky integrate-and-fire model useful for exploring input summation, event timing and network activity at low computational cost. Its fixed parameters and reset omit the evolving voltage-gated ion channels of the Hodgkin-Huxley model, spatial integration in dendrites and mechanisms of neural adaptation. Treating a synapse as a prescribed $I(t)$ also omits its voltage-dependent driving force unless a conductance term is added. See 1.4 Limitations of the Leaky Integrate-and-Fire Model and constant-input firing rate of a leaky integrate-and-fire neuron.

A neuronal refractory period is the change in excitability following an action potential: recent firing changes whether, and how readily, a neuron can fire again. The absolute neuronal refractory period prevents a second regenerative action potential; during the relative neuronal refractory period, some stimuli that work after full recovery still fail. These are useful distinctions, but their durations are not universal constants. The preparation, temperature, starting membrane potential, stimulation protocol and criterion for detecting an action potential must accompany a measured interval.

The mechanism continues the opposing sodium and potassium electric currents described under action potential and the Hodgkin-Huxley model. After a spike, ion-channel inactivation leaves fewer sodium-selective voltage-gated ion channels available for regenerative depolarization, while persisting potassium electrical conductance opposes that depolarization. In the Hodgkin-Huxley model, recovery of $h$ restores sodium availability and relaxation of $n$ reduces potassium electrical conductance. Closed but available channels differ from inactivated channels: opening and recovery from ion-channel inactivation are different processes.^[ref]

This short memory matters for spike trains: a stimulus-only Poisson process can predict very short interspike intervals that a refractory neuron cannot produce. Neuronal paired-pulse recovery makes that memory visible. The phenomenological paired-pulse recovery example below separates a chosen recovery rule, pulse strength and event readout; refractory limits on neuronal firing then shows why an absolute bound is not a prediction of sustained firing rate.

Alan Hodgkin and Andrew Huxley recorded the membrane potential directly between the inside of a squid axon and the external solution. The crucial result was an action potential that passed through zero: the interior became positive, rather than merely losing its negative resting membrane potential. A theory in which excitation simply abolishes membrane selectivity and shorts out the resting membrane potential cannot account for that reversal. Their original report establishes the historical interpretation independently of the unavailable examination picture.^[ref]

The later Hodgkin-Huxley model explains the rising phase through increased sodium permeability and the falling phase through sodium ion-channel inactivation and increased potassium permeability. Distinct Nernst equilibrium potentials make a selective conductance change capable of reversing the membrane potential. The recording established the phenomenon; it did not by itself reveal individual ion channel structures or all their gating kinetics.

Source illustration TODO: the source page's recording is missing in both the held PDF and its archive parent. No amplitudes, time intervals, or trace layout are reconstructed here; the surviving historical request is answered above.

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.

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.

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.

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.

At rest, the cell has relatively high intracellular potassium and high extracellular sodium. The membrane potential is negative inside, largely because selective resting permeability, particularly potassium leak, combines with these electrochemical gradients. The sodium-potassium pump maintains the gradients over time; it is not the immediate regenerative current of an action potential.

Depolarization rapidly activates sodium voltage-gated ion channels, causing inward sodium current and further depolarization. Slower sodium inactivation and delayed potassium activation then reduce inward current and increase outward potassium current, repolarizing the membrane. Persisting potassium conductance can produce an after-hyperpolarization; recovery of sodium availability contributes to the neuronal refractory period. A spike transfers only a small fraction of the bulk ions, so it changes charge separation much more than it changes the overall concentration gradients.

For two-state ion-channel gating, let $X$ be the fraction of permissive gates. Opening transfers probability from $1-X$ at rate ${\alpha }_X(V)$, and closing transfers it back at rate ${\beta }_X(V)$:Independent subunits that must all be permissive multiply their probabilities. The Hodgkin-Huxley model therefore uses $g_{\mathrm{Na}}={\bar{g}}_{\mathrm{Na}}{m}^{3}h$ and $g_{\mathrm{K}}={\bar{g}}_{\mathrm{K}}{n}^4$: $m$ describes sodium activation, $h$ sodium availability, and $n$ potassium activation. Their voltage-dependent time scales produce the sequence of inward and outward current. This channel-level construction is the mechanism in the primary Hodgkin–Huxley membrane-current model.

The leaky integrate-and-fire model treats the membrane as a capacitance $C_m$. The leak conductance $g_L$ drives voltage towards its resting reversal $V_L$; the excitatory conductance $g_E$ drives it towards its reversal $V_E$; and $I_e$ is injected current, positive inward. A sufficiently high $V_E$ makes increasing $g_E$ depolarizing over the relevant voltage range, while also increasing total conductance and shortening the integration time.

