CalciumDetailed#
- class brainpy.dyn.CalciumDetailed(size, keep_size=False, T=36.0, d=1.0, C_rest=0.00024, tau=5.0, C0=2.0, C_initializer=OneInit(value=0.00024), method='exp_auto', name=None, mode=None, **channels)[source]#
Dynamical Calcium model proposed.
1. The dynamics of intracellular \(Ca^{2+}\)
The dynamics of intracellular \(Ca^{2+}\) were determined by two contributions [1] :
(i) Influx of \(Ca^{2+}\) due to Calcium currents
\(Ca^{2+}\) ions enter through \(Ca^{2+}\) channels and diffuse into the interior of the cell. Only the \(Ca^{2+}\) concentration in a thin shell beneath the membrane was modeled. The influx of \(Ca^{2+}\) into such a thin shell followed:
\[[Ca]_{i}=-\frac{k}{2 F d} I_{Ca}\]where \(F=96489\, \mathrm{C\, mol^{-1}}\) is the Faraday constant, \(d=1\, \mathrm{\mu m}\) is the depth of the shell beneath the membrane, the unit conversion constant is \(k=0.1\) for \(I_T\) in \(\mathrm{\mu A/cm^{2}}\) and \([Ca]_{i}\) in millimolar, and \(I_{Ca}\) is the summation of all \(Ca^{2+}\) currents.
(ii) Efflux of \(Ca^{2+}\) due to an active pump
In a thin shell beneath the membrane, \(Ca^{2+}\) retrieval usually consists of a combination of several processes, such as binding to \(Ca^{2+}\) buffers, calcium efflux due to \(Ca^{2+}\) ATPase pump activity and diffusion to neighboring shells. Only the \(Ca^{2+}\) pump was modeled here. We adopted the following kinetic scheme:
\[Ca _{i}^{2+}+ P \overset{c_1}{\underset{c_2}{\rightleftharpoons}} CaP \xrightarrow{c_3} P+ Ca _{0}^{2+}\]where P represents the \(Ca^{2+}\) pump, CaP is an intermediate state, \(Ca _{ o }^{2+}\) is the extracellular \(Ca^{2+}\) concentration, and \(c_{1}, c_{2}\) and \(c_{3}\) are rate constants. \(Ca^{2+}\) ions have a high affinity for the pump \(P\), whereas extrusion of \(Ca^{2+}\) follows a slower process (Blaustein, 1988 ). Therefore, \(c_{3}\) is low compared to \(c_{1}\) and \(c_{2}\) and the Michaelis-Menten approximation can be used for describing the kinetics of the pump. According to such a scheme, the kinetic equation for the \(Ca^{2+}\) pump is:
\[\frac{[Ca^{2+}]_{i}}{dt}=-\frac{K_{T}[Ca]_{i}}{[Ca]_{i}+K_{d}}\]where \(K_{T}=10^{-4}\, \mathrm{mM\, ms^{-1}}\) is the product of \(c_{3}\) with the total concentration of \(P\) and \(K_{d}=c_{2} / c_{1}=10^{-4}\, \mathrm{mM}\) is the dissociation constant, which can be interpreted here as the value of \([Ca]_{i}\) at which the pump is half activated (if \([Ca]_{i} \ll K_{d}\) then the efflux is negligible).
2.A simple first-order model
While, in (Bazhenov, et al., 1998) [2], the \(Ca^{2+}\) dynamics is described by a simple first-order model,
\[\frac{d\left[Ca^{2+}\right]_{i}}{d t}=-\frac{I_{Ca}}{z F d}+\frac{\left[Ca^{2+}\right]_{rest}-\left[C a^{2+}\right]_{i}}{\tau_{Ca}}\]where \(I_{Ca}\) is the summation of all \(Ca ^{2+}\) currents, \(d\) is the thickness of the perimembrane “shell” in which calcium is able to affect membrane properties \((1.\, \mathrm{\mu M})\), \(z=2\) is the valence of the \(Ca ^{2+}\) ion, \(F\) is the Faraday constant, and \(\tau_{C a}\) is the \(Ca ^{2+}\) removal rate. The resting \(Ca ^{2+}\) concentration was set to be \(\left[ Ca ^{2+}\right]_{\text {rest}}=.05\, \mathrm{\mu M}\) .
3. The reversal potential
The reversal potential of calcium \(Ca ^{2+}\) is calculated according to the Nernst equation:
\[E = k'{RT \over 2F} log{[Ca^{2+}]_0 \over [Ca^{2+}]_i}\]where \(R=8.31441 \, \mathrm{J} /(\mathrm{mol}^{\circ} \mathrm{K})\), \(T=309.15^{\circ} \mathrm{K}\), \(F=96,489 \mathrm{C} / \mathrm{mol}\), and \(\left[\mathrm{Ca}^{2+}\right]_{0}=2 \mathrm{mM}\).
- Parameters:
d (float) – The thickness of the peri-membrane “shell”.
F (float) – The Faraday constant. (\(C*mmol^{-1}\))
tau (float) – The time constant of the \(Ca ^{2+}\) removal rate. (ms)
C_rest (float) – The resting \(Ca ^{2+}\) concentration.
C0 (float) – The \(Ca ^{2+}\) concentration outside of the membrane.
R (float) – The gas constant. (:math:` J*mol^{-1}*K^{-1}`)
References
- master_type#
alias of
HHTypedNeuron