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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: 2019-04-03 16:33:28
import numpy as np
from ..core import PointNeuron
from ..constants import Z_Ca
class Thalamic(PointNeuron):
''' Generic thalamic neuron
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)
ENa = 50.0 # Sodium Nernst potential (mV)
EK = -90.0 # Potassium Nernst potential (mV)
ECa = 120.0 # Calcium Nernst potential (mV)
def __init__(self):
self.states = ['m', 'h', 'n', 's', 'u']
self.rates = self.getRatesNames(self.states)
def alpham(self, Vm):
''' Voltage-dependent activation rate of m-gate
:param Vm: membrane potential (mV)
:return: rate (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):
''' Voltage-dependent inactivation rate of m-gate
:param Vm: membrane potential (mV)
:return: rate (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):
''' Voltage-dependent activation rate of h-gate
:param Vm: membrane potential (mV)
:return: rate (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):
''' Voltage-dependent inactivation rate of h-gate
:param Vm: membrane potential (mV)
:return: rate (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):
''' Voltage-dependent activation rate of n-gate
:param Vm: membrane potential (mV)
:return: rate (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):
''' Voltage-dependent inactivation rate of n-gate
:param Vm: membrane potential (mV)
:return: rate (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):
''' Evolution of m-gate open-probability
:param Vm: membrane potential (mV)
:param m: open-probability of m-gate (-)
:return: time derivative of m-gate open-probability (s-1)
'''
return self.alpham(Vm) * (1 - m) - self.betam(Vm) * m
def derH(self, Vm, h):
''' Evolution of h-gate open-probability
:param Vm: membrane potential (mV)
:param h: open-probability of h-gate (-)
:return: time derivative of h-gate open-probability (s-1)
'''
return self.alphah(Vm) * (1 - h) - self.betah(Vm) * h
def derN(self, Vm, n):
''' Evolution of n-gate open-probability
:param Vm: membrane potential (mV)
:param n: open-probability of n-gate (-)
:return: time derivative of n-gate open-probability (s-1)
'''
return self.alphan(Vm) * (1 - n) - self.betan(Vm) * n
def derS(self, Vm, s):
''' Evolution of s-gate open-probability
:param Vm: membrane potential (mV)
:param s: open-probability of s-gate (-)
:return: time derivative of s-gate open-probability (s-1)
'''
return (self.sinf(Vm) - s) / self.taus(Vm)
def derU(self, Vm, u):
''' Evolution of u-gate open-probability
:param Vm: membrane potential (mV)
:param u: open-probability of u-gate (-)
:return: time derivative of u-gate open-probability (s-1)
'''
return (self.uinf(Vm) - u) / self.tauu(Vm)
def iNa(self, m, h, Vm):
''' Sodium current
:param m: open-probability of m-gate (-)
:param h: open-probability of h-gate (-)
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.gNabar * m**3 * h * (Vm - self.ENa)
def iKd(self, n, Vm):
''' delayed-rectifier Potassium current
:param n: open-probability of n-gate (-)
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.gKdbar * n**4 * (Vm - self.EK)
def iCaT(self, s, u, Vm):
''' low-threshold (Ts-type) Calcium current
:param s: open-probability of s-gate (-)
:param u: open-probability of u-gate (-)
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.gCaTbar * s**2 * u * (Vm - self.ECa)
def iLeak(self, Vm):
''' non-specific leakage current
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.gLeak * (Vm - self.ELeak)
def currents(self, Vm, states):
''' Overriding of abstract parent method. '''
m, h, n, s, u = states
return {
'iNa': self.iNa(m, h, Vm),
'iKd': self.iKd(n, Vm),
'iCaT': self.iCaT(s, u, Vm),
'iLeak': self.iLeak(Vm)
} # mA/m2
def steadyStates(self, Vm):
''' Overriding of abstract parent method. '''
# Voltage-gated steady-states
m_eq = self.alpham(Vm) / (self.alpham(Vm) + self.betam(Vm))
h_eq = self.alphah(Vm) / (self.alphah(Vm) + self.betah(Vm))
n_eq = self.alphan(Vm) / (self.alphan(Vm) + self.betan(Vm))
s_eq = self.sinf(Vm)
u_eq = self.uinf(Vm)
return np.array([m_eq, h_eq, n_eq, s_eq, u_eq])
def derStates(self, Vm, states):
