# -*- coding: utf-8 -*-
"""
This module implements voltage-dependent calcium channels.
"""
from typing import Union, Callable
import brainpy.math as bm
from brainpy._src.dynsys import Channel
from brainpy._src.initialize import OneInit, Initializer, parameter, variable
from brainpy._src.integrators.joint_eq import JointEq
from brainpy._src.integrators.ode.generic import odeint
from brainpy.types import Shape, ArrayType
from .base import Calcium, CalciumChannel
__all__ = [
'CalciumFixed',
'CalciumDyna',
'CalciumDetailed',
'CalciumFirstOrder',
'_ICa_p2q_ss', '_ICa_p2q_markov',
'ICaN_IS2008',
'ICaT_HM1992',
'ICaT_HP1992',
'ICaHT_HM1992',
'ICaL_IS2008',
]
[docs]class CalciumFixed(Calcium):
"""Fixed Calcium dynamics.
This calcium model has no dynamics. It holds fixed reversal
potential :math:`E` and concentration :math:`C`.
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = 120.,
C: Union[float, ArrayType, Initializer, Callable] = 2.4e-4,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
**channels
):
super(CalciumFixed, self).__init__(size,
keep_size=keep_size,
method=method,
name=name,
mode=mode,
**channels)
self.E = parameter(E, self.varshape, allow_none=False)
self.C = parameter(C, self.varshape, allow_none=False)
def update(self, tdi, V):
for node in self.implicit_nodes.values():
node.update(tdi, V, self.C, self.E)
def reset_state(self, V, C_Ca=None, E_Ca=None, batch_size=None):
C_Ca = self.C if C_Ca is None else C_Ca
E_Ca = self.E if E_Ca is None else E_Ca
for node in self.nodes(level=1, include_self=False).unique().subset(Channel).values():
node.reset_state(V, C_Ca, E_Ca, batch_size=batch_size)
[docs]class CalciumDyna(Calcium):
"""Calcium ion flow with dynamics.
Parameters
----------
size: int, tuple of int
The ion size.
keep_size: bool
Keep the geometry size.
C0: float, ArrayType, Initializer, Callable
The Calcium concentration outside of membrane.
T: float, ArrayType, Initializer, Callable
The temperature.
C_initializer: Initializer, Callable, ArrayType
The initializer for Calcium concentration.
method: str
The numerical method.
name: str
The ion name.
"""
R = 8.31441 # gas constant, J*mol-1*K-1
F = 96.489 # the Faraday constant
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
C0: Union[float, ArrayType, Initializer, Callable] = 2.,
T: Union[float, ArrayType, Initializer, Callable] = 36.,
C_initializer: Union[Initializer, Callable, ArrayType] = OneInit(2.4e-4),
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
**channels
):
super(CalciumDyna, self).__init__(size,
keep_size=keep_size,
method=method,
name=name,
mode=mode,
**channels)
# parameters
self.C0 = parameter(C0, self.varshape, allow_none=False)
self.T = parameter(T, self.varshape, allow_none=False) # temperature
self._C_initializer = C_initializer
self._constant = self.R / (2 * self.F) * (273.15 + self.T)
# variables
self.C = variable(C_initializer, self.mode, self.varshape) # Calcium concentration
self.E = bm.Variable(self._reversal_potential(self.C),
batch_axis=0 if isinstance(self.mode, bm.BatchingMode) else None) # Reversal potential
# function
self.integral = odeint(self.derivative, method=method)
def derivative(self, C, t, V):
raise NotImplementedError
def reset_state(self, V, C_Ca=None, E_Ca=None, batch_size=None):
self.C.value = variable(self._C_initializer, batch_size, self.varshape) if (C_Ca is None) else C_Ca
self.E.value = self._reversal_potential(self.C)
for node in self.nodes(level=1, include_self=False).unique().subset(Channel).values():
node.reset(V, self.C, self.E, batch_size=batch_size)
def update(self, tdi, V):
for node in self.nodes(level=1, include_self=False).unique().subset(Channel).values():
node.update(tdi, V, self.C.value, self.E.value)
self.C.value = self.integral(self.C.value, tdi['t'], V, tdi['dt'])
self.E.value = self._reversal_potential(self.C.value)
def _reversal_potential(self, C):
return self._constant * bm.log(self.C0 / C)
[docs]class CalciumDetailed(CalciumDyna):
r"""Dynamical Calcium model proposed.
