# -*- coding: utf-8 -*-
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# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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# http://www.apache.org/licenses/LICENSE-2.0
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# ==============================================================================
"""
This module implements voltage-dependent sodium channels.
"""
from typing import Union, Callable
import brainpy.math as bm
from brainpy.context import share
from brainpy.dyn.ions.sodium import Sodium
from brainpy.initialize import Initializer, parameter, variable
from brainpy.integrators import odeint, JointEq
from brainpy.types import ArrayType, Shape
from .base import IonChannel
__all__ = [
'SodiumChannel',
'INa_Ba2002v2',
'INa_TM1991v2',
'INa_HH1952v2',
]
[docs]
class SodiumChannel(IonChannel):
"""Base class for sodium channel dynamics."""
master_type = Sodium
def update(self, V, C, E):
raise NotImplementedError
def current(self, V, C, E):
raise NotImplementedError
def reset(self, V, C, E, batch_size: int = None):
self.reset_state(V, C, E, batch_size)
def reset_state(self, V, C, E, batch_size: int = None):
raise NotImplementedError('Must be implemented by the subclass.')
class _INa_p3q_markov_v2(SodiumChannel):
r"""The sodium current model of :math:`p^3q` current which described with first-order Markov chain.
The general model can be used to model the dynamics with:
.. math::
\begin{aligned}
I_{\mathrm{Na}} &= g_{\mathrm{max}} * p^3 * q \\
\frac{dp}{dt} &= \phi ( \alpha_p (1-p) - \beta_p p) \\
\frac{dq}{dt} & = \phi ( \alpha_q (1-h) - \beta_q h) \\
\end{aligned}
where :math:`\phi` is a temperature-dependent factor.
Parameters::
g_max : float, ArrayType, Callable, Initializer
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Callable, Initializer
The reversal potential (mV).
phi : float, ArrayType, Callable, Initializer
The temperature-dependent factor.
method: str
The numerical method
name: str
The name of the object.
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
g_max: Union[int, float, ArrayType, Initializer, Callable] = 90.,
phi: Union[int, float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size=size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
self.phi = parameter(phi, 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)
# function
self.integral = odeint(JointEq([self.dp, self.dq]), method=method)
def reset_state(self, V, C, E, batch_size=None):
alpha = self.f_p_alpha(V)
beta = self.f_p_beta(V)
self.p.value = alpha / (alpha + beta)
alpha = self.f_q_alpha(V)
beta = self.f_q_beta(V)
self.q.value = alpha / (alpha + beta)
if isinstance(batch_size, int):
assert self.p.shape[0] == batch_size
assert self.q.shape[0] == batch_size
def dp(self, p, t, V):
return self.phi * (self.f_p_alpha(V) * (1. - p) - self.f_p_beta(V) * p)
def dq(self, q, t, V):
return self.phi * (self.f_q_alpha(V) * (1. - q) - self.f_q_beta(V) * q)
def update(self, V, C, E):
p, q = self.integral(self.p, self.q, share['t'], V, share['dt'])
self.p.value, self.q.value = p, q
def current(self, V, C, E):
return self.g_max * self.p ** 3 * self.q * (E - V)
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 INa_Ba2002v2(_INa_p3q_markov_v2):
r"""The sodium current model.
The sodium current model is adopted from (Bazhenov, et, al. 2002) [1]_.
It's dynamics is given by:
.. math::
\begin{aligned}
I_{\mathrm{Na}} &= g_{\mathrm{max}} * p^3 * q \\
\frac{dp}{dt} &= \phi ( \alpha_p (1-p) - \beta_p p) \\
\alpha_{p} &=\frac{0.32\left(V-V_{sh}-13\right)}{1-\exp \left(-\left(V-V_{sh}-13\right) / 4\right)} \\
\beta_{p} &=\frac{-0.28\left(V-V_{sh}-40\right)}{1-\exp \left(\left(V-V_{sh}-40\right) / 5\right)} \\
\frac{dq}{dt} & = \phi ( \alpha_q (1-h) - \beta_q h) \\
\alpha_q &=0.128 \exp \left(-\left(V-V_{sh}-17\right) / 18\right) \\
\beta_q &= \frac{4}{1+\exp \left(-\left(V-V_{sh}-40\right) / 5\right)}
\end{aligned}
where :math:`\phi` is a temperature-dependent factor, which is given by
:math:`\phi=3^{\frac{T-36}{10}}` (:math:`T` is the temperature in Celsius).
Parameters::
g_max : float, ArrayType, Callable, Initializer
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Callable, Initializer
The reversal potential (mV).
T : float, ArrayType
The temperature (Celsius, :math:`^{\circ}C`).
V_sh : float, ArrayType, Callable, Initializer
The shift of the membrane potential to spike.
References::
.. [1] Bazhenov, Maxim, et al. "Model of thalamocortical slow-wave sleep oscillations
and transitions to activated states." Journal of neuroscience 22.19 (2002): 8691-8704.
