HindmarshRose

Systems / ODEs / Chemical & biological systems

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

continuous · ODE3 dimensionsburstingchaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - \frac{a y_{0}^{3}}{tx} + \frac{b y_{0}^{2}}{tx} - y_{0} + \frac{y_{1}}{tx} + \frac{y_{2}}{tx} \\ \dot{y_{1}} &= - a y_{0}^{3} - y_{0}^{2} \left(- b + d\right) + y_{2} \\ \dot{y_{2}} &= \frac{c}{tz} - \frac{s y_{0}}{tz} - \frac{y_{2}}{tz} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.1758,\; +0.002052,\; -6.232$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.029$

Divergence ∇·f

$\nabla\!\cdot f = - \frac{3 a y_{0}^{2}}{tx} + \frac{2 b y_{0}}{tx} - 1 - \frac{1}{tz}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.HindmarshRose()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Hindmarsh & Rose (1984), Proc. R. Soc. Lond. B 221, 87-102

BibTeX

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