Model Mathematics

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

i_Kdr = g_Kdr (gamma_KSS + (xi1 + xi2) (1 - gamma_KSS)) {xa}^{2} (V - E_K)

\frac{d}{d time} (xa) = \frac{xa_infinity - xa}{tau_xa} xa_infinity = \frac{1}{1 + ⅇ^{\frac{5.5 - V}{6}}} tau_xa = 0.003262 - (0.00003552 V)

\frac{d}{d time} (xi1) = \frac{xi1_infinity - xi1}{tau_xi1} xi1_infinity = \frac{1}{1 + ⅇ^{\frac{V + 4.3}{7.5}}}

\frac{d}{d time} (xi2) = \frac{xi2_infinity - xi2}{tau_xi2} xi2_infinity = \frac{1}{1 + ⅇ^{\frac{V + 4.3}{7.5}}}

i_CaL = 4 P_CaL d f ICaL_perm2 ICaL_perm2 = ICaL_perm1 (Cai ⅇ^{\frac{V_surf}{\frac{0.5 R T}{F}}} - (Cao ⅇ^{\frac{- (V - V_surf)}{\frac{0.5 R T}{F}}})) ICaL_perm1 = \frac{(V - V_surf) F}{R T (1 - (ⅇ^{\frac{(- (V - V_surf)) 2 F}{R T}}))}

alpha_d = \frac{30 (V + 18)}{1 - (ⅇ^{(- 0.25) (V + 18)})} beta_d = \frac{12 (V + 18)}{ⅇ^{0.1 (V + 18)} - 1} \frac{d}{d time} (d) = alpha_d - ((alpha_d + beta_d) d)

f_infinity = \frac{1}{1 + ⅇ^{\frac{23 + V}{6.6}}} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

i_KCa = g_KCa xCa1 B (V - E_K)

\frac{d}{d time} (xCa1) = \frac{xCa1_infinity - xCa1}{tau_xCa1} xCa1_infinity = \frac{{Cai}^{2} + K4 Cai}{{Cai}^{2} + K4 Cai (1 + \frac{alpha}{beta}) + \frac{K4 K2 alpha}{beta}} K4 = 0.0000125 ⅇ^{\frac{(- 1.99) V F}{R T}} K2 = 0.000275 ⅇ^{\frac{(- 1.51) V F}{R T}}

K1 = 0.85 ⅇ^{0.04 V} K_1 = 0.24 ⅇ^{(- 0.012) V} \frac{d}{d time} (B) = \{\begin{matrix} K1 Cai xCa1 - (K_1 B) if Cai \geq 0.1 \\ 1 otherwise \end{matrix}

i_ClCa = \frac{g_Cl (V - E_Cl) 1}{1 + {(\frac{CaCT}{Cai})}^{h}}

i_cationic = i_nsNa + i_nsK + i_nsCa i_nsNa = \frac{\frac{ns alpha_Ca P_nsCa V F^{2}}{R T} (0.75 Nai ⅇ^{\frac{V F}{R T}} - (0.75 Nao))}{ⅇ^{\frac{V F}{R T}} - 1} i_nsK = \frac{\frac{ns alpha_Ca P_nsCa V F^{2}}{R T} (0.75 Ki ⅇ^{\frac{V F}{R T}} - (0.75 Ko))}{ⅇ^{\frac{V F}{R T}} - 1} i_nsCa = \frac{\frac{ns alpha_Ca P_nsCa 4 V F^{2}}{R T} (1.5 Cai ⅇ^{\frac{V F}{R T}} - (1.5 Cao))}{ⅇ^{\frac{V F}{R T}} - 1} alpha_Ca = \frac{1}{1 + {(\frac{Km_nsCa}{Cai})}^{3}}

ns_infinity = \frac{1}{1 + ⅇ^{\frac{V - 69.8}{- 11.9}}} \frac{d}{d time} (ns) = \frac{ns_infinity - ns}{tau_ns} tau_ns = 0.20025 - (0.00089875 V)

i_pCa = \frac{i_pCamax Cai}{Kmp_Ca + Cai}

i_NaK = \frac{\frac{i_NaKmax Ko}{Ko + Km_K} Nai}{Nai + Km_Na}

i_bCa = g_bCa (V - E_Ca)

i_bK = g_bK (V - E_K)

i_bNa = g_bNa (V - E_Na)

\frac{d}{d time} (V) = \frac{- 1}{cap} (i_CaL + i_Kdr + i_KCa + i_ClCa + i_cationic + i_NaK + i_pCa + i_bCa + i_bK + i_bNa)

Cai = C + upeak + peak + plat upeak = \{\begin{matrix} up if time < x1 + x2 \\ 0 otherwise \end{matrix} up = \{\begin{matrix} 0 if time < x1 \\ \frac{(time - x1) A}{x2} otherwise \end{matrix} peak = \{\begin{matrix} 0 if time < x0 \\ A ⅇ^{\frac{- (time - x0)}{t36}} otherwise \end{matrix} x0 = x1 + x2 plat = \frac{B}{1 + ⅇ^{\frac{time - (x1 + p + ds)}{n}}} - (\frac{B}{1 + ⅇ^{\frac{time - (x1 + p)}{n}}})