# -*- coding: utf-8 -*-
"""
This module implements voltage-dependent sodium channels.
"""
from typing import Union, Callable, Sequence
import brainpy.math as bm
from brainpy._src.context import share
from brainpy._src.dyn.neurons.hh import HHTypedNeuron
from brainpy._src.initialize import Initializer, parameter, variable
from brainpy._src.integrators import odeint, JointEq
from brainpy.types import ArrayType
from .base import IonChannel
__all__ = [
'INa_Ba2002',
'INa_TM1991',
'INa_HH1952',
]
class _INa_p3q_markov(IonChannel):
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.
"""
master_type = HHTypedNeuron
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[int, float, ArrayType, Initializer, Callable] = None,
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.E = parameter(E, self.varshape, allow_none=False)
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, 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):
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):
return self.g_max * self.p ** 3 * self.q * (self.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_Ba2002(_INa_p3q_markov):
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: Union[int, Sequence[int]],
keep_size: bool = False,
T: Union[int, float, ArrayType] = 36.,
E: Union[int, float, ArrayType, Initializer, Callable] = 50.,
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,
E=E,
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_TM1991(_INa_p3q_markov):
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: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[int, float, ArrayType, Initializer, Callable] = 50.,
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,
E=E,
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_HH1952(_INa_p3q_markov):
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: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[int, float, ArrayType, Initializer, Callable] = 50.,
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,
E=E,
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))