''' Overriding of abstract parent 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):
''' Overriding of abstract parent 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):
''' Overriding of abstract parent method. '''
rates = {rn: np.interp(Qm, interp_data['Q'], interp_data[rn]) for rn in self.rates}
m, h, n, s, u = states
dmdt = rates['alpham'] * (1 - m) - rates['betam'] * m
dhdt = rates['alphah'] * (1 - h) - rates['betah'] * h
dndt = rates['alphan'] * (1 - n) - rates['betan'] * n
dsdt = rates['alphas'] * (1 - s) - rates['betas'] * s
dudt = rates['alphau'] * (1 - u) - rates['betau'] * u
return [dmdt, dhdt, dndt, dsdt, dudt]
class ThalamicRE(Thalamic):
''' 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)
gNabar = 2000.0 # Max. conductance of Sodium current (S/m^2)
gKdbar = 200.0 # Max. conductance of Potassium current (S/m^2)
gCaTbar = 30.0 # Max. conductance of low-threshold Calcium current (S/m^2)
gLeak = 0.5 # Conductance of non-specific leakage current (S/m^2)
ELeak = -90.0 # Non-specific leakage Nernst potential (mV)
VT = -67.0 # Spike threshold adjustment parameter (mV)
def __init__(self):
super().__init__()
def sinf(self, Vm):
''' Voltage-dependent steady-state opening of s-gate
:param Vm: membrane potential (mV)
:return: steady-state opening (-)
'''
return 1.0 / (1.0 + np.exp(-(Vm + 52.0) / 7.4)) # prob
def taus(self, Vm):
''' Voltage-dependent adaptation time for adaptation of s-gate
:param Vm: membrane potential (mV)
:return: adaptation time (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):
''' Voltage-dependent steady-state opening of u-gate
:param Vm: membrane potential (mV)
:return: steady-state opening (-)
'''
return 1.0 / (1.0 + np.exp((Vm + 80.0) / 5.0)) # prob
def tauu(self, Vm):
''' Voltage-dependent adaptation time for adaptation of u-gate
:param Vm: membrane potential (mV)
:return: adaptation time (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):
''' Thalamo-cortical neuron, with a specific
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)
gNabar = 900.0 # bar. conductance of Sodium current (S/m^2)
gKdbar = 100.0 # bar. conductance of Potassium current (S/m^2)
gCaTbar = 20.0 # Max. conductance of low-threshold Calcium current (S/m^2)
gKLeak = 0.138 # Conductance of leakage Potassium current (S/m^2)
gHbar = 0.175 # Max. conductance of mixed cationic current (S/m^2)
gLeak = 0.1 # Conductance of non-specific leakage current (S/m^2)
EH = -40.0 # Mixed cationic current reversal potential (mV)
ELeak = -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)
taur_Cai = 5e-3 # decay time constant for intracellular Ca2+ dissolution (s)
Cai_min = 50e-9 # minimal intracellular Calcium concentration (M)
deff = 100e-9 # effective depth beneath membrane for intracellular [Ca2+] calculation
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)
def __init__(self):
super().__init__()
self.iCa_to_Cai_rate = self.currentToConcentrationRate(Z_Ca, self.deff)
self.states += ['O', 'C', 'P0', 'Cai']
self.rates += self.getRatesNames(['O'])
def getPltScheme(self):
pltscheme = super().getPltScheme()
pltscheme['i_{H}\\ kin.'] = ['O', 'OL', 'P0']
pltscheme['[Ca^{2+}]_i'] = ['Cai']
return pltscheme
def getPltVars(self, wrapleft='df["', wrapright='"]'):
pltvars = super().getPltVars(wrapleft, wrapright)
pltvars.update({
'Cai': {
'desc': 'sumbmembrane Ca2+ concentration',
'label': '[Ca^{2+}]_i',
'unit': 'uM',
'factor': 1e6
},
'OL': {
'desc': 'iH O-gate locked-opening',
'label': 'O_L',
'bounds': (-0.1, 1.1),
'func': 'OL({0}O{1}, {0}C{1})'.format(wrapleft, wrapright)
},
'P0': {
'desc': 'iH regulating factor activation',
'label': 'P_0',
'bounds': (-0.1, 1.1)
}
})
return pltvars
def sinf(self, Vm):
''' Voltage-dependent steady-state opening of s-gate
:param Vm: membrane potential (mV)
:return: steady-state opening (-)
'''
return 1.0 / (1.0 + np.exp(-(Vm + self.Vx + 57.0) / 6.2)) # prob
def taus(self, Vm):
''' Voltage-dependent adaptation time for adaptation of s-gate
:param Vm: membrane potential (mV)
:return: adaptation time (s)
'''
x = 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 / x) * 1e-3 # s
def uinf(self, Vm):
''' Voltage-dependent steady-state opening of u-gate
:param Vm: membrane potential (mV)
