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= FitzHugh-Nagumo Equations =

{{Audience_Sidebar
| difficulty = Beginner
| reading_time = 5 minutes
| prerequisites = ODEs; phase-plane analysis.
| if_too_advanced_see = [[Wilson-Cowan_Model]]
| if_too_basic_see = [[Hodgkin-Huxley_Equations]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Mostly-plus (irrelevant here).
| units = Dimensionless throughout (model is rescaled).
}}

The '''FitzHugh-Nagumo equations''' (Richard FitzHugh 1961, Jin-Ichi Nagumo et al. 1962) are a two-variable simplification of the [[Hodgkin-Huxley_Equations|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 ==

{| class="wikitable"
! 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 ==

# 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.
# '''Quasi-steady-state for m''': replace m by its instantaneous voltage-dependent value m∞(V).
# '''Linear combination of h and n''': introduce a single recovery variable w that captures both inactivation (h decreasing) and K+-channel activation (n increasing).
# '''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).
# '''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''':

:<math>\frac{\partial v}{\partial t} = D\,\nabla^2 v + v - \frac{v^3}{3} - w + I</math>

:<math>\frac{\partial w}{\partial t} = \varepsilon\,(v + a - b w)</math>

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_Model|Wilson-Cowan]] / [[Amari_Neural_Field|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.

[[Category:Psionics]]
[[Category:Equations]]
[[Category:Neuroscience]]

FitzHugh-Nagumo Equations - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$