HindmarshRose

Systems / ODEs / Chemical & biological systems

A three-variable neuron model reproducing spiking and bursting with chaotic interspike patterns.

continuous · ODE3 dimensionsburstingchaotic

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Definition

\[ \begin{aligned} \dot{x} &= - \frac{a x^{3}}{tx} + \frac{b x^{2}}{tx} - x + \frac{y}{tx} + \frac{z}{tx} \\ \dot{y} &= - a x^{3} - x^{2} \left(- b + d\right) + z \\ \dot{z} &= \frac{c}{tz} - \frac{s x}{tz} - \frac{z}{tz} \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.49	coefficient of the cubic spike term
b	1	coefficient of the quadratic spike term
c	0.0322	constant of the fast recovery dynamics
d	1	constant of the fast recovery dynamics
s	1	strength of the slow adaptation feedback
tx	0.03	timescale constant of the fast (x) variable
tz	0.8	timescale constant of the slow (z) variable

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - \frac{3 a x^{2}}{tx} + \frac{2 b x}{tx} - 1 - \frac{1}{tz}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HindmarshRose()
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

Hindmarsh & Rose (1984), Proc. R. Soc. Lond. B 221, 87-102
doi:10.1098/rspb.1984.0024

BibTeX

@misc{hindmarshrose,
  title = {HindmarshRose system},
  note = {Hindmarsh & Rose (1984), Proc. R. Soc. Lond. B 221, 87-102},
  doi = {10.1098/rspb.1984.0024}
}