Model Mathematics

RTONF = \frac{R T}{F} \frac{d}{d time} (V) = \frac{- (i_f + i_K + i_K1 + i_Na_b + i_Ca_b + i_p + i_NaCa + i_Na + i_si)}{C}

E_Na = RTONF \ln (\frac{Nao}{Nai}) E_K = RTONF \ln (\frac{Kc}{Ki}) i_fNa = \frac{y^{2} Kc}{Kc + Km_f} g_f_Na (V - E_Na) i_fK = \frac{y^{2} Kc}{Kc + Km_f} g_f_K (V - E_K) i_f = i_fNa + i_fK

alpha_y = 0.014 ⅇ^{\frac{- (V + shifty)}{16}} beta_y = 9.75 ⅇ^{\frac{V + shifty}{19}} \frac{d}{d time} (y) = speed_y (alpha_y (1 - y) - (beta_y y))

I_K = \frac{i_K_max (Ki - (Kc ⅇ^{\frac{- V}{RTONF}}))}{140} i_K = x I_K

alpha_x = 2.1 ⅇ^{\frac{V}{28}} beta_x = 0.96 ⅇ^{\frac{- V}{24}} \frac{d}{d time} (x) = alpha_x (1 - x) - (beta_x x)

i_K1 = \frac{\frac{g_K1 Kc}{Kc + Km_K1} (V - E_K)}{1 + ⅇ^{\frac{(V + 10 - E_K) 2}{RTONF}}}

i_Na_b = g_Nab (V - E_Na)

E_Ca = 0.5 RTONF \ln (\frac{Cao}{Cai}) i_Ca_b = g_Cab (V - E_Ca)

i_p = \frac{\frac{I_p Kc}{K_mK + Kc} Nai}{K_mNa + Nai}

i_NaCa = \frac{K_NaCa (ⅇ^{\frac{gamma (n_NaCa - 2) V}{RTONF}} {Nai}^{n_NaCa} Cao - (ⅇ^{\frac{(gamma - 1) (n_NaCa - 2) V}{RTONF}} {Nao}^{n_NaCa} Cai))}{(1 + d_NaCa (Cai {Nao}^{n_NaCa} + Cao {Nai}^{n_NaCa})) (1 + \frac{Cai}{0.0069})}

E_mh = RTONF \ln (\frac{Nao + 0.12 Kc}{Nai + 0.12 Ki}) i_Na = g_Na m^{3} h (V - E_mh)

E0_m = V + 41 alpha_m = \{\begin{matrix} 2000 if | E0_m | < delta_m \\ \frac{200 E0_m}{1 - (ⅇ^{(- 0.1) E0_m})} otherwise \end{matrix} beta_m = 8000 ⅇ^{(- 0.056) (V + 66)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = 20 ⅇ^{(- 0.125) (V + 75)} beta_h = \frac{2000}{320 ⅇ^{(- 0.1) (V + 75)} + 1} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

i_siCa = \frac{4 P_si (V - 50)}{RTONF (1 - (ⅇ^{\frac{(- 1) (V - 50) 2}{RTONF}}))} (Cai ⅇ^{\frac{100}{RTONF}} - (Cao ⅇ^{\frac{(- 2) (V - 50)}{RTONF}})) d f f2 i_siK = \frac{0.01 P_si (V - 50)}{RTONF (1 - (ⅇ^{\frac{(- 1) (V - 50)}{RTONF}}))} (Ki ⅇ^{\frac{50}{RTONF}} - (Kc ⅇ^{\frac{(- 1) (V - 50)}{RTONF}})) d f f2 i_siNa = \frac{0.01 P_si (V - 50)}{RTONF (1 - (ⅇ^{\frac{(- 1) (V - 50)}{RTONF}}))} (Nai ⅇ^{\frac{50}{RTONF}} - (Nao ⅇ^{\frac{(- 1) (V - 50)}{RTONF}})) d f f2 i_si = i_siCa + i_siK + i_siNa

E0_d = V + 24 - 5 alpha_d = \{\begin{matrix} 120 if | E0_d | < delta_d \\ \frac{30 E0_d}{1 - (ⅇ^{\frac{(- 1) E0_d}{4}})} otherwise \end{matrix} beta_d = \{\begin{matrix} 120 if | E0_d | < delta_d \\ \frac{12 E0_d}{ⅇ^{\frac{E0_d}{10}} - 1} otherwise \end{matrix} \frac{d}{d time} (d) = alpha_d (1 - d) - (beta_d d)

E0_f = V + 34 alpha_f = \{\begin{matrix} 25 if | E0_f | < delta_f \\ \frac{6.25 E0_f}{ⅇ^{\frac{E0_f}{4}} - 1} otherwise \end{matrix} beta_f = \frac{50}{1 + ⅇ^{\frac{(- 1) (V + 34)}{4}}} \frac{d}{d time} (f) = alpha_f (1 - f) - (beta_f f)

beta_f2 = \frac{Cai alpha_f2}{K_mf2} \frac{d}{d time} (f2) = alpha_f2 - (f2 (alpha_f2 + beta_f2))

V_Cell = 3.141592654 {radius}^{2} length Vi = V_Cell (1 - V_e_ratio) \frac{d}{d time} (Nai) = \frac{(- 1) (i_Na + i_Na_b + i_fNa + i_siNa + i_p 3 + \frac{i_NaCa n_NaCa}{n_NaCa - 2})}{1 Vi F}

i_up = \frac{2 1 Vi F}{1 tau_up Ca_up_max} Cai (Ca_up_max - Ca_up) i_tr = \frac{2 1 V_rel F}{1 tau_rep} p (Ca_up - Ca_rel) E0_p = V + 34 - (- 30) alpha_p = \frac{0.625 E0_p}{ⅇ^{\frac{E0_p}{4}} - 1} beta_p = \frac{5}{1 + ⅇ^{\frac{(- 1) E0_p}{4}}} i_rel = \frac{\frac{2 1 V_rel F}{1 tau_rel} Ca_rel {Cai}^{rCa}}{{Cai}^{rCa} + {K_mCa}^{rCa}} \frac{d}{d time} (p) = alpha_p (1 - p) - (beta_p p) \frac{d}{d time} (Ca_up) = \frac{1 (i_up - i_tr)}{2 1 V_up F} V_up = Vi 0.05 \frac{d}{d time} (Ca_rel) = \frac{1 (i_tr - i_rel)}{2 1 V_rel F} V_rel = Vi 0.02 \frac{d}{d time} (Cai) = \frac{(- 1) ((i_siCa + i_Ca_b - (\frac{2 i_NaCa}{n_NaCa - 2}) - i_rel) + i_up)}{2 1 Vi F}

i_mK = i_K1 + i_K + i_fK + i_siK - (2 i_p) \frac{d}{d time} (Ki) = \frac{(- 1) i_mK}{1 Vi F}