Model Mathematics

\frac{d}{d time} (V) = \frac{I_stim - (i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_NaCa + i_K1 + i_Kp + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_to + i_Cl_Ca)}{C}

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

alpha_m = \frac{0.32 (V + 47.13)}{1.0 - (ⅇ^{(-0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11.0}} \frac{d}{d time} (m) = alpha_m (1.0 - m) - (beta_m m)

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{80.0 + V}{-6.8}} if V < -40.0 \\ 0.0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000.0 ⅇ^{0.35 V} if V < -40.0 \\ \frac{1.0}{0.13 (1.0 + ⅇ^{\frac{- (V + 10.66)}{11.1}})} otherwise \end{matrix} \frac{d}{d time} (h) = alpha_h (1.0 - h) - (beta_h h)

alpha_j = \{\begin{matrix} ((-127140.0) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(-0.04391) V})) \frac{V + 37.78}{1.0 + ⅇ^{0.311 (V + 79.23)}} if V < -40.0 \\ 0.0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.1212 ⅇ^{(-0.01052) V}}{1.0 + ⅇ^{(-0.1378) (V + 40.14)}} if V < -40.0 \\ \frac{0.3 ⅇ^{(-0.0000002535) V}}{1.0 + ⅇ^{(-0.1) (V + 32.0)}} otherwise \end{matrix} \frac{d}{d time} (j) = alpha_j (1.0 - j) - (beta_j j)

i_CaCa = d f f_Ca I_CaCa i_CaNa = d f f_Ca I_CaNa i_CaK = d f f_Ca I_CaK I_CaCa = P_Ca {2.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Cai Cai ⅇ^{\frac{2.0 V F}{R T}} - (gamma_Cao Cao)}{ⅇ^{\frac{2.0 V F}{R T}} - 1.0} I_CaNa = P_Na {1.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Nai Nai ⅇ^{\frac{1.0 V F}{R T}} - (gamma_Nao Nao)}{ⅇ^{\frac{1.0 V F}{R T}} - 1.0} I_CaK = P_K {1.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Ki Ki ⅇ^{\frac{1.0 V F}{R T}} - (gamma_Ko Ko)}{ⅇ^{\frac{1.0 V F}{R T}} - 1.0} i_Ca_L = i_CaCa + i_CaK + i_CaNa

alpha_d = \frac{d_infinity}{tau_d} d_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V + 10.0}{6.24})}} tau_d = d_infinity \frac{1.0 - (ⅇ^{- (\frac{V + 10.0}{6.24})})}{0.035 (V + 10.0)} beta_d = \frac{1.0 - d_infinity}{tau_d} \frac{d}{d time} (d) = alpha_d (1.0 - d) - (beta_d d)

alpha_f = \frac{f_infinity}{tau_f} f_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 35.06}{8.6}}} + \frac{0.6}{1.0 + ⅇ^{\frac{50.0 - V}{20.0}}} tau_f = \frac{1.0}{0.0197 ⅇ^{- ({(0.0337 (V + 10.0))}^{2.0})} + 0.02} beta_f = \frac{1.0 - f_infinity}{tau_f} \frac{d}{d time} (f) = alpha_f (1.0 - f) - (beta_f f)

f_Ca = \frac{1.0}{1.0 + {(\frac{Cai}{Km_Ca})}^{2.0}}

i_Ca_T = g_Ca_T b g (V - E_Ca) E_Ca = \frac{R T}{2.0 F} \ln (\frac{Cao}{Cai})

\frac{d}{d time} (b) = \frac{b_infinity - b}{tau_b} b_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 48.0)}{6.1}}} tau_b = 0.1 + \frac{5.4}{1.0 + ⅇ^{\frac{V + 100.0}{33.0}}}

\frac{d}{d time} (g) = \frac{g_infinity - g}{tau_g} g_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 66.0}{6.6}}} tau_g = 8.0 + \frac{32.0}{1.0 + ⅇ^{\frac{V + 65.0}{5.0}}}

g_Kr = 0.02612 \sqrt{\frac{Ko}{5.4}} E_Kr = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_Kr = g_Kr Xr Rr (V - E_Kr)

\frac{d}{d time} (Xr) = \frac{Xr_infinity - Xr}{tau_Xr} Xr_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 50.0)}{7.5}}} tau_Xr = \frac{1.0}{\frac{0.00138 (V + 7.0)}{1.0 - (ⅇ^{(-0.123) (V + 7.0)})} + \frac{0.00061 (V + 10.0)}{ⅇ^{0.145 (V + 10.0) 0.145 - 1.0}}}

Rr = \frac{1.0}{1.0 + ⅇ^{\frac{V + 33.0}{22.4}}}

g_Ks = \frac{0.057 + 0.19}{1.0 + ⅇ^{\frac{(-7.2) + p_Ca}{0.6}}} p_Ca = (- (\log_{10} Cai)) + 3.0 E_Ks = \frac{R T}{F} \ln (\frac{Ko + P_NaK Nao}{Ki + P_NaK Nai}) i_Ks = g_Ks {Xs}^{2.0} (V - E_Ks)

