# -*- 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.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
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# ==============================================================================
"""
This module implements voltage-dependent potassium channels.
"""
from typing import Union, Callable, Optional, Sequence
import brainpy.math as bm
from brainpy.context import share
from brainpy.dyn.ions.potassium import Potassium
from brainpy.dyn.neurons.hh import HHTypedNeuron
from brainpy.initialize import Initializer, parameter, variable
from brainpy.integrators import odeint, JointEq
from brainpy.types import ArrayType
from .base import IonChannel
__all__ = [
'PotassiumChannel',
'IKDR_Ba2002v2',
'IK_TM1991v2',
'IK_HH1952v2',
'IKA1_HM1992v2',
'IKA2_HM1992v2',
'IKK2A_HM1992v2',
'IKK2B_HM1992v2',
'IKNI_Ya1989v2',
'IK_Leak',
]
[docs]
class PotassiumChannel(IonChannel):
"""Base class for sodium channel dynamics."""
master_type = Potassium
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 _IK_p4_markov_v2(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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
phi : float, ArrayType, Initializer, Callable
The temperature-dependent factor.
method: str
The numerical integration method.
name: str
The object name.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
mode=mode)
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, self.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, V, C, E):
self.p.value = self.integral(self.p.value, share['t'], V, share['dt'])
def current(self, V, C, E):
return self.g_max * self.p ** 4 * (E - V)
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)
if isinstance(batch_size, int):
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_Ba2002v2(_IK_p4_markov_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
T_base : float, ArrayType
The brainpy_object of temperature factor.
T : float, ArrayType, Initializer, Callable
The temperature (Celsius, :math:`^{\circ}C`).
V_sh : float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
V_sh: Union[float, ArrayType, Initializer, Callable] = -50.,
T_base: Union[float, ArrayType] = 3.,
T: Union[float, ArrayType] = 36.,
phi: Optional[Union[float, ArrayType, Initializer, Callable]] = None,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
phi = T_base ** ((T - 36) / 10) if phi is None else phi
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
g_max=g_max,
phi=phi,
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_TM1991v2(_IK_p4_markov_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi: Union[float, ArrayType, Initializer, Callable] = 1.,
V_sh: Union[int, float, ArrayType, Initializer, Callable] = -60.,
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):
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_HH1952v2(_IK_p4_markov_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi: Union[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 + 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)
class _IKA_p4q_ss_v2(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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
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, 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 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, V, C, E):
self.p.value, self.q.value = self.integral(self.p.value, self.q.value, share['t'], V, share['dt'])
def current(self, V, C, E):
return self.g_max * self.p ** 4 * self.q * (E - V)
def reset_state(self, V, C, E, batch_size=None):
self.p.value = self.f_p_inf(V)
self.q.value = self.f_q_inf(V)
if isinstance(batch_size, int):
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_HM1992v2(_IKA_p4q_ss_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 30.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
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_HM1992v2(_IKA_p4q_ss_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 20.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
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.)
class _IKK2_pq_ss_v2(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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
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, 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 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, V, C, E):
self.p.value, self.q.value = self.integral(self.p.value, self.q.value, share['t'], V, share['dt'])
def current(self, V, C, E):
return self.g_max * self.p * self.q * (E - V)
def reset_state(self, V, C, E, batch_size=None):
self.p.value = self.f_p_inf(V)
self.q.value = self.f_q_inf(V)
if isinstance(batch_size, int):
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_HM1992v2(_IKK2_pq_ss_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
phi_p=phi_p,
phi_q=phi_q,
g_max=g_max,
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 + 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):
return 1. / (1. + bm.exp((V - self.V_sh + 58.) / 10.6))
def f_q_tau(self, V):
return 1. / (bm.exp((V - self.V_sh - 1329.) / 200.) +
bm.exp(-(V - self.V_sh + 130.) / 7.1))
[docs]
class IKK2B_HM1992v2(_IKK2_pq_ss_v2):
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
method=method,
phi_p=phi_p,
phi_q=phi_q,
g_max=g_max,
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 + 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):
return 1. / (1. + bm.exp((V - self.V_sh + 58.) / 10.6))
def f_q_tau(self, V):
return 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_Ya1989v2(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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
tau_max: float, ArrayType, Callable, Initializer
The :math:`tau_{\max}` parameter.
References::
.. [1] Yamada, Walter M. "Multiple channels and calcium dynamics." Methods in neuronal modeling (1989): 97-133.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[float, ArrayType, Initializer, Callable] = 0.004,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
tau_max: Union[float, ArrayType, Initializer, Callable] = 4e3,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size,
keep_size=keep_size,
name=name,
mode=mode)
# parameters
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, self.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, V, C, E):
self.p.value = self.integral(self.p.value, share['t'], V, share['dt'])
def current(self, V, C, E):
return self.g_max * self.p * (E - V)
def reset_state(self, V, C, E, batch_size=None):
self.p.value = self.f_p_inf(V)
if isinstance(batch_size, int):
assert self.p.shape[0] == batch_size
def f_p_inf(self, V):
return 1. / (1. + bm.exp(-(V - self.V_sh + 35.) / 10.))
def f_p_tau(self, V):
temp = V - self.V_sh + 35.
return self.tau_max / (3.3 * bm.exp(temp / 20.) + bm.exp(-temp / 20.))
