FitzHugh-Nagumo Equations

FitzHugh-Nagumo Equations

Audience

Difficulty	Beginner

Notation on this page

The FitzHugh-Nagumo equations (Richard FitzHugh 1961, Jin-Ichi Nagumo et al. 1962) are a two-variable simplification of the Hodgkin-Huxley equations. They retain HH's essential qualitative behaviour — threshold excitation, action potentials, refractoriness — while being much easier to analyse and visualise.

FN is the pedagogical workhorse of neural-excitability theory: every neuroscience textbook reduces HH to FN-like form for phase-plane intuition.

Statement

 dv/dt = v − v3/3 − w + I
 dw/dt = ε (v + a − b w)

Components

Symbol	Meaning	Typical value
v	Voltage-like fast variable	dimensionless
w	Recovery slow variable (analogue of Hodgkin-Huxley n and h)	dimensionless
ε	Time-scale separation	~ 0.08 (small ⇒ fast-slow dynamics)
a	Recovery-nullcline offset	~ 0.7
b	Recovery-nullcline slope	~ 0.8
I	External stimulus	dimensionless

Derivation sketch

1. Take HH and notice that the fast variables (V, m) and slow variables (h, n) have a clear time-scale separation: m equilibrates much faster than V, while h and n evolve on a slower timescale.
2. Quasi-steady-state for m: replace m by its instantaneous voltage-dependent value m_∞(V).
3. Linear combination of h and n: introduce a single recovery variable w that captures both inactivation (h decreasing) and K⁺-channel activation (n increasing).
4. Cubic approximation: fit the resulting V-nullcline by a cubic polynomial (the v − v³/3 form is the simplest cubic with the right qualitative shape).
5. Linear w-dynamics: approximate the recovery by linear relaxation toward a linear nullcline.

This is a standard model-reduction exercise; FN preserves all the qualitative dynamics that matter for excitability and oscillation.

Phase-plane analysis

The nullclines:

• v-nullcline: w = v − v³/3 + I (cubic, with three roots for I in an excitable range).
• w-nullcline: w = (v + a) / b (straight line).

Depending on where these nullclines intersect:

• Single fixed point on left branch (stable) — excitable resting state. A super-threshold perturbation drives a large excursion (the analogue of an action potential) before returning to rest.
• Single fixed point on middle branch (unstable) — Hopf bifurcation produces limit-cycle oscillations (repetitive firing).
• Three fixed points — bistability.

The cubic + line structure is the simplest possible mathematics that produces threshold excitability — and is therefore the canonical "minimal model" of excitable systems generally.

Wave propagation

Adding spatial diffusion gives the Nagumo equation:

$ {\frac {\partial v}{\partial t}}=D\,\nabla ^{2}v+v-{\frac {v^{3}}{3}}-w+I $

$ {\frac {\partial w}{\partial t}}=\varepsilon \,(v+a-bw) $

This supports travelling-wave solutions — the canonical model for action-potential propagation along an axon, and for excitation waves in cardiac tissue.

Sanity-check limits

• I = 0, subthreshold perturbation: v decays back to rest. ✓
• I = 0, superthreshold perturbation: full excursion (action-potential-like). ✓
• Large I: repetitive firing (limit cycle). ✓
• ε → 0 (extreme time-scale separation): relaxation oscillations with sharp transitions. ✓
• Cubic → linear (small v): linear damped oscillator. ✓

Connection to ψ

Like Hodgkin-Huxley, FN is single-neuron level; ψ effects emerge at the population level via Wilson-Cowan / Amari extensions. FN is most useful in the framework as a pedagogical tool for explaining how excitable dynamics produce population-level oscillations and travelling waves that ultimately source ψ.

Experimental status

FitzHugh-Nagumo is universally accepted as the qualitative reduction of HH. It is not used for quantitative single-neuron modelling (HH is preferred), but it is used everywhere for:

• Pedagogy in neuroscience and dynamical-systems courses.
• Cardiac-tissue excitation models (where its travelling-wave behaviour matches cardiac action-potential propagation).
• General excitable-media theory in chemistry (e.g. Belousov-Zhabotinsky reaction).

See Also

• Hodgkin-Huxley_Equations
• Wilson-Cowan_Model
• Amari_Neural_Field
• Neural_Field_Equations

References

• FitzHugh, R. (1961). "Impulses and physiological states in theoretical models of nerve membrane." Biophysical Journal 1: 445–466.
• Nagumo, J., Arimoto, S., Yoshizawa, S. (1962). "An active pulse transmission line simulating nerve axon." Proceedings of the IRE 50: 2061–2070.
• Murray, J. D. (2002). Mathematical Biology (3rd ed.), Springer — chapter on excitable dynamics.