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thalamic.py
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#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Author: Theo Lemaire
# @Date: 2017-07-31 15:20:54
# @Email: theo.lemaire@epfl.ch
# @Last Modified by: Theo Lemaire
# @Last Modified time: 2018-09-28 14:06:20
import numpy as np
from ..core import PointNeuron
class Thalamic(PointNeuron):
''' Class defining the generic membrane channel dynamics of a thalamic neuron
with 4 different current types:
- Inward Sodium current
- Outward Potassium current
- Inward Calcium current
- Non-specific leakage current
This generic class cannot be used directly as it does not contain any specific parameters.
Reference:
*Plaksin, M., Kimmel, E., and Shoham, S. (2016). Cell-Type-Selective Effects of
Intramembrane Cavitation as a Unifying Theoretical Framework for Ultrasonic
Neuromodulation. eNeuro 3.*
'''
# Generic biophysical parameters of thalamic cells
Cm0 = 1e-2 # Cell membrane resting capacitance (F/m2)
Vm0 = 0.0 # Dummy value for membrane potential (mV)
VNa = 50.0 # Sodium Nernst potential (mV)
VK = -90.0 # Potassium Nernst potential (mV)
VCa = 120.0 # Calcium Nernst potential (mV)
def __init__(self):
''' Constructor of the class '''
# Names and initial states of the channels state probabilities
self.states_names = ['m', 'h', 'n', 's', 'u']
self.states0 = np.array([])
# Names of the different coefficients to be averaged in a lookup table.
self.coeff_names = ['alpham', 'betam', 'alphah', 'betah', 'alphan', 'betan',
'alphas', 'betas', 'alphau', 'betau']
# Charge interval bounds for lookup creation
self.Qbounds = (np.round(self.Vm0 - 25.0) * 1e-5, 50.0e-5)
def alpham(self, Vm):
''' Compute the alpha rate for the open-probability of Sodium channels.
:param Vm: membrane potential (mV)
:return: rate constant (s-1)
'''
Vdiff = Vm - self.VT
alpha = 0.32 * self.vtrap(13 - Vdiff, 4) # ms-1
return alpha * 1e3 # s-1
def betam(self, Vm):
''' Compute the beta rate for the open-probability of Sodium channels.
:param Vm: membrane potential (mV)
:return: rate constant (s-1)
'''
Vdiff = Vm - self.VT
beta = 0.28 * self.vtrap(Vdiff - 40, 5) # ms-1
return beta * 1e3 # s-1
def alphah(self, Vm):
''' Compute the alpha rate for the inactivation-probability of Sodium channels.
:param Vm: membrane potential (mV)
:return: rate constant (s-1)
'''
Vdiff = Vm - self.VT
alpha = (0.128 * np.exp(-(Vdiff - 17) / 18)) # ms-1
return alpha * 1e3 # s-1
def betah(self, Vm):
''' Compute the beta rate for the inactivation-probability of Sodium channels.
:param Vm: membrane potential (mV)
:return: rate constant (s-1)
'''
Vdiff = Vm - self.VT
beta = (4 / (1 + np.exp(-(Vdiff - 40) / 5))) # ms-1
return beta * 1e3 # s-1
def alphan(self, Vm):
''' Compute the alpha rate for the open-probability of delayed-rectifier Potassium channels.
:param Vm: membrane potential (mV)
:return: rate constant (s-1)
'''
Vdiff = Vm - self.VT
alpha = 0.032 * self.vtrap(15 - Vdiff, 5) # ms-1
return alpha * 1e3 # s-1
def betan(self, Vm):
''' Compute the beta rate for the open-probability of delayed-rectifier Potassium channels.
:param Vm: membrane potential (mV)
:return: rate constant (s-1)
'''
Vdiff = Vm - self.VT
beta = (0.5 * np.exp(-(Vdiff - 10) / 40)) # ms-1
return beta * 1e3 # s-1
def derM(self, Vm, m):
''' Compute the evolution of the open-probability of Sodium channels.