See Also::
INa_TM1991
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
T: Union[int, float, ArrayType] = 36.,
g_max: Union[int, float, ArrayType, Initializer, Callable] = 90.,
V_sh: Union[int, float, ArrayType, Initializer, Callable] = -50.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
phi=3 ** ((T - 36) / 10),
g_max=g_max,
mode=mode)
self.T = parameter(T, self.varshape, allow_none=False)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_alpha(self, V):
temp = V - self.V_sh - 13.
return 0.32 * temp / (1. - bm.exp(-temp / 4.))
def f_p_beta(self, V):
temp = V - self.V_sh - 40.
return -0.28 * temp / (1. - bm.exp(temp / 5.))
def f_q_alpha(self, V):
return 0.128 * bm.exp(-(V - self.V_sh - 17.) / 18.)
def f_q_beta(self, V):
return 4. / (1. + bm.exp(-(V - self.V_sh - 40.) / 5.))
[docs]
class INa_TM1991v2(_INa_p3q_markov_v2):
r"""The sodium current model described by (Traub and Miles, 1991) [1]_.
The dynamics of this sodium current model is given by:
.. math::
\begin{split}
\begin{aligned}
I_{\mathrm{Na}} &= g_{\mathrm{max}} m^3 h \\
\frac {dm} {dt} &= \phi(\alpha_m (1-x) - \beta_m) \\
&\alpha_m(V) = 0.32 \frac{(13 - V + V_{sh})}{\exp((13 - V +V_{sh}) / 4) - 1.} \\
&\beta_m(V) = 0.28 \frac{(V - V_{sh} - 40)}{(\exp((V - V_{sh} - 40) / 5) - 1)} \\
\frac {dh} {dt} &= \phi(\alpha_h (1-x) - \beta_h) \\
&\alpha_h(V) = 0.128 * \exp((17 - V + V_{sh}) / 18) \\
&\beta_h(V) = 4. / (1 + \exp(-(V - V_{sh} - 40) / 5)) \\
\end{aligned}
\end{split}
where :math:`V_{sh}` is the membrane shift (default -63 mV), and
:math:`\phi` is the temperature-dependent factor (default 1.).
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 maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Callable, Initializer
The reversal potential (mV).
V_sh: float, ArrayType, Callable, Initializer
The membrane shift.
References::
.. [1] Traub, Roger D., and Richard Miles. Neuronal networks of the hippocampus.
Vol. 777. Cambridge University Press, 1991.
See Also::
INa_Ba2002
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
g_max: Union[int, float, ArrayType, Initializer, Callable] = 120.,
phi: Union[int, float, ArrayType, Initializer, Callable] = 1.,
V_sh: Union[int, float, ArrayType, Initializer, Callable] = -63.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
phi=phi,
g_max=g_max,
mode=mode)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_alpha(self, V):
temp = 13 - V + self.V_sh
return 0.32 * temp / (bm.exp(temp / 4) - 1.)
def f_p_beta(self, V):
temp = V - self.V_sh - 40
return 0.28 * temp / (bm.exp(temp / 5) - 1)
def f_q_alpha(self, V):
return 0.128 * bm.exp((17 - V + self.V_sh) / 18)
def f_q_beta(self, V):
return 4. / (1 + bm.exp(-(V - self.V_sh - 40) / 5))
[docs]
class INa_HH1952v2(_INa_p3q_markov_v2):
r"""The sodium current model described by Hodgkin–Huxley model [1]_.
The dynamics of this sodium current model is given by:
.. math::
\begin{split}
\begin{aligned}
I_{\mathrm{Na}} &= g_{\mathrm{max}} m^3 h \\
\frac {dm} {dt} &= \phi (\alpha_m (1-x) - \beta_m) \\
&\alpha_m(V) = \frac {0.1(V-V_{sh}-5)}{1-\exp(\frac{-(V -V_{sh} -5)} {10})} \\
&\beta_m(V) = 4.0 \exp(\frac{-(V -V_{sh}+ 20)} {18}) \\
\frac {dh} {dt} &= \phi (\alpha_h (1-x) - \beta_h) \\
&\alpha_h(V) = 0.07 \exp(\frac{-(V-V_{sh}+20)}{20}) \\
&\beta_h(V) = \frac 1 {1 + \exp(\frac{-(V -V_{sh}-10)} {10})} \\
\end{aligned}
\end{split}
where :math:`V_{sh}` is the membrane shift (default -45 mV), and
:math:`\phi` is the temperature-dependent factor (default 1.).
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 maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Callable, Initializer
The reversal potential (mV).
V_sh: float, ArrayType, Callable, Initializer
The membrane shift.
References::
.. [1] Hodgkin, Alan L., and Andrew F. Huxley. "A quantitative description of
membrane current and its application to conduction and excitation in
nerve." The Journal of physiology 117.4 (1952): 500.
See Also::
IK_HH1952
"""
def __init__(
self,
size: Shape,
keep_size: bool = False,
g_max: Union[int, float, ArrayType, Initializer, Callable] = 120.,
phi: Union[int, float, ArrayType, Initializer, Callable] = 1.,
V_sh: Union[int, float, ArrayType, Initializer, Callable] = -45.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
phi=phi,
g_max=g_max,
mode=mode)
self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_alpha(self, V):
temp = V - self.V_sh - 5
return 0.1 * temp / (1 - bm.exp(-temp / 10))
def f_p_beta(self, V):
return 4.0 * bm.exp(-(V - self.V_sh + 20) / 18)
def f_q_alpha(self, V):
return 0.07 * bm.exp(-(V - self.V_sh + 20) / 20.)
def f_q_beta(self, V):
return 1 / (1 + bm.exp(-(V - self.V_sh - 10) / 10))
