An action potential: Hodgkin and Huxley's channels, and the ions they let through

A squid axon's membrane with Hodgkin and Huxley's voltage-gated Na⁺ and K⁺ channels, stimulated by a brief current every so often. Past threshold, Na⁺ channels open, which depolarizes the membrane and opens more of them; then they inactivate, K⁺ channels open, and the membrane swings back below rest. Underneath, the ions themselves are solved for, on both sides: the level diagram shows each one's driving force across the membrane, through the spike. Vary the stimulus, the temperature, and block channels as tetrodotoxin (Na⁺) and tetraethylammonium (K⁺) do; or put the membrane behind the narrow space a squid axon really has outside it, where the K⁺ of each spike piles up. (More about the model below.)

About this simulation

A slice across the axon's membrane: 5 µm of unstirred sea water outside, held at a fixed composition beyond that, the membrane, and 20 µm of axoplasm, closed at its far end. Both solutions are strictly neutral; the membrane is a capacitor (1 µF/cm²) whose charge sits in the solutions beside it. The concentrations are a squid axon's (mM, outside / inside: Na⁺ 440 / 50, K⁺ 20 / 400, Cl⁻ 450 / 52, with anions that don't cross making up the rest; sea water's Mg²⁺ and Ca²⁺ are left out, so there's less Cl⁻ than its 560 mM or so). The stimulus is K⁺ injected into the axoplasm from 15 µm in, as from a micropipette, for 1 ms each time.

The channels are Hodgkin and Huxley's (1952), from their voltage-clamp measurements at 6.3 °C: the Na⁺ channel open with probability m³h and the K⁺ channel with n⁴, each gate relaxing toward its own voltage-dependent level at its own rate. They're gates on the membrane's permeabilities, so each ion still crosses by electrodiffusion in the field across the membrane (Goldman–Hodgkin–Katz), down its own electrochemical potential, rather than through Hodgkin and Huxley's linear conductances. Those differ, so the permeabilities are the ones that, fully open at rest (−65 mV), give their maximum conductances (Na⁺ 120, K⁺ 36 mS/cm², and their leak, 0.3 mS/cm², carried here by Cl⁻). GHK's currents rectify, each ion conducting more easily from the side where it's plentiful: at +12 mV, Na⁺'s conductance is two thirds of its value at rest and K⁺'s is double, so the spike peaks near +12 mV rather than the +40 mV of Hodgkin and Huxley's linear version. A real squid spike overshoots to about +40 mV, and Hodgkin and Huxley found their open channels' instantaneous currents nearly linear in the voltage, so the smaller spike here is a cost of treating the channels by electrodiffusion. Matched instead where each channel opens, the rest destabilizes and the axon fires on its own: the two models can't agree at every voltage. Warmer, every gate runs faster (three times for each 10 °C, as they measured), and the spike gets briefer and smaller, as K⁺ channels and Na⁺ inactivation catch up with Na⁺ activation: a 1 ms stimulus must be stronger to fire it, about 6 µA/cm² at 6.3 °C, 30 at 25 °C and 90 at 30 °C, and there the spike barely reaches 0 mV. That's Hodgkin and Huxley's heat block, coming on.

The level diagram draws each ion's species voltage V_i = μ̄_i/(z_iF), its electrochemical potential in volts, against its value outside, and φ the same way. Each line is flat through each solution and steps at the membrane by V_m − E_i, that ion's driving force; φ steps by V_m itself. At rest Na⁺ has a big step: it would rush in if it could. As the Na⁺ channels open its step shrinks, as V_m heads toward E_Na, while K⁺'s grows; then the K⁺ channels open and bring V_m back. The ions that cross in a spike are few (V_m swings by charging the membrane), so the levels in each solution barely move.

The K⁺ that leaves in each spike has to go somewhere. Frankenhaeuser and Hodgkin (1956) found that a squid axon's membrane faces not the open sea water but a space some 30 nm thick, between it and the Schwann cells wrapped around it, which ions cross only slowly; K⁺ piles up there with every spike and clears over tens of milliseconds. Choose that outside, and the space is a 30 nm layer behind a barrier that every ion crosses with one permeability, set to clear it in 50 ms (θ/P, as they estimated). Here each spike adds about 3.5 mM (some 13 pmol/cm² of K⁺ leaves per spike in this model; they put it at about 1.6 mM a spike), and K⁺ just outside the membrane sits at 24 mM even at rest, where it leaks out steadily. E_K, which counts only the concentrations either side of the membrane, follows it: the rest depolarizes by a few millivolts, and each spike's undershoot is shallower than the last, as they saw. In the level diagram, zoomed onto the space, K⁺'s line steps up across the barrier: the driving force that clears it. Nothing in Hodgkin and Huxley's equations, which take E_K as fixed, can do this.

There's no Na⁺/K⁺ pump, so this axon has no true steady state: over hours it would run down to Donnan equilibrium, as the resting-potential demo shows. Over the milliseconds watched here that doesn't show. The gating currents of the channels' voltage sensors are left out, as Hodgkin and Huxley left them out. Against the space-clamped Hodgkin–Huxley equations with GHK currents, solved separately, the spike agrees to within 1 mV (the gates test).