**1. The dynamics of intracellular** :math:`Ca^{2+}`
The dynamics of intracellular :math:`Ca^{2+}` were determined by two contributions [1]_ :
*(i) Influx of* :math:`Ca^{2+}` *due to Calcium currents*
:math:`Ca^{2+}` ions enter through :math:`Ca^{2+}` channels and diffuse into the
interior of the cell. Only the :math:`Ca^{2+}` concentration in a thin shell beneath
the membrane was modeled. The influx of :math:`Ca^{2+}` into such a thin shell followed:
.. math::
[Ca]_{i}=-\frac{k}{2 F d} I_{Ca}
where :math:`F=96489\, \mathrm{C\, mol^{-1}}` is the Faraday constant,
:math:`d=1\, \mathrm{\mu m}` is the depth of the shell beneath the membrane,
the unit conversion constant is :math:`k=0.1` for :math:`I_T` in
:math:`\mathrm{\mu A/cm^{2}}` and :math:`[Ca]_{i}` in millimolar,
and :math:`I_{Ca}` is the summation of all :math:`Ca^{2+}` currents.
*(ii) Efflux of* :math:`Ca^{2+}` *due to an active pump*
In a thin shell beneath the membrane, :math:`Ca^{2+}` retrieval usually consists of a
combination of several processes, such as binding to :math:`Ca^{2+}` buffers, calcium
efflux due to :math:`Ca^{2+}` ATPase pump activity and diffusion to neighboring shells.
Only the :math:`Ca^{2+}` pump was modeled here. We adopted the following kinetic scheme:
.. math::
Ca _{i}^{2+}+ P \overset{c_1}{\underset{c_2}{\rightleftharpoons}} CaP \xrightarrow{c_3} P+ Ca _{0}^{2+}
where P represents the :math:`Ca^{2+}` pump, CaP is an intermediate state,
:math:`Ca _{ o }^{2+}` is the extracellular :math:`Ca^{2+}` concentration,
and :math:`c_{1}, c_{2}` and :math:`c_{3}` are rate constants. :math:`Ca^{2+}`
ions have a high affinity for the pump :math:`P`, whereas extrusion of
:math:`Ca^{2+}` follows a slower process (Blaustein, 1988 ). Therefore,
:math:`c_{3}` is low compared to :math:`c_{1}` and :math:`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 :math:`Ca^{2+}` pump is:
.. math::
\frac{[Ca^{2+}]_{i}}{dt}=-\frac{K_{T}[Ca]_{i}}{[Ca]_{i}+K_{d}}
where :math:`K_{T}=10^{-4}\, \mathrm{mM\, ms^{-1}}` is the product of :math:`c_{3}`
with the total concentration of :math:`P` and :math:`K_{d}=c_{2} / c_{1}=10^{-4}\, \mathrm{mM}`
is the dissociation constant, which can be interpreted here as the value of
:math:`[Ca]_{i}` at which the pump is half activated (if :math:`[Ca]_{i} \ll K_{d}`
then the efflux is negligible).
**2.A simple first-order model**
While, in (Bazhenov, et al., 1998) [2]_, the :math:`Ca^{2+}` dynamics is
described by a simple first-order model,
.. math::
\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 :math:`I_{Ca}` is the summation of all :math:`Ca ^{2+}` currents, :math:`d`
is the thickness of the perimembrane "shell" in which calcium is able to affect
membrane properties :math:`(1.\, \mathrm{\mu M})`, :math:`z=2` is the valence of the
:math:`Ca ^{2+}` ion, :math:`F` is the Faraday constant, and :math:`\tau_{C a}` is
the :math:`Ca ^{2+}` removal rate. The resting :math:`Ca ^{2+}` concentration was
set to be :math:`\left[ Ca ^{2+}\right]_{\text {rest}}=.05\, \mathrm{\mu M}` .
**3. The reversal potential**
The reversal potential of calcium :math:`Ca ^{2+}` is calculated according to the
Nernst equation:
.. math::
E = k'{RT \over 2F} log{[Ca^{2+}]_0 \over [Ca^{2+}]_i}
where :math:`R=8.31441 \, \mathrm{J} /(\mathrm{mol}^{\circ} \mathrm{K})`,
:math:`T=309.15^{\circ} \mathrm{K}`,
:math:`F=96,489 \mathrm{C} / \mathrm{mol}`,
and :math:`\left[\mathrm{Ca}^{2+}\right]_{0}=2 \mathrm{mM}`.
Parameters
----------
d : float
The thickness of the peri-membrane "shell".
F : float
The Faraday constant. (:math:`C*mmol^{-1}`)
tau : float
The time constant of the :math:`Ca ^{2+}` removal rate. (ms)
C_rest : float
The resting :math:`Ca ^{2+}` concentration.
C0 : float
The :math:`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.
