Two-neuron conductance-based model with dynamic ion concentrations to study NaV1.1 channel mutations

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Accession:267047
Gain of function mutations of SCN1A, the gene coding for the voltage-gated sodium channel NaV1.1, cause familial hemiplegic migraine type 3 (FHM-3), whereas loss of function mutations cause different types of epilepsy. To study those mutations, we developed a two-neuron conductance-based model of interconnected GABAergic and pyramidal glutamatergic neurons, with dynamic ion concentrations. We modeled FHM-3 mutations with persistent sodium current in the GABAergic neuron and epileptogenic mutations by decreasing the fast-inactivating sodium conductance in the GABAergic neuron.
Reference:
1 . Lemaire L, Desroches M, Krupa M, Pizzamiglio L, Scalmani P, Mantegazza M (2021) Modeling NaV1.1/SCN1A sodium channel mutations in a microcircuit with realistic ion concentration dynamics suggests differential GABAergic mechanisms leading to hyperexcitability in epilepsy and hemiplegic migraine PLoS Computational Biology 17(7):e1009239
Model Information (Click on a link to find other models with that property)
Model Type: Neuron or other electrically excitable cell;
Brain Region(s)/Organism:
Cell Type(s):
Channel(s): I K; I Na,t; I_AHP; KCC2; NKCC1; I K,leak; I Cl, leak; I Na, leak; I Na,p; Na/K pump;
Gap Junctions:
Receptor(s): GabaA; AMPA;
Gene(s): Nav1.1 SCN1A;
Transmitter(s): Gaba; Glutamate;
Simulation Environment: XPP;
Model Concept(s): Spreading depolarization; Epilepsy;
Implementer(s):
Search NeuronDB for information about:  GabaA; AMPA; I Na,p; I Na,t; I K; I K,leak; Na/K pump; I_AHP; I Cl, leak; I Na, leak; KCC2; NKCC1; Gaba; Glutamate;
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2_neurons_NaV1_1
ic
par
screenshots
readme.html
2_neurons.ode
                            
################################################################################
# CONDUCTANCE-BASED MODEL OF A PAIR OF NEURONS: 
# A GABAERGIC NEURON AND A PYRAMIDAL NEURON
################################################################################


#-------------------------------------------------------------------------------
# INITIAL CONDITIONS
#-------------------------------------------------------------------------------

# initial conditions given in .ic file


#-------------------------------------------------------------------------------
# MODEL PARAMETERS
#-------------------------------------------------------------------------------

# parameter values given in .par file


# EXTERNAL GLUTAERGIC CONDUCTANCES FOR EACH NEURON
par g_D_e
par g_D_i

# CONSERVATION OF MASS
par Na_sum
par Cl_sum

# RATIO OF TOTAL INTRACELLULAR VOLUME OVER EXTRACELLULAR VOLUME
par beta_1

# RATIO OF GABAERGIC NEURON VOLUME OVER PYRAMIDAL NEURON VOLUME
par beta_2

# PUMP MAXIMAL RATE AT -70 mV
par rho_pump
# PUMP HALF ACTIVATION INTRACELLULAR SODIUM
par K_pump_Na
# PUMP HALF ACTIVATION INTRACELLULAR POTASSIUM
par K_pump_K
# PARAMETERS FOR VOLTAGE DEPENDENCE OF THE PUMP
par a
par b
# EXTRACELLULAR POTASSIUM DIFFUSION RATE
par epsilon
# POTASSIUM BATH CONCENTRATION
par K_bath


#### PYRAMIDAL NEURON

# FIRST INTEGRAL
par H_1
# CONVERSION FACTOR FROM CURRENT DENSITY TO CONCENTRATION PER TIME
par gamma_e

# EXCITATORY SYNAPSE
global 1 v_e-vethres {s_e=1}
# TIME CONSTANT FOR DECAY
par tau_e
# VOLTAGE THRESHOLD
par vethres

# FAST INACTIVATING SODIUM MAXIMAL CONDUCTANCE
par g_Na_FI_e
# DELAYED RECTIFIER POTASSIUM MAXIMAL CONDUCTANCE
par g_K_DR_e

