Model Mathematics

\frac{d}{d time} (Vm) = - (i_K1 + i_to + i_Kr + i_Ks + i_CaL + i_NaK + i_Na + i_NaCa + i_PCa + i_f + i_b_Na + i_b_Ca)

E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai}) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) E_Ks = \frac{R T}{F} \ln (\frac{Ko + PkNa Nao}{Ki + PkNa Nai}) E_Ca = \frac{0.5 R T}{F} \ln (\frac{Cao}{Cai})

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

m_inf = \frac{1}{{(1 + ⅇ^{\frac{(- Vm) 1000 - 34.1}{5.9}})}^{\frac{1}{3}}} alpha_m = \frac{1}{1 + ⅇ^{\frac{(- Vm) 1000 - 60}{5}}} beta_m = \frac{0.1}{1 + ⅇ^{\frac{Vm 1000 + 35}{5}}} + \frac{0.1}{1 + ⅇ^{\frac{Vm 1000 - 50}{200}}} tau_m = \frac{1 alpha_m beta_m}{1000} \frac{d}{d time} (m) = \frac{m_inf - m}{tau_m}

h_inf = \frac{1}{\sqrt{1 + ⅇ^{\frac{Vm 1000 + 72.1}{5.7}}}} alpha_h = \{\begin{matrix} 0.057 ⅇ^{\frac{- (Vm 1000 + 80)}{6.8}} if Vm < - 0.04 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 2.7 ⅇ^{0.079 Vm 1000} + 3.1 10^{5} ⅇ^{0.3485 Vm 1000} if Vm < - 0.04 \\ \frac{0.77}{0.13 (1 + ⅇ^{\frac{Vm 1000 + 10.66}{- 11.1}})} otherwise \end{matrix} tau_h = \{\begin{matrix} \frac{1.5}{(alpha_h + beta_h) 1000} if Vm < - 0.04 \\ \frac{2.542}{1000} otherwise \end{matrix} \frac{d}{d time} (h) = \frac{h_inf - h}{tau_h}

j_inf = \frac{1}{\sqrt{1 + ⅇ^{\frac{Vm 1000 + 72.1}{5.7}}}} alpha_j = \{\begin{matrix} \frac{((- 25428) ⅇ^{0.2444 Vm 1000} - (6.948 10^{- 6} ⅇ^{(- 0.04391) Vm 1000})) (Vm 1000 + 37.78)}{1 + ⅇ^{0.311 (Vm 1000 + 79.23)}} if Vm < - 0.04 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.02424 ⅇ^{(- 0.01052) Vm 1000}}{1 + ⅇ^{(- 0.1378) (Vm 1000 + 40.14)}} if Vm < - 0.04 \\ \frac{0.6 ⅇ^{0.057 Vm 1000}}{1 + ⅇ^{(- 0.1) (Vm 1000 + 32)}} otherwise \end{matrix} tau_j = \frac{7}{(alpha_j + beta_j) 1000} \frac{d}{d time} (j) = \frac{j_inf - j}{tau_j}

i_CaL = \frac{\frac{g_CaL 4 Vm F^{2}}{R T} (Cai ⅇ^{\frac{2 Vm F}{R T}} - (0.341 Cao))}{ⅇ^{\frac{2 Vm F}{R T}} - 1} d f1 f2 fCa

d_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm 1000 + 5.986)}{7}}} alpha_d = 0.25 + \frac{1.4}{1 + ⅇ^{\frac{(- Vm) 1000 - 35}{13}}} beta_d = \frac{1.4}{1 + ⅇ^{\frac{Vm 1000 + 5}{5}}} gamma_d = \frac{1}{1 + ⅇ^{\frac{(- Vm) 1000 + 50}{20}}} tau_d = \frac{(alpha_d beta_d + gamma_d) 1}{1000} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

f1_inf = \frac{1}{1 + ⅇ^{\frac{Vm 1000 + 25.226}{3}}} tau_f1 = \frac{(20 + 1102.5 ⅇ^{- ({(\frac{{(Vm 1000 + 27)}^{2}}{15})}^{2})} + \frac{200}{1 + ⅇ^{\frac{13 - (Vm 1000)}{10}}} + \frac{180}{1 + ⅇ^{\frac{30 + Vm 1000}{10}}}) constf1}{1000} constf1 = \{\begin{matrix} 1 + 1433 (Cai - (50 1 e -6)) if f1_inf - f1 > 0 \\ 1 otherwise \end{matrix} \frac{d}{d time} (f1) = \frac{f1_inf - f1}{tau_f1}

