# Model Mathematics

### Component: membrane

$dd time Em =- i_f + i_Kr + i_Ks + i_bNa + i_Na + i_NaK + i_NaCa + i_to + i_CaL + i_CaT + i_bK + i_bCl Cm$

### Component: reversal_potentials

$E_Na = R ⁢ T F ⁢ln⁡ Nao Nai E_Ca_rev = R ⁢ T 2.0 ⁢ F ⁢ln⁡ Cao Cai E_Cl = R ⁢ T -1.0 ⁢ F ⁢ln⁡ Clo Cli E_K = R ⁢ T F ⁢ln⁡ Ko Ki$

### Component: L_type_calcium_current

$i_CaL = g_CaL ⁢ A_CaL ⁢ Em - E_Ca_rev dd time A_CaL = alpha1 ⁢ I3 + beta1 ⁢ I1 - beta1 + alpha2 ⁢ A_CaL dd time I1 = alpha2 ⁢ A_CaL + beta3 ⁢ I2 - beta2 + alpha3 + Cai 2.0 ⁢ I1 dd time I2 = alpha3 ⁢ Cai 2.0 ⁢ I1 + beta4 ⁢ Cai 2.0 ⁢ I3 - beta3 + alpha4 ⁢ I2 dd time I3 = beta1 ⁢ A_CaL + alpha4 ⁢ I2 - alpha1 + beta4 ⁢ Cai 2.0 ⁢ I3 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope beta2 = beta2_infinity + beta2_0 - beta2_infinity 1.0 +ⅇ Em - beta2_E50 beta2_Eslope$

### Component: T_type_calcium_current

$i_CaT = g_CaT ⁢ A_CaT ⁢ Em - E_Ca_rev dd time A_CaT = alpha1 ⁢ I2 - beta1 + alpha2 ⁢ A_CaT dd time I1 = alpha2 ⁢ A_CaT + beta3 ⁢ I2 - alpha3 ⁢ I1 dd time I2 = alpha3 ⁢ I1 + beta1 ⁢ A_CaT - beta3 + alpha1 ⁢ I2 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope alpha3 = alpha3_infinity + alpha3_0 - alpha3_infinity 1.0 +ⅇ Em - alpha3_E50 alpha3_Eslope beta3 = beta3_infinity + beta3_0 - beta3_infinity 1.0 +ⅇ Em - beta3_E50 beta3_Eslope$

### Component: rapid_delayed_rectifying_potassium_current

$i_Kr = i_Kr_Na + i_Kr_K i_Kr_Na = g_Kr ⁢ A_Kr ⁢ PKNa_r ⁢ Em - E_Na i_Kr_K = g_Kr ⁢ A_Kr ⁢ Em - E_K dd time A_Kr = alpha1 ⁢ I3 - beta1 + alpha2 ⁢ A_Kr dd time I1 = alpha2 ⁢ A_Kr - alpha3 ⁢ I1 dd time I2 = alpha3 ⁢ I1 + beta4 ⁢ I3 - alpha4 ⁢ I2 dd time I3 = beta1 ⁢ A_Kr + alpha4 ⁢ I2 - alpha1 + beta4 ⁢ I3 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope alpha3 = alpha3_infinity + alpha3_0 - alpha3_infinity 1.0 +ⅇ Em - alpha3_E50 alpha3_Eslope alpha4 = alpha4_infinity + alpha4_0 - alpha4_infinity 1.0 +ⅇ Em - alpha4_E50 alpha4_Eslope beta4 = beta4_infinity + beta4_0 - beta4_infinity 1.0 +ⅇ Em - beta4_E50 beta4_Eslope$

### Component: slow_delayed_rectifying_potassium_current

$i_Ks = i_Ks_Na + i_Ks_K i_Ks_Na = g_Ks ⁢ A_Ks ⁢ PKNa_s ⁢ Em - E_Na i_Ks_K = g_Ks ⁢ A_Ks ⁢ Em - E_K dd time A_Ks = alpha1 ⁢ I2 - beta1 + alpha2 ⁢ A_Ks dd time I1 = alpha2 ⁢ A_Ks + beta3 ⁢ I2 - alpha3 ⁢ I1 dd time I2 = alpha3 ⁢ I1 + beta1 ⁢ A_Ks - alpha1 + beta3 ⁢ I2 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope alpha3 = alpha3_infinity + alpha3_0 - alpha3_infinity 1.0 +ⅇ Em - alpha3_E50 alpha3_Eslope beta3 = beta3_infinity + beta3_0 - beta3_infinity 1.0 +ⅇ Em - beta3_E50 beta3_Eslope$

