Source code for brainpy.dyn.channels.K

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

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

"""

from typing import Union, Callable, Optional

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

__all__ = [
  'IK_p4_markov',
  'IKDR_Ba2002',
  'IK_TM1991',
  'IK_HH1952',

  'IKA_p4q_ss',
  'IKA1_HM1992',
  'IKA2_HM1992',

  'IKK2_pq_ss',
  'IKK2A_HM1992',
  'IKK2B_HM1992',

  'IKNI_Ya1989',
]


[docs]class IK_p4_markov(PotassiumChannel): r"""The delayed rectifier potassium channel of :math:`p^4` current which described with first-order Markov chain. This general potassium current model should have the form of .. math:: \begin{aligned} I_{\mathrm{K}} &= g_{\mathrm{max}} * p^4 \\ \frac{dp}{dt} &= \phi * (\alpha_p (1-p) - \beta_p p) \end{aligned} where :math:`\phi` is a temperature-dependent factor. Parameters ---------- size: int, sequence of int The object size. keep_size: bool Whether we use `size` to initialize the variable. Otherwise, variable shape will be initialized as `num`. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). phi : float, JaxArray, ndarray, Initializer, Callable The temperature-dependent factor. method: str The numerical integration method. name: str The object name. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., phi: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IK_p4_markov, self).__init__(size, keep_size=keep_size, name=name, mode=mode) 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, mode, self.varshape) # function self.integral = odeint(self.derivative, method=method)
def derivative(self, p, t, V): return self.phi * (self.f_p_alpha(V) * (1. - p) - self.f_p_beta(V) * p) def update(self, tdi, V): self.p.value = self.integral(self.p, tdi['t'], V, tdi['dt']) def current(self, V): return self.g_max * self.p ** 4 * (self.E - V) 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) if batch_size is not None: assert self.p.shape[0] == batch_size def f_p_alpha(self, V): raise NotImplementedError def f_p_beta(self, V): raise NotImplementedError
[docs]class IKDR_Ba2002(IK_p4_markov): r"""The delayed rectifier potassium channel current. The potassium current model is adopted from (Bazhenov, et, al. 2002) [1]_. It's dynamics is given by: .. math:: \begin{aligned} I_{\mathrm{K}} &= g_{\mathrm{max}} * p^4 \\ \frac{dp}{dt} &= \phi * (\alpha_p (1-p) - \beta_p p) \\ \alpha_{p} &=\frac{0.032\left(V-V_{sh}-15\right)}{1-\exp \left(-\left(V-V_{sh}-15\right) / 5\right)} \\ \beta_p &= 0.5 \exp \left(-\left(V-V_{sh}-10\right) / 40\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 ---------- size: int, sequence of int The object size. keep_size: bool Whether we use `size` to initialize the variable. Otherwise, variable shape will be initialized as `num`. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). T_base : float, JaxArray, ndarray The base of temperature factor. T : float, JaxArray, ndarray, Initializer, Callable The temperature (Celsius, :math:`^{\circ}C`). V_sh : float, JaxArray, ndarray, Initializer, Callable The shift of the membrane potential to spike. method: str The numerical integration method. name: str The object name. 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. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., V_sh: Union[float, Array, Initializer, Callable] = -50., T_base: Union[float, Array] = 3., T: Union[float, Array] = 36., phi: Optional[Union[float, Array, Initializer, Callable]] = None, method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): phi = T_base ** ((T - 36) / 10) if phi is None else phi super(IKDR_Ba2002, self).__init__(size, keep_size=keep_size, name=name, method=method, g_max=g_max, phi=phi, E=E, mode=mode) # parameters self.T = parameter(T, self.varshape, allow_none=False) self.T_base = parameter(T_base, self.varshape, allow_none=False) self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_alpha(self, V): tmp = V - self.V_sh - 15. return 0.032 * tmp / (1. - bm.exp(-tmp / 5.)) def f_p_beta(self, V): return 0.5 * bm.exp(-(V - self.V_sh - 10.) / 40.)
