Derivation of the Hodgkin–Huxley Model

We derive the Hodgkin–Huxley model from a membrane–circuit analogy and extend it to the one-dimensional cable equation. The membrane is treated as a parallel plate capacitor with conductances for Na+, K+, and leak. The Na+ and K+ pathways use m, h, n voltage-dependent gating. The leak term is ungated with a fixed reversal potential. The aim is a concise, consistent derivation that explicitly links the biophysics to the equations. While many graduate-level texts discuss elements of this derivation, in my experience it is difficult to find a contemporary derivation that is both complete—including the cable extension—and conceptually accessible to undergraduate readers.