:param Vm: membrane potential (mV)
:param m: open-probability of Sodium channels (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return self.alpham(Vm) * (1 - m) - self.betam(Vm) * m
def derH(self, Vm, h):
''' Compute the evolution of the inactivation-probability of Sodium channels.
:param Vm: membrane potential (mV)
:param h: inactivation-probability of Sodium channels (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return self.alphah(Vm) * (1 - h) - self.betah(Vm) * h
def derN(self, Vm, n):
''' Compute the evolution of the open-probability of delayed-rectifier Potassium channels.
:param Vm: membrane potential (mV)
:param n: open-probability of delayed-rectifier Potassium channels (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return self.alphan(Vm) * (1 - n) - self.betan(Vm) * n
def derS(self, Vm, s):
''' Compute the evolution of the open-probability of the S-type,
activation gate of Calcium channels.
:param Vm: membrane potential (mV)
:param s: open-probability of S-type Calcium activation gates (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return (self.sinf(Vm) - s) / self.taus(Vm)
def derU(self, Vm, u):
''' Compute the evolution of the open-probability of the U-type,
inactivation gate of Calcium channels.
:param Vm: membrane potential (mV)
:param u: open-probability of U-type Calcium inactivation gates (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return (self.uinf(Vm) - u) / self.tauu(Vm)
def currNa(self, m, h, Vm):
''' Compute the inward Sodium current per unit area.
:param m: open-probability of Sodium channels
:param h: inactivation-probability of Sodium channels
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
GNa = self.GNaMax * m**3 * h
return GNa * (Vm - self.VNa)
def currK(self, n, Vm):
''' Compute the outward delayed-rectifier Potassium current per unit area.
:param n: open-probability of delayed-rectifier Potassium channels
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
GK = self.GKMax * n**4
return GK * (Vm - self.VK)
def currCa(self, s, u, Vm):
''' Compute the inward Calcium current per unit area.
:param s: open-probability of the S-type activation gate of Calcium channels
:param u: open-probability of the U-type inactivation gate of Calcium channels
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
GT = self.GTMax * s**2 * u
return GT * (Vm - self.VCa)
def currL(self, Vm):
''' Compute the non-specific leakage current per unit area.
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.GL * (Vm - self.VL)
def currNet(self, Vm, states):
''' Concrete implementation of the abstract API method. '''
m, h, n, s, u = states
return (self.currNa(m, h, Vm) + self.currK(n, Vm) +
self.currCa(s, u, Vm) + self.currL(Vm)) # mA/m2
def steadyStates(self, Vm):
''' Concrete implementation of the abstract API method. '''
# Solve the equation dx/dt = 0 at Vm for each x-state
meq = self.alpham(Vm) / (self.alpham(Vm) + self.betam(Vm))
heq = self.alphah(Vm) / (self.alphah(Vm) + self.betah(Vm))
neq = self.alphan(Vm) / (self.alphan(Vm) + self.betan(Vm))
seq = self.sinf(Vm)
ueq = self.uinf(Vm)
return np.array([meq, heq, neq, seq, ueq])
def derStates(self, Vm, states):
''' Concrete implementation of the abstract API method. '''
m, h, n, s, u = states
dmdt = self.derM(Vm, m)
dhdt = self.derH(Vm, h)
dndt = self.derN(Vm, n)
dsdt = self.derS(Vm, s)
dudt = self.derU(Vm, u)
return [dmdt, dhdt, dndt, dsdt, dudt]
def getEffRates(self, Vm):
''' Concrete implementation of the abstract API method. '''
# Compute average cycle value for rate constants
am_avg = np.mean(self.alpham(Vm))
bm_avg = np.mean(self.betam(Vm))
ah_avg = np.mean(self.alphah(Vm))
bh_avg = np.mean(self.betah(Vm))
an_avg = np.mean(self.alphan(Vm))
bn_avg = np.mean(self.betan(Vm))
Ts = self.taus(Vm)
as_avg = np.mean(self.sinf(Vm) / Ts)
bs_avg = np.mean(1 / Ts) - as_avg
Tu = np.array([self.tauu(v) for v in Vm])
au_avg = np.mean(self.uinf(Vm) / Tu)
bu_avg = np.mean(1 / Tu) - au_avg
# Return array of coefficients
return np.array([am_avg, bm_avg, ah_avg, bh_avg, an_avg, bn_avg,
as_avg, bs_avg, au_avg, bu_avg])
def derStatesEff(self, Qm, states, interp_data):
''' Concrete implementation of the abstract API method. '''
rates = np.array([np.interp(Qm, interp_data['Q'], interp_data[rn])
for rn in self.coeff_names])
m, h, n, s, u = states
dmdt = rates[0] * (1 - m) - rates[1] * m
dhdt = rates[2] * (1 - h) - rates[3] * h
dndt = rates[4] * (1 - n) - rates[5] * n
dsdt = rates[6] * (1 - s) - rates[7] * s
dudt = rates[8] * (1 - u) - rates[9] * u
return [dmdt, dhdt, dndt, dsdt, dudt]
class ThalamicRE(Thalamic):
''' Specific membrane channel dynamics of a thalamic reticular neuron.