# CALCIUM ACTIVATED POTASSIUM MAXIMAL CONDUCTANCE
par g_K_AHP_e
# CALCIUM ACTIVATED POTASSIUM CURRENT HALF ACTIVATION CALCIUM CONCENTRATION
par K_Ca

# SODIUM LEAK CONDUCTANCE
par g_Na_L_e
# POTASSIUM CONDUCTANCE
par g_K_L_e
# CHLORIDE LEAK CONDUCTANCE
par g_Cl_L_e

# KCC2 COTRANSPORTER STRENGTH
par rho_KCC
# NKCC1 COTRANSPORTER STRENGTH
par rho_NKCC
# NKCC1 HALF ACTIVATION POTASSIUM CONCENTRATION
par K_NKCC_K

# GLUAMATERGIC SYNAPSE MAXIMAL CONDUCTANCE
par g_GLU_e
# GABAERGIC SYNAPSE MAXIMAL CONDUCTANCE
par g_GABA_e

# CALCIUM MAXIMAL CONDUCTANCE
par g_Ca_e
# CALCIUM REVERSAL POTENTIAL
par E_Ca_e
# TIME CONSTANT FOR CALCIUM EXTRUSION AND BUFFERING 
par tau_Ca


#### GABAERGIC NEURON

# FIRST INTEGRAL
par H_2
# CONVERSION FACTOR FROM CURRENT DENSITY TO CONCENTRATION PER TIME
par gamma_i

# INHIBITORY SYNAPSE
global 1 v_i-vithres {s_i=1}
# TIME CONSTANT FOR DECAY
par tau_i
# VOLTAGE THRESHOLD
par vithres

# FAST INACTIVATING SODIUM MAXIMAL CONDUCTANCE
par g_Na_FI_i
# PERSISTENT SODIUM MAXIMAL CONDUCTANCE
par g_Na_P_i
# SHIFT OF VOLTAGE DEPENDENCE OF ACTIVATION FOR PERSITENT SODIUM CURRENT
par shift_P
# DELAYED RECTIFIER POTASSIUM MAXIMAL CONDUCTANCE
par g_K_DR_i

# SODIUM LEAK CONDUCTANCE
par g_Na_L_i
# POTASSIUM LEAK CONDUCTANCE
par g_K_L_i

# GLUTAMATERGIC SYNAPSE MAXIMAL CONDUCTANCE
par g_GLU_i


#-------------------------------------------------------------------------------
# CONSERVATIONS, REVERSAL POTENTIALS, (IN)ACTIVATIONS FUNCTIONS AND CURRENTS
#-------------------------------------------------------------------------------

#### CONSERVATIONS

Na_o = Na_sum - beta_1*(1/(1+beta_2)*Na_e + beta_2/(1+beta_2)*Na_i )
Cl_o = Cl_sum - beta_1/(1+beta_2)*Cl_e
aux Na_o_ = Na_o
aux Cl_o_ = Cl_o

K_e = gamma_e*(v_e - H_1) - Na_e + Cl_e
K_i = gamma_i*(v_i - H_2) - Na_i
aux K_e_ = K_e
aux K_i_ = K_i


#### PYRAMIDAL NEURON

# REVERSAL POTENTIALS
E_K_e = 26.64*log(K_o/K_e)
E_Na_e = 26.64*log(Na_o/Na_e)
E_Cl_e = 26.64*log(Cl_e/Cl_o)
aux E_K_e_ = E_K_e
aux E_Na_e_ = E_Na_e
aux E_Cl_e_ = E_Cl_e

# (IN)ACTIVATION FUNCTIONS
am_e = 0.32*(v_e + 54)/(1-exp(-(v_e + 54)/4))
bm_e = 0.28*(v_e + 27)/(exp((v_e + 27)/5) - 1)
ah_e = 0.128*exp( - (v_e + 50)/18)
bh_e = 4/(1 + exp( - (v_e + 27)/5))
an_e = 0.032*(v_e + 52)/(1 - exp( - (v_e + 52)/5))
bn_e = 0.5*exp( - (v_e + 57)/40)
minf_Ca = 1/(1 + exp(-(v_e + 25)/2.5))

