FitzHugh–Nagumo Model

Excitability, Spikes, and Fast-Slow Dynamics

The FitzHugh–Nagumo model is a simplified neuron model that captures key features of excitability and spiking behavior (FitzHugh 1961; Nagumo, Arimoto, and Yoshizawa 1962). It consists of two coupled ODEs:

\[\begin{aligned} \epsilon \dot v &= f(v) - w + I_{\text{app}} \\ \dot w &= v - \gamma w \end{aligned} \tag{1}\]

Where \(v\) is the fast variable representing the membrane potential, \(w\) is a recovery variable, \(I_{\text{app}}\) is an applied current, and \(\epsilon \ll 1\) controls the timescale separation between the fast variable \(v\) and the slow variable \(w\). \(f(v)\) is a cubic nonlinearity, often taken as

\[ f(v) = v (1 - v)(v - \alpha),\quad \text{for } 0 < \alpha < 1 \tag{2}\]

Here \(v\) is the fast variable and \(w\) is the slow recovery variable. That time-scale split is what makes the model a useful example of excitability and spike-like excursions.

Nullclines and Geometry

The nullclines are

\[ \dot v = 0 \quad \Longrightarrow \quad w = f(v) + I_{\text{app}}, \qquad \dot w = 0 \quad \Longrightarrow \quad w = \frac{v}{\gamma}. \]

The cubic-shaped \(v\)-nullcline and the straight-line \(w\)-nullcline organize the phase-plane flow. Depending on the parameter values, trajectories may relax to rest, produce a large excursion, or settle into sustained oscillations.

Figure 1: Phase plane of the FitzHugh–Nagumo system with epsilon = 0.01, gamma = 0.5, I_app = 0.5 and alpha = 0.1. Trajectories are shown for different initial conditions (IC as v-w, marked black). Generated with scipy.integrate.solve_ivp() and matplotlib.

Implement the Vector Field

def fitzhugh_nagumo(t, vw, i_app=0.5, gamma=0.5, alpha=0.1, epsilon=0.01):
    v, w = vw
    fv = v * (1 - v) * (v - alpha)
    dvdt = (fv - w + i_app) / epsilon
    dwdt = v - gamma * w
    return np.array([dvdt, dwdt])

Follow the CDIMA pipeline again: solve trajectories, inspect the phase plane, compute nullclines, and then animate the motion.

Time Series and Excitability

In the excitable regime, trajectories typically stay near the resting equilibrium, but sufficiently large perturbations can trigger a large excursion (a “spike”) before returning to rest.

Below we plot the fast variable \(v(t)\) for a fixed parameter set and initial condition.

Figure 2: Time series \(v(t)\) for the FitzHugh–Nagumo system (same parameters as the phase-plane plot). Depending on the initial condition, the solution may relax to rest or exhibit a spike/oscillation.

Interpret the Parameters

  • Smaller \(\epsilon\) strengthens the fast-slow separation.
  • Changing \(I_{\text{app}}\) shifts the system between rest and stronger activation.
  • The parameters \(\gamma\) and \(\alpha\) reshape the nullclines and therefore the trajectory geometry.

What to Look For

  • Some initial conditions return directly to equilibrium.
  • Others produce a large transient spike before recovery.
  • The time trace and the phase-plane picture explain the same phenomenon from two complementary viewpoints.

Try changing the initial condition and parameters to see how the time series changes. Then connect those changes back to the nullclines and the phase-plane trajectories.

What’s Next?

Once the fast-slow geometry is clear, use the animation pattern to turn the static phase plane into an interactive exploration tool.

Animation Template Assignment

References

FitzHugh, Richard. 1961. “Impulses and Physiological States in Theoretical Models of Nerve Membrane.” Biophysical Journal 1 (6): 445–66. https://doi.org/10.1016/S0006-3495(61)86902-6.
Nagumo, Jin-ichi, Suguru Arimoto, and Shuji Yoshizawa. 1962. “An Active Pulse Transmission Line Simulating Nerve Axon.” Proceedings of the IRE 50 (10): 2061–70. https://doi.org/10.1109/JRPROC.1962.288235.