= Two-state ion-channel gating
{title2=$\dot X=\alpha_X(V)(1-X)-\beta_X(V)X$}
<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 https://www.ncbi.nlm.nih.gov/books/NBK10841/[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>:
$$
\frac{dp}{dt}=\alpha(1-p)-\beta p=\frac{p_\infty-p}{\tau},\qquad
p_\infty=\frac{\alpha}{\alpha+\beta},\qquad \tau=\frac{1}{\alpha+\beta}.
$$
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> $p(0)=p_0\in[0,1]$,
$$
p(t)=p_\infty+(p_0-p_\infty)e^{-t/\tau}.
$$
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 https://neuronaldynamics.epfl.ch/online/Ch2.S2.html[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 $p_\infty=0.5$, $\tau=0.01\,\mathrm{s}=10\,\mathrm{ms}$, and
$$
p(10\,\mathrm{ms})=0.5-0.4e^{-1}\approx0.352848.
$$
In 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 $p(10\,\mathrm{ms})=0.8-0.7e^{-2.5}\approx0.742540$. 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.
\Image[/ion-channel-gating-relaxation.png]
{title=<Two-state ion-channel gating>: two constant-rate relaxation conditions}
{description=Original computed <probability> curves, both starting at $p_0=0.1$. Dashed lines show the limiting open <probabilities> $0.5$ and $0.8$; dotted lines mark <time constants> $10$ and $4\,\mathrm{ms}$. The dot marks condition A at $10\,\mathrm{ms}$, where $p\approx0.352848$. Smooth curves represent open <probability> or population averages, not an individual gate's random state trace or an ionic <electric current>.}
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 $\bar g_{\mathrm{Na}}m^3h$ and $\bar g_{\mathrm K}n^4$ for <sodium> and <potassium> <electrical conductances>, respectively.
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