The resting neuron
A neuron sits between two ion batteries: potassium at $E_K = -90$ mV and sodium at $E_{Na} = +60$ mV. It opens MORE potassium channels. Its voltage…
At the default $g_K = 7$ the marker rests at $V = -71.25$ mV — far closer to $E_K$ than to $E_{Na}$. Push $g_K$ to $20$ and it slides to $-82.86$ mV, creeping toward the $-90$ line without ever reaching it. Drop $g_K$ to $1$, so the two conductances tie, and $V$ jumps to $-15.00$ mV: the exact midpoint of $-90$ and $+60$.
In this model, the membrane settles at a conductance-weighted average of the ion batteries:
Each battery pulls $V$ toward its own value, and conductance is the strength of its grip — a tug-of-war where opening channels adds rope on one side. With $g_K = 7$ and $g_{Na} = 1$ the average lands at $\frac{7(-90) + 60}{8} = -71.25$ mV. This is the core reason a typical neuron rests near $-70$ mV: at rest, potassium conductance dominates, so the average sits near $E_K$, while a small sodium leak keeps it from getting all the way there. The full biological account adds more players — chloride, the sodium–potassium pump — but the weighting logic is the same.
About this model
What it represents: the steady-state voltage of a patch of membrane with two conductances, potassium and sodium, each pulling toward its own battery — the parallel-conductance picture behind the resting potential.
What it leaves out: membrane capacitance and time — the equation says where the voltage settles, never how fast it gets there, so it cannot describe a changing signal. Also missing: the kinetics of the sodium–potassium pump that maintains the concentration gradients, chloride and calcium conductances, and all spatial structure — the cell here is a single point.
Source: Gerstner, Kistler, Naud and Paninski, Neuronal Dynamics, Ch. 2.2 (EPFL, free online) — the full conductance-based membrane equation, including the capacitance this lesson omits.
Why does the resting neuron sit near $-70$ mV instead of exactly at $E_K = -90$ mV?
Set $g_K = 3$ with $g_{Na} = 1$. What voltage does the model predict?
Transfer: suppose sodium conductance suddenly dominates — $g_{Na}$ becomes huge while $g_K$ stays at $7$. What does the weighted average do?
Membrane voltage in this model is a conductance-weighted average of ion batteries: $V$ obeys whichever conductance dominates. The fraction gives the steady state — where the voltage settles, not how it moves. Changing voltage takes time, because membranes have capacitance, and those dynamics are exactly what this model omits and lesson 2 adds.