Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (V) = (- (\frac{1.0}{Cm})) (i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_K1 + i_K_ATP + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca + I_st)

Component: fast_sodium_current

i_Na = g_Na m^{3.0} h j (V - E_Na) E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai})

Component: fast_sodium_current_m_gate

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)

Component: fast_sodium_current_h_gate

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)

Component: fast_sodium_current_j_gate

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)

Component: L_type_Ca_channel

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

Component: L_type_Ca_channel_d_gate

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)

Component: L_type_Ca_channel_f_gate

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)

Component: L_type_Ca_channel_f_Ca_gate

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

Component: T_type_Ca_channel

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

Component: T_type_Ca_channel_b_gate

\frac{d}{d time} (b) = \frac{b_infinity - b}{tau_b} b_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V + 14.0}{10.8})}} tau_b = \frac{3.7 + 6.1}{1.0 + ⅇ^{\frac{25.0 + V}{4.5}}}

Component: T_type_Ca_channel_g_gate

\frac{d}{d time} (g) = \frac{g_infinity - g}{tau_g} g_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 60.0}{5.6}}} tau_g = \{\begin{matrix} 12.0 if V > 0.0 \\ (-0.875) V + 12.0 otherwise \end{matrix}

Component: rapid_time_dependent_potassium_current

g_Kr = 0.02614 \sqrt{\frac{Ko}{5.4}} E_Kr = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_Kr = g_Kr Xr Rr (V - E_Kr)

Component: rapid_time_dependent_potassium_current_Xr_gate

\frac{d}{d time} (Xr) = \frac{Xr_infinity - Xr}{tau_Xr} Xr_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V + 21.5}{7.5})}} tau_Xr = \frac{1.0}{\frac{0.00138 (V + 14.2)}{1.0 - (ⅇ^{(-0.123) (V + 14.2)})} + \frac{0.00061 (V + 38.9)}{ⅇ^{0.145 (V + 38.9)} - 1.0}}

Component: rapid_time_dependent_potassium_current_Rr_gate

Rr = \frac{1.0}{1.0 + ⅇ^{\frac{V + 9.0}{22.4}}}

Component: slow_time_dependent_potassium_current

g_Ks = 0.433 (1.0 + \frac{0.6}{1.0 + {(\frac{0.000038}{Cai})}^{1.4}}) E_Ks = \frac{R T}{F} \ln (\frac{Ko + P_NaK Nao}{Ki + P_NaK Nai}) i_Ks = g_Ks Xs1 Xs2 (V - E_Ks)

Component: slow_time_dependent_potassium_current_Xs1_gate

\frac{d}{d time} (Xs1) = \frac{Xs_infinity - Xs1}{tau_Xs1} Xs_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V - 1.5}{16.7})}} tau_Xs1 = \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}}

Component: slow_time_dependent_potassium_current_Xs2_gate

\frac{d}{d time} (Xs2) = \frac{Xs_infinity - Xs2}{tau_Xs2} tau_Xs2 = 4.0 tau_Xs1

Component: time_independent_potassium_current

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)

Component: time_independent_potassium_current_K1_gate

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}

Component: plateau_potassium_current

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)

Component: sarcolemmal_calcium_pump

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

Component: sodium_background_current

E_NaN = E_Na i_Na_b = g_Nab (V - E_NaN)

Component: calcium_background_current

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

Component: sodium_potassium_pump

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}

Component: non_specific_calcium_activated_current

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{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_ns_K = P_ns_Ca {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}

Component: ATP_dependent_potassium_current

i_K_ATP = g_K_ATP (V - E_K) g_K_ATP = G_K_ATP P_ATP {(\frac{Ko}{Ko_normal})}^{n} E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) G_K_ATP = \frac{0.000195}{Nichols_area} P_ATP = \frac{1.0}{1.0 + {(\frac{ATP_i}{K_05})}^{H}}

Component: Na_Ca_exchanger

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))

Component: calcium_buffers_in_the_myoplasm

TRPN_buff = TRPN_max \frac{Cai}{Cai + K_mTRPN} CMDN_buff = CMDN_max \frac{Cai}{Cai + K_mCMDN} Ca_JSR_new = \frac{\sqrt{{b1}^{2.0} + 4.0 c1} - b1}{2.0} delta_Ca_JSR = Ca_JSR_old - Ca_JSR_new delta_Cai = Cai_old - Cai b1 = K_mCSQN - (CSQN_max + CSQN_buff + delta_Ca_JSR + Ca_JSR_old) c1 = K_mCSQN (CSQN_buff + delta_Ca_JSR + Ca_JSR_old) Cai = \frac{2.0}{3.0} \sqrt{b^{2.0} - (3.0 c)} \cos (\frac{\arccos (\frac{9.0 b c - (2.0 b^{3.0} + 27.0 d)}{2.0 {(b^{2.0} - (3.0 c))}^{\frac{3.0}{2.0}}})}{3.0}) - (\frac{b}{3.0}) b = CMDN_max + TRPN_max + K_mTRPN + K_mCMDN - Ca_total c = K_mCMDN K_mTRPN + TRPN_max K_mCMDN + CMDN_max K_mTRPN - (Ca_total (K_mTRPN + K_mCMDN)) d = (- (K_mTRPN K_mCMDN)) Ca_total Ca_total = TRPN_buff + CMDN_buff + delta_Cai + Cai_old

Component: calcium_fluxes_in_the_SR

i_rel = G_rel (Ca_JSR_new - Cai) G_rel = \{\begin{matrix} 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})} if calcium_overload = 0.0 \\ G_rel_max (1.0 - (ⅇ^{- (\frac{t}{tau_on})})) ⅇ^{- (\frac{t}{tau_off})} otherwise \end{matrix} G_rel_max = \{\begin{matrix} \{\begin{matrix} 0.0 if delta_Ca_i2 < delta_Ca_ith \\ 60.0 otherwise \end{matrix} if calcium_overload = 0.0 \\ \{\begin{matrix} 0.0 if CSQN_buff < CSQN_th \\ 4.0 otherwise \end{matrix} otherwise \end{matrix} CSQN_buff = CSQN_max \frac{Ca_JSR_new}{Ca_JSR_new + 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_new}{tau_tr}

Component: ionic_concentrations

\frac{d}{d time} (Nai) = (- (i_Na + i_CaNa + i_Na_b + i_ns_Na + i_NaCa 3.0 + i_NaK 3.0)) \frac{A_cap}{V_myo F} \frac{d}{d time} (Ki) = (- (i_CaK + i_Kr + i_Ks + i_K1 + i_K_ATP + i_Kp + i_ns_K + (- (i_NaK 2.0)))) \frac{A_cap}{V_myo F} \frac{d}{d time} (Ko) = (i_CaK + i_Kr + i_Ks + i_K1 + i_K_ATP + i_Kp + i_ns_K + (- (i_NaK 2.0))) \frac{A_cap}{V_cleft F} \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