For constant input and conductances, putThenIf this trajectory reaches $V_{\theta }$, register a spike and reset the voltage to $V_0<V_{\theta }$, optionally enforcing a specified neuronal refractory period. The reset makes repeated threshold crossings possible under a constant suprathreshold drive. If $V_{\infty }\le V_{\theta }$ and the initial voltage is below threshold, no finite-time crossing occurs; equality gives asymptotic approach. For time-varying currents or conductances, integrate the subthreshold differential equation until the next crossing.

Use this model when spike timing, spike counts and network integration matter more than the detailed waveform: it is inexpensive for large recurrent networks or comparisons of current-driven and conductance-driven input. It omits ionic spike initiation, channel-specific adaptation and many dendritic effects, for which a Hodgkin-Huxley model or a spatial model may be needed.

Computational models and biological experiments answer complementary questions. Experiments constrain what a nervous system actually does; a model makes a proposed explanation precise enough to test whether its mechanisms can produce those observations. The useful comparison is between competing, experimentally constrained explanations, rather than between modelling and experimentation as alternatives. A good model exposes assumptions, predicts new observations and identifies measurements that would distinguish it from rivals.

A Hodgkin-Huxley model links action potentials to voltage-dependent conductances and channel kinetics. It is more informative than merely fitting a spike waveform because changes in its conductances predict changes in excitability. Yet a model calibrated for one cell or preparation does not automatically represent another: different channel combinations and spatial geometry can yield similar outputs. Detailed parameters must be constrained by experiments, and apparent agreement does not make them uniquely identifiable. A leaky integrate-and-fire model deliberately sacrifices channel detail to study the effects of connectivity, timing and synapses in a large network. It is useful for such questions even though it cannot explain the microscopic origin of the action-potential waveform.

At a functional level, an LNP model tests which stimulus features predict a neuron's response and how those features map to firing rate. Independent testing, including response correlations rather than only mean rate, reveals what its assumptions miss. It can characterize a receptive field without claiming that the brain explicitly performs its particular fitted calculation. A Hopfield network likewise demonstrates that recurrent interactions can implement content-addressable memory: corrupted cues relax toward stored patterns. Its usefulness is an existence proof and a quantitative account of interference, not evidence that biological memory literally uses symmetric binary connections and the same update schedule.

Developmental models connect observed input statistics with synaptic plasticity. Hebbian learning with normalization or synaptic competition can produce selective weights and segregation of competing inputs. Models of correlated ON and OFF inputs can explain the formation of oriented simple-cell subregions and predict how changing the correlations changes their organization. That makes altered visual experience or patterned stimulation informative tests. Such models still need anatomical, developmental and plasticity constraints; several learning rules may reproduce similar final receptive fields. A primary model of activity-dependent ON/OFF competition illustrates this approach.

Normative models instead ask what representation would perform a specified task efficiently. For example, sparse coding of natural images can produce localized oriented filters resembling simple cell receptive fields. This links environmental statistics to a useful coding objective, but resemblance alone neither proves that the objective is the only explanation nor specifies the biological learning mechanism. Predictions about altered stimulus ensembles, population activity and synaptic changes provide stronger tests. The original sparse-code receptive-field model provides an example.

These distinctions also explain the dangers of models. Excessive flexibility can fit noise; unrealistic assumptions can generate a desired effect by construction; and multiple parameter sets can have nearly indistinguishable outputs. Simulation without analysis may obscure rather than explain a mechanism. Cross-validation, uncertainty analysis, comparisons with simpler models and controlled perturbations are therefore important. Conversely, collecting more measurements without a discriminating hypothesis can leave the same explanatory ambiguity unresolved.

The productive cycle is to propose a constrained model, determine which observations it predicts beyond the fitting data, perform the relevant experiment, and revise or reject the model. Models help select experiments and interpret results; experiments establish their scope and reveal omitted mechanisms. The appropriate level of detail depends on the question: spike-generation mechanisms, circuit computation and developmental organization need not be explained by the same model.

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.

Replacing the fast activation gate by $m=m_{\infty }(V)$, taking ${\tau }_n(V)={\tau }_h(V)$ and assuming $n_{\infty }(V)+h_{\infty }(V)=K$ constant reduces the Hodgkin-Huxley model to equations for $V,n$ when $n(0)+h(0)=K$. Indeed $q=n+h$ satisfies $\tau (V)\dot{q}=K-q$, so $q=K$ and $h=K-n$. The FitzHugh-Nagumo model additionally simplifies the two resulting response functions to a cubic activation equation and a linear recovery equation.