f2_inf = 0.33 + \frac{0.67}{1 + ⅇ^{\frac{Vm 1000 + 31.226}{4}}} constf2 = 2 tau_f2 = \frac{(600 ⅇ^{\frac{- ({(Vm 1000 + 25)}^{2})}{170}} + \frac{31}{1 + ⅇ^{\frac{25 - (Vm 1000)}{10}}} + \frac{16}{1 + ⅇ^{\frac{30 + Vm 1000}{10}}}) constf2}{1000} \frac{d}{d time} (f2) = \frac{f2_inf - f2}{tau_f2}

alpha_fCa = \frac{1}{1 + {(\frac{Cai}{0.0006})}^{8}} beta_fCa = \frac{0.1}{1 + ⅇ^{\frac{Cai - 0.0009}{0.0001}}} gamma_fCa = \frac{0.3}{1 + ⅇ^{\frac{Cai - 0.00075}{0.0008}}} fCa_inf = \frac{alpha_fCa + beta_fCa + gamma_fCa}{1.3156} constfCa = \{\begin{matrix} 0 if Vm > - 0.06 \land fCa_inf > fCa \\ 1 otherwise \end{matrix} \frac{d}{d time} (fCa) = \frac{constfCa (fCa_inf - fCa)}{tau_fCa}

i_Kr = g_Kr (Vm - E_K) Xr1 Xr2 \sqrt{\frac{Ko}{5.4}}

V_half = 1000 (\frac{(- R) T}{F Q} \ln (\frac{{(1 + \frac{Cao}{2.6})}^{4}}{L0 {(1 + \frac{Cao}{0.58})}^{4}}) - 0.019) Xr1_inf = \frac{1}{1 + ⅇ^{\frac{V_half - (Vm 1000)}{4.9}}} alpha_Xr1 = \frac{450}{1 + ⅇ^{\frac{- 45 - (Vm 1000)}{10}}} beta_Xr1 = \frac{6}{1 + ⅇ^{\frac{30 + Vm 1000}{11.5}}} tau_Xr1 = \frac{1 alpha_Xr1 beta_Xr1}{1000} \frac{d}{d time} (Xr1) = \frac{Xr1_inf - Xr1}{tau_Xr1}

Xr2_infinity = \frac{1}{1 + ⅇ^{\frac{Vm 1000 + 88}{50}}} alpha_Xr2 = \frac{3}{1 + ⅇ^{\frac{- 60 - (Vm 1000)}{20}}} beta_Xr2 = \frac{1.12}{1 + ⅇ^{\frac{(- 60) + Vm 1000}{20}}} tau_Xr2 = \frac{1 alpha_Xr2 beta_Xr2}{1000} \frac{d}{d time} (Xr2) = \frac{Xr2_infinity - Xr2}{tau_Xr2}

i_Ks = g_Ks (Vm - E_Ks) {Xs}^{2} (1 + \frac{0.6}{1 + {(\frac{3.8 0.00001}{Cai})}^{1.4}})

Xs_infinity = \frac{1}{1 + ⅇ^{\frac{(- Vm) 1000 - 20}{16}}} alpha_Xs = \frac{1100}{\sqrt{1 + ⅇ^{\frac{- 10 - (Vm 1000)}{6}}}} beta_Xs = \frac{1}{1 + ⅇ^{\frac{(- 60) + Vm 1000}{20}}} tau_Xs = \frac{1 alpha_Xs beta_Xs}{1000} \frac{d}{d time} (Xs) = \frac{Xs_infinity - Xs}{tau_Xs}

alpha_K1 = \frac{3.91}{1 + ⅇ^{0.5942 (Vm 1000 - (E_K 1000) - 200)}} beta_K1 = \frac{(- 1.509) ⅇ^{0.0002 ((Vm 1000 - (E_K 1000)) + 100)} + ⅇ^{0.5886 (Vm 1000 - (E_K 1000) - 10)}}{1 + ⅇ^{0.4547 (Vm 1000 - (E_K 1000))}} XK1_inf = \frac{alpha_K1}{alpha_K1 + beta_K1} i_K1 = g_K1 XK1_inf (Vm - E_K) \sqrt{\frac{Ko}{5.4}}

i_f = g_f Xf (Vm - E_f)

