FitzHughNagumo¶

Systems / ODEs / Chemical & biological systems

The two-variable nerve membrane — a fast cubic voltage coupled to a slow recovery current: excitable at rest, spiking under an applied current.

continuous · ODE2 dimensionsexcitablerelaxation

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{v} &= curr - \frac{v^{3}}{3} + v - w \\ \dot{w} &= \gamma \left(a - b w + v\right) \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`0.7`	offset of the linear recovery nullcline
`b`	`0.8`	slope of the linear recovery nullcline
`curr`	`0.5`	applied stimulus current (0 = excitable, 0.5 = oscillatory)
`gamma`	`0.08`	recovery timescale (small ⇒ slow–fast dynamics)

State variables: v, w

Properties¶

Lyapunov spectrum

≥ 0 positive exponents (literature — analytic: Poincare-Bendixson forbids chaos in a planar autonomous flow)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - b \gamma - v^{2} + 1$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.FitzHughNagumo()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

FitzHugh (1961), Biophys. J. 1, 445-466
doi:10.1016/s0006-3495(61)86902-6

BibTeX

@misc{fitzhughnagumo,
  title = {FitzHughNagumo system},
  note = {FitzHugh (1961), Biophys. J. 1, 445-466},
  doi = {10.1016/s0006-3495(61)86902-6}
}