:return: steady-state opening (-)
'''
return 1.0 / (1.0 + np.exp((Vm + self.Vx + 81.0) / 4.0)) # prob
def tauu(self, Vm):
''' Voltage-dependent adaptation time for adaptation of u-gate
:param Vm: membrane potential (mV)
:return: adaptation time (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):
''' Evolution of s-gate open-probability
:param Vm: membrane potential (mV)
:param s: open-probability of s-gate (-)
:return: time derivative of s-gate open-probability (s-1)
'''
return (self.sinf(Vm) - s) / self.taus(Vm)
def derU(self, Vm, u):
''' Evolution of u-gate open-probability
:param Vm: membrane potential (mV)
:param u: open-probability of u-gate (-)
:return: time derivative of u-gate open-probability (s-1)
'''
return (self.uinf(Vm) - u) / self.tauu(Vm)
def oinf(self, Vm):
''' Voltage-dependent steady-state opening of O-gate
:param Vm: membrane potential (mV)
:return: steady-state opening (-)
'''
return 1.0 / (1.0 + np.exp((Vm + 75.0) / 5.5))
def tauo(self, Vm):
''' Voltage-dependent adaptation time for adaptation of O-gate
:param Vm: membrane potential (mV)
:return: adaptation time (s)
'''
return 1 / (np.exp(-14.59 - 0.086 * Vm) + np.exp(-1.87 + 0.0701 * Vm)) * 1e-3
def alphao(self, Vm):
''' Voltage-dependent transition rate between closed and open forms of O-gate
:param Vm: membrane potential (mV)
:return: rate (s-1)
'''
return self.oinf(Vm) / self.tauo(Vm)
def betao(self, Vm):
''' Voltage-dependent transition rate between open and closed forms of O-gate
:param Vm: membrane potential (mV)
:return: rate (s-1)
'''
return (1 - self.oinf(Vm)) / self.tauo(Vm)
def derC(self, C, O, Vm):
''' Evolution of O-gate closed-probability
:param C: closed-probability of O-gate (-)
:param O: open-probability of O-gate (-)
:param Vm: membrane potential (mV)
:return: time derivative of O-gate closed-probability (s-1)
'''
return self.betao(Vm) * O - self.alphao(Vm) * C
def derO(self, C, O, P0, Vm):
''' Evolution of O-gate open-probability
:param C: closed-probability of O-gate (-)
:param O: open-probability of O-gate (-)
:param P0: proportion of Ih channels regulating factor in unbound state (-)
:param Vm: membrane potential (mV)
:return: time derivative of O-gate open-probability (s-1)
'''
return - self.derC(C, O, Vm) - self.k3 * O * (1 - P0) + self.k4 * (1 - O - C)
def OL(self, O, C):
''' O-gate locked-open probability.
:param O: open-probability of O-gate (-)
:param C: closed-probability of O-gate (-)
:return: loked-open-probability of O-gate (-)
'''
return 1 - O - C
def derP0(self, P0, Cai):
''' Evolution of unbound probability of Ih regulating factor.
:param P0: unbound probability of Ih regulating factor (-)
:param Cai: submembrane Calcium concentration (M)
:return: time derivative of ubnound probability (s-1)
'''
return self.k2 * (1 - P0) - self.k1 * P0 * Cai**self.nCa
def derCai(self, Cai, s, u, Vm):
''' Evolution of submembrane Calcium concentration.
Model of Ca2+ buffering and contribution from iCaT 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 Cai: submembrane Calcium concentration (M)
:param s: open-probability of s-gate (-)
:param u: open-probability of u-gate (-)
:param Vm: membrane potential (mV)
:return: time derivative of submembrane Calcium concentration (M/s)
'''
return (self.Cai_min - Cai) / self.taur_Cai - self.iCa_to_Cai_rate * self.iCaT(s, u, Vm)
def iKLeak(self, Vm):
''' Potassium leakage current
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.gKLeak * (Vm - self.EK)
def iH(self, O, C, Vm):
''' outward mixed cationic current
:param C: closed-probability of O-gate (-)
:param O: open-probability of O-gate (-)
:param Vm: membrane potential (mV)
:return: current per unit area (mA/m2)
'''
return self.gHbar * (O + 2 * self.OL(O, C)) * (Vm - self.EH)
def currents(self, Vm, states):
''' Overriding of abstract parent method. '''
m, h, n, s, u, O, C, _, _ = states
currents = super().currents(Vm, [m, h, n, s, u])
currents['iKLeak'] = self.iKLeak(Vm) # mA/m2
currents['iH'] = self.iH(O, C, Vm) # mA/m2
return currents
def Caiinf(self, Vm, s, u):
''' Find the steady-state intracellular Calcium concentration for a
specific membrane potential and voltage-gated channel states.