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType, Initializer, Callable] = 36.,
d: Union[float, ArrayType, Initializer, Callable] = 1.,
C_rest: Union[float, ArrayType, Initializer, Callable] = 2.4e-4,
tau: Union[float, ArrayType, Initializer, Callable] = 5.,
C0: Union[float, ArrayType, Initializer, Callable] = 2.,
C_initializer: Union[Initializer, Callable, ArrayType] = OneInit(2.4e-4),
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
**channels
):
super(CalciumDetailed, self).__init__(size,
keep_size=keep_size,
method=method,
name=name,
T=T,
C0=C0,
C_initializer=C_initializer,
mode=mode,
**channels)
# parameters
self.d = parameter(d, self.varshape, allow_none=False)
self.tau = parameter(tau, self.varshape, allow_none=False)
self.C_rest = parameter(C_rest, self.varshape, allow_none=False)
def derivative(self, C, t, V):
ICa = self.current(V, C, self.E)
drive = bm.maximum(- ICa / (2 * self.F * self.d), 0.)
return drive + (self.C_rest - C) / self.tau
[docs]class CalciumFirstOrder(CalciumDyna):
r"""The first-order calcium concentration model.
.. math::
Ca' = -\alpha I_{Ca} + -\beta Ca
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType, Initializer, Callable] = 36.,
alpha: Union[float, ArrayType, Initializer, Callable] = 0.13,
beta: Union[float, ArrayType, Initializer, Callable] = 0.075,
C0: Union[float, ArrayType, Initializer, Callable] = 2.,
C_initializer: Union[Initializer, Callable, ArrayType] = OneInit(2.4e-4),
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
**channels
):
super(CalciumFirstOrder, self).__init__(size,
keep_size=keep_size,
method=method,
name=name,
T=T,
C0=C0,
C_initializer=C_initializer,
mode=mode,
**channels)
# parameters
self.alpha = parameter(alpha, self.varshape, allow_none=False)
self.beta = parameter(beta, self.varshape, allow_none=False)
def derivative(self, C, t, V):
ICa = self.current(V, C, self.E)
drive = bm.maximum(- self.alpha * ICa, 0.)
return drive - self.beta * C
# -------------------------
class _ICa_p2q_ss(CalciumChannel):
r"""The calcium current model of :math:`p^2q` current which described with steady-state format.
The dynamics of this generalized calcium current model is given by:
.. math::
I_{CaT} &= g_{max} p^2 q(V-E_{Ca}) \\
{dp \over dt} &= {\phi_p \cdot (p_{\infty}-p)\over \tau_p} \\
{dq \over dt} &= {\phi_q \cdot (q_{\infty}-q) \over \tau_q} \\
where :math:`\phi_p` and :math:`\phi_q` are temperature-dependent factors,
:math:`E_{Ca}` is the reversal potential of Calcium channel.
Parameters
----------
size: int, tuple of int
The size of the simulation target.
keep_size: bool
Keep size or flatten the size?
method: str
The numerical method
name: str
The name of the object.
g_max : float, ArrayType, Callable, Initializer
The maximum conductance.
phi_p : float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`q`.
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
phi_p: Union[float, ArrayType, Initializer, Callable] = 3.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 3.,
g_max: Union[float, ArrayType, Initializer, Callable] = 2.,
method: str = 'exp_auto',
mode: bm.Mode = None,
name: str = None
):
super(_ICa_p2q_ss, self).__init__(size,
keep_size=keep_size,
name=name,
mode=mode, )
# parameters
self.phi_p = parameter(phi_p, self.varshape, allow_none=False)
self.phi_q = parameter(phi_q, self.varshape, allow_none=False)
self.g_max = parameter(g_max, self.varshape, allow_none=False)
# variables
self.p = variable(bm.zeros, self.mode, self.varshape)
self.q = variable(bm.zeros, self.mode, self.varshape)
# functions
self.integral = odeint(JointEq([self.dp, self.dq]), method=method)
def dp(self, p, t, V):
return self.phi_p * (self.f_p_inf(V) - p) / self.f_p_tau(V)
def dq(self, q, t, V):
return self.phi_q * (self.f_q_inf(V) - q) / self.f_q_tau(V)
def update(self, tdi, V, C_Ca, E_Ca):
self.p.value, self.q.value = self.integral(self.p, self.q, tdi['t'], V, tdi['dt'])
def current(self, V, C_Ca, E_Ca):
return self.g_max * self.p * self.p * self.q * (E_Ca - V)
def reset_state(self, V, C_Ca, E_Ca, batch_size=None):
self.p.value = self.f_p_inf(V)
self.q.value = self.f_q_inf(V)
if batch_size is not None:
assert self.p.shape[0] == batch_size
assert self.q.shape[0] == batch_size
def f_p_inf(self, V):
raise NotImplementedError
def f_p_tau(self, V):
raise NotImplementedError
def f_q_inf(self, V):
raise NotImplementedError
def f_q_tau(self, V):
raise NotImplementedError
class _ICa_p2q_markov(CalciumChannel):
r"""The calcium current model of :math:`p^2q` current which described with first-order Markov chain.
