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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.

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.

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.