Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (Em) = \frac{- (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 = \frac{R T}{F} \ln (\frac{Nao}{Nai}) E_Ca_rev = \frac{R T}{2.0 F} \ln (\frac{Cao}{Cai}) E_Cl = \frac{R T}{(-1.0) F} \ln (\frac{Clo}{Cli}) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki})

Component: L_type_calcium_current

i_CaL = g_CaL A_CaL (Em - E_Ca_rev) \frac{d}{d time} (A_CaL) = alpha1 I3 + beta1 I1 - ((beta1 + alpha2) A_CaL) \frac{d}{d time} (I1) = alpha2 A_CaL + beta3 I2 - ((beta2 + alpha3 + {Cai}^{2.0}) I1) \frac{d}{d time} (I2) = alpha3 {Cai}^{2.0} I1 + beta4 {Cai}^{2.0} I3 - ((beta3 + alpha4) I2) \frac{d}{d time} (I3) = beta1 A_CaL + alpha4 I2 - ((alpha1 + beta4 {Cai}^{2.0}) I3) alpha1 = alpha1_infinity + \frac{alpha1_0 - alpha1_infinity}{1.0 + ⅇ^{\frac{Em - alpha1_E50}{alpha1_Eslope}}} beta1 = beta1_infinity + \frac{beta1_0 - beta1_infinity}{1.0 + ⅇ^{\frac{Em - beta1_E50}{beta1_Eslope}}} alpha2 = alpha2_infinity + \frac{alpha2_0 - alpha2_infinity}{1.0 + ⅇ^{\frac{Em - alpha2_E50}{alpha2_Eslope}}} beta2 = beta2_infinity + \frac{beta2_0 - beta2_infinity}{1.0 + ⅇ^{\frac{Em - beta2_E50}{beta2_Eslope}}}

