# brainpy.dyn.channels.Ca.CalciumDetailed#

class brainpy.dyn.channels.Ca.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=NormalMode, **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

1

Destexhe, Alain, Agnessa Babloyantz, and Terrence J. Sejnowski. “Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons.” Biophysical journal 65, no. 4 (1993): 1538-1552.

2

Bazhenov, Maxim, Igor Timofeev, Mircea Steriade, and Terrence J. Sejnowski. “Cellular and network models for intrathalamic augmenting responses during 10-Hz stimulation.” Journal of neurophysiology 79, no. 5 (1998): 2730-2748.

__init__(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=NormalMode, **channels)[source]#

Methods

 __init__(size[, keep_size, T, d, C_rest, ...]) clear_input() current(V[, C_Ca, E_Ca]) derivative(C, t, V) get_delay_data(identifier, delay_step, *indices) Get delay data according to the provided delay steps. load_states(filename[, verbose]) Load the model states. nodes([method, level, include_self]) Collect all children nodes. offline_fit(target, fit_record) offline_init() online_fit(target, fit_record) online_init() register_delay(identifier, delay_step, ...) Register delay variable. register_implicit_nodes(*channels, ...) register_implicit_vars(*variables, ...) reset(V[, batch_size]) Reset function which reset the whole variables in the model. reset_local_delays([nodes]) Reset local delay variables. reset_state(V[, C_Ca, E_Ca, batch_size]) Reset function which reset the states in the model. save_states(filename[, variables]) Save the model states. train_vars([method, level, include_self]) The shortcut for retrieving all trainable variables. unique_name([name, type_]) Get the unique name for this object. update(tdi, V) Update function of a container. update_local_delays([nodes]) Update local delay variables. vars([method, level, include_self]) Collect all variables in this node and the children nodes.

Attributes

 F R global_delay_data mode Mode of the model, which is useful to control the multiple behaviors of the model. name Name of the model. varshape