References:
*Destexhe, A., Contreras, D., Steriade, M., Sejnowski, T.J., and Huguenard, J.R. (1996).
In vivo, in vitro, and computational analysis of dendritic calcium currents in thalamic
reticular neurons. J. Neurosci. 16, 169–185.*
*Huguenard, J.R., and Prince, D.A. (1992). A novel T-type current underlies prolonged
Ca(2+)-dependent burst firing in GABAergic neurons of rat thalamic reticular nucleus.
J. Neurosci. 12, 3804–3817.*
'''
# Name of channel mechanism
name = 'RE'
# Cell-specific biophysical parameters
Vm0 = -89.5 # Cell membrane resting potential (mV)
GNaMax = 2000.0 # Max. conductance of Sodium current (S/m^2)
GKMax = 200.0 # Max. conductance of Potassium current (S/m^2)
GTMax = 30.0 # Max. conductance of low-threshold Calcium current (S/m^2)
GL = 0.5 # Conductance of non-specific leakage current (S/m^2)
VL = -90.0 # Non-specific leakage Nernst potential (mV)
VT = -67.0 # Spike threshold adjustment parameter (mV)
# Default plotting scheme
pltvars_scheme = {
'i_{Na}\ kin.': ['m', 'h', 'm3h'],
'i_K\ kin.': ['n'],
'i_{TS}\ kin.': ['s', 'u', 's2u'],
'I': ['iNa', 'iK', 'iTs', 'iL', 'iNet']
}
def __init__(self):
''' Constructor of the class. '''
# Instantiate parent class
super().__init__()
# Define initial channel probabilities (solving dx/dt = 0 at resting potential)
self.states0 = self.steadyStates(self.Vm0)
def sinf(self, Vm):
''' Compute the asymptotic value of the open-probability of the S-type,
activation gate of Calcium channels.
:param Vm: membrane potential (mV)
:return: asymptotic probability (-)
'''
return 1.0 / (1.0 + np.exp(-(Vm + 52.0) / 7.4)) # prob
def taus(self, Vm):
''' Compute the decay time constant for adaptation of S-type,
activation gate of Calcium channels.
:param Vm: membrane potential (mV)
:return: decayed time constant (s)
'''
return (1 + 0.33 / (np.exp((Vm + 27.0) / 10.0) + np.exp(-(Vm + 102.0) / 15.0))) * 1e-3 # s
def uinf(self, Vm):
''' Compute the asymptotic value of the open-probability of the U-type,
inactivation gate of Calcium channels.
:param Vm: membrane potential (mV)
:return: asymptotic probability (-)
'''
return 1.0 / (1.0 + np.exp((Vm + 80.0) / 5.0)) # prob
def tauu(self, Vm):
''' Compute the decay time constant for adaptation of U-type,
inactivation gate of Calcium channels.
