= Reversal potential
{title2=$I=g(E_{\mathrm{rev}}-V)$}
{wiki}
A <reversal potential> $E_{\mathrm{rev}}$ is a <membrane potential> at which a specified open <ion channel>, or specified combination of <ion channels>, carries zero net <electric current>, with the surrounding conditions fixed; the <electric current> changes sign across that voltage. Specifying which current matters: an individual <ion> contribution and a sum of several contributions can reverse at different voltages. Zero net <electric current> does not mean that all <ions> have stopped moving. See https://www.ncbi.nlm.nih.gov/books/NBK11013/[Purves et al., Neuroscience, 2nd edition (2001): reversal of a mixed-ion channel current].
Throughout these examples, <membrane potential> is $V=V_{\mathrm{inside}}-V_{\mathrm{outside}}$, and \b[<conventional current> is inward-positive]. Positive <electric current> corresponds to net positive <electric charge> entering or net negative <electric charge> leaving. Negative <electric current> corresponds to the reverse charge transfer. A convenient Ohmic approximation is
$$
I_{\mathrm{in}}=g(E_{\mathrm{rev}}-V),\qquad g\geq0,
$$
where $g$ is the available <electrical conductance>. For fixed $g>0$, $V<E_{\mathrm{rev}}$ gives positive <electric current>, $V=E_{\mathrm{rev}}$ gives zero, and $V>E_{\mathrm{rev}}$ gives negative <electric current>; the line has slope $-g$. If $g=0$, the modeled pathway carries zero <electric current> at every voltage: closing it does not locate its <reversal potential>. <Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition by Wulfram Gerstner and others/3.1 Synapses> (§3.1: conductance-based synaptic current) uses outward-positive $g(V-E)$; the formula here reverses that sign to match this wiki's inward-positive convention.
For the stipulated single-pathway example from <ion channel>, $E_{\mathrm{rev}}=-80\,\mathrm{mV}$ and $g=10\,\mathrm{nS}$ give
$$
\begin{array}{c|rrr}
V\ (\mathrm{mV})&-100&-80&-60\\
E_{\mathrm{rev}}-V\ (\mathrm{mV})&20&0&-20\\
I_{\mathrm{in}}\ (\mathrm{pA})&200&0&-200
\end{array}
$$
The open <ion channel> carries zero net <electric current> at $-80\,\mathrm{mV}$ even though $g$ remains positive. Near that point a change of <membrane potential> changes the <electric current>, which is why zero current at one voltage does not make an open pathway electrically irrelevant; see <shunting inhibition>.
\Image[/ion-channel-current-and-reversal.png]
{title=Inward-positive <electric current>: one reversal and two-channel cancellation}
{description=Original computed curves for stipulated constant <electrical conductances> and fixed <reversal potentials>. Left: $g=10\,\mathrm{nS}$, $E_{\mathrm{rev}}=-80\,\mathrm{mV}$. Right: a <potassium> pathway with $g_{\mathrm K}=10\,\mathrm{nS}$, $E_{\mathrm K}=-80\,\mathrm{mV}$ and a <sodium> pathway with $g_{\mathrm{Na}}=2\,\mathrm{nS}$, $E_{\mathrm{Na}}=+50\,\mathrm{mV}$. Their net <electric current> vanishes at $-58.3333\,\mathrm{mV}$ while both individual <electric currents> remain nonzero. These are model values, not empirical measurements.}
For an <ion channel> passing only one <ion> species by <passive transport>, its <reversal potential> can equal that species' <Nernst equilibrium potential>, where the chemical and electrical contributions to <electrochemical potential> balance. At that <thermodynamic equilibrium>, passage in both directions can continue with no net transport. For an <ion channel> admitting several <ion> species, opposing contributions can cancel while each species continues to have net transport. Such a <reversal potential> is not generally any one species' <Nernst equilibrium potential>. See the Purves reference above.
To work out that cancellation explicitly, put two Ohmic <ion channel> populations across the same <cell membrane>. Stipulate constant <electrical conductances> and fixed <reversal potentials>:
$$
g_{\mathrm K}=10\,\mathrm{nS},\quad E_{\mathrm K}=-80\,\mathrm{mV},\qquad g_{\mathrm{Na}}=2\,\mathrm{nS},\quad E_{\mathrm{Na}}=+50\,\mathrm{mV}.
$$
The <potassium> and <sodium> contributions add because both pathways experience the same <membrane potential>:
$$
\begin{aligned}
I_{\mathrm{total}}&=g_{\mathrm K}(E_{\mathrm K}-V)+g_{\mathrm{Na}}(E_{\mathrm{Na}}-V)\\
&=(g_{\mathrm K}+g_{\mathrm{Na}})(E_{\mathrm{eff}}-V),\\
E_{\mathrm{eff}}&=\frac{g_{\mathrm K}E_{\mathrm K}+g_{\mathrm{Na}}E_{\mathrm{Na}}}{g_{\mathrm K}+g_{\mathrm{Na}}}
=\frac{-700}{12}\,\mathrm{mV}\approx-58.3333\,\mathrm{mV}.
\end{aligned}
$$
Thus $I_{\mathrm{total}}=12\,\mathrm{nS}(E_{\mathrm{eff}}-V)$. This <weighted mean> follows simply by expanding and regrouping the two current terms; it assumes additive Ohmic contributions and a positive total <electrical conductance>. At $V=E_{\mathrm{eff}}$, the inward-positive individual <electric currents> are
$$
I_{\mathrm K}=-\frac{650}{3}\,\mathrm{pA}\approx-216.6667\,\mathrm{pA},\qquad
I_{\mathrm{Na}}=+\frac{650}{3}\,\mathrm{pA}\approx+216.6667\,\mathrm{pA}.
$$
The <potassium> contribution is outward and the <sodium> contribution inward. Equal and opposite <electric currents> give $I_{\mathrm{total}}=0$; neither pathway has stopped transporting its <ions>. The calculation treats two distinct pathways. Applying it to one mixed-species <ion channel> would additionally require a justified decomposition into independent Ohmic contributions; a general mixed-channel <reversal potential> cannot be obtained by assuming those hypotheses silently.
This is an instantaneous balance of the specified passive <electric currents>. It would be the stationary voltage of a passive <capacitance> model containing only these constant pathways and no applied <electric current>, but it is not a complete prediction of a living cell's resting <membrane potential>. Other <electric currents>, voltage-dependent gating, and pumps involved in <active transport> can matter. Pumps that transfer net <electric charge> are electrogenic and contribute their own <electric current>. See https://www.ncbi.nlm.nih.gov/books/NBK10857/[Purves et al.: functional properties of the sodium/potassium pump].
The toy model holds <ion> <concentrations> and therefore the stipulated <reversal potentials> fixed. Continuing inward <sodium> and outward <potassium> transport would change finite reservoirs unless <active transport> or external control maintained them. Transport can be a small fraction of the reservoir's <ions> over a short interval, making fixed <concentrations> a useful approximation; it is not a claim that the reservoirs never change. See https://neuronaldynamics.epfl.ch/online/Ch2.S1.html[Gerstner et al., Neuronal Dynamics (2014), §2.1: equilibrium potentials, channel currents and maintained gradients].
A <reversal potential> makes the effect of opening an <ion channel> predictable from the present <membrane potential>. The same open pathway can carry inward or outward <electric current> as $V$ crosses its <reversal potential>. The two-pathway example reveals a particularly useful distinction: zero net charge transfer can coexist with continuing exchange of different substances. That distinction connects <membrane transport> to <electric current> measurements without mistaking electrical balance for <thermodynamic equilibrium>.
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