[docs]class IK_TM1991(IK_p4_markov): r"""The potassium channel described by (Traub and Miles, 1991) [1]_. The dynamics of this channel is given by: .. math:: \begin{aligned} I_{\mathrm{K}} &= g_{\mathrm{max}} * p^4 \\ \frac{dp}{dt} &= \phi * (\alpha_p (1-p) - \beta_p p) \\ \alpha_{p} &= 0.032 \frac{(15 - V + V_{sh})}{(\exp((15 - V + V_{sh}) / 5) - 1.)} \\ \beta_p &= 0.5 * \exp((10 - V + V_{sh}) / 40) \end{aligned} where :math:`V_{sh}` is the membrane shift (default -63 mV), and :math:`\phi` is the temperature-dependent factor (default 1.). Parameters ---------- size: int, sequence of int The geometry size. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). method: str The numerical integration method. name: str The object name. References ---------- .. [1] Traub, Roger D., and Richard Miles. Neuronal networks of the hippocampus. Vol. 777. Cambridge University Press, 1991. See Also -------- INa_TM1991 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., phi: Union[float, Array, Initializer, Callable] = 1., V_sh: Union[int, float, Array, Initializer, Callable] = -60., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IK_TM1991, self).__init__(size, keep_size=keep_size, name=name, method=method, phi=phi, E=E, g_max=g_max, mode=mode) self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_alpha(self, V): c = 15 - V + self.V_sh return 0.032 * c / (bm.exp(c / 5) - 1.) def f_p_beta(self, V): return 0.5 * bm.exp((10 - V + self.V_sh) / 40)
[docs]class IK_HH1952(IK_p4_markov): r"""The potassium channel described by Hodgkin–Huxley model [1]_. The dynamics of this channel is given by: .. math:: \begin{aligned} I_{\mathrm{K}} &= g_{\mathrm{max}} * p^4 \\ \frac{dp}{dt} &= \phi * (\alpha_p (1-p) - \beta_p p) \\ \alpha_{p} &= \frac{0.01 (V -V_{sh} + 10)}{1-\exp \left(-\left(V-V_{sh}+ 10\right) / 10\right)} \\ \beta_p &= 0.125 \exp \left(-\left(V-V_{sh}+20\right) / 80\right) \end{aligned} where :math:`V_{sh}` is the membrane shift (default -45 mV), and :math:`\phi` is the temperature-dependent factor (default 1.). Parameters ---------- size: int, sequence of int The geometry size. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). method: str The numerical integration method. name: str The object name. 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 -------- INa_HH1952 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., phi: Union[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(IK_HH1952, self).__init__(size, keep_size=keep_size, name=name, method=method, phi=phi, E=E, 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 + 10 return 0.01 * temp / (1 - bm.exp(-temp / 10)) def f_p_beta(self, V): return 0.125 * bm.exp(-(V - self.V_sh + 20) / 80)
[docs]class IKA_p4q_ss(PotassiumChannel): r"""The rapidly inactivating Potassium channel of :math:`p^4q` current which described with steady-state format. This model is developed according to the average behavior of rapidly inactivating Potassium channel in Thalamus relay neurons [2]_ [3]_. .. math:: &IA = g_{\mathrm{max}} p^4 q (E-V) \\ &\frac{dp}{dt} = \phi_p \frac{p_{\infty} - p}{\tau_p} \\ &\frac{dq}{dt} = \phi_q \frac{q_{\infty} - q}{\tau_q} \\ where :math:`\phi_p` and :math:`\phi_q` are the temperature dependent factors (default 1.). Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. phi_q : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`q`. References ---------- .. [2] Huguenard, John R., and David A. McCormick. "Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons." Journal of neurophysiology 68.4 (1992): 1373-1383. .. [3] Huguenard, J. R., and D. A. Prince. "Slow inactivation of a TEA-sensitive K current in acutely isolated rat thalamic relay neurons." Journal of neurophysiology 66.4 (1991): 1316-1328. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKA_p4q_ss, 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_p = parameter(phi_p, self.varshape, allow_none=False) self.phi_q = parameter(phi_q, 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 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): t, dt = tdi['t'], tdi['dt'] self.p.value, self.q.value = self.integral(self.p.value, self.q.value, t, V, dt) def current(self, V): return self.g_max * self.p ** 4 * self.q * (self.E - V) def reset_state(self, V, 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
[docs]class IKA1_HM1992(IKA_p4q_ss): r"""The rapidly inactivating Potassium channel (IA1) model proposed by (Huguenard & McCormick, 1992) [2]_. This model is developed according to the average behavior of rapidly inactivating Potassium channel in Thalamus relay neurons [2]_ [1]_. .. math:: &IA = g_{\mathrm{max}} p^4 q (E-V) \\ &\frac{dp}{dt} = \phi_p \frac{p_{\infty} - p}{\tau_p} \\ &p_{\infty} = \frac{1}{1+ \exp[-(V -V_{sh}+ 60)/8.5]} \\ &\tau_{p}=\frac{1}{\exp \left(\frac{V -V_{sh}+35.8}{19.7}\right)+ \exp \left(\frac{V -V_{sh}+79.7}{-12.7}\right)}+0.37 \\ &\frac{dq}{dt} = \phi_q \frac{q_{\infty} - q}{\tau_q} \\ &q_{\infty} = \frac{1}{1+ \exp[(V -V_{sh} + 78)/6]} \\ &\begin{array}{l} \tau_{q} = \frac{1}{\exp((V -V_{sh}+46)/5.) + \exp((V -V_{sh}+238)/-37.5)} \quad V<(-63+V_{sh})\, mV \\ \tau_{q} = 19 \quad V \geq (-63 + V_{sh})\, mV \end{array} where :math:`\phi_p` and :math:`\phi_q` are the temperature dependent factors (default 1.). Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). V_sh : float, Array, Callable, Initializer The membrane potential shift. phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. phi_q : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`q`. References ---------- .. [2] Huguenard, John R., and David A. McCormick. "Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons." Journal of neurophysiology 68.4 (1992): 1373-1383. .. [1] Huguenard, J. R., and D. A. Prince. "Slow inactivation of a TEA-sensitive K current in acutely isolated rat thalamic relay neurons." Journal of neurophysiology 66.4 (1991): 1316-1328. See Also -------- IKA2_HM1992 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 30., V_sh: Union[float, Array, Initializer, Callable] = 0., phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKA1_HM1992, self).__init__(size, keep_size=keep_size, name=name, method=method, E=E, g_max=g_max, phi_p=phi_p, phi_q=phi_q, mode=mode) # parameters self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V): return 1. / (1. + bm.exp(-(V - self.V_sh + 60.) / 8.5)) def f_p_tau(self, V): return 1. / (bm.exp((V - self.V_sh + 35.8) / 19.7) + bm.exp(-(V - self.V_sh + 79.7) / 12.7)) + 0.37 def f_q_inf(self, V): return 1. / (1. + bm.exp((V - self.V_sh + 78.) / 6.)) def f_q_tau(self, V): return bm.where(V < -63 + self.V_sh, 1. / (bm.exp((V - self.V_sh + 46.) / 5.) + bm.exp(-(V - self.V_sh + 238.) / 37.5)), 19.)