:param Vm: membrane potential (mV)
:return: decayed time constant (s)
'''
return (28.3 + 0.33 / (np.exp((Vm + 48.0) / 4.0) + np.exp(-(Vm + 407.0) / 50.0))) * 1e-3 # s
class ThalamoCortical(Thalamic):
''' Specific membrane channel dynamics of a thalamo-cortical neuron, with a specific
hyperpolarization-activated, mixed cationic current and a leakage Potassium current.
References:
*Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Frégnac, Y.,
Markram, H., and Destexhe, A. (2008). Minimal Hodgkin-Huxley type models for different
classes of cortical and thalamic neurons. Biol Cybern 99, 427–441.*
*Destexhe, A., Bal, T., McCormick, D.A., and Sejnowski, T.J. (1996). Ionic mechanisms
underlying synchronized oscillations and propagating waves in a model of ferret
thalamic slices. J. Neurophysiol. 76, 2049–2070.*
*McCormick, D.A., and Huguenard, J.R. (1992). A model of the electrophysiological
properties of thalamocortical relay neurons. J. Neurophysiol. 68, 1384–1400.*
'''
# Name of channel mechanism
name = 'TC'
# Cell-specific biophysical parameters
# Vm0 = -63.4 # Cell membrane resting potential (mV)
Vm0 = -61.93 # Cell membrane resting potential (mV)
GNaMax = 900.0 # Max. conductance of Sodium current (S/m^2)
GKMax = 100.0 # Max. conductance of Potassium current (S/m^2)
GTMax = 20.0 # Max. conductance of low-threshold Calcium current (S/m^2)
GKL = 0.138 # Conductance of leakage Potassium current (S/m^2)
GhMax = 0.175 # Max. conductance of mixed cationic current (S/m^2)
GL = 0.1 # Conductance of non-specific leakage current (S/m^2)
Vh = -40.0 # Mixed cationic current reversal potential (mV)
VL = -70.0 # Non-specific leakage Nernst potential (mV)
VT = -52.0 # Spike threshold adjustment parameter (mV)
Vx = 0.0 # Voltage-dependence uniform shift factor at 36°C (mV)
tau_Ca_removal = 5e-3 # decay time constant for intracellular Ca2+ dissolution (s)
CCa_min = 50e-9 # minimal intracellular Calcium concentration (M)
deff = 100e-9 # effective depth beneath membrane for intracellular [Ca2+] calculation
F_Ca = 1.92988e5 # Faraday constant for bivalent ion (Coulomb / mole)
nCa = 4 # number of Calcium binding sites on regulating factor
k1 = 2.5e22 # intracellular Ca2+ regulation factor (M-4 s-1)
k2 = 0.4 # intracellular Ca2+ regulation factor (s-1)
k3 = 100.0 # intracellular Ca2+ regulation factor (s-1)
k4 = 1.0 # intracellular Ca2+ regulation factor (s-1)
# Default plotting scheme
pltvars_scheme = {
'i_{Na}\ kin.': ['m', 'h'],
'i_K\ kin.': ['n'],
'i_{T}\ kin.': ['s', 'u'],
'i_{H}\ kin.': ['O', 'OL', 'O + 2OL'],
'I': ['iNa', 'iK', 'iT', 'iH', 'iKL', 'iL', 'iNet']
}
def __init__(self):
''' Constructor of the class. '''
# Instantiate parent class
super().__init__()
# Compute current to concentration conversion constant
self.iT_2_CCa = 1e-6 / (self.deff * self.F_Ca)
# Define names of the channels state probabilities
self.states_names += ['O', 'C', 'P0', 'C_Ca']
# Define the names of the different coefficients to be averaged in a lookup table.
self.coeff_names += ['alphao', 'betao']
# Define initial channel probabilities (solving dx/dt = 0 at resting potential)
self.states0 = self.steadyStates(self.Vm0)
def sinf(self, Vm):
''' Compute the asymptotic value of the open-probability of the S-type,
activation gate of Calcium channels.