# SODIUM FAST INACTIVATING CURRENT
I_Na_FI_e = g_Na_FI_e*m_e^3*h_e*(v_e - E_Na_e)
aux I_Na_FI_e_ = I_Na_FI_e
# POTASSIUM DELAYED RECTIFIER CURRENT
I_K_DR_e = g_K_DR_e*n_e^4*(v_e - E_K_e)
aux I_K_DR_e_ = I_K_DR_e

# CALCIUM DEPENDENT POTASSIUM CURRENT
I_K_AHP_e = g_K_AHP_e*(Ca_e/(Ca_e + K_Ca))*(v_e - E_K_e)
aux I_K_AHP_e_ = I_K_AHP_e

# LEAK CURRENTS
I_Na_L_e = g_Na_L_e*(v_e - E_Na_e)
aux I_Na_L_e_ = I_Na_L_e
I_K_L_e = g_K_L_e*(v_e - E_K_e)
aux I_K_L_e_ = I_K_L_e
I_Cl_L_e = g_Cl_L_e*(v_e - E_Cl_e)
aux I_Cl_L_e_ = I_Cl_L_e
I_L_e = I_Na_L_e + I_K_L_e + I_Cl_L_e
aux I_L_e_ = I_L_e

# KCC2 COTRANSPORTER 
I_KCC = rho_KCC*log((K_e*Cl_e)/(K_o*Cl_o))
aux I_KCC_ = I_KCC
# NKCC1 COTRANSPORTER 
I_NKCC = rho_NKCC/(1+exp(K_NKCC_K-K_o))*(log((K_e*Cl_e)/(K_o*Cl_o))+log((Na_e*Cl_e)/(Na_o*Cl_o)))
aux I_NKCC_ = I_NKCC

# PUMP CURRENT
I_pump_e = rho_pump * (1+tanh(a/26.64*v_e + b))/(1+tanh(a/26.64*(-70) + b)) * (Na_e/(Na_e + K_pump_Na))^3*(K_o/(K_o + K_pump_K))^2
aux I_pump_e_ = I_pump_e

# SYNAPTIC EXCITATORY CURRENT
I_Na_GLU_e = g_GLU_e/2*s_e*(v_e - E_Na_e)
I_K_GLU_e = g_GLU_e/2*s_e*(v_e - E_K_e)
I_GLU_e = I_Na_GLU_e + I_K_GLU_e
aux I_GLU_e_ = I_GLU_e
# SYNAPTIC INHIBITORY CURRENT
I_GABA_e = g_GABA_e*s_i*(v_e - E_Cl_e)
aux I_GABA_e_ = I_GABA_e

# EXTERNAL INPUT
I_Na_D_e = g_D_e/2*(v_e - E_Na_e)
I_K_D_e = g_D_e/2*(v_e - E_K_e)
I_D_e = I_Na_D_e + I_K_D_e
aux I_D_e_ = I_D_e


#### GABAERGIC NEURON

# REVERSAL POTENTIALS
E_K_i = 26.64*log(K_o/K_i)
E_Na_i = 26.64*log(Na_o/Na_i)
aux E_K_i_ = E_K_i
aux E_Na_i_ = E_Na_i

# (IN)ACTIVATION FUNCTIONS
minf_i = 1/(1+exp(-(v_i-(-24))/11.5))
minf_P_i = 1/(1+exp(-(v_i+shift_P-(-24))/11.5))
h_inf_i = 1/(1+exp(-(v_i-(-58.3))/(-6.7)))
tau_h_i = 0.5 + 14/(1 + exp(-(v_i-(-60))/(-12)))
n_inf_i = 1/(1 + exp(-(v_i - (-12.4))/6.8))
tau_n_i = (0.087 + 11.4/(1 + exp((v_i + 14.6)/8.6)))*(0.087 + 11.4 * 1/(1 + exp(-(v_i - 1.3)/18.7)))

# SODIUM FAST INACTIVATING CURRENT
I_Na_FI_i = g_Na_FI_i*minf_i^3*h_i*(v_i - E_Na_i)
aux I_Na_FI_i_ = I_Na_FI_i
# SODIUM PERSISTENT CURRENT
I_Na_P_i = g_Na_P_i*minf_P_i^3*(v_i - E_Na_i)
aux I_Na_P_i_ = I_Na_P_i
# POTASSIUM DELAYED RECTIFIER CURRENT
I_K_DR_i = g_K_DR_i*n_i^2*(v_i - E_K_i)
aux I_K_DR_i_ = I_K_DR_i

