Model Mathematics

dV_dt = \frac{I_st - (i_Na + i_Ca_L + i_K + i_K1 + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca)}{Cm}

I_st = \{\begin{matrix} stimCurrent if time mod stimPeriod < stimDuration \\ 0.0 otherwise \end{matrix}

\frac{d}{d time} (V) = dV_dt

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

E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai})

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}}

g_K = g_K_max \sqrt{\frac{Ko}{5.4}}

E_K = \frac{R T}{F} \ln (\frac{Ko + PR_NaK Nao}{Ki + PR_NaK Nai})

i_K = g_K X^{2.0} Xi (V - E_K)

alpha_X = \frac{0.0000719 (V + 30.0)}{1.0 - (ⅇ^{(-0.148) (V + 30.0)})}

beta_X = \frac{0.000131 (V + 30.0)}{(-1.0) + ⅇ^{0.0687 (V + 30.0)}}

\frac{d}{d time} (X) = alpha_X (1.0 - X) - (beta_X X)

Xi = \frac{1.0}{1.0 + ⅇ^{\frac{V - 56.26}{32.1}}}

g_K1 = g_K1_max \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^{0.08032 (V + 5.476 - E_K1)} + ⅇ^{0.06175 (V - (E_K1 + 594.31))}}{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 + {(\frac{K_mNai}{Nai})}^{1.5}} \frac{Ko}{Ko + K_mKo}

EnsCa = \frac{R T}{F} \ln (\frac{Ko + Nao}{Ki + Nai})

Vns = V - EnsCa

i_ns_Na = I_ns_Na \frac{1.0}{1.0 + {(\frac{K_m_ns_Ca}{Cai})}^{3.0}}

i_ns_K = I_ns_K \frac{1.0}{1.0 + {(\frac{K_m_ns_Ca}{Cai})}^{3.0}}

i_ns_Ca = i_ns_Na + i_ns_K

I_ns_Na = P_ns_Ca {1.0}^{2.0} \frac{Vns F^{2.0}}{R T} \frac{gamma_Nai Nai ⅇ^{\frac{1.0 Vns F}{R T}} - (gamma_Nao Nao)}{ⅇ^{\frac{1.0 Vns F}{R T}} - 1.0}

I_ns_K = P_ns_Ca {1.0}^{2.0} \frac{Vns F^{2.0}}{R T} \frac{gamma_Ki Ki ⅇ^{\frac{1.0 Vns F}{R T}} - (gamma_Ko Ko)}{ⅇ^{\frac{1.0 Vns F}{R T}} - 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))

i_rel = G_rel (Ca_JSR - Cai)

G_rel = G_rel_peak \frac{delta_Ca_i2 - delta_Ca_ith}{K_mrel + delta_Ca_i2 - delta_Ca_ith} (1.0 - (ⅇ^{- (\frac{t_CICR}{tau_on})})) ⅇ^{- (\frac{t_CICR}{tau_off})}

G_rel_peak = \{\begin{matrix} 0.0 if delta_Ca_i2 < delta_Ca_ith \\ G_rel_max otherwise \end{matrix}

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_ns_Na + i_NaCa 3.0 + i_NaK 3.0)) \frac{Am}{V_myo F}

\frac{d}{d time} (Cai) = (- (i_CaCa + i_p_Ca + i_Ca_b - i_NaCa)) \frac{Am}{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_K + i_K1 + i_Kp + i_ns_K + (- (i_NaK 2.0)))) \frac{Am}{V_myo 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)