E.1 The electrochemical potential and the Nernst equation

The cochlea is a mechanical device — fluid in a bony tube, membranes that vibrate, microscopic levers — but it runs on electrochemical gradients. This reference chapter supplies the bioelectric physics the cochlea and auditory-nerve chapters draw on, building on the electromagnetism of the physics book. It begins with the quantity that governs which way an ion moves across a membrane.

The electrochemical potential

A charged species in solution carries two kinds of potential energy. The chemical part comes from its concentration — a dilute species gains free energy by spreading from high concentration to low — and the electrical part comes from the local voltage acting on its charge. Their sum is the electrochemical potential,

\tilde\mu(\mathbf{r}) \;=\; \mu_0 \;+\; k_B T \ln c(\mathbf{r}) \;+\; z e\,V(\mathbf{r}).

where

$\tilde\mu$: electrochemical potential per ion J
$c$: local ion concentration mol/m³
$z$: ion valence (sign included) —
$e$: elementary charge C
$V$: local electric potential V
$k_B T$: thermal energy J

An ion flows until its electrochemical potential is uniform. Two compartments at different concentrations can still be in equilibrium if a voltage difference exactly offsets the concentration difference — and that balancing voltage is the Nernst potential.

The Nernst equation

▶ Nernst potential from electrochemical equilibrium Derivation

For an ion permeable across a membrane, equilibrium requires equal electrochemical potential on the two sides, $\tilde\mu^\text{in} = \tilde\mu^\text{out}$ :

k_B T \ln c^\text{in} + z e V^\text{in} \;=\; k_B T \ln c^\text{out} + z e V^\text{out}.

Solving for the membrane voltage $V_\text{Nernst} = V^\text{in} - V^\text{out}$ ,

V_\text{Nernst} \;=\; \frac{k_B T}{z e}\ln\frac{c^\text{out}}{c^\text{in}} \;=\; \frac{RT}{zF}\ln\frac{c^\text{out}}{c^\text{in}}.

The Nernst potential is the voltage at which the electrical and chemical driving forces on an ion exactly cancel. At body temperature ( $T = 310\,\text{K}$ ), $RT/F = 26.7\,\text{mV}$ ; multiplying by $\ln 10$ gives the useful rule of thumb — a tenfold concentration ratio corresponds to $\pm 61\,\text{mV}$ per unit charge.

E_K (Nernst)-89.0 mV

E_Na60.6 mV

E_Cl-73.6 mV

V_rest (GHK)-69.8 mV

K⁺ [in] 140 mM [out] 5 mM P 1.00

Na⁺ [in] 15 mM [out] 145 mM P 0.04

Cl⁻ [in] 7 mM [out] 110 mM P 0.45

Presets:

Each ion's Nernst potential is the voltage at which its electrical and chemical gradients balance. The GHK resting potential is the weighted average — heavily weighted toward whichever species the membrane is most permeable to. In a resting neuron, P_K ≫ P_Na, so V_rest sits near E_K ≈ −90 mV. At the hair-cell apex bathed in endolymph (K⁺ ≈ 150 mM both sides), E_K ≈ 0 — and the +80 mV endocochlear potential becomes the driver of MET-channel current.

If the actual membrane voltage differs from an ion’s Nernst potential, that ion is out of equilibrium and flows — down its electrochemical gradient — carrying current across the membrane. A real membrane, permeable to several ions at once, settles at a compromise voltage between their individual Nernst potentials, the subject of the next lesson.