Xf_infinity = \frac{1}{1 + ⅇ^{\frac{Vm 1000 + 77.85}{5}}} tau_Xf = \frac{\frac{1900}{1 + ⅇ^{\frac{Vm 1000 + 15}{10}}}}{1000} \frac{d}{d time} (Xf) = \frac{Xf_infinity - Xf}{tau_Xf}

i_b_Na = g_b_Na (Vm - E_Na)

i_b_Ca = g_b_Ca (Vm - E_Ca)

i_NaK = \frac{\frac{\frac{PNaK Ko}{Ko + Km_K} Nai}{Nai + Km_Na}}{1 + 0.1245 ⅇ^{\frac{(- 0.1) Vm F}{R T}} + 0.0353 ⅇ^{\frac{(- Vm) F}{R T}}}

i_NaCa = \frac{kNaCa (ⅇ^{\frac{gamma Vm F}{R T}} {Nai}^{3} Cao - (ⅇ^{\frac{(gamma - 1) Vm F}{R T}} {Nao}^{3} Cai alpha))}{({KmNai}^{3} + {Nao}^{3}) (KmCa + Cao) (1 + Ksat ⅇ^{\frac{(gamma - 1) Vm F}{R T}})}

i_PCa = \frac{g_PCa Cai}{Cai + KPCa}

i_to = g_to (Vm - E_K) q r

q_inf = \frac{1}{1 + ⅇ^{\frac{Vm 1000 + 53}{13}}} tau_q = \frac{6.06 + \frac{39.102}{0.57 ⅇ^{(- 0.08) (Vm 1000 + 44)} + 0.065 ⅇ^{0.1 (Vm 1000 + 45.93)}}}{1000} \frac{d}{d time} (q) = \frac{q_inf - q}{tau_q}

r_inf = \frac{1}{1 + ⅇ^{\frac{- (Vm 1000 - 22.3)}{18.75}}} tau_r = \frac{2.75352 + \frac{14.40516}{1.037 ⅇ^{0.09 (Vm 1000 + 30.61)} + 0.369 ⅇ^{(- 0.12) (Vm 1000 + 23.84)}}}{1000} \frac{d}{d time} (r) = \frac{r_inf - r}{tau_r}

\frac{d}{d time} (Nai) = \frac{(- Cm) (i_Na + i_b_Na + 3 i_NaK + 3 i_NaCa)}{F Vc 1 e -18}

i_rel = (c_rel + \frac{a_rel {Ca_SR}^{2}}{{b_rel}^{2} + {Ca_SR}^{2}}) d g 0.0556 i_up = \frac{VmaxUp}{1 + \frac{{Kup}^{2}}{{Cai}^{2}}} i_leak = (Ca_SR - Cai) V_leak g_inf = \{\begin{matrix} \frac{1}{1 + {(\frac{Cai}{0.00035})}^{6}} if Cai ≦ 0.00035 \\ \frac{1}{1 + {(\frac{Cai}{0.00035})}^{16}} otherwise \end{matrix} const2 = \{\begin{matrix} 0 if g_inf > g \land Vm > - 0.06 \\ 1 otherwise \end{matrix} \frac{d}{d time} (g) = \frac{const2 (g_inf - g)}{tau_g} Cai_bufc = \frac{1}{1 + \frac{Buf_C Kbuf_C}{{(Cai + Kbuf_C)}^{2}}} Ca_SR_bufSR = \frac{1}{1 + \frac{Buf_SR Kbuf_SR}{{(Ca_SR + Kbuf_SR)}^{2}}} \frac{d}{d time} (Cai) = Cai_bufc ((i_leak - i_up) + i_rel - (\frac{(i_CaL + i_b_Ca + i_PCa - (2 i_NaCa)) Cm}{2 Vc F 1 e -18})) \frac{d}{d time} (Ca_SR) = \frac{Ca_SR_bufSR Vc}{V_SR} (i_up - (i_rel + i_leak))