FitzHugh-Nagumo Excitable Neuron

What you are seeing: a two-variable reduction of the Hodgkin-Huxley neuron model: $\dot v = v - v^3/3 - w + I$ , $\dot w = \epsilon (v + a - b w)$ with $a = 0.7$ , $b = 0.8$ , $\epsilon = 0.08$ . The fast variable $v$ is voltage, the slow variable $w$ is recovery. External input $I$ tunes the system from excitable rest (one spike per suprathreshold kick) to limit-cycle oscillation.

Press kick to jump $v$ to 0 from rest. If the kick is subthreshold (small), $v$ returns to rest. If suprathreshold (large enough $I$ or $v$ jump), the system fires one full action potential before returning. Slide $I$ above 0.4 and the system goes through a Hopf bifurcation into periodic firing. The phase portrait shows the trajectory relative to the cubic nullcline.

Figure 1. FitzHugh-Nagumo model. Method: RK4 with explicit nullcline overlay.

I (input)0.00

speed3

WHAT TO TRY

It starts with I = 0.5, in the oscillatory band: watch v spike and crash while the slow recovery w trails behind, and the phase trajectory settle onto the limit cycle that loops around the unstable fixed point.
Drop I below 0.4: the fixed point becomes stable and the neuron falls quiet. Now hit Kick to inject a perturbation and watch a single all-or-none spike fire before it relaxes back, the signature of excitability.
Read the phase plane: spikes happen when the trajectory rounds the fold of the cubic v-nullcline; the w-nullcline sets where the fixed point sits and whether it is stable.