Source code for brainpy.dyn.channels.Na

# -*- coding: utf-8 -*-

"""
This module implements voltage-dependent sodium channels.

"""

from typing import Union, Callable

import brainpy.math as bm
from brainpy.initialize import Initializer, parameter, variable
from brainpy.integrators import odeint, JointEq
from brainpy.types import Array, Shape
from brainpy.modes import Mode, BatchingMode, normal
from .base import SodiumChannel

__all__ = [
  'INa_p3q_markov',
  'INa_Ba2002',
  'INa_TM1991',
  'INa_HH1952',
]


[docs]class INa_p3q_markov(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, Array, Callable, Initializer The maximal conductance density (:math:`mS/cm^2`). E : float, Array, Callable, Initializer The reversal potential (mV). phi : float, Array, Callable, Initializer The temperature-dependent factor. method: str The numerical method name: str The name of the object. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[int, float, Array, Initializer, Callable] = 50., g_max: Union[int, float, Array, Initializer, Callable] = 90., phi: Union[int, float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(INa_p3q_markov, self).__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, mode, self.varshape) self.q = variable(bm.zeros, 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 batch_size is not None: 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, tdi, V): t, dt = tdi['t'], tdi['dt'] p, q = self.integral(self.p, self.q, t, V, 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, Array, Callable, Initializer The maximal conductance density (:math:`mS/cm^2`). E : float, Array, Callable, Initializer The reversal potential (mV). T : float, Array The temperature (Celsius, :math:`^{\circ}C`). V_sh : float, Array, 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 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, T: Union[int, float, Array] = 36., E: Union[int, float, Array, Initializer, Callable] = 50., g_max: Union[int, float, Array, Initializer, Callable] = 90., V_sh: Union[int, float, Array, Initializer, Callable] = -50., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(INa_Ba2002, self).__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, Array, Callable, Initializer The maximal conductance density (:math:`mS/cm^2`). E : float, Array, Callable, Initializer The reversal potential (mV). V_sh: float, Array, 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 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[int, float, Array, Initializer, Callable] = 50., g_max: Union[int, float, Array, Initializer, Callable] = 120., phi: Union[int, float, Array, Initializer, Callable] = 1., V_sh: Union[int, float, Array, Initializer, Callable] = -63., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(INa_TM1991, self).__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, Array, Callable, Initializer The maximal conductance density (:math:`mS/cm^2`). E : float, Array, Callable, Initializer The reversal potential (mV). V_sh: float, Array, 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 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[int, float, Array, Initializer, Callable] = 50., g_max: Union[int, float, Array, Initializer, Callable] = 120., phi: Union[int, float, Array, Initializer, Callable] = 1., V_sh: Union[int, float, Array, Initializer, Callable] = -45., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(INa_HH1952, self).__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))