Component: T_type_calcium_current

i_CaT = g_CaT A_CaT (Em - E_Ca_rev) \frac{d}{d time} (A_CaT) = alpha1 I2 - ((beta1 + alpha2) A_CaT) \frac{d}{d time} (I1) = alpha2 A_CaT + beta3 I2 - (alpha3 I1) \frac{d}{d time} (I2) = alpha3 I1 + beta1 A_CaT - ((beta3 + alpha1) I2) alpha1 = alpha1_infinity + \frac{alpha1_0 - alpha1_infinity}{1.0 + ⅇ^{\frac{Em - alpha1_E50}{alpha1_Eslope}}} beta1 = beta1_infinity + \frac{beta1_0 - beta1_infinity}{1.0 + ⅇ^{\frac{Em - beta1_E50}{beta1_Eslope}}} alpha2 = alpha2_infinity + \frac{alpha2_0 - alpha2_infinity}{1.0 + ⅇ^{\frac{Em - alpha2_E50}{alpha2_Eslope}}} alpha3 = alpha3_infinity + \frac{alpha3_0 - alpha3_infinity}{1.0 + ⅇ^{\frac{Em - alpha3_E50}{alpha3_Eslope}}} beta3 = beta3_infinity + \frac{beta3_0 - beta3_infinity}{1.0 + ⅇ^{\frac{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) \frac{d}{d time} (A_Kr) = alpha1 I3 - ((beta1 + alpha2) A_Kr) \frac{d}{d time} (I1) = alpha2 A_Kr - (alpha3 I1) \frac{d}{d time} (I2) = alpha3 I1 + beta4 I3 - (alpha4 I2) \frac{d}{d time} (I3) = beta1 A_Kr + alpha4 I2 - ((alpha1 + beta4) I3) alpha1 = alpha1_infinity + \frac{alpha1_0 - alpha1_infinity}{1.0 + ⅇ^{\frac{Em - alpha1_E50}{alpha1_Eslope}}} beta1 = beta1_infinity + \frac{beta1_0 - beta1_infinity}{1.0 + ⅇ^{\frac{Em - beta1_E50}{beta1_Eslope}}} alpha2 = alpha2_infinity + \frac{alpha2_0 - alpha2_infinity}{1.0 + ⅇ^{\frac{Em - alpha2_E50}{alpha2_Eslope}}} alpha3 = alpha3_infinity + \frac{alpha3_0 - alpha3_infinity}{1.0 + ⅇ^{\frac{Em - alpha3_E50}{alpha3_Eslope}}} alpha4 = alpha4_infinity + \frac{alpha4_0 - alpha4_infinity}{1.0 + ⅇ^{\frac{Em - alpha4_E50}{alpha4_Eslope}}} beta4 = beta4_infinity + \frac{beta4_0 - beta4_infinity}{1.0 + ⅇ^{\frac{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) \frac{d}{d time} (A_Ks) = alpha1 I2 - ((beta1 + alpha2) A_Ks) \frac{d}{d time} (I1) = alpha2 A_Ks + beta3 I2 - (alpha3 I1) \frac{d}{d time} (I2) = alpha3 I1 + beta1 A_Ks - ((alpha1 + beta3) I2) alpha1 = alpha1_infinity + \frac{alpha1_0 - alpha1_infinity}{1.0 + ⅇ^{\frac{Em - alpha1_E50}{alpha1_Eslope}}} beta1 = beta1_infinity + \frac{beta1_0 - beta1_infinity}{1.0 + ⅇ^{\frac{Em - beta1_E50}{beta1_Eslope}}} alpha2 = alpha2_infinity + \frac{alpha2_0 - alpha2_infinity}{1.0 + ⅇ^{\frac{Em - alpha2_E50}{alpha2_Eslope}}} alpha3 = alpha3_infinity + \frac{alpha3_0 - alpha3_infinity}{1.0 + ⅇ^{\frac{Em - alpha3_E50}{alpha3_Eslope}}} beta3 = beta3_infinity + \frac{beta3_0 - beta3_infinity}{1.0 + ⅇ^{\frac{Em - beta3_E50}{beta3_Eslope}}}

Component: TTX_sensitive_sodium_current

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

Component: sodium_calcium_exchange_current

i_NaCa = kNaCa \frac{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 = \frac{Nai}{K3ni + Nai} k12 = \frac{Cai ⅇ^{- (\frac{Em Qci F}{R T})}}{Kci D1} k14 = \frac{{Nai}^{2.0} (1.0 + \frac{Nai}{K3ni}) ⅇ^{\frac{Qn Em F}{2.0 R T}}}{K1ni K2ni D1} k41 = ⅇ^{- (\frac{Em Qn F}{2.0 R T})} D1 = 1.0 + \frac{Cai}{Kci} (1.0 + ⅇ^{- (\frac{Em Qci F}{R T})}) \frac{Cai Nai}{Kci Kcni} + \frac{Nai}{K1ni} (1.0 + \frac{Nai}{K2ni} (1.0 + \frac{Nai}{K3ni})) k34 = \frac{Nao}{K3no + Nao} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{Qco Em F}{R T}}}{D2} k23 = \frac{{Nao}^{2.0} (1.0 + \frac{Nao}{K3no}) ⅇ^{- (\frac{Qn Em F}{2.0 R T})}}{K1no K2no D2} k32 = ⅇ^{\frac{Qn Em F}{2.0 R T}} D2 = 1.0 + \frac{Cao}{Kco} (1.0 + ⅇ^{\frac{Qco Em F}{R T}}) + \frac{Nao}{K1no} (1.0 + \frac{Nao}{K2no}) (1.0 + \frac{Nao}{K3no})

Component: hyperpolarisation_activated_current

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

Component: sodium_potassium_pump_current

i_NaK = Ip_max \frac{Nai}{Nai + Km_Na} \frac{Ko}{Ko + Km_K} (1.0 - ({(\frac{Em - 40.0}{211.0})}^{2.0}))