:param Vm: membrane potential (mV)
:param s: open-probability of s-gate
:param u: open-probability of u-gate
:return: steady-state Calcium concentration in submembrane space (M)
'''
return self.Cai_min - self.taur_Cai * self.iCa_to_Cai_rate * self.iCaT(s, u, Vm)
def P0inf(self, Cai):
''' Find the steady-state unbound probability of Ih regulating factor
for a specific intracellular Calcium concentration.
:param Cai : Calcium concentration in submembrane space (M)
:return: steady-state unbound probability of Ih regulating factor
'''
return self.k2 / (self.k2 + self.k1 * Cai**self.nCa)
def Oinf(self, Cai, Vm):
''' Find the steady-state O-gate open-probability for specific
membrane potential and intracellular Calcium concentration.
:param Cai : Calcium concentration in submembrane space (M)
:param Vm: membrane potential (mV)
:return: steady-state O-gate open-probability
'''
BA = self.betao(Vm) / self.alphao(Vm)
return self.k4 / (self.k3 * (1 - self.P0inf(Cai)) + self.k4 * (1 + BA))
def Cinf(self, Cai, Vm):
''' Find the steady-state O-gate closed-probability for specific
membrane potential and intracellular Calcium concentration.
:param Cai : Calcium concentration in submembrane space (M)
:param Vm: membrane potential (mV)
:return: steady-state O-gate closed-probability
'''
BA = self.betao(Vm) / self.alphao(Vm)
return BA * self.Oinf(Cai, Vm)
def steadyStates(self, Vm):
''' Overriding of abstract parent method. '''
# Voltage-gated steady-states
NaKCa_eq = super().steadyStates(Vm)
seq = NaKCa_eq[3]
ueq = NaKCa_eq[4]
# Other steady-states
Cai_eq = self.Caiinf(Vm, seq, ueq)
P0_eq = self.P0inf(Cai_eq)
O_eq = self.Oinf(Cai_eq, Vm)
C_eq = self.Cinf(Cai_eq, Vm)
kin_eq = np.array([O_eq, C_eq, P0_eq, Cai_eq])
return np.concatenate((NaKCa_eq, kin_eq))
def derStates(self, Vm, states):
''' Overriding of abstract parent method. '''
m, h, n, s, u, O, C, P0, Cai = states
NaKCa_states = [m, h, n, s, u]
NaKCa_derstates = super().derStates(Vm, NaKCa_states)
dOdt = self.derO(C, O, P0, Vm)
dCdt = self.derC(C, O, Vm)
dP0dt = self.derP0(P0, Cai)
dCaidt = self.derCai(Cai, s, u, Vm)
return NaKCa_derstates + [dOdt, dCdt, dP0dt, dCaidt]
def getEffRates(self, Vm):
''' Overriding of abstract parent 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):
''' Overriding of abstract parent method. '''
rates = {rn: np.interp(Qm, interp_data['Q'], interp_data[rn]) for rn in self.rates}
Vmeff = np.interp(Qm, interp_data['Q'], interp_data['V'])
# Unpack states
m, h, n, s, u, O, C, P0, Cai = states
# INa, IK, iCaT effective states derivatives
dmdt = rates['alpham'] * (1 - m) - rates['betam'] * m
dhdt = rates['alphah'] * (1 - h) - rates['betah'] * h
dndt = rates['alphan'] * (1 - n) - rates['betan'] * n
dsdt = rates['alphas'] * (1 - s) - rates['betas'] * s
dudt = rates['alphau'] * (1 - u) - rates['betau'] * u
# Ih effective states derivatives
dCdt = rates['betao'] * O - rates['alphao'] * C
dOdt = - dCdt - self.k3 * O * (1 - P0) + self.k4 * (1 - O - C)
dP0dt = self.derP0(P0, Cai)
dCaidt = self.derCai(Cai, s, u, Vmeff)
# Merge derivatives and return
return [dmdt, dhdt, dndt, dsdt, dudt, dOdt, dCdt, dP0dt, dCaidt]

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