# LEAK CURRENTS
I_Na_L_i = g_Na_L_i*(v_i - E_Na_i)
aux I_Na_L_i_ = I_Na_L_i
I_K_L_i = g_K_L_i *(v_i - E_K_i)
aux I_K_L_i_ = I_K_L_i
I_L_i = I_Na_L_i + I_K_L_i
aux I_L_i_ = I_L_i

# PUMP CURRENT
I_pump_i = rho_pump * (1+tanh(a/26.64*v_i + b))/(1+tanh(a/26.64*(-70) + b)) * (Na_i/(Na_i + K_pump_Na))^3*(K_o/(K_o + K_pump_K))^2
aux I_pump_i_ = I_pump_i

# SYNAPTIC EXCITATORY CURRENT
I_Na_GLU_i = g_GLU_i/2*s_e*(v_i - E_Na_i)
I_K_GLU_i = g_GLU_i/2*s_e*(v_i - E_K_i)
I_GLU_i = I_Na_GLU_i + I_K_GLU_i
aux I_GLU_i_ = I_GLU_i

# EXTERNAL INPUT
I_Na_D_i = g_D_i/2*(v_i - E_Na_i)
I_K_D_i = g_D_i/2*(v_i - E_K_i)
I_D_i = I_Na_D_i + I_K_D_i
aux I_D_i_ = I_D_i 


#### EXTRACELLULAR POTASSIUM DIFFUSION
I_diff = epsilon*(K_o-K_bath)
aux I_diff_ = I_diff


#-------------------------------------------------------------------------------
# ODEs
#-------------------------------------------------------------------------------

#### PYRAMIDAL NEURON
v_e' = - I_Na_FI_e - I_K_DR_e - I_K_AHP_e - I_L_e - I_pump_e - I_GLU_e - I_GABA_e - I_D_e
m_e' = am_e*(1 - m_e) - bm_e*m_e
h_e' = ah_e*(1 - h_e) - bh_e*h_e
n_e' = an_e*(1 - n_e) - bn_e*n_e

Na_e' = - gamma_e*(I_Na_FI_e + I_Na_L_e + 3*I_pump_e + I_Na_GLU_e + I_Na_D_e) - I_NKCC
Cl_e' = gamma_e*(I_Cl_L_e + I_GABA_e) - I_KCC - 2*I_NKCC
Ca_e' = - gamma_e/2*g_Ca_e*minf_Ca*(v_e - E_Ca_e) - Ca_e/tau_Ca

s_e' = - (1/tau_e)*s_e


#### GABAERGIC NEURON
v_i' = - I_Na_FI_i - I_Na_P_i - I_K_DR_i - I_L_i - I_pump_i - I_GLU_i - I_D_i
h_i' = (h_inf_i - h_i)/tau_h_i
n_i' = (n_inf_i - n_i)/tau_n_i

Na_i' = - gamma_i*( I_Na_FI_i + I_Na_P_i + I_Na_L_i + 3*I_pump_i + I_Na_GLU_i + I_Na_D_i )

s_i' = - (1/tau_i)*s_i


#### EXTRACELLULAR POTASSIUM
K_o' = beta_1 * ( 1/(1+beta_2)*(gamma_e*(I_K_DR_e + I_K_AHP_e +  I_K_L_e - 2*I_pump_e + I_K_GLU_e + I_K_D_e) + I_KCC + I_NKCC) + beta_2/(1+beta_2)*gamma_i*(I_K_DR_i + I_K_L_i - 2*I_pump_i + I_K_GLU_i + I_K_D_i) ) - I_diff


#-------------------------------------------------------------------------------
# NUMERICS
#-------------------------------------------------------------------------------

@ method=rk4, dt=0.01, total=30000, nout=1
@ xp=t, yp=v_e, xlo=0, xhi=30000, ylo=-100, yhi=100
@ bound=9000000
@ maxstor=9000000


done



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