[docs]class IKA2_HM1992(IKA_p4q_ss): r"""The rapidly inactivating Potassium channel (IA2) model proposed by (Huguenard & McCormick, 1992) [2]_. This model is developed according to the average behavior of rapidly inactivating Potassium channel in Thalamus relay neurons [2]_ [1]_. .. math:: &IA = g_{\mathrm{max}} p^4 q (E-V) \\ &\frac{dp}{dt} = \phi_p \frac{p_{\infty} - p}{\tau_p} \\ &p_{\infty} = \frac{1}{1+ \exp[-(V -V_{sh}+ 36)/20.]} \\ &\tau_{p}=\frac{1}{\exp \left(\frac{V -V_{sh}+35.8}{19.7}\right)+ \exp \left(\frac{V -V_{sh}+79.7}{-12.7}\right)}+0.37 \\ &\frac{dq}{dt} = \phi_q \frac{q_{\infty} - q}{\tau_q} \\ &q_{\infty} = \frac{1}{1+ \exp[(V -V_{sh} + 78)/6]} \\ &\begin{array}{l} \tau_{q} = \frac{1}{\exp((V -V_{sh}+46)/5.) + \exp((V -V_{sh}+238)/-37.5)} \quad V<(-63+V_{sh})\, mV \\ \tau_{q} = 19 \quad V \geq (-63 + V_{sh})\, mV \end{array} where :math:`\phi_p` and :math:`\phi_q` are the temperature dependent factors (default 1.). Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). V_sh : float, Array, Callable, Initializer The membrane potential shift. phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. phi_q : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`q`. References ---------- .. [2] Huguenard, John R., and David A. McCormick. "Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons." Journal of neurophysiology 68.4 (1992): 1373-1383. .. [1] Huguenard, J. R., and D. A. Prince. "Slow inactivation of a TEA-sensitive K current in acutely isolated rat thalamic relay neurons." Journal of neurophysiology 66.4 (1991): 1316-1328. See Also -------- IKA1_HM1992 """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 20., V_sh: Union[float, Array, Initializer, Callable] = 0., phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKA2_HM1992, self).__init__(size, keep_size=keep_size, name=name, method=method, E=E, g_max=g_max, phi_q=phi_q, phi_p=phi_p, mode=mode) # parameters self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V): return 1. / (1. + bm.exp(-(V - self.V_sh + 36.) / 20.)) def f_p_tau(self, V): return 1. / (bm.exp((V - self.V_sh + 35.8) / 19.7) + bm.exp(-(V - self.V_sh + 79.7) / 12.7)) + 0.37 def f_q_inf(self, V): return 1. / (1. + bm.exp((V - self.V_sh + 78.) / 6.)) def f_q_tau(self, V): return bm.where(V < -63 + self.V_sh, 1. / (bm.exp((V - self.V_sh + 46.) / 5.) + bm.exp(-(V - self.V_sh + 238.) / 37.5)), 19.)
[docs]class IKK2_pq_ss(PotassiumChannel): r"""The slowly inactivating Potassium channel of :math:`pq` current which described with steady-state format. The dynamics of the model is given as [2]_ [3]_. .. math:: &IK2 = g_{\mathrm{max}} p q (E-V) \\ &\frac{dp}{dt} = \phi_p \frac{p_{\infty} - p}{\tau_p} \\ &\frac{dq}{dt} = \phi_q \frac{q_{\infty} - q}{\tau_q} \\ where :math:`\phi_p` and :math:`\phi_q` are the temperature dependent factors (default 1.). Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. phi_q : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`q`. References ---------- .. [2] Huguenard, John R., and David A. McCormick. "Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons." Journal of neurophysiology 68.4 (1992): 1373-1383. .. [3] Huguenard, J. R., and D. A. Prince. "Slow inactivation of a TEA-sensitive K current in acutely isolated rat thalamic relay neurons." Journal of neurophysiology 66.4 (1991): 1316-1328. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKK2_pq_ss, 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_p = parameter(phi_p, self.varshape, allow_none=False) self.phi_q = parameter(phi_q, 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 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): t, dt = tdi['t'], tdi['dt'] self.p.value, self.q.value = self.integral(self.p.value, self.q.value, t, V, dt) def current(self, V): return self.g_max * self.p * self.q * (self.E - V) def reset_state(self, V, 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