The dynamics of this generalized calcium current model is given by:
.. math::
I_{CaT} &= g_{max} p^2 q(V-E_{Ca}) \\
{dp \over dt} &= \phi_p (\alpha_p(V)(1-p) - \beta_p(V)p) \\
{dq \over dt} &= \phi_q (\alpha_q(V)(1-q) - \beta_q(V)q) \\
where :math:`\phi_p` and :math:`\phi_q` are temperature-dependent factors,
:math:`E_{Ca}` is the reversal potential of Calcium channel.
Parameters
----------
size: int, tuple of int
The size of the simulation target.
keep_size: bool
Keep size or flatten the size?
method: str
The numerical method
name: str
The name of the object.
g_max : float, ArrayType, Callable, Initializer
The maximum conductance.
phi_p : float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`q`.
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
phi_p: Union[float, ArrayType, Initializer, Callable] = 3.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 3.,
g_max: Union[float, ArrayType, Initializer, Callable] = 2.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super(_ICa_p2q_markov, self).__init__(size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
self.phi_p = parameter(phi_p, self.varshape, allow_none=False)
self.phi_q = parameter(phi_q, self.varshape, allow_none=False)
self.g_max = parameter(g_max, self.varshape, allow_none=False)
# variables
self.p = variable(bm.zeros, self.mode, self.varshape)
self.q = variable(bm.zeros, self.mode, self.varshape)
# functions
self.integral = odeint(JointEq([self.dp, self.dq]), method=method)
def dp(self, p, t, V):
return self.phi_p * (self.f_p_alpha(V) * (1 - p) - self.f_p_beta(V) * p)
def dq(self, q, t, V):
return self.phi_q * (self.f_q_alpha(V) * (1 - q) - self.f_q_beta(V) * q)
def update(self, tdi, V, C_Ca, E_Ca):
self.p.value, self.q.value = self.integral(self.p, self.q, tdi['t'], V, tdi['dt'])
def current(self, V, C_Ca, E_Ca):
return self.g_max * self.p * self.p * self.q * (E_Ca - V)
def reset_state(self, V, C_Ca, E_Ca, batch_size=None):
alpha, beta = self.f_p_alpha(V), self.f_p_beta(V)
self.p.value = alpha / (alpha + beta)
alpha, beta = self.f_q_alpha(V), self.f_q_beta(V)
self.q.value = alpha / (alpha + beta)
if batch_size is not None:
assert self.p.shape[0] == batch_size
assert self.q.shape[0] == batch_size
def f_p_alpha(self, V):
raise NotImplementedError
def f_p_beta(self, V):
raise NotImplementedError
def f_q_alpha(self, V):
raise NotImplementedError
def f_q_beta(self, V):
raise NotImplementedError
[docs]class ICaN_IS2008(CalciumChannel):
r"""The calcium-activated non-selective cation channel model
proposed by (Inoue & Strowbridge, 2008) [2]_.
The dynamics of the calcium-activated non-selective cation channel model [1]_ [2]_ is given by:
.. math::
\begin{aligned}
I_{CAN} &=g_{\mathrm{max}} M\left([Ca^{2+}]_{i}\right) p \left(V-E\right)\\
&M\left([Ca^{2+}]_{i}\right) ={[Ca^{2+}]_{i} \over 0.2+[Ca^{2+}]_{i}} \\
&{dp \over dt} = {\phi \cdot (p_{\infty}-p)\over \tau_p} \\
&p_{\infty} = {1.0 \over 1 + \exp(-(V + 43) / 5.2)} \\
&\tau_{p} = {2.7 \over \exp(-(V + 55) / 15) + \exp((V + 55) / 15)} + 1.6
\end{aligned}
where :math:`\phi` is the temperature factor.
Parameters
----------
g_max : float
The maximal conductance density (:math:`mS/cm^2`).
E : float
The reversal potential (mV).
phi : float
The temperature factor.
References
----------
.. [1] Destexhe, Alain, et al. "A model of spindle rhythmicity in the isolated
thalamic reticular nucleus." Journal of neurophysiology 72.2 (1994): 803-818.
.. [2] Inoue T, Strowbridge BW (2008) Transient activity induces a long-lasting
increase in the excitability of olfactory bulb interneurons.
J Neurophysiol 99: 187–199.
"""
'''The type of the master object.'''
master_type = CalciumDyna
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = 10.,
g_max: Union[float, ArrayType, Initializer, Callable] = 1.,
phi: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super(ICaN_IS2008, self).__init__(size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
self.E = parameter(E, self.varshape, allow_none=False)
self.g_max = parameter(g_max, self.varshape, allow_none=False)
self.phi = parameter(phi, self.varshape, allow_none=False)
# variables
self.p = variable(bm.zeros, self.mode, self.varshape)
# function
self.integral = odeint(self.derivative, method=method)
def derivative(self, p, t, V):
phi_p = 1.0 / (1 + bm.exp(-(V + 43.) / 5.2))
p_inf = 2.7 / (bm.exp(-(V + 55.) / 15.) + bm.exp((V + 55.) / 15.)) + 1.6
return self.phi * (phi_p - p) / p_inf
def update(self, tdi, V, C_Ca, E_Ca):
self.p.value = self.integral(self.p.value, tdi['t'], V, tdi['dt'])
def current(self, V, C_Ca, E_Ca):
M = C_Ca / (C_Ca + 0.2)
g = self.g_max * M * self.p
return g * (self.E - V)
def reset_state(self, V, C_Ca, E_Ca, batch_size=None):
self.p.value = 1.0 / (1 + bm.exp(-(V + 43.) / 5.2))
if batch_size is not None:
assert self.p.shape[0] == batch_size
[docs]class ICaT_HM1992(_ICa_p2q_ss):
r"""The low-threshold T-type calcium current model proposed by (Huguenard & McCormick, 1992) [1]_.
