CaTwoPlusQuasiperiodic

Systems / ODEs / Chemical & biological systems

The Houart–Dupont–Goldbeter intracellular Ca²⁺ oscillator in its quasiperiodic/chaotic regime, driven by self-modulated InsP₃ feedback.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{z} &= V_{0} + V_{1} \beta - \frac{Vm_{2} z^{2}}{K_{2}^{2} + z^{2}} + \frac{Vm_{3} a^{4} y^{2} z^{m}}{\left(Ka^{4} + a^{4}\right) \left(Ky^{2} + y^{2}\right) \left(Kz^{m} + z^{m}\right)} - k z + kf y \\ \dot{y} &= \frac{Vm_{2} z^{2}}{K_{2}^{2} + z^{2}} - \frac{Vm_{3} a^{4} y^{2} z^{m}}{\left(Ka^{4} + a^{4}\right) \left(Ky^{2} + y^{2}\right) \left(Kz^{m} + z^{m}\right)} - kf y \\ \dot{a} &= V_{4} \beta - \frac{Vm_{5} a^{p} z^{n}}{\left(K_{5}^{p} + a^{p}\right) \left(Kd^{n} + z^{n}\right)} - a eps \end{aligned} \]

Parameters

Symbol	Default	Role
K2	0.1
K5	0.3
Ka	0.2
Kd	0.5
Ky	0.2
Kz	0.5
V0	2	constant Ca²⁺ influx
V1	2
V4	5
Vm2	6
Vm3	20
Vm5	30
beta	0.51	external stimulus level
eps	0.1	InsP₃ degradation rate
k	10	cytosolic Ca²⁺ removal
kf	1	store leak rate
m	2
n	4
p	2

State variables: z, y, a

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - eps - k - kf$
constant part shown — state-dependent remainder omitted (too large)

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.CaTwoPlusQuasiperiodic()
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

Houart, Dupont & Goldbeter (1999), Bull. Math. Biol. 61, 507-530
doi:10.1006/bulm.1999.0095

BibTeX

@misc{catwoplusquasiperiodic,
  title = {CaTwoPlusQuasiperiodic system},
  note = {Houart, Dupont & Goldbeter (1999), Bull. Math. Biol. 61, 507-530},
  doi = {10.1006/bulm.1999.0095}
}