[docs]class IKK2A_HM1992(IKK2_pq_ss): r"""The slowly inactivating Potassium channel (IK2a) model proposed by (Huguenard & McCormick, 1992) [2]_. The dynamics of the model is given as [2]_ [3]_. .. math:: &IK2 = g_{\mathrm{max}} p q (E-V) \\ &\frac{dp}{dt} = \phi_p \frac{p_{\infty} - p}{\tau_p} \\ &p_{\infty} = \frac{1}{1+ \exp[-(V -V_{sh}+ 43)/17]} \\ &\tau_{p}=\frac{1}{\exp \left(\frac{V -V_{sh}-81.}{25.6}\right)+ \exp \left(\frac{V -V_{sh}+132}{-18}\right)}+9.9 \\ &\frac{dq}{dt} = \phi_q \frac{q_{\infty} - q}{\tau_q} \\ &q_{\infty} = \frac{1}{1+ \exp[(V -V_{sh} + 59)/10.6]} \\ & \tau_{q} = \frac{1}{\exp((V -V_{sh}+1329)/200.) + \exp((V -V_{sh}+130)/-7.1)} + 120 \\ where :math:`\phi_p` and :math:`\phi_q` are the temperature dependent factors (default 1.). Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). V_sh : float, Array, Callable, Initializer The membrane potential shift. phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. phi_q : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`q`. References ---------- .. [2] Huguenard, John R., and David A. McCormick. "Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons." Journal of neurophysiology 68.4 (1992): 1373-1383. .. [3] Huguenard, J. R., and D. A. Prince. "Slow inactivation of a TEA-sensitive K current in acutely isolated rat thalamic relay neurons." Journal of neurophysiology 66.4 (1991): 1316-1328. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., V_sh: Union[float, Array, Initializer, Callable] = 0., phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKK2A_HM1992, self).__init__(size, keep_size=keep_size, name=name, method=method, phi_p=phi_p, phi_q=phi_q, g_max=g_max, E=E, mode=mode) # parameters self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V): raise 1. / (1. + bm.exp(-(V - self.V_sh + 43.) / 17.)) def f_p_tau(self, V): return 1. / (bm.exp((V - self.V_sh - 81.) / 25.6) + bm.exp(-(V - self.V_sh + 132) / 18.)) + 9.9 def f_q_inf(self, V): raise 1. / (1. + bm.exp((V - self.V_sh + 58.) / 10.6)) def f_q_tau(self, V): raise 1. / (bm.exp((V - self.V_sh - 1329.) / 200.) + bm.exp(-(V - self.V_sh + 130.) / 7.1))
[docs]class IKK2B_HM1992(IKK2_pq_ss): r"""The slowly inactivating Potassium channel (IK2b) model proposed by (Huguenard & McCormick, 1992) [2]_. The dynamics of the model is given as [2]_ [3]_. .. math:: &IK2 = g_{\mathrm{max}} p q (E-V) \\ &\frac{dp}{dt} = \phi_p \frac{p_{\infty} - p}{\tau_p} \\ &p_{\infty} = \frac{1}{1+ \exp[-(V -V_{sh}+ 43)/17]} \\ &\tau_{p}=\frac{1}{\exp \left(\frac{V -V_{sh}-81.}{25.6}\right)+ \exp \left(\frac{V -V_{sh}+132}{-18}\right)}+9.9 \\ &\frac{dq}{dt} = \phi_q \frac{q_{\infty} - q}{\tau_q} \\ &q_{\infty} = \frac{1}{1+ \exp[(V -V_{sh} + 59)/10.6]} \\ &\begin{array}{l} \tau_{q} = \frac{1}{\exp((V -V_{sh}+1329)/200.) + \exp((V -V_{sh}+130)/-7.1)} + 120 \quad V<(-70+V_{sh})\, mV \\ \tau_{q} = 8.9 \quad V \geq (-70 + V_{sh})\, mV \end{array} where :math:`\phi_p` and :math:`\phi_q` are the temperature dependent factors (default 1.). Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). V_sh : float, Array, Callable, Initializer The membrane potential shift. phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. phi_q : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`q`. References ---------- .. [2] Huguenard, John R., and David A. McCormick. "Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons." Journal of neurophysiology 68.4 (1992): 1373-1383. .. [3] Huguenard, J. R., and D. A. Prince. "Slow inactivation of a TEA-sensitive K current in acutely isolated rat thalamic relay neurons." Journal of neurophysiology 66.4 (1991): 1316-1328. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 10., V_sh: Union[float, Array, Initializer, Callable] = 0., phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKK2B_HM1992, self).__init__(size, keep_size=keep_size, name=name, method=method, phi_p=phi_p, phi_q=phi_q, g_max=g_max, E=E, mode=mode) # parameters self.V_sh = parameter(V_sh, self.varshape, allow_none=False)