Reference:
*Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Frégnac, Y.,
Markram, H., and Destexhe, A. (2008). Minimal Hodgkin-Huxley type models for different
classes of cortical and thalamic neurons. Biol Cybern 99, 427–441.*
:param Vm: membrane potential (mV)
:return: asymptotic probability (-)
'''
return 1.0 / (1.0 + np.exp(-(Vm + self.Vx + 57.0) / 6.2)) # prob
def taus(self, Vm):
''' Compute the decay time constant for adaptation of S-type,
activation gate of Calcium channels.
Reference:
*Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Frégnac, Y.,
Markram, H., and Destexhe, A. (2008). Minimal Hodgkin-Huxley type models for different
classes of cortical and thalamic neurons. Biol Cybern 99, 427–441.*
:param Vm: membrane potential (mV)
:return: decayed time constant (s)
'''
tmp = np.exp(-(Vm + self.Vx + 132.0) / 16.7) + np.exp((Vm + self.Vx + 16.8) / 18.2)
return 1.0 / 3.7 * (0.612 + 1.0 / tmp) * 1e-3 # s
def uinf(self, Vm):
''' Compute the asymptotic value of the open-probability of the U-type,
inactivation gate of Calcium channels.
Reference:
*Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Frégnac, Y.,
Markram, H., and Destexhe, A. (2008). Minimal Hodgkin-Huxley type models for different
classes of cortical and thalamic neurons. Biol Cybern 99, 427–441.*
:param Vm: membrane potential (mV)
:return: asymptotic probability (-)
'''
return 1.0 / (1.0 + np.exp((Vm + self.Vx + 81.0) / 4.0)) # prob
def tauu(self, Vm):
''' Compute the decay time constant for adaptation of U-type,
inactivation gate of Calcium channels.
Reference:
*Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Frégnac, Y.,
Markram, H., and Destexhe, A. (2008). Minimal Hodgkin-Huxley type models for different
classes of cortical and thalamic neurons. Biol Cybern 99, 427–441.*
:param Vm: membrane potential (mV)
:return: decayed time constant (s)
'''
if Vm + self.Vx < -80.0:
return 1.0 / 3.7 * np.exp((Vm + self.Vx + 467.0) / 66.6) * 1e-3 # s
else:
return 1 / 3.7 * (np.exp(-(Vm + self.Vx + 22) / 10.5) + 28.0) * 1e-3 # s
def derS(self, Vm, s):
''' Compute the evolution of the open-probability of the S-type,
activation gate of Calcium channels.
:param Vm: membrane potential (mV)
:param s: open-probability of S-type Calcium activation gates (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return (self.sinf(Vm) - s) / self.taus(Vm)
def derU(self, Vm, u):
''' Compute the evolution of the open-probability of the U-type,
inactivation gate of Calcium channels.
:param Vm: membrane potential (mV)
:param u: open-probability of U-type Calcium inactivation gates (prob)
:return: derivative of open-probability w.r.t. time (prob/s)
'''
return (self.uinf(Vm) - u) / self.tauu(Vm)
def oinf(self, Vm):
''' Voltage-dependent steady-state activation of hyperpolarization-activated
cation current channels.
Reference:
*Huguenard, J.R., and McCormick, D.A. (1992). Simulation of the currents involved in
rhythmic oscillations in thalamic relay neurons. J. Neurophysiol. 68, 1373–1383.*
:param Vm: membrane potential (mV)
:return: steady-state activation (-)
'''
return 1.0 / (1.0 + np.exp((Vm + 75.0) / 5.5))
def tauo(self, Vm):
''' Time constant for activation of hyperpolarization-activated cation current channels.
Reference:
*Huguenard, J.R., and McCormick, D.A. (1992). Simulation of the currents involved in
rhythmic oscillations in thalamic relay neurons. J. Neurophysiol. 68, 1373–1383.*
:param Vm: membrane potential (mV)
:return: time constant (s)
'''
return 1 / (np.exp(-14.59 - 0.086 * Vm) + np.exp(-1.87 + 0.0701 * Vm)) * 1e-3
def alphao(self, Vm):
''' Transition rate between closed and open form of hyperpolarization-activated
cation current channels.
