CalciumDetailed

CalciumDetailed#

class braincell.ion.CalciumDetailed(size, temp=Quantity(309.15, "K"), d=Quantity(1., "um"), tau=Quantity(5., "ms"), C_rest=Quantity(0.00024, "mM"), Co=None, Ci_initializer=Constant(value=0.00024 mM), name=None, **channels)#

Dynamic calcium concentration with a Nernst-computed reversal.

derivative() implements only the first-order relaxation model of section 2 below. Section 1 reproduces, for background only, the fuller ATP-pump model this family of models is drawn from: the pump kinetics in section 1 are NOT implemented by this class. derivative is three lines long and contains no Michaelis-Menten term and no pump parameter.

1. Background: the dynamics of intracellular \(Ca^{2+}\) (not implemented)

The dynamics of intracellular \(Ca^{2+}\) were determined by two contributions [3] :

(i) Influx of \(Ca^{2+}\) due to Calcium currents

\(Ca^{2+}\) ion enter through \(Ca^{2+}\) channel 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:

\[[Ca]_{i}=-\frac{I_{Ca}}{2 F d}\]

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, \(I_T\) 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. \(Ca^{2+}\) ion 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). None of this pump exists in derivative() below – no saturating term and no pump parameter appears anywhere in this class.

2. The implemented model: a simple first-order relaxation

What derivative() actually computes is the first-order model of (Bazhenov, et al., 1998) [2], which that paper’s own Appendix credits to (Destexhe et al., 1994) [1]:

\[\frac{d\left[Ca^{2+}\right]_{i}}{d t}=-\frac{I_{Ca}}{z F d}+\frac{\left[Ca^{2+}\right]_{rest}-\left[C a^{2+}\right]_{i}}{\tau_{Ca}}\]

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 \(\left[ Ca ^{2+}\right]_{\text {rest}}=2.4\times 10^{-4}\, \mathrm{mM}\) (\(0.24\, \mathrm{\mu M}\)), matching the default C_rest below exactly. BrainCell additionally clamps the influx term at zero before adding it (see Notes), which neither paper does.

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:

size (int | Sequence[int] | integer | Sequence[integer]) – The size of the simulation target, typically the number of neurons or compartments. Forwarded unchanged to Calcium.
temp (Array | ndarray | bool | number | bool | int | float | complex | Quantity | Callable) – Absolute temperature used by the Nernst equation in E. Defaults to 36 degrees Celsius, converted to kelvin via u.celsius2kelvin before being stored.
d (Array | ndarray | bool | number | bool | int | float | complex | Quantity | Callable) – Thickness \(d\) of the peri-membrane calcium shell. Defaults to 1.0 um.
tau (Array | ndarray | bool | number | bool | int | float | complex | Quantity | Callable) – Time constant \(\tau_{Ca}\) of the calcium removal rate. Defaults to 5.0 ms.
C_rest (Array | ndarray | bool | number | bool | int | float | complex | Quantity | Callable) – Resting intracellular calcium concentration \([Ca^{2+}]_{rest}\) that Ci relaxes toward. Defaults to 2.4e-4 mM (0.24 uM).
Co (Array | ndarray | bool | number | bool | int | float | complex | Quantity | Callable | None) – Extracellular calcium concentration. Defaults to None, which falls back to Calcium.default_Co inside DynamicNernstIon._init_dynamic_nernst_ion().
Ci_initializer (Array | ndarray | bool | number | bool | int | float | complex | Quantity | Callable) – Initializer for the dynamic Ci state. Defaults to a constant 2.4e-4 mM initializer, matching C_rest.
name (str | None) – Runtime ion instance name. Defaults to None.
**channels – Channel instances to attach to this ion, forwarded unchanged to Calcium.

Raises:

ValueError – If temp is explicitly passed as None. DynamicNernstIon._init_dynamic_nernst_ion() requires an explicit temperature and does not fall back to a class default.

CalciumDetailed

Contents

CalciumDetailed#

Modeling

Infrastructure

Compilation