Cardiac Reentry: How a FitzHugh-Nagumo Wave Becomes an Arrhythmia

A minimal model of an excitable cell

Real cardiac cells fire action potentials through the full Hodgkin-Huxley machinery of multiple ion channels (fast sodium, L-type calcium, several potassium currents), which is accurate but expensive to simulate across a whole sheet of tissue. The FitzHugh-Nagumo model (1961-62) strips that down to the two qualitative ingredients that actually matter for tissue-scale wave behaviour: a fast excitation variable v (standing in for membrane voltage) and a slow recovery variable w (standing in for the combined effect of the slower ionic currents that shut the excitation back off).

dv/dt = v - v³/3 - w + I         fast variable (voltage-like)
dw/dt = ε(v + a - b·w)            slow recovery variable, ε ≪ 1

tissue (reaction-diffusion) form, for a 2D sheet of cells:
∂v/∂t = D∇²v + v - v³/3 - w + I
∂w/∂t = ε(v + a - b·w)

The v - v³/3 term gives the model a cubic, N-shaped nullcline: below a threshold, a small stimulus decays straight back to rest, but push v past the threshold and the fast dynamics take over, driving v sharply upward — the upstroke of an action potential — before the slow variable w catches up and pulls it back down. That threshold behaviour is exactly what makes cardiac tissue an excitable medium: all-or-nothing response to a stimulus, followed by a refractory period while w is still elevated, during which the cell cannot be re-excited no matter how strong the next stimulus is.

Why a wave, not just a single cell

Coupling many FitzHugh-Nagumo cells together with a diffusion term (current leaking between adjacent heart cells through gap junctions) turns individual excitability into a travelling wave: an excited cell drives its still-resting neighbours past threshold, they fire and drive the next ring outward, and the excitation propagates outward from the stimulus point as an expanding ring, exactly like a normal heartbeat's electrical wavefront sweeping from the sinoatrial node across the atria and ventricles. Behind the wavefront, cells sit in their refractory period and cannot be re-excited until they have recovered — which is precisely what keeps a normal heartbeat as one clean wave rather than chaotic re-firing.

Reentry: when the wave outruns its own recovery

Cardiac arrhythmia enters the model when a wavefront encounters tissue that has only partially recovered — for instance a stimulus arriving from an unusual direction, or a patch of tissue with abnormally slow recovery. Instead of the wave dying out cleanly against a refractory wall, part of the wavefront can curl around the edge of the still-refractory patch, and by the time it curls all the way around, the tissue it started from has recovered again. The wave now has a free end feeding into recovered tissue and can propagate in a tight, self-sustaining spiral, decoupled from the heart's normal pacemaker entirely. That spiral wave is the mechanistic picture of reentrant arrhythmia — the same core instability behind atrial flutter and, when the spiral itself breaks up into multiple competing spirals, ventricular fibrillation.

normal propagation:   stimulus → expanding ring → dies at fully refractory tissue
reentry:               stimulus meets partially-refractory patch
                       → wavefront breaks, curls around the patch
                       → tissue behind it recovers before the curl completes
                       → self-sustaining spiral, independent of the SA node

Why a toy model is still clinically useful

FitzHugh-Nagumo throws away almost all of the biophysical detail of a real cardiac action potential — there is no separate sodium spike, no calcium plateau, no distinction between atrial and ventricular cell types — yet it reproduces the two qualitative behaviours that actually govern tissue-scale rhythm: an all-or-nothing excitation threshold and a refractory recovery period. Because those two properties, not the exact ion-channel kinetics, are what determine whether a wave propagates cleanly or degenerates into reentry, simplified excitable-media models like this one remain a standard tool for studying arrhythmia mechanisms and testing ablation strategies before committing to a full ionic-current model.

Frequently asked questions

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