The dynamics of the low-threshold T-type calcium current model [1]_ is given by:
.. math::
I_{CaT} &= g_{max} p^2 q(V-E_{Ca}) \\
{dp \over dt} &= {\phi_p \cdot (p_{\infty}-p)\over \tau_p} \\
&p_{\infty} = {1 \over 1+\exp [-(V+59-V_{sh}) / 6.2]} \\
&\tau_{p} = 0.612 + {1 \over \exp [-(V+132.-V_{sh}) / 16.7]+\exp [(V+16.8-V_{sh}) / 18.2]} \\
{dq \over dt} &= {\phi_q \cdot (q_{\infty}-q) \over \tau_q} \\
&q_{\infty} = {1 \over 1+\exp [(V+83-V_{sh}) / 4]} \\
& \begin{array}{l} \tau_{q} = \exp \left(\frac{V+467-V_{sh}}{66.6}\right) \quad V< (-80 +V_{sh})\, mV \\
\tau_{q} = \exp \left(\frac{V+22-V_{sh}}{-10.5}\right)+28 \quad V \geq (-80 + V_{sh})\, mV \end{array}
where :math:`\phi_p = 3.55^{\frac{T-24}{10}}` and :math:`\phi_q = 3^{\frac{T-24}{10}}`
are temperature-dependent factors (:math:`T` is the temperature in Celsius),
:math:`E_{Ca}` is the reversal potential of Calcium channel.
Parameters
----------
T : float, ArrayType
The temperature.
T_base_p : float, ArrayType
The brainpy_object temperature factor of :math:`p` channel.
T_base_q : float, ArrayType
The brainpy_object temperature factor of :math:`q` channel.
g_max : float, ArrayType, Callable, Initializer
The maximum conductance.
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`q`.
References
----------
.. [1] Huguenard JR, McCormick DA (1992) Simulation of the currents involved in
rhythmic oscillations in thalamic relay neurons. J Neurophysiol 68:1373–1383.
See Also
--------
ICa_p2q_form
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType] = 36.,
T_base_p: Union[float, ArrayType] = 3.55,
T_base_q: Union[float, ArrayType] = 3.,
g_max: Union[float, ArrayType, Initializer, Callable] = 2.,
V_sh: Union[float, ArrayType, Initializer, Callable] = -3.,
phi_p: Union[float, ArrayType, Initializer, Callable] = None,
phi_q: Union[float, ArrayType, Initializer, Callable] = None,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
phi_p = T_base_p ** ((T - 24) / 10) if phi_p is None else phi_p
phi_q = T_base_q ** ((T - 24) / 10) if phi_q is None else phi_q
super(ICaT_HM1992, self).__init__(size,
keep_size=keep_size,
name=name,
method=method,
g_max=g_max,
phi_p=phi_p,
phi_q=phi_q,
mode=mode)
# parameters
self.T = parameter(T, self.varshape, allow_none=False)
self.T_base_p = parameter(T_base_p, self.varshape, allow_none=False)
self.T_base_q = parameter(T_base_q, self.varshape, allow_none=False)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V):
return 1. / (1 + bm.exp(-(V + 59. - self.V_sh) / 6.2))
def f_p_tau(self, V):
return 1. / (bm.exp(-(V + 132. - self.V_sh) / 16.7) +
bm.exp((V + 16.8 - self.V_sh) / 18.2)) + 0.612
def f_q_inf(self, V):
return 1. / (1. + bm.exp((V + 83. - self.V_sh) / 4.0))
def f_q_tau(self, V):
return bm.where(V >= (-80. + self.V_sh),
bm.exp(-(V + 22. - self.V_sh) / 10.5) + 28.,
bm.exp((V + 467. - self.V_sh) / 66.6))
[docs]class ICaT_HP1992(_ICa_p2q_ss):
r"""The low-threshold T-type calcium current model for thalamic
reticular nucleus proposed by (Huguenard & Prince, 1992) [1]_.
