= Hodgkin-Huxley model
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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> are
$$
g_{\mathrm{Na}}=\bar g_{\mathrm{Na}}m^3h,\qquad g_{\mathrm K}=\bar g_{\mathrm K}n^4.
$$
The 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 https://neuronaldynamics.epfl.ch/online/Ch2.S2.html[Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics (2014), §2.2: model definition and stochastic opening].
With inward-positive <electric current>, a single-compartment form is
$$
C\frac{dV}{dt}=I_{\mathrm{ext}}+g_{\mathrm{Na}}(E_{\mathrm{Na}}-V)+g_{\mathrm K}(E_{\mathrm K}-V)+g_L(E_L-V).
$$
Here $C$ 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 https://pmc.ncbi.nlm.nih.gov/articles/PMC1392413/[Hodgkin and Huxley (1952), A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology 117, 500–544, doi:10.1113/jphysiol.1952.sp004764]{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.
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