ExcitableCell

Systems / ODEs / Chemical & biological systems

The Chay conductance model of a bursting pancreatic β-cell — fast spiking gated by a slow Ca²⁺ wave, with period doubling and chaos across the beating–bursting transition.

continuous · ODE3 dimensionsbursting

time series

Definition

\[ \begin{aligned} \dot{y_{0}} &= \frac{5.74594990367292 \cdot 10^{-6} gi \left(vi - y_{0}\right) \left(y_{0} + 25\right)^{3} e^{- 0.05 y_{0}}}{\left(1 - 0.0820849986238988 e^{- 0.1 y_{0}}\right)^{3} \left(0.00574594990367292 e^{- 0.05 y_{0}} + \frac{1}{e^{- 0.1 y_{0} - 2} + 1}\right) \left(4 e^{- \frac{y_{0}}{18} - \frac{25}{9}} + \frac{0.1 \left(y_{0} + 25\right)}{1 - 0.0820849986238988 e^{- 0.1 y_{0}}}\right)^{3}} + \frac{gkc y_{2} \left(vk - y_{0}\right)}{y_{2} + 1} + gkv y_{1}^{4} \left(vk - y_{0}\right) + gl \left(vl - y_{0}\right) \\ \dot{y_{1}} &= 230 \left(- y_{1} + \frac{0.01 \left(y_{0} + 20\right)}{\left(1 - e^{- 0.1 y_{0} - 2}\right) \left(0.125 e^{- \frac{y_{0}}{80} - \frac{3}{8}} + \frac{0.01 \left(y_{0} + 20\right)}{1 - e^{- 0.1 y_{0} - 2}}\right)}\right) \left(0.125 e^{- \frac{y_{0}}{80} - \frac{3}{8}} + \frac{0.01 \left(y_{0} + 20\right)}{1 - e^{- 0.1 y_{0} - 2}}\right) \\ \dot{y_{2}} &= \rho \left(- kc y_{2} + \frac{5.74594990367292 \cdot 10^{-6} \left(vc - y_{0}\right) \left(y_{0} + 25\right)^{3} e^{- 0.05 y_{0}}}{\left(1 - 0.0820849986238988 e^{- 0.1 y_{0}}\right)^{3} \left(0.00574594990367292 e^{- 0.05 y_{0}} + \frac{1}{e^{- 0.1 y_{0} - 2} + 1}\right) \left(4 e^{- \frac{y_{0}}{18} - \frac{25}{9}} + \frac{0.1 \left(y_{0} + 25\right)}{1 - 0.0820849986238988 e^{- 0.1 y_{0}}}\right)^{3}}\right) \end{aligned} \]

Parameters

Symbol	Default	Role
gi	1800	maximum mixed Na⁺/Ca²⁺ inward conductance
gkc	11	maximum Ca²⁺-activated K⁺ conductance
gkv	1700	maximum voltage-gated K⁺ conductance
gl	7	leak conductance
kc	0.183333	intracellular Ca²⁺ removal rate
rho	0.27	slow-timescale factor of the Ca²⁺ dynamics
vc	100	reversal potential driving Ca²⁺ entry
vi	100	inward-current reversal potential
vk	-75	K⁺ reversal potential
vl	-40	leak reversal potential
vm	-50	half-activation voltage of the gating rate
vn	-30	half-activation voltage of the gating rate

Properties

Lyapunov spectrum

$+0.3996,\; +0.01067,\; -28.94$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.014$

Divergence ∇·f

$\nabla\!\cdot f = - gl - kc \rho$
constant part shown — state-dependent remainder omitted (too large)

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Chay (1985), Physica D 16, 233-242

BibTeX

@misc{excitablecell,
  title = {ExcitableCell system},
  note = {Chay (1985), Physica D 16, 233-242}
}