Component: transient_outward_current

i_to = g_to A_to (Em - E_K) \frac{d}{d time} (A_to) = alpha1 I3 - ((beta1 + alpha2) A_to) \frac{d}{d time} (I1) = alpha2 A_to - (alpha3 I1) \frac{d}{d time} (I2) = alpha3 I1 + beta4 I3 - (alpha4 I2) \frac{d}{d time} (I3) = beta1 A_to + alpha4 I2 - ((alpha1 + beta4) I3) alpha1 = alpha1_infinity + \frac{alpha1_0 - alpha1_infinity}{1.0 + ⅇ^{\frac{Em - alpha1_E50}{alpha1_Eslope}}} beta1 = beta1_infinity + \frac{beta1_0 - beta1_infinity}{1.0 + ⅇ^{\frac{Em - beta1_E50}{beta1_Eslope}}} alpha2 = alpha2_infinity + \frac{alpha2_0 - alpha2_infinity}{1.0 + ⅇ^{\frac{Em - alpha2_E50}{alpha2_Eslope}}} alpha3 = alpha3_infinity + \frac{alpha3_0 - alpha3_infinity}{1.0 + ⅇ^{\frac{Em - alpha3_E50}{alpha3_Eslope}}} alpha4 = alpha4_infinity + \frac{alpha4_0 - alpha4_infinity}{1.0 + ⅇ^{\frac{Em - alpha4_E50}{alpha4_Eslope}}} beta4 = beta4_infinity + \frac{beta4_0 - beta4_infinity}{1.0 + ⅇ^{\frac{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^{- (\frac{Em F}{R T})})) i_bCl = g_bCl (Em - E_Cl)

Component: calcium_dynamics

i_Ca_up = Iup_max \frac{1.0}{1.0 + {(\frac{Km_Caup}{Cai})}^{2.0}} i_Ca_rel = \frac{2.0 F Vrel}{tau_rel} Carel \frac{1.0}{1.0 + {(\frac{Km_Carel}{Cai})}^{2.0}} i_Ca_tr = \frac{2.0 F Vrel}{tau_tr} Caup

Component: ion_concentrations

\frac{d}{d time} (Cai) = - (\frac{i_CaL + i_CaT + i_Ca_up - (2.0 i_NaCa + i_Ca_rel)}{2.0 F Vi}) \frac{d}{d time} (Caup) = \frac{i_Ca_up - i_Ca_tr}{2.0 F Vup} \frac{d}{d time} (Carel) = \frac{i_Ca_tr - i_Ca_rel}{2.0 F Vrel} \frac{d}{d time} (Cao) = \frac{i_CaL + i_CaT - (2.0 i_NaCa)}{2.0 F Ve} + \frac{Cab - Cao}{tau_b} \frac{d}{d time} (Nai) = - (\frac{i_Kr_Na + i_Ks_Na + i_Na + 3.0 i_NaCa + i_bNa + i_fNa + 3.0 i_NaK}{F Vi}) \frac{d}{d time} (Nao) = \frac{i_Kr_Na + i_Ks_Na + i_Na + 3.0 i_NaCa + i_bNa + i_fNa + 3.0 i_NaK}{F Ve} + \frac{Nab - Nao}{tau_b} \frac{d}{d time} (Ki) = - (\frac{i_Kr_K + i_Ks_K + i_bK + i_fK + i_to - (2.0 i_NaK)}{F Vi}) \frac{d}{d time} (Ko) = \frac{i_Kr_K + i_Ks_K + i_bK + i_fK + i_to - (2.0 i_NaK)}{F Ve} + \frac{Kb - Ko}{tau_b} \frac{d}{d time} (Cli) = \frac{i_bCl}{F Vi} \frac{d}{d time} (Clo) = \frac{i_bCl}{F Ve} + \frac{Clb - Clo}{tau_b}