The dynamics of the low-threshold T-type calcium current model in thalamic
reticular nucleus neurons [1]_ is given by:
.. math::
I_{CaT} &= g_{max} p^2 q(V-E_{Ca}) \\
{dp \over dt} &= {\phi_p \cdot (p_{\infty}-p)\over \tau_p} \\
&p_{\infty} = {1 \over 1+\exp [-(V+52-V_{sh}) / 7.4]} \\
&\tau_{p} = 3+{1 \over \exp [(V+27-V_{sh}) / 10]+\exp [-(V+102-V_{sh}) / 15]} \\
{dq \over dt} &= {\phi_q \cdot (q_{\infty}-q) \over \tau_q} \\
&q_{\infty} = {1 \over 1+\exp [(V+80-V_{sh}) / 5]} \\
& \tau_q = 85+ {1 \over \exp [(V+48-V_{sh}) / 4]+\exp [-(V+407-V_{sh}) / 50]}
where :math:`\phi_p = 5^{\frac{T-24}{10}}` and :math:`\phi_q = 3^{\frac{T-24}{10}}`
are temperature-dependent factors (:math:`T` is the temperature in Celsius),
:math:`E_{Ca}` is the reversal potential of Calcium channel.
Parameters
----------
T : float, ArrayType
The temperature.
T_base_p : float, ArrayType
The brainpy_object temperature factor of :math:`p` channel.
T_base_q : float, ArrayType
The brainpy_object temperature factor of :math:`q` channel.
g_max : float, ArrayType, Callable, Initializer
The maximum conductance.
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`q`.
References
----------
.. [1] Huguenard JR, Prince DA (1992) A novel T-type current underlies
prolonged Ca2+- dependent burst firing in GABAergic neurons of rat
thalamic reticular nucleus. J Neurosci 12: 3804–3817.
See Also
--------
ICa_p2q_form
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType] = 36.,
T_base_p: Union[float, ArrayType] = 5.,
T_base_q: Union[float, ArrayType] = 3.,
g_max: Union[float, ArrayType, Initializer, Callable] = 1.75,
V_sh: Union[float, ArrayType, Initializer, Callable] = -3.,
phi_p: Union[float, ArrayType, Initializer, Callable] = None,
phi_q: Union[float, ArrayType, Initializer, Callable] = None,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
phi_p = T_base_p ** ((T - 24) / 10) if phi_p is None else phi_p
phi_q = T_base_q ** ((T - 24) / 10) if phi_q is None else phi_q
super(ICaT_HP1992, self).__init__(size,
keep_size=keep_size,
name=name,
method=method,
g_max=g_max,
phi_p=phi_p,
phi_q=phi_q,
mode=mode)
# parameters
self.T = parameter(T, self.varshape, allow_none=False)
self.T_base_p = parameter(T_base_p, self.varshape, allow_none=False)
self.T_base_q = parameter(T_base_q, self.varshape, allow_none=False)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V):
return 1. / (1. + bm.exp(-(V + 52. - self.V_sh) / 7.4))
def f_p_tau(self, V):
return 3. + 1. / (bm.exp((V + 27. - self.V_sh) / 10.) +
bm.exp(-(V + 102. - self.V_sh) / 15.))
def f_q_inf(self, V):
return 1. / (1. + bm.exp((V + 80. - self.V_sh) / 5.))
def f_q_tau(self, V):
return 85. + 1. / (bm.exp((V + 48. - self.V_sh) / 4.) +
bm.exp(-(V + 407. - self.V_sh) / 50.))
[docs]class ICaHT_HM1992(_ICa_p2q_ss):
r"""The high-threshold T-type calcium current model proposed by (Huguenard & McCormick, 1992) [1]_.
The high-threshold T-type calcium current model is adopted from [1]_.
Its dynamics is given by
.. math::
\begin{aligned}
I_{\mathrm{Ca/HT}} &= g_{\mathrm{max}} p^2 q (V-E_{Ca})
\\
{dp \over dt} &= {\phi_{p} \cdot (p_{\infty} - p) \over \tau_{p}} \\
&\tau_{p} =\frac{1}{\exp \left(\frac{V+132-V_{sh}}{-16.7}\right)+\exp \left(\frac{V+16.8-V_{sh}}{18.2}\right)}+0.612 \\
& p_{\infty} = {1 \over 1+exp[-(V+59-V_{sh}) / 6.2]}
\\
{dq \over dt} &= {\phi_{q} \cdot (q_{\infty} - h) \over \tau_{q}} \\
& \begin{array}{l} \tau_q = \exp \left(\frac{V+467-V_{sh}}{66.6}\right) \quad V< (-80 +V_{sh})\, mV \\
\tau_q = \exp \left(\frac{V+22-V_{sh}}{-10.5}\right)+28 \quad V \geq (-80 + V_{sh})\, mV \end{array} \\
&q_{\infty} = {1 \over 1+exp[(V+83 -V_{shift})/4]}
\end{aligned}
where :math:`phi_p = 3.55^{\frac{T-24}{10}}` and :math:`phi_q = 3^{\frac{T-24}{10}}`
are temperature-dependent factors (:math:`T` is the temperature in Celsius),
:math:`E_{Ca}` is the reversal potential of Calcium channel.
