FitzHugh-Nagumo Neuron Model

The FitzHugh-Nagumo model is a 2D reduction of the Hodgkin-Huxley equations capturing the essential dynamics of neuronal excitability. The phase plane shows nullclines (v-nullcline: cubic, w-nullcline: linear) and the animated trajectory. Increasing the external current I drives the system from rest to periodic spiking.

Resting

Phase plane (v, w)

Membrane voltage v(t)

External current I_ext 0.00

Recovery rate a 0.70

Recovery timescale ε 0.080

b (w-nullcline slope) 0.50

FitzHugh-Nagumo equations:
dv/dt = v − v³/3 − w + I_ext
dw/dt = ε(v + a − b·w)

v-nullcline: w = v − v³/3 + I_ext (cubic, fast manifold)
w-nullcline: w = (v + a)/b (linear, slow manifold)

When I_ext is small, the intersections form a stable fixed point (rest). As I_ext increases past the Hopf bifurcation threshold (~0.33), the fixed point loses stability and a limit cycle emerges — the neuron fires periodically. The cubic nullcline shifts up with I_ext.