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

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

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

State variables: v, n, c

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.ExcitableCell()
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

Chay (1985), Physica D 16, 233-242
doi:10.1016/0167-2789(85)90060-0

BibTeX

@misc{excitablecell,
  title = {ExcitableCell system},
  note = {Chay (1985), Physica D 16, 233-242},
  doi = {10.1016/0167-2789(85)90060-0}
}