The Hodgkin–Huxley model

This is the entry that explains why this site has two directions rather than one. The Hodgkin–Huxley model is the clearest case of an idea crossing the border, being transformed, and crossing back.

Borrowed: the axon as a circuit

Hodgkin and Huxley did not set out to invent a branch of biophysics. They had voltage-clamp data from a squid giant axon and needed a language to describe it, so they reached for the one already available in electrical engineering. The membrane became a capacitor. Each ion species became a conductance in series with a battery, the battery being its Nernst potential. Current conservation gave them

$$C_m \frac{dV}{dt} = -\bar{g}_{\mathrm{Na}} m^3 h (V - E_{\mathrm{Na}}) - \bar{g}_{\mathrm{K}} n^4 (V - E_{\mathrm{K}}) - g_L (V - E_L) + I_{\text{ext}}$$

The move that made it work was letting the conductances depend on voltage, through gating variables $m$, $h$, and $n$ that each relax toward a voltage-dependent steady state with a voltage-dependent time constant. Sodium conductance rises with depolarisation, which depolarises further — positive feedback, and the regenerative upstroke of the action potential. Potassium conductance rises more slowly and pulls the voltage back down. Negative feedback, delayed. An excitable system.

Why it mattered on the biology side

The fit was extraordinarily good, and it was predictive rather than descriptive: the model implied conduction velocities and threshold behaviour the authors had not fitted to.

It also implied machinery. The exponents on $m^3h$ and $n^4$ are not curve-fitting decoration — they suggest independent gating particles that must all be in place for conduction to occur. That inference pointed straight at discrete molecular gates, twenty-four years before the patch clamp let anyone observe one. A model built out of borrowed circuit theory told biologists what to go looking for.

Handed back: silicon neurons

Then the traffic reversed. If a neuron is a capacitor with voltage-dependent conductances, and a subthreshold MOS transistor has an exponential current–voltage relationship of roughly the right shape, you can build the equations rather than integrate them. The gating dynamics become transistor physics; the membrane capacitor becomes a capacitor.

This is the foundation of analog neuromorphic design, and the appeal is not that it is a tidy analogy. It is that the silicon runs the dynamics in real time at a tiny fraction of the energy a numerical solver would need for the same equations. The model that biology borrowed from circuits came back as a circuit.

What the model leaves out

Worth being clear about, since the model’s success has made it something of a default:

• It is a point neuron. Real dendrites have geometry, and geometry does computational work that a single compartment cannot represent.
• The gating variables are phenomenological. They fit the data beautifully without describing any actual channel protein’s conformational states.
• Squid axon is not cortex. Channel complements differ enormously across cell types, and the parameter set does not transfer.

None of which detracts from the point. A description good enough to predict unobserved molecular machinery, and later to be fabricated, has earned its place twice over.