:param Vm: membrane potential (mV)
:return: transition rate (s-1)
'''
return self.oinf(Vm) / self.tauo(Vm)
def betao(self, Vm):
''' Transition rate between open and closed form of hyperpolarization-activated
cation current channels.
:param Vm: membrane potential (mV)
:return: transition rate (s-1)
'''
return (1 - self.oinf(Vm)) / self.tauo(Vm)
def derC(self, C, O, Vm):
''' Compute the evolution of the proportion of hyperpolarization-activated
cation current channels in closed state.
Kinetics scheme of Calcium dependent activation derived from:
*Destexhe, A., Bal, T., McCormick, D.A., and Sejnowski, T.J. (1996). Ionic mechanisms
underlying synchronized oscillations and propagating waves in a model of ferret
thalamic slices. J. Neurophysiol. 76, 2049–2070.*
:param Vm: membrane potential (mV)
:param C: proportion of Ih channels in closed state (-)
:param O: proportion of Ih channels in open state (-)
:return: derivative of proportion w.r.t. time (s-1)
'''
return self.betao(Vm) * O - self.alphao(Vm) * C
def derO(self, C, O, P0, Vm):
''' Compute the evolution of the proportion of hyperpolarization-activated
cation current channels in open state.
Kinetics scheme of Calcium dependent activation derived from:
*Destexhe, A., Bal, T., McCormick, D.A., and Sejnowski, T.J. (1996). Ionic mechanisms
underlying synchronized oscillations and propagating waves in a model of ferret
thalamic slices. J. Neurophysiol. 76, 2049–2070.*
:param Vm: membrane potential (mV)
:param C: proportion of Ih channels in closed state (-)
:param O: proportion of Ih channels in open state (-)
:param P0: proportion of Ih channels regulating factor in unbound state (-)
:return: derivative of proportion w.r.t. time (s-1)
'''
return - self.derC(C, O, Vm) - self.k3 * O * (1 - P0) + self.k4 * (1 - O - C)
def derP0(self, P0, C_Ca):
''' Compute the evolution of the proportion of Ih channels regulating factor
in unbound state.
Kinetics scheme of Calcium dependent activation derived from:
*Destexhe, A., Bal, T., McCormick, D.A., and Sejnowski, T.J. (1996). Ionic mechanisms
underlying synchronized oscillations and propagating waves in a model of ferret
thalamic slices. J. Neurophysiol. 76, 2049–2070.*
:param Vm: membrane potential (mV)
:param P0: proportion of Ih channels regulating factor in unbound state (-)
:param C_Ca: Calcium concentration in effective submembranal space (M)
:return: derivative of proportion w.r.t. time (s-1)
'''
return self.k2 * (1 - P0) - self.k1 * P0 * C_Ca**self.nCa
def derC_Ca(self, C_Ca, ICa):
''' Compute the evolution of the Calcium concentration in submembranal space.
Model of Ca2+ buffering and contribution from iCa derived from:
*McCormick, D.A., and Huguenard, J.R. (1992). A model of the electrophysiological
properties of thalamocortical relay neurons. J. Neurophysiol. 68, 1384–1400.*
:param Vm: membrane potential (mV)
:param C_Ca: Calcium concentration in submembranal space (M)
:param ICa: inward Calcium current filling up the submembranal space with Ca2+ (mA/m2)
:return: derivative of Calcium concentration in submembranal space w.r.t. time (s-1)
'''
return (self.CCa_min - C_Ca) / self.tau_Ca_removal - self.iT_2_CCa * ICa
def currKL(self, Vm):
''' Compute the voltage-dependent leak Potassium current per unit area.
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.GKL * (Vm - self.VK)
def currH(self, O, C, Vm):
''' Compute the outward mixed cationic current per unit area.