During a relative neuronal refractory period, another action potential is possible, but incomplete recovery changes the excitation needed to obtain it. Remaining sodium ion-channel inactivation and elevated potassium electrical conductance both matter. A hyperpolarization after an action potential can increase the voltage excursion needed for firing, while extra electrical conductance can drain an injected pulse faster. Thus a single fixed neuronal firing threshold measured from the instantaneous voltage misses relevant state variables.^[ref]

Recovery can depend on earlier spikes as well as the last one; the Hodgkin-Huxley model and neural adaptation provide richer histories than an imposed reset. A fixed hold in a leaky integrate-and-fire model is a useful simplification with a specified domain, rather than a complete explanation of a relative neuronal refractory period.^[ref]

Two-state ion-channel gating models an individual gate as a random switch between closed and open states. Write $p(t)$ for its open probability; this is the $X$ in the header equation. An individual gate's state is always closed or open, rather than partly open. A population of identical gates can have a fraction open near $p(t)$, and averaging repeated identically prepared trials estimates the same probability. A finite population's observed fraction fluctuates. See Purves and colleagues, Neuroscience, 2nd edition (2001): Ion Channels Underlying Action Potentials for single-channel steps and their population averages.

At fixed membrane potential $V$, let constant rates $\alpha ,\beta >0$ govern closed-to-open and open-to-closed transitions respectively, in ${\mathrm{s}}^{-1}$. This is a two-state continuous-time Markov chain. Over a short interval $\Delta t$, an initially closed gate opens with probability $\alpha \Delta t+o(\Delta t)$, and an initially open gate closes with probability $\beta \Delta t+o(\Delta t)$. Gain minus loss gives the deterministic ordinary differential equation for the probability:Here $p_{\infty }$ is the fixed-voltage limiting open probability, and $\tau$ is a time constant. The ratio of the rates fixes the limit, while their sum fixes the speed. With initial probability $p(0)=p_0\in [0,1]$,For a short derivation, subtract $p_{\infty }$: the deviation obeys $d(p-p_{\infty })/dt=-(p-p_{\infty })/\tau$, yielding the exponential function. After one time constant, $1/e$ of the initial deviation remains. The solution stays between $p_0$ and $p_{\infty }$, so it remains a valid probability. The rate and time constant formulations are described in Gerstner and colleagues, Neuronal Dynamics, §2.2.2: transition rates and channel kinetics.

For a worked toy condition A, choose $\alpha =\beta =50{\mathrm{s}}^{-1}$ and $p_0=0.1$. Then $p_{\infty }=0.5$, $\tau =0.01\mathrm{s}=10\mathrm{ms}$, andIn a separately prepared condition B, choose $\alpha =200{\mathrm{s}}^{-1}$, $\beta =50{\mathrm{s}}^{-1}$ and the same $p_0=0.1$. Now $p_{\infty }=0.8$, $\tau =4\mathrm{ms}$, and $p(10\mathrm{ms})=0.8-0.7e^{-2.5}\approx 0.742540$. Condition B approaches a higher limit more quickly. These are two stipulated constant-rate conditions with $V$ held fixed in each experiment, not an empirical function relating rates to voltage or a sequential switch between conditions. The initial $p_0=0.1$ need not be the limiting probability at the newly imposed voltage.

In the Hodgkin-Huxley model, the rates depend on membrane potential: ${\alpha }_X(V)$ and ${\beta }_X(V)$. Its $X_{\infty }(V)={\alpha }_X(V)/({\alpha }_X(V)+{\beta }_X(V))$ is the limit for voltage held at that $V$. During an action potential, changing $V$ changes the rates and gates feed back onto $V$ through ionic electric currents; substituting the present voltage into the constant-rate solution does not solve that coupled system. Gating describes availability of a conducting pathway. Even an open ion channel carries zero net ionic electric current at its reversal potential.

The useful bridge is from abrupt random molecular events to a smooth population prediction. A channel requiring several independent permissive gates in an ion channel has an open probability equal to their product; this can make channel availability much smaller than each gate's availability. The Hodgkin-Huxley model uses ${\bar{g}}_{\mathrm{Na}}{m}^{3}h$ and ${\bar{g}}_{\mathrm{K}}{n}^4$ for sodium and potassium electrical conductances, respectively.