Model Mathematics

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

E_Ca = \frac{R T}{2.0 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 = w g_CaL d f f2 (E + (- E_Ca) + 75.0)

d_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{E + 6.6}{-6.6}}} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

f_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{E + 25.0}{6.0}}} tau_f = 0.031 + \frac{1.0}{1.0 + ⅇ^{\frac{E + 37.6}{8.1}}} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

\frac{d}{d time} (f2) = alpha_f2 (1.0 - f2) - (beta_f2 Cai f2)

alpha_w = \frac{0.0033 ACh + 0.0000006}{ACh + 0.000036} beta_w = \frac{0.0037 ACh}{ACh + 0.000036} \frac{d}{d time} (w) = alpha_w (1.0 - w) - (beta_w w)

i_CaT = g_CaT d f (E + (- E_Ca) + 75.0)

d_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{E + 23.0}{-6.1}}} tau_d = 0.0006 + \frac{0.0054}{1.0 + ⅇ^{0.03 (E + 100.0)}} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

f_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{E + 75.0}{6.6}}} tau_f = 0.001 + \frac{0.04}{1.0 + ⅇ^{0.08 (E + 65.0)}} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

i_Na = g_Na m^{3.0} h (E - E_Na)

alpha_m = \frac{200.0 (E + 34.3)}{1.0 - (ⅇ^{(-0.09) (E + 34.3)})} beta_m = 8000.0 ⅇ^{(-0.15) (E + 56.2)} \frac{d}{d time} (m) = alpha_m (1.0 - m) - (beta_m m)

alpha_h = 32.4 ⅇ^{(-0.14) (E + 93.4)} beta_h = \frac{709.0}{1.0 + 4.2 ⅇ^{(-0.06) (E + 45.4)}} \frac{d}{d time} (h) = alpha_h (1.0 - h) - (beta_h h)

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

x_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{E + 25.1}{-7.4}}} tau_x = \frac{1.0}{17.0 ⅇ^{0.0398 E} + 0.211 ⅇ^{(-0.051) E}} \frac{d}{d time} (x) = \frac{x_infinity - x}{tau_x}

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

alpha_v = \frac{4.0 ACh}{ACh + 0.00008} beta_v = \frac{0.00001}{ACh + 0.00008} \frac{d}{d time} (v) = alpha_v (1.0 - v) - (beta_v v) alpha_y = \frac{0.36 (E + 137.8 + v^{4.0} E_shift_max)}{ⅇ^{0.0666 (E + 137.8 + v^{4.0} E_shift_max)} - 1.0} beta_y = \frac{0.1 (E + 76.3 + v^{4.0} E_shift_max)}{1.0 - (ⅇ^{(-0.21) (E + 76.3 + v^{4.0} E_shift_max)})} \frac{d}{d time} (y) = alpha_y (1.0 - y) - (beta_y y)

i_p = i_pmax \frac{Nai}{Nai + KmNa} \frac{Ko}{Ko + KmK} (1.0 - ({(\frac{E - 40.0}{211.0})}^{2.0}))

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{\frac{Cai}{Kci} ⅇ^{\frac{(- Qci) E F}{R T}}}{di} k14 = \frac{(\frac{{Nai}^{2.0}}{K1ni K2ni} + \frac{{Nai}^{3.0}}{K1ni K2ni K3ni}) ⅇ^{\frac{Qn E F}{R T}}}{di} k41 = ⅇ^{\frac{- (Qn E F)}{2.0 R T}} di = 1.0 + \frac{Cai}{Kci} + \frac{Cai}{Kci} ⅇ^{\frac{- (Qci E F)}{R T}} + \frac{Cai Nai}{Kci Kcni} + \frac{Nai}{K1ni} + \frac{{Nai}^{2.0}}{K1ni K2ni} + \frac{{Nai}^{3.0}}{K1ni K2ni K3ni} k34 = \frac{Nao}{K3no + Nao} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{Qco E F}{R T}}}{do} k23 = \frac{\frac{{Nao}^{2.0}}{K1no K2no} \frac{{Nao}^{3.0}}{K1no K2no K3no} ⅇ^{\frac{- (Qn E F)}{2.0 R T}}}{do} k32 = ⅇ^{\frac{Qn E F}{2.0 R T}} do = 1.0 + \frac{Cao}{Kco} + \frac{Cao}{Kco} ⅇ^{\frac{- (Qco E F)}{R T}} + \frac{Nao}{K1no} + \frac{{Nao}^{2.0}}{K1no K2no} + \frac{{Nao}^{3.0}}{K1no K2no K3no}

i_bNa = g_Nab (E - E_Na)

i_KACh = (KbK + u^{4.0} KKACh) {Ko}^{0.41} (Ki - (Ko ⅇ^{\frac{(- E) F}{R T}}))

alpha_u = \frac{4.7 ACh}{ACh + 0.00012} beta_u = \frac{0.00027}{ACh + 0.00012} \frac{d}{d time} (u) = alpha_u (1.0 - u) - (beta_u u)

\frac{d}{d time} (ACh_ms) = \frac{k_H}{F_ms} ACh + \frac{k_E}{F_ms} ACh_ex \frac{d}{d time} (ACh) = (- k_H) ACh - (k_D (ACh - ACh_ex)) \frac{d}{d time} (ACh_ex) = (- (\frac{k_E}{F_ex})) ACh - (\frac{k_D}{F_ms} (ACh_ex - ACh))

i_up = i_up_max \frac{{Cai}^{2.0}}{{Cai}^{2.0} + {KmCaup}^{2.0}} i_tr = alpha_tr Caup i_rel = alpha_rel Carel \frac{{Cai}^{2.0}}{{Cai}^{2.0} + {KmCarel}^{2.0}} alpha_rel = \frac{2.0 V_rel F}{tau_rel} alpha_tr = \frac{2.0 V_rel F}{tau_rel}

\frac{d}{d time} (Nai) = \frac{- (i_bNa + i_fNa + i_Na + 3.0 i_p + 3.0 i_NaCa + i_KNa)}{F V_i} \frac{d}{d time} (Nao) = \frac{i_bNa + i_fNa + i_Na + 3.0 i_p + 3.0 i_NaCa + i_KNa}{F V_e} + \frac{Nab - Nao}{tau_b} \frac{d}{d time} (Ki) = \frac{- (i_KK + i_fK + (-2.0) i_p + i_KACh)}{F V_i} \frac{d}{d time} (Ko) = \frac{i_KK + i_fK + (-2.0) i_p + i_KACh}{F V_e} + \frac{Kb - Ko}{tau_b} \frac{d}{d time} (Cai) = \frac{- (i_CaL + i_CaT + (-2.0) i_NaCa + i_up + (- i_rel))}{2.0 F V_i} \frac{d}{d time} (Cao) = \frac{i_CaL + i_CaT + (-2.0) i_NaCa}{2.0 F V_e} + \frac{Cab - Cao}{tau_b} \frac{d}{d time} (Caup) = \frac{i_up - i_tr}{2.0 V_up F} \frac{d}{d time} (Carel) = \frac{i_tr - i_rel}{2.0 V_rel F}