:param O: proportion of the channels in open form
:param OL: proportion of the channels in locked-open form
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
OL = 1 - O - C
return self.GhMax * (O + 2 * OL) * (Vm - self.Vh)
def currNet(self, Vm, states):
''' Concrete implementation of the abstract API method. '''
m, h, n, s, u, O, C, _, _ = states
return (self.currNa(m, h, Vm) + self.currK(n, Vm) + self.currCa(s, u, Vm) +
self.currKL(Vm) + self.currH(O, C, Vm) + self.currL(Vm)) # mA/m2
def steadyStates(self, Vm):
''' Concrete implementation of the abstract API method. '''
# Call parent method to compute Sodium, Potassium and Calcium channels gates steady-states
NaKCa_eqstates = super().steadyStates(Vm)
# Compute steady-state Calcium current
seq = NaKCa_eqstates[3]
ueq = NaKCa_eqstates[4]
iTeq = self.currCa(seq, ueq, Vm)
# Compute steady-state variables for the kinetics system of Ih
CCa_eq = self.CCa_min - self.tau_Ca_removal * self.iT_2_CCa * iTeq
P0_eq = self.k2 / (self.k2 + self.k1 * CCa_eq**self.nCa)
BA = self.betao(Vm) / self.alphao(Vm)
O_eq = self.k4 / (self.k3 * (1 - P0_eq) + self.k4 * (1 + BA))
C_eq = BA * O_eq
kin_eqstates = np.array([O_eq, C_eq, P0_eq, CCa_eq])
# Merge all steady-states and return
return np.concatenate((NaKCa_eqstates, kin_eqstates))
def derStates(self, Vm, states):
''' Concrete implementation of the abstract API method. '''
m, h, n, s, u, O, C, P0, C_Ca = states
NaKCa_states = [m, h, n, s, u]
NaKCa_derstates = super().derStates(Vm, NaKCa_states)
dO_dt = self.derO(C, O, P0, Vm)
dC_dt = self.derC(C, O, Vm)
dP0_dt = self.derP0(P0, C_Ca)
ICa = self.currCa(s, u, Vm)
dCCa_dt = self.derC_Ca(C_Ca, ICa)
return NaKCa_derstates + [dO_dt, dC_dt, dP0_dt, dCCa_dt]
def getEffRates(self, Vm):
''' Concrete implementation of the abstract API method. '''
# Compute effective coefficients for Sodium, Potassium and Calcium conductances
NaKCa_effrates = super().getEffRates(Vm)
# Compute effective coefficients for Ih conductance
ao_avg = np.mean(self.alphao(Vm))
bo_avg = np.mean(self.betao(Vm))
iH_effrates = np.array([ao_avg, bo_avg])
# Return array of coefficients
return np.concatenate((NaKCa_effrates, iH_effrates))
def derStatesEff(self, Qm, states, interp_data):
''' Concrete implementation of the abstract API method. '''
rates = np.array([np.interp(Qm, interp_data['Q'], interp_data[rn])
for rn in self.coeff_names])
Vmeff = np.interp(Qm, interp_data['Q'], interp_data['V'])
# Unpack states
m, h, n, s, u, O, C, P0, C_Ca = states
# INa, IK, ICa effective states derivatives
dmdt = rates[0] * (1 - m) - rates[1] * m
dhdt = rates[2] * (1 - h) - rates[3] * h
dndt = rates[4] * (1 - n) - rates[5] * n
dsdt = rates[6] * (1 - s) - rates[7] * s
dudt = rates[8] * (1 - u) - rates[9] * u
# Ih effective states derivatives
dC_dt = rates[11] * O - rates[10] * C
dO_dt = - dC_dt - self.k3 * O * (1 - P0) + self.k4 * (1 - O - C)
dP0_dt = self.derP0(P0, C_Ca)
ICa_eff = self.currCa(s, u, Vmeff)
dCCa_dt = self.derC_Ca(C_Ca, ICa_eff)
# Merge derivatives and return
return [dmdt, dhdt, dndt, dsdt, dudt, dO_dt, dC_dt, dP0_dt, dCCa_dt]

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