\frac{d}{d time} (Xs) = \frac{Xs_infinity - Xs}{tau_Xs} Xs_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + (-1.5))}{16.7}}} tau_Xs = \frac{1.0}{\frac{0.0000719 (V + 30.0)}{1.0 - (ⅇ^{(-0.148) (V + 30.0)})} + \frac{0.000131 (V + 30.0)}{ⅇ^{0.0687 (V + 30.0)} - 1.0}}

i_NaCa = K_NaCa \frac{1.0}{{K_mNa}^{3.0} + {Nao}^{3.0}} \frac{1.0}{K_mCa + Cao} \frac{1.0}{1.0 + K_sat ⅇ^{(eta - 1.0) V \frac{F}{R T}}} (ⅇ^{eta V \frac{F}{R T}} {Nai}^{3.0} Cao - (ⅇ^{(eta - 1.0) V \frac{F}{R T}} {Nao}^{3.0} Cai))

g_K1 = 0.75 \sqrt{\frac{Ko}{5.4}} E_K1 = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_K1 = g_K1 K1_infinity (V - E_K1)

alpha_K1 = \frac{1.02}{1.0 + ⅇ^{0.2385 (V - E_K1 - 59.215)}} beta_K1 = \frac{0.49124 ⅇ^{\frac{V + 5.476 - E_K1}{12.45}} + ⅇ^{\frac{V - (E_K1 + 594.31)}{16.2}}}{1.0 + ⅇ^{(-0.5143) (V - (E_K1 + 4.753))}} K1_infinity = \frac{alpha_K1}{alpha_K1 + beta_K1}

E_Kp = E_K1 Kp = \frac{1.0}{1.0 + ⅇ^{\frac{7.488 - V}{5.98}}} i_Kp = g_Kp Kp (V - E_Kp)

i_p_Ca = I_pCa \frac{Cai}{K_mpCa + Cai}

E_NaN = E_Na i_Na_b = g_Nab (V - E_NaN)

E_CaN = \frac{R T}{2.0 F} \ln (\frac{Cao}{Cai}) i_Ca_b = g_Cab (V - E_CaN)

f_NaK = \frac{1.0}{1.0 + 0.1245 ⅇ^{(-0.1) \frac{V F}{R T}} + 0.0365 sigma ⅇ^{- (\frac{V F}{R T})}} sigma = \frac{1.0}{7.0} (ⅇ^{\frac{Nao}{67.3}} - 1.0) i_NaK = I_NaK f_NaK \frac{1.0}{1.0 + \sqrt{{(\frac{K_mNai}{Nai})}^{3.0}}} \frac{Ko}{Ko + K_mKo}

i_Cl_Ca = g_Cl \frac{V - E_Cl}{1.0 + \frac{K_mCa}{Cai}}

i_to = g_to X_to Y_to (V - E_K)

alpha_X_to = 0.04561 ⅇ^{0.03577 V} beta_X_to = 0.0989 ⅇ^{(-0.06237) V}

alpha_Y_to = 0.005415 ⅇ^{\frac{- (\frac{V + 33.5}{5.0})}{1.0 + 0.051335 ⅇ^{- (\frac{V + 33.5}{5.0})}}} beta_Y_to = 0.005415 ⅇ^{\frac{\frac{V + 33.5}{5.0}}{1.0 + 0.051335 ⅇ^{\frac{V + 33.5}{5.0}}}}

TRPN_buff = TRPN_max \frac{Cai}{Cai + K_mTRPN} CMDN_buff = CMDN_max \frac{Cai}{Cai + K_mCMDN}

i_rel = G_rel (Ca_JSR - Cai) G_rel = G_rel_max \frac{delta_Ca_i2 - delta_Ca_ith}{K_mrel + delta_Ca_i2 - delta_Ca_ith} (1.0 - (ⅇ^{- (\frac{t}{tau_on})})) ⅇ^{- (\frac{t}{tau_off})} G_rel_max = \{\begin{matrix} 0.0 if delta_Ca_i2 < delta_Ca_ith \\ 60.0 otherwise \end{matrix} CSQN_buff = CSQN_max \frac{Ca_JSR}{Ca_JSR + K_mCSQN} i_up = I_up \frac{Cai}{Cai + K_mup} i_leak = K_leak Ca_NSR K_leak = \frac{I_up}{Ca_NSR_max} i_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr}

\frac{d}{d time} (Nai) = (- (i_Na + i_CaNa + i_Na_b + i_NaCa 3.0 + i_NaK 3.0)) \frac{A_cap}{V_myo F} \frac{d}{d time} (Cai) = (i_CaCa + i_p_Ca + i_Ca_b + i_Ca_T - i_NaCa) \frac{A_cap}{2.0 V_myo F} + i_rel \frac{V_JSR}{V_myo} + (i_leak - i_up) \frac{V_NSR}{V_myo} \frac{d}{d time} (Ki) = (- (i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_to + (- (i_NaK 2.0)))) \frac{A_cap}{V_myo F} \frac{d}{d time} (Ko) = (i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_to + (- (i_NaK 2.0))) \frac{A_cap}{V_cleft F} \frac{d}{d time} (Ca_JSR) = - (i_rel - (i_tr \frac{V_NSR}{V_JSR})) \frac{d}{d time} (Ca_NSR) = - (i_leak + i_tr - i_up) \frac{d}{d time} (Ca_foot) = (- i_CaCa) \frac{A_cap}{2.0 V_myo F} R_A_V