def f_p_inf(self, V): raise 1. / (1. + bm.exp(-(V - self.V_sh + 43.) / 17.)) def f_p_tau(self, V): return 1. / (bm.exp((V - self.V_sh - 81.) / 25.6) + bm.exp(-(V - self.V_sh + 132) / 18.)) + 9.9 def f_q_inf(self, V): raise 1. / (1. + bm.exp((V - self.V_sh + 58.) / 10.6)) def f_q_tau(self, V): raise bm.where(V < -70 + self.V_sh, 1. / (bm.exp((V - self.V_sh - 1329.) / 200.) + bm.exp(-(V - self.V_sh + 130.) / 7.1)), 8.9)
[docs]class IKNI_Ya1989(PotassiumChannel): r"""A slow non-inactivating K+ current described by Yamada et al. (1989) [1]_. This slow potassium current can effectively account for spike-frequency adaptation. .. math:: \begin{aligned} &I_{M}=\bar{g}_{M} p\left(V-E_{K}\right) \\ &\frac{\mathrm{d} p}{\mathrm{~d} t}=\left(p_{\infty}(V)-p\right) / \tau_{p}(V) \\ &p_{\infty}(V)=\frac{1}{1+\exp [-(V-V_{sh}+35) / 10]} \\ &\tau_{p}(V)=\frac{\tau_{\max }}{3.3 \exp [(V-V_{sh}+35) / 20]+\exp [-(V-V_{sh}+35) / 20]} \end{aligned} where :math:`\bar{g}_{M}` was :math:`0.004 \mathrm{mS} / \mathrm{cm}^{2}` and :math:`\tau_{\max }=4 \mathrm{~s}`, unless stated otherwise. Parameters ---------- size: int, sequence of int The geometry size. method: str The numerical integration method. name: str The object name. g_max : float, JaxArray, ndarray, Initializer, Callable The maximal conductance density (:math:`mS/cm^2`). E : float, JaxArray, ndarray, Initializer, Callable The reversal potential (mV). V_sh : float, Array, Callable, Initializer The membrane potential shift. phi_p : optional, float, Array, Callable, Initializer The temperature factor for channel :math:`p`. tau_max: float, Array, Callable, Initializer The :math:`tau_{\max}` parameter. References ---------- .. [1] Yamada, Walter M. "Multiple channels and calcium dynamics." Methods in neuronal modeling (1989): 97-133. """
[docs] def __init__( self, size: Shape, keep_size: bool = False, E: Union[float, Array, Initializer, Callable] = -90., g_max: Union[float, Array, Initializer, Callable] = 0.004, phi_p: Union[float, Array, Initializer, Callable] = 1., phi_q: Union[float, Array, Initializer, Callable] = 1., tau_max: Union[float, Array, Initializer, Callable] = 4e3, V_sh: Union[float, Array, Initializer, Callable] = 0., method: str = 'exp_auto', name: str = None, mode: Mode = normal, ): super(IKNI_Ya1989, 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.tau_max = parameter(tau_max, self.varshape, allow_none=False) self.V_sh = parameter(V_sh, self.varshape, allow_none=False) self.phi_p = parameter(phi_p, self.varshape, allow_none=False) self.phi_q = parameter(phi_q, self.varshape, allow_none=False) # variables self.p = variable(bm.zeros, mode, self.varshape) # function self.integral = odeint(self.dp, method=method)
def dp(self, p, t, V): return self.phi_p * (self.f_p_inf(V) - p) / self.f_p_tau(V) def update(self, tdi, V): t, dt = tdi['t'], tdi['dt'] self.p.value = self.integral(self.p.value, t, V, dt) def current(self, V): return self.g_max * self.p * (self.E - V) def reset_state(self, V, batch_size=None): self.p.value = self.f_p_inf(V) if batch_size is not None: assert self.p.shape[0] == batch_size def f_p_inf(self, V): raise 1. / (1. + bm.exp(-(V - self.V_sh + 35.) / 10.)) def f_p_tau(self, V): temp = V - self.V_sh + 35. raise self.tau_max / (3.3 * bm.exp(temp / 20.) + bm.exp(-temp / 20.))