An action potential travels: cable theory out of the ions

Six centimetres of a squid giant axon, along its length. The axoplasm is a salt solution whose ions drift in the field along it; the membrane, all along, is a capacitor with Hodgkin and Huxley's channels to the sea water outside. Stimulate one end and the spike runs down the axon at 18.7 m/s, within half a percent of the 18.8 m/s Hodgkin and Huxley computed, though nothing of cable theory is put in. Watch what carries it: the current along the axon is mostly K⁺, not the Na⁺ that enters. Vary the axon's radius (speed goes as its square root), the temperature, and block Na⁺ channels. (More about the model below.)

About this simulation

x runs along the axon, so the axoplasm is one region 6 cm long, sealed at both ends, and the membrane is a port through all of it: a capacitance (1 µF/cm²) and Hodgkin and Huxley's Na⁺, K⁺ and leak channels, each a conductance gated by m³h, n⁴ or nothing, from the voltage across the membrane at that point to the sea water outside. A cylinder of radius a has 2/a of membrane per volume of axoplasm. The sea water is held at its composition and at φ = 0: it's big enough that its resistance doesn't matter (Hodgkin and Huxley's assumption too). The axoplasm is strictly neutral, its ions (K⁺ 400, Na⁺ 50, Cl⁻ 52 mM, and large impermeant anions, at about half the small ions' mobility, as isethionate and aspartate have) diffusing and drifting along x, with diffusivities that give Hodgkin and Huxley's resistivity, 35.4 Ω·cm. The concentrations put each ion's Nernst potential at their reversal potentials (Na⁺ +50, K⁺ −77, leak −54 mV) at 18.5 °C, their axon's temperature. The stimulus is K⁺ injected into the first millimetre for 0.2 ms.

Nothing here is the cable equation, but it comes out. Where the membrane is depolarised, current flows on along the axoplasm and out through the membrane ahead, charging it (the capacitive current); past threshold there, Na⁺ channels open and the spike moves on. Behind it, Na⁺ inactivation and open K⁺ channels leave the membrane refractory, so it can't turn back. Its speed matches the cable equation, solved separately, to 2e-3 (18.73 m/s at their 238 µm; they computed 18.8 by hand), and goes as the square root of the radius, which is why the squid's axon is giant and ours are wrapped in myelin instead. Warmer, every gate is faster (three times per 10 °C) and so is the spike, but it shrinks: at 30 °C it travels at 24 m/s and barely reaches 0 mV, at 34.5 °C it still travels, peaking at −20 mV, and at 35 °C it dies out within a few centimetres: heat block. With 60% of the Na⁺ channels it slows to 15 m/s; with 40% it can't propagate at all.

What the cable equation can't say is which ions carry the current. Along the axoplasm it's carried by every ion in proportion to its conductivity: more than half (57%) by K⁺, the most abundant, and 28% by the large anions, whatever crosses the membrane (Na⁺ and Cl⁻, with one diffusivity and nearly one concentration here, carry 7% each). The Na⁺ that enters at the front mostly stays near where it came in, and the concentrations hardly change: by a few parts per million per spike in the giant axon, and about two in ten thousand in one of 10 µm, whose membrane is 24 times larger for its volume. The level diagram draws each ion's species voltage V_i = μ̄_i/(z_iF) inside, its electrochemical potential in volts, against its level outside: each line is that ion's driving force across the membrane, V_m − E_i, all along the axon, and φ's is V_m itself. At the spike's peak Na⁺'s line comes halfway up toward zero, its equilibrium (the peak, about +25 mV, is 25 mV short of E_Na), while K⁺'s is driven furthest from its own, ready to pull the membrane back.