Hodgkin–Huxley boiled down to two variables you can draw in a plane. A cubic voltage nullcline and a slow recovery line give you excitability — small nudges decay, a nudge past threshold fires a full spike — and, over a window of drive current, a Hopf limit cycle: the neuron fires again and again. Rendered, not quoted.
source FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J. 1 (1961) 445–466 · PMC1366333 · doi:10.1016/S0006-3495(61)86902-6 · Nagumo, Arimoto & Yoshizawa, Proc. IRE 50 (1962) 2061–2070. params illustrative
Two state variables: fast voltage-like v and slow recovery w. The dynamics:
The cubic in v makes the voltage nullcline an N-shape; the fast–slow split (small ε) makes v jump between the outer branches. That N-shape is the whole trick: it is what lets a threshold exist.
Illustrative constants a=0.7, b=0.8, ε=0.08 (FitzHugh's BVP values). AMBER: dimensionless model, not a fit to a specific cell. No medical advice.
The spike in two dimensions. Hodgkin–Huxley has four coupled variables (V, m, h, n); FitzHugh collapsed m→instantaneous and merged h,n into one recovery w, keeping the excitable–oscillatory essence.
Neighbour: the-hodgkin-huxley — the full ionic spike — reduced to a drawable phase plane. Same all-or-none threshold, same refractory shadow, now visible as a loop.
Re-runs the full selfcheck() live against the current engine: fixed point on both nullclines, sub-threshold decay vs supra-threshold spike, the Hopf window, and the cubic N-shape.
If the Red tamper (window 6) flips the nullcline to linear, this badge turns red and names the broken invariant.
A drive current I and an initial kick to the resting state. Fixed constants a=0.7, b=0.8, ε=0.08. Integrated with fixed-step RK4 — a deterministic seed, no randomness in the verdict.
White N-curve = v-nullcline (w = v − v³/3 + I). Purple line = w-nullcline. Green × = fixed point. The trajectory is the live spike / limit cycle.
Proven, at boot, over the pure functions: excitability with a real threshold, and a Hopf window I≈[0.34, 1.40] where a stable limit cycle exists — repetitive firing — and nowhere outside it.
“It is not a real neuron.” True. FHN is a caricature: v is not millivolts, there are no separate Na/K currents, and the constants are illustrative, not fitted. It buys geometric clarity by dropping ionic detail.
“The Hopf window depends on the constants.” True — move a,b,ε and the window moves. The structure (excitability, an N-nullcline, a Hopf onset of oscillation) is what is claimed, and that is invariant; the exact bounds are AMBER.
“A bigger kick always gives a bigger spike.” → No — it is all-or-none. Below threshold the response decays; above it, the spike height barely changes with kick size.
“More current always means faster firing forever.” → No — past the upper Hopf point the fixed point re-stabilises (depolarisation block): firing stops.
“You need all four HH variables to spike.” → No — two suffice for excitability; the plane shows why.
Drop the cubic: make the v-nullcline linear (dv/dt = v − w + I). No N-fold → no threshold, no bounded excursion — trajectories diverge. The Witness (7) catches it.
Engine intact.