# -*- coding: utf-8 -*-
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"""
This module implements calcium-dependent potassium channels.
"""
from typing import Union, Callable, Optional
import brainpy.math as bm
from brainpy.context import share
from brainpy.dyn.ions.calcium import Calcium
from brainpy.dyn.ions.potassium import Potassium
from brainpy.initialize import Initializer, parameter, variable
from brainpy.integrators.ode.generic import odeint
from brainpy.mixin import JointType
from brainpy.types import Shape, ArrayType
from .calcium import CalciumChannel
from .potassium import PotassiumChannel
__all__ = [
'IAHP_De1994v2',
]
class KCaChannel(PotassiumChannel, CalciumChannel):
pass
[docs]
class IAHP_De1994v2(KCaChannel):
r"""The calcium-dependent potassium current model proposed by (Destexhe, et al., 1994) [1]_.
Both in vivo (Contreras et al. 1993; Mulle et al. 1986) and in
vitro recordings (Avanzini et al. 1989) show the presence of a
marked after-hyper-polarization (AHP) after each burst of the RE
cell. This slow AHP is mediated by a slow :math:`Ca^{2+}`-dependent K+
current (Bal and McCormick 1993). (Destexhe, et al., 1994) adopted a
modified version of a model of :math:`I_{KCa}` introduced previously (Yamada et al.
1989) that requires the binding of :math:`nCa^{2+}` to open the channel
.. math::
(\text { closed })+n \mathrm{Ca}_{i}^{2+} \underset{\beta}{\stackrel{\alpha}{\rightleftharpoons}(\text { open })
where :math:`Ca_i^{2+}` is the intracellular calcium and :math:`\alpha` and
:math:`\beta` are rate constants. The ionic current is then given by
.. math::
\begin{aligned}
I_{AHP} &= g_{\mathrm{max}} p^2 (V - E_K) \\
{dp \over dt} &= \phi {p_{\infty}(V, [Ca^{2+}]_i) - p \over \tau_p(V, [Ca^{2+}]_i)} \\
p_{\infty} &=\frac{\alpha[Ca^{2+}]_i^n}{\left(\alpha[Ca^{2+}]_i^n + \beta\right)} \\
\tau_p &=\frac{1}{\left(\alpha[Ca^{2+}]_i +\beta\right)}
\end{aligned}
where :math:`E` is the reversal potential, :math:`g_{max}` is the maximum conductance,
:math:`[Ca^{2+}]_i` is the intracellular Calcium concentration.
The values :math:`n=2, \alpha=48 \mathrm{~ms}^{-1} \mathrm{mM}^{-2}` and
:math:`\beta=0.03 \mathrm{~ms}^{-1}` yielded AHPs very similar to those RE cells
recorded in vivo and in vitro.
Parameters::
g_max : float
The maximal conductance density (:math:`mS/cm^2`).
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.
"""
'''The type of the master object.'''
master_type = JointType[Calcium, Potassium]
def __init__(
self,
size: Shape,
keep_size: bool = False,
n: Union[float, ArrayType, Initializer, Callable] = 2,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
alpha: Union[float, ArrayType, Initializer, Callable] = 48.,
beta: Union[float, ArrayType, Initializer, Callable] = 0.09,
phi: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: Optional[str] = None,
mode: Optional[bm.Mode] = None,
):
super().__init__(size=size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
self.g_max = parameter(g_max, self.varshape, allow_none=False)
self.n = parameter(n, self.varshape, allow_none=False)
self.alpha = parameter(alpha, self.varshape, allow_none=False)
self.beta = parameter(beta, 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.dp, method=method)
def dp(self, p, t, C_Ca):
C2 = self.alpha * bm.power(C_Ca, self.n)
C3 = C2 + self.beta
return self.phi * (C2 / C3 - p) * C3
def update(self, V, Ca_info, K_info):
self.p.value = self.integral(self.p.value, share['t'], C_Ca=Ca_info['C'], dt=share['dt'])
def current(self, V, Ca_info, K_info):
return self.g_max * self.p * self.p * (K_info['E'] - V)
def reset_state(self, V, Ca_info, K_info, batch_size=None):
C2 = self.alpha * bm.power(Ca_info['C'], self.n)
C3 = C2 + self.beta
if batch_size is None:
self.p.value = bm.broadcast_to(C2 / C3, self.varshape)
else:
self.p.value = bm.broadcast_to(C2 / C3, (batch_size,) + self.varshape)
assert self.p.shape[0] == batch_size
