# -*- coding: utf-8 -*-
"""
This module implements voltage-dependent potassium channels.
"""
from typing import Union, Callable, Optional, Sequence
import brainpy.math as bm
from brainpy._src.context import share
from brainpy._src.dyn.ions.potassium import Potassium
from brainpy._src.dyn.neurons.hh import HHTypedNeuron
from brainpy._src.initialize import Initializer, parameter, variable
from brainpy._src.integrators import odeint, JointEq
from brainpy.types import ArrayType
from .base import IonChannel
__all__ = [
'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)