### Component: TTX_sensitive_sodium_current

$i_Na = g_Na ⁢ A_Na ⁢ Em - E_Na dd time A_Na = alpha1 ⁢ I3 - beta1 + alpha2 ⁢ A_Na dd time I1 = alpha2 ⁢ A_Na - alpha3 ⁢ I1 dd time I2 = alpha3 ⁢ I1 + beta4 ⁢ I3 - alpha4 ⁢ I2 dd time I3 = beta1 ⁢ A_Na + alpha4 ⁢ I2 - alpha1 + beta4 ⁢ I3 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope alpha3 = alpha3_infinity + alpha3_0 - alpha3_infinity 1.0 +ⅇ Em - alpha3_E50 alpha3_Eslope alpha4 = alpha4_infinity + alpha4_0 - alpha4_infinity 1.0 +ⅇ Em - alpha4_E50 alpha4_Eslope beta4 = beta4_infinity + beta4_0 - beta4_infinity 1.0 +ⅇ Em - beta4_E50 beta4_Eslope$

### Component: sodium_calcium_exchange_current

$i_NaCa = kNaCa ⁢ x2 ⁢ k21 - x1 ⁢ k12 x1 + x2 + x3 + x4 x1 = k41 ⁢ k34 ⁢ k23 + k21 + k21 ⁢ k32 ⁢ k43 + k41 x2 = k32 ⁢ k43 ⁢ k14 + k12 + k41 ⁢ k12 ⁢ k34 + k32 x3 = k14 ⁢ k43 ⁢ k23 + k21 + k12 ⁢ k23 ⁢ k43 + k41 x4 = k23 ⁢ k34 ⁢ k14 + k12 + k14 ⁢ k21 ⁢ k34 + k32 k43 = Nai K3ni + Nai k12 = Cai ⁢ⅇ- Em ⁢ Qci ⁢ F R ⁢ T Kci ⁢ D1 k14 = Nai 2.0 ⁢ 1.0 + Nai K3ni ⁢ⅇ Qn ⁢ Em ⁢ F 2.0 ⁢ R ⁢ T K1ni ⁢ K2ni ⁢ D1 k41 =ⅇ- Em ⁢ Qn ⁢ F 2.0 ⁢ R ⁢ T D1 = 1.0 + Cai Kci ⁢ 1.0 +ⅇ- Em ⁢ Qci ⁢ F R ⁢ T ⁢ Cai ⁢ Nai Kci ⁢ Kcni + Nai K1ni ⁢ 1.0 + Nai K2ni ⁢ 1.0 + Nai K3ni k34 = Nao K3no + Nao k21 = Cao Kco ⁢ⅇ Qco ⁢ Em ⁢ F R ⁢ T D2 k23 = Nao 2.0 ⁢ 1.0 + Nao K3no ⁢ⅇ- Qn ⁢ Em ⁢ F 2.0 ⁢ R ⁢ T K1no ⁢ K2no ⁢ D2 k32 =ⅇ Qn ⁢ Em ⁢ F 2.0 ⁢ R ⁢ T D2 = 1.0 + Cao Kco ⁢ 1.0 +ⅇ Qco ⁢ Em ⁢ F R ⁢ T + Nao K1no ⁢ 1.0 + Nao K2no ⁢ 1.0 + Nao K3no$

### Component: hyperpolarisation_activated_current

$i_f = i_fNa + i_fK i_fNa = A_f ⁢ Ko 1.83 Ko 1.83 + Km_f 1.83 ⁢ g_f_Na ⁢ Em - E_Na i_fK = A_f ⁢ Ko 1.83 Ko 1.83 + Km_f 1.83 ⁢ P_f_K ⁢ g_f_K ⁢ Em - E_K dd time A_f = alpha1 ⁢ I2 - beta1 + alpha2 ⁢ A_f dd time I1 = alpha2 ⁢ A_f + beta3 ⁢ I2 - alpha3 ⁢ I1 dd time I2 = alpha3 ⁢ I1 + beta1 ⁢ A_f - beta3 + alpha1 ⁢ I2 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope alpha3 = alpha3_infinity + alpha3_0 - alpha3_infinity 1.0 +ⅇ Em - alpha3_E50 alpha3_Eslope beta3 = beta3_infinity + beta3_0 - beta3_infinity 1.0 +ⅇ Em - beta3_E50 beta3_Eslope$