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
phi : float, ArrayType, Initializer, Callable
The temperature-dependent factor.
method: str
The numerical integration method.
name: str
The object name.
"""
master_type = HHTypedNeuron
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__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, self.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, V):
self.p.value = self.integral(self.p.value, share['t'], V, share['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 isinstance(batch_size, int):
assert self.p.shape[0] == batch_size
def f_p_alpha(self, V):
raise NotImplementedError
def f_p_beta(self, V):
raise NotImplementedError
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
T_base : float, ArrayType
The brainpy_object of temperature factor.
T : float, ArrayType, Initializer, Callable
The temperature (Celsius, :math:`^{\circ}C`).
V_sh : float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
V_sh: Union[float, ArrayType, Initializer, Callable] = -50.,
T_base: Union[float, ArrayType] = 3.,
T: Union[float, ArrayType] = 36.,
phi: Optional[Union[float, ArrayType, Initializer, Callable]] = None,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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.)
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi: Union[float, ArrayType, Initializer, Callable] = 1.,
V_sh: Union[int, float, ArrayType, Initializer, Callable] = -60.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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)
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi: Union[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(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)
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
master_type = HHTypedNeuron
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__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, 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 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, V):
self.p.value, self.q.value = self.integral(self.p.value, self.q.value, share['t'], V, share['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 isinstance(batch_size, int):
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
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 30.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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.)
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 20.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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.)
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
master_type = HHTypedNeuron
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
super().__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, 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 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, V):
self.p.value, self.q.value = self.integral(self.p.value, self.q.value, share['t'], V, share['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 isinstance(batch_size, int):
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
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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):
return 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):
return 1. / (1. + bm.exp((V - self.V_sh + 58.) / 10.6))
def f_q_tau(self, V):
return 1. / (bm.exp((V - self.V_sh - 1329.) / 200.) +
bm.exp(-(V - self.V_sh + 130.) / 7.1))
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
phi_q : optional, float, ArrayType, 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.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 10.,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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):
return 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):
return 1. / (1. + bm.exp((V - self.V_sh + 58.) / 10.6))
def f_q_tau(self, V):
return 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)
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, ArrayType, Initializer, Callable
The maximal conductance density (:math:`mS/cm^2`).
E : float, ArrayType, Initializer, Callable
The reversal potential (mV).
V_sh : float, ArrayType, Callable, Initializer
The membrane potential shift.
phi_p : optional, float, ArrayType, Callable, Initializer
The temperature factor for channel :math:`p`.
tau_max: float, ArrayType, Callable, Initializer
The :math:`tau_{\max}` parameter.
References::
.. [1] Yamada, Walter M. "Multiple channels and calcium dynamics." Methods in neuronal modeling (1989): 97-133.
"""
master_type = HHTypedNeuron
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
E: Union[float, ArrayType, Initializer, Callable] = -90.,
g_max: Union[float, ArrayType, Initializer, Callable] = 0.004,
phi_p: Union[float, ArrayType, Initializer, Callable] = 1.,
phi_q: Union[float, ArrayType, Initializer, Callable] = 1.,
tau_max: Union[float, ArrayType, Initializer, Callable] = 4e3,
V_sh: Union[float, ArrayType, Initializer, Callable] = 0.,
method: str = 'exp_auto',
name: str = None,
mode: bm.Mode = None,
):
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, self.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, V):
self.p.value = self.integral(self.p.value, share['t'], V, share['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 isinstance(batch_size, int):
assert self.p.shape[0] == batch_size
def f_p_inf(self, V):
return 1. / (1. + bm.exp(-(V - self.V_sh + 35.) / 10.))
def f_p_tau(self, V):
temp = V - self.V_sh + 35.
return self.tau_max / (3.3 * bm.exp(temp / 20.) + bm.exp(-temp / 20.))
[docs]
class IK_Leak(PotassiumChannel):
"""The potassium leak channel current.
Parameters::
g_max : float
The potassium leakage conductance which is modulated by both
acetylcholine and norepinephrine.
"""
def __init__(
self,
size: Union[int, Sequence[int]],
keep_size: bool = False,
g_max: Union[int, float, ArrayType, Initializer, Callable] = 0.005,
method: str = None,
name: str = None,
mode: bm.Mode = None,
):
super().__init__(size=size,
keep_size=keep_size,
method=method,
name=name,
mode=mode)
self.g_max = self.init_param(g_max, self.varshape)
def reset_state(self, V, C, E, batch_size: int = None):
pass
def update(self, V, C, E):
pass
def current(self, V, C, E):
return self.g_max * (E - V)