Parameters
----------
T : float, ArrayType
The temperature.
T_base_p : float, ArrayType
The brainpy_object temperature factor of :math:`p` channel.
T_base_q : float, ArrayType
The brainpy_object temperature factor of :math:`q` channel.
g_max : float, ArrayType, Initializer, Callable
The maximum conductance.
V_sh : float, ArrayType, Initializer, Callable
The membrane potential shift.
References
----------
.. [1] Huguenard JR, McCormick DA (1992) Simulation of the currents involved in
rhythmic oscillations in thalamic relay neurons. J Neurophysiol 68:1373–1383.
See Also
--------
ICa_p2q_form
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType] = 36.,
T_base_p: Union[float, ArrayType] = 3.55,
T_base_q: Union[float, ArrayType] = 3.,
g_max: Union[float, ArrayType, Initializer, Callable] = 2.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 25.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super(ICaHT_HM1992, self).__init__(size,
keep_size=keep_size,
name=name,
method=method,
g_max=g_max,
phi_p=T_base_p ** ((T - 24) / 10),
phi_q=T_base_q ** ((T - 24) / 10),
mode=mode)
# parameters
self.T = parameter(T, self.varshape, allow_none=False)
self.T_base_p = parameter(T_base_p, self.varshape, allow_none=False)
self.T_base_q = parameter(T_base_q, self.varshape, allow_none=False)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
# variables
self.p = variable(bm.zeros, self.mode, self.varshape)
self.q = variable(bm.zeros, self.mode, self.varshape)
# function
self.integral = odeint(JointEq([self.dp, self.dq]), method=method)
def f_p_inf(self, V):
return 1. / (1. + bm.exp(-(V + 59. - self.V_sh) / 6.2))
def f_p_tau(self, V):
return 1. / (bm.exp(-(V + 132. - self.V_sh) / 16.7) +
bm.exp((V + 16.8 - self.V_sh) / 18.2)) + 0.612
def f_q_inf(self, V):
return 1. / (1. + bm.exp((V + 83. - self.V_sh) / 4.))
def f_q_tau(self, V):
return bm.where(V >= (-80. + self.V_sh),
bm.exp(-(V + 22. - self.V_sh) / 10.5) + 28.,
bm.exp((V + 467. - self.V_sh) / 66.6))
class ICaHT_Re1993(_ICa_p2q_markov):
r"""The high-threshold T-type calcium current model proposed by (Reuveni, et al., 1993) [1]_.
HVA Calcium current was described for neocortical neurons by Sayer et al. (1990).
Its dynamics is given by (the rate functions are measured under 36 Celsius):
.. math::
\begin{aligned}
I_{L} &=\bar{g}_{L} q^{2} r\left(V-E_{\mathrm{Ca}}\right) \\
\frac{\mathrm{d} q}{\mathrm{~d} t} &= \phi_p (\alpha_{q}(V)(1-q)-\beta_{q}(V) q) \\
\frac{\mathrm{d} r}{\mathrm{~d} t} &= \phi_q (\alpha_{r}(V)(1-r)-\beta_{r}(V) r) \\
\alpha_{q} &=\frac{0.055(-27-V+V_{sh})}{\exp [(-27-V+V_{sh}) / 3.8]-1} \\
\beta_{q} &=0.94 \exp [(-75-V+V_{sh}) / 17] \\
\alpha_{r} &=0.000457 \exp [(-13-V+V_{sh}) / 50] \\
\beta_{r} &=\frac{0.0065}{\exp [(-15-V+V_{sh}) / 28]+1},
\end{aligned}
Parameters
----------
size: int, tuple of int
The size of the simulation target.
keep_size: bool
Keep size or flatten the size?
method: str
The numerical method
name: str
The name of the object.
g_max : float, ArrayType, Callable, Initializer
The maximum conductance.
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
T : float, ArrayType
The temperature.
T_base_p : float, ArrayType
The brainpy_object temperature factor of :math:`p` channel.
T_base_q : float, ArrayType
The brainpy_object temperature factor of :math:`q` channel.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
If `None`, :math:`\phi_p = \mathrm{T_base_p}^{\frac{T-23}{10}}`.
phi_q : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`q`.
If `None`, :math:`\phi_q = \mathrm{T_base_q}^{\frac{T-23}{10}}`.
References
----------
.. [1] Reuveni, I., et al. "Stepwise repolarization from Ca2+ plateaus
in neocortical pyramidal cells: evidence for nonhomogeneous
distribution of HVA Ca2+ channels in dendrites." Journal of
Neuroscience 13.11 (1993): 4609-4621.