### Component: sodium_potassium_pump_current

$i_NaK = Ip_max ⁢ Nai Nai + Km_Na ⁢ Ko Ko + Km_K ⁢ 1.0 - Em - 40.0 211.0 2.0$

### Component: transient_outward_current

$i_to = g_to ⁢ A_to ⁢ Em - E_K dd time A_to = alpha1 ⁢ I3 - beta1 + alpha2 ⁢ A_to dd time I1 = alpha2 ⁢ A_to - alpha3 ⁢ I1 dd time I2 = alpha3 ⁢ I1 + beta4 ⁢ I3 - alpha4 ⁢ I2 dd time I3 = beta1 ⁢ A_to + alpha4 ⁢ I2 - alpha1 + beta4 ⁢ I3 alpha1 = alpha1_infinity + alpha1_0 - alpha1_infinity 1.0 +ⅇ Em - alpha1_E50 alpha1_Eslope beta1 = beta1_infinity + beta1_0 - beta1_infinity 1.0 +ⅇ Em - beta1_E50 beta1_Eslope alpha2 = alpha2_infinity + alpha2_0 - alpha2_infinity 1.0 +ⅇ Em - alpha2_E50 alpha2_Eslope alpha3 = alpha3_infinity + alpha3_0 - alpha3_infinity 1.0 +ⅇ Em - alpha3_E50 alpha3_Eslope alpha4 = alpha4_infinity + alpha4_0 - alpha4_infinity 1.0 +ⅇ Em - alpha4_E50 alpha4_Eslope beta4 = beta4_infinity + beta4_0 - beta4_infinity 1.0 +ⅇ Em - beta4_E50 beta4_Eslope$

### Component: background_currents

$i_bNa = g_Na ⁢ Em - E_Na i_bK = Kb_K ⁢ Ko 0.41 ⁢ Ki - Ko ⁢ⅇ- Em ⁢ F R ⁢ T i_bCl = g_bCl ⁢ Em - E_Cl$

### Component: calcium_dynamics

$i_Ca_up = Iup_max ⁢ 1.0 1.0 + Km_Caup Cai 2.0 i_Ca_rel = 2.0 ⁢ F ⁢ Vrel tau_rel ⁢ Carel ⁢ 1.0 1.0 + Km_Carel Cai 2.0 i_Ca_tr = 2.0 ⁢ F ⁢ Vrel tau_tr ⁢ Caup$

### Component: ion_concentrations

$dd time Cai =- i_CaL + i_CaT + i_Ca_up - 2.0 ⁢ i_NaCa + i_Ca_rel 2.0 ⁢ F ⁢ Vi dd time Caup = i_Ca_up - i_Ca_tr 2.0 ⁢ F ⁢ Vup dd time Carel = i_Ca_tr - i_Ca_rel 2.0 ⁢ F ⁢ Vrel dd time Cao = i_CaL + i_CaT - 2.0 ⁢ i_NaCa 2.0 ⁢ F ⁢ Ve + Cab - Cao tau_b dd time Nai =- i_Kr_Na + i_Ks_Na + i_Na + 3.0 ⁢ i_NaCa + i_bNa + i_fNa + 3.0 ⁢ i_NaK F ⁢ Vi dd time Nao = i_Kr_Na + i_Ks_Na + i_Na + 3.0 ⁢ i_NaCa + i_bNa + i_fNa + 3.0 ⁢ i_NaK F ⁢ Ve + Nab - Nao tau_b dd time Ki =- i_Kr_K + i_Ks_K + i_bK + i_fK + i_to - 2.0 ⁢ i_NaK F ⁢ Vi dd time Ko = i_Kr_K + i_Ks_K + i_bK + i_fK + i_to - 2.0 ⁢ i_NaK F ⁢ Ve + Kb - Ko tau_b dd time Cli = i_bCl F ⁢ Vi dd time Clo = i_bCl F ⁢ Ve + Clb - Clo tau_b$
Source
Derived from workspace Lovell, Cloherty, Celler, Dokos, 2004 at changeset 0b0b899b03f4.
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