FitzHugh–Nagumo Phase Plane & Hopf Bifurcation

How a single control parameter — injected current I — flips a quiet neuron into a self-sustained spiking oscillator, and back again.

Model

\[ \dot V = V - \dfrac{V^{3}}{3} - W + I \qquad\quad \dot W = \epsilon\,(V + a - bW) \]

Jacobian & eigenvalues at (V^*, W^*)

\[ J=\begin{bmatrix}1-(V^{*})^{2} & -1\\[2pt] \epsilon & -\epsilon b\end{bmatrix},\quad \lambda_{\pm}=\tfrac12\!\left(\mathrm{tr}J \pm \sqrt{(\mathrm{tr}J)^2-4\det J}\right) \]

Phase plane

V-nullcline W-nullcline stable fixed point unstable fixed point trajectory current state

Click anywhere on the plane to drop a new initial condition (V₀, W₀) and watch it integrate live. When the fixed point is unstable, trajectories spiral onto a stable limit cycle (repetitive spiking).

Applied current I

0.33 1.42

I =

a, b set the nullcline geometry; ε is the recovery-timescale ratio. b is kept below 1 so det(J) stays strictly positive — a single fixed point, never a saddle.

RESTING

V^*

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W^*

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λ₁

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λ₂

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classification

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Membrane potential V(t)

Bifurcation: Re(λ) vs I

Shaded band = unstable window where Re(λ) > 0 (repetitive spiking / limit cycle). Dashed lines mark the two Hopf bifurcations.

Challenges

Drive the simulation to tick these off automatically.

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◼ Pause & Think

There’s an unstable window 0.33 < I < 1.42. Confirm it by watching Re(λ) cross zero. Why does spiking STOP again at high current?