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType] = 36.,
T_base_p: Union[float, ArrayType] = 2.3,
T_base_q: Union[float, ArrayType] = 2.3,
phi_p: Union[float, ArrayType, Initializer, Callable] = None,
phi_q: Union[float, ArrayType, Initializer, Callable] = None,
g_max: Union[float, ArrayType, Initializer, Callable] = 1.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
phi_p = T_base_p ** ((T - 23.) / 10.) if phi_p is None else phi_p
phi_q = T_base_q ** ((T - 23.) / 10.) if phi_q is None else phi_q
super(ICaHT_Re1993, self).__init__(size,
keep_size=keep_size,
name=name,
method=method,
g_max=g_max,
phi_p=phi_p,
phi_q=phi_q,
mode=mode)
self.T = parameter(T, self.varshape, allow_none=False)
self.T_base_p = parameter(T_base_p, self.varshape, allow_none=False)
self.T_base_q = parameter(T_base_q, self.varshape, allow_none=False)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_alpha(self, V):
temp = -27 - V + self.V_sh
return 0.055 * temp / (bm.exp(temp / 3.8) - 1)
def f_p_beta(self, V):
return 0.94 * bm.exp((-75. - V + self.V_sh) / 17.)
def f_q_alpha(self, V):
return 0.000457 * bm.exp((-13. - V + self.V_sh) / 50.)
def f_q_beta(self, V):
return 0.0065 / (bm.exp((-15. - V + self.V_sh) / 28.) + 1.)
[docs]class ICaL_IS2008(_ICa_p2q_ss):
r"""The L-type calcium channel model proposed by (Inoue & Strowbridge, 2008) [1]_.
The L-type calcium channel model is adopted from (Inoue, et, al., 2008) [1]_.
Its dynamics is given by:
.. math::
I_{CaL} &= g_{max} p^2 q(V-E_{Ca}) \\
{dp \over dt} &= {\phi_p \cdot (p_{\infty}-p)\over \tau_p} \\
& p_{\infty} = {1 \over 1+\exp [-(V+10-V_{sh}) / 4.]} \\
& \tau_{p} = 0.4+{0.7 \over \exp [(V+5-V_{sh}) / 15]+\exp [-(V+5-V_{sh}) / 15]} \\
{dq \over dt} &= {\phi_q \cdot (q_{\infty}-q) \over \tau_q} \\
& q_{\infty} = {1 \over 1+\exp [(V+25-V_{sh}) / 2]} \\
& \tau_q = 300 + {100 \over \exp [(V+40-V_{sh}) / 9.5]+\exp [-(V+40-V_{sh}) / 9.5]}
where :math:`phi_p = 3.55^{\frac{T-24}{10}}` and :math:`phi_q = 3^{\frac{T-24}{10}}`
are temperature-dependent factors (:math:`T` is the temperature in Celsius),
:math:`E_{Ca}` is the reversal potential of Calcium channel.
Parameters
----------
T : float
The temperature.
T_base_p : float
The brainpy_object temperature factor of :math:`p` channel.
T_base_q : float
The brainpy_object temperature factor of :math:`q` channel.
g_max : float
The maximum conductance.
V_sh : float
The membrane potential shift.
References
----------
.. [1] Inoue, Tsuyoshi, and Ben W. Strowbridge. "Transient activity induces a long-lasting
increase in the excitability of olfactory bulb interneurons." Journal of
neurophysiology 99, no. 1 (2008): 187-199.
See Also
--------
ICa_p2q_form
"""
[docs] def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[float, ArrayType, Initializer, Callable] = 36.,
T_base_p: Union[float, ArrayType, Initializer, Callable] = 3.55,
T_base_q: Union[float, ArrayType, Initializer, Callable] = 3.,
g_max: Union[float, ArrayType, Initializer, Callable] = 1.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super(ICaL_IS2008, self).__init__(size,
keep_size=keep_size,
name=name,
method=method,
g_max=g_max,
phi_p=T_base_p ** ((T - 24) / 10),
phi_q=T_base_q ** ((T - 24) / 10),
mode=mode)
# parameters
self.T = parameter(T, self.varshape, allow_none=False)
self.T_base_p = parameter(T_base_p, self.varshape, allow_none=False)
self.T_base_q = parameter(T_base_q, self.varshape, allow_none=False)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V):
return 1. / (1 + bm.exp(-(V + 10. - self.V_sh) / 4.))
def f_p_tau(self, V):
return 0.4 + .7 / (bm.exp(-(V + 5. - self.V_sh) / 15.) +
bm.exp((V + 5. - self.V_sh) / 15.))
def f_q_inf(self, V):
return 1. / (1. + bm.exp((V + 25. - self.V_sh) / 2.))
def f_q_tau(self, V):
return 300. + 100. / (bm.exp((V + 40 - self.V_sh) / 9.5) +
bm.exp(-(V + 40 - self.V_sh) / 9.5))
