Model Mathematics

i_tot = i_CaL + i_CaT + i_Na + i_K + i_f + i_p + i_NaCa + i_bNa + i_bK \frac{d}{d time} (E) = \frac{- i_tot}{C}

E_Ca = \frac{R T}{2 F} \ln (\frac{Cao}{Cai}) E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai}) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki})

i_CaL = g_CaL dL fL fL2 ((E - E_Ca) + 75)

dL_infinity = \frac{1}{1 + ⅇ^{\frac{E + 6.6}{- 6.6}}} \frac{d}{d time} (dL) = \frac{dL_infinity - dL}{tau_dL}

fL_infinity = \frac{1}{1 + ⅇ^{\frac{E + 25}{6}}} tau_fL = 0.031 + \frac{1}{1 + ⅇ^{\frac{E + 37.6}{8.1}}} \frac{d}{d time} (fL) = \frac{fL_infinity - fL}{tau_fL}

\frac{d}{d time} (fL2) = alpha_fL2 (1 - fL2) - (beta_fL2 Cai fL2)

i_CaT = g_CaT dT fT ((E - E_Ca) + 75)

dT_infinity = \frac{1}{1 + ⅇ^{\frac{E + 23}{- 6.1}}} tau_dT = 0.0006 + \frac{0.0054}{1 + ⅇ^{0.03 (E + 100)}} \frac{d}{d time} (dT) = \frac{dT_infinity - dT}{tau_dT}

fT_infinity = \frac{1}{1 + ⅇ^{\frac{E + 75}{6.6}}} tau_fT = 0.001 + \frac{0.04}{1 + ⅇ^{0.08 (E + 65)}} \frac{d}{d time} (fT) = \frac{fT_infinity - fT}{tau_fT}

i_Na = g_Na m^{3} h (E - E_Na)

alpha_m = \frac{200 (E + 34.3)}{1 - (ⅇ^{(- 0.09) (E + 34.3)})} beta_m = 8000 ⅇ^{(- 0.15) (E + 56.2)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = 32.4 ⅇ^{(- 0.14) (E + 93.4)} beta_h = \frac{709}{1 + 4.2 ⅇ^{(- 0.06) (E + 45.4)}} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

i_KK = x Kk {(\frac{Ko}{1})}^{0.59} (Ki - (Ko ⅇ^{\frac{(- E) F}{R T}})) i_KNa = x Kk P_KNa {(\frac{Ko}{1})}^{0.59} (Nai - (Nao ⅇ^{\frac{(- E) F}{R T}})) i_K = i_KK + i_KNa

x_infinity = \frac{1}{1 + ⅇ^{\frac{E + 25.1}{- 7.4}}} tau_x = \frac{1}{17 ⅇ^{0.0398 E} + 0.211 ⅇ^{(- 0.051) E}} \frac{d}{d time} (x) = \frac{x_infinity - x}{tau_x}

i_fNa = \frac{y {Ko}^{1.83}}{{Ko}^{1.83} + {Kmf}^{1.83}} g_fNa (E - E_Na) i_fK = \frac{y {Ko}^{1.83}}{{Ko}^{1.83} + {Kmf}^{1.83}} g_fK (E - E_K) i_f = i_fK + i_fNa

alpha_y = \frac{0.36 (E + 137.8)}{ⅇ^{0.066 (E + 137.8)} - 1} beta_y = \frac{0.1 (E + 76.3)}{1 - (ⅇ^{(- 0.21) (E + 76.3)})} \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

i_p = \frac{\frac{i_pmax Nai}{Nai + KmNa} Ko}{Ko + KmK} (1 - ({(\frac{E - 40}{211})}^{2}))

do = 1 + \frac{Cao}{Kco} + \frac{Cao}{Kco} ⅇ^{\frac{Qco E F}{R T}} + \frac{Nao}{K1no} + \frac{{Nao}^{2}}{K1no K2no} + \frac{{Nao}^{3}}{K1no K2no K3no} k32 = ⅇ^{\frac{Qn E F}{2 R T}} k23 = \frac{(\frac{{Nao}^{2}}{K1no K2no} + \frac{{Nao}^{3}}{K1no K2no K3no}) ⅇ^{\frac{(- Qn) E F}{2 R T}}}{do} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{(- Qco) E F}{R T}}}{do} k34 = \frac{Nao}{K3no + Nao} di = 1 + \frac{Cai}{Kci} + \frac{Cai}{Kci} ⅇ^{\frac{(- Qci) E F}{R T}} + \frac{Cai Nai}{Kci Kcni} + \frac{Nai}{K1ni} + \frac{{Nai}^{2}}{K1ni K2ni} + \frac{{Nai}^{3}}{K1ni K2ni K3ni} k41 = ⅇ^{\frac{(- Qn) E F}{2 R T}} k14 = \frac{(\frac{{Nai}^{2}}{K1ni K2ni} + \frac{{Nai}^{3}}{K1ni K2ni K3ni}) ⅇ^{\frac{Qn E F}{2 R T}}}{di} k12 = \frac{\frac{Cai}{Kci} ⅇ^{\frac{(- Qci) E F}{R T}}}{di} k43 = \frac{Nai}{K3ni + Nai} 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) i_NaCa = \frac{kNaCa (x2 k21 - (x1 k12))}{x1 + x2 + x3 + x4}

i_bNa = g_Nab (E - E_Na)

i_bK = KbK {(\frac{Ko}{1})}^{0.41} (Ki - (Ko ⅇ^{\frac{(- E) F}{R T}}))

V_rel = 0.006 V_i V_up = 0.014 V_i i_up = \frac{i_up_max {Cai}^{2}}{{Cai}^{2} + {KmCaup}^{2}} i_tr = \frac{2 1 V_rel F}{1 tau_tr} Caup i_rel = \frac{\frac{2 1 V_rel F}{1 tau_rel} Carel {Cai}^{2}}{{Cai}^{2} + {KmCarel}^{2}}

V_e = 0.2 V_i \frac{d}{d time} (Nai) = \frac{(- 1) (i_bNa + i_fNa + i_Na + 3 i_p + 3 i_NaCa + i_KNa)}{F 1 V_i} \frac{d}{d time} (Nao) = \frac{1 (i_bNa + i_fNa + i_Na + 3 i_p + 3 i_NaCa + i_KNa)}{F 1 V_e} + \frac{Nab - Nao}{tau_b} \frac{d}{d time} (Ki) = \frac{(- 1) ((i_KK + i_fK - (2 i_p)) + i_bK)}{F 1 V_i} \frac{d}{d time} (Ko) = \frac{1 ((i_KK + i_fK - (2 i_p)) + i_bK)}{F 1 V_e} + \frac{Kb - Ko}{tau_b} \frac{d}{d time} (Cai) = \frac{(- 1) ((i_CaL + i_CaT - (2 i_NaCa)) + i_up + (- i_rel))}{2 F 1 V_i} \frac{d}{d time} (Cao) = \frac{1 (i_CaL + i_CaT - (2 i_NaCa))}{2 F 1 V_e} + \frac{Cab - Cao}{tau_b} \frac{d}{d time} (Caup) = \frac{1 (i_up - i_tr)}{2 1 V_up F} \frac{d}{d time} (Carel) = \frac{1 (i_tr - i_rel)}{2 1 V_rel F}