Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Na + i_Ca_L + i_Ca_T + i_Kv + i_BK + i_NCX + i_NaK + i_NsK + i_NsNa + I_stim)

Component: I_stim

stim_start = \{\begin{matrix} period 0 if time \geq period 0 \land time ≦ period 1 \\ period 1 if time \geq period 1 \land time ≦ period 2 \\ period 2 if time \geq period 2 \land time < period 3 \\ period 3 if time \geq period 3 \land time < period 4 \\ 0 otherwise \end{matrix} local_time = time - stim_start I_stim = \{\begin{matrix} Gcouple (V - (V_ICCrest + \frac{V_ICCamp local_time}{f_2})) if local_time < t_ICCpeak \\ Gcouple (V - (V_ICCrest + \frac{V_ICCamp (1 + ⅇ^{\frac{- f_1}{2 t_slope}}) 1}{1 + ⅇ^{\frac{local_time - f_2 - (0.5 f_1)}{t_slope}}})) otherwise \end{matrix}

Component: L_type_Ca_channel

E_Ca = \frac{R T}{2 F} \ln (\frac{Cao}{Cai}) i_Ca_L = g_CaL O_CaL (V - E_Ca)

Component: L_type_Ca_channel_states

T_correction_Ca = {Q10Ca}^{\frac{T - 310}{10}} alpha = T_correction_Ca 0.731 ⅇ^{\frac{V}{30}} beta = T_correction_Ca 0.2149 ⅇ^{\frac{- V}{40}} alpha_0 = 4 alpha alpha_1 = 3 alpha alpha_2 = 2 alpha alpha_3 = 1 alpha beta_0 = 1 beta beta_1 = 2 beta beta_2 = 3 beta beta_3 = 4 beta phi_f = T_correction_Ca 0.4742 ⅇ^{\frac{V}{10}} phi_s = T_correction_Ca 0.05956 ⅇ^{\frac{- V}{40}} xi_f = T_correction_Ca 0.01407 ⅇ^{\frac{- V}{300}} xi_s = T_correction_Ca 0.01213 ⅇ^{\frac{V}{500}} psi_f = T_correction_Ca 0.02197 ⅇ^{\frac{V}{500}} psi_s = T_correction_Ca 0.00232 ⅇ^{\frac{- V}{280}} omega_f = \frac{beta_3 xi_f phi_f}{alpha_3 psi_f} omega_s = \frac{beta_3 xi_s phi_s}{alpha_3 psi_s} omega_sf = \frac{xi_s psi_f}{xi_f} omega_fs = psi_s delta = T_correction_Ca 0.01 theta = \frac{T_correction_Ca 4}{1 + \frac{1}{Cai}} norm = C3 + C2 + C1 + C0 + C3Ca + C2Ca + C1Ca + C0Ca + O_CaL + ICa + IVs + IVf + IVsCa + IVfCa \frac{d}{d time} (C3) = (\frac{alpha_2 C2}{norm} - (\frac{(alpha_3 + beta_2 + phi_s + phi_f + theta) C3}{norm})) + \frac{beta_3 O_CaL}{norm} + \frac{omega_f IVf}{norm} + \frac{omega_s IVs}{norm} + \frac{delta C3Ca}{norm} \frac{d}{d time} (C2) = (\frac{alpha_1 C1}{norm} - (\frac{(alpha_2 + beta_1 + theta) C2}{norm})) + \frac{beta_2 C3}{norm} + \frac{delta C2Ca}{norm} \frac{d}{d time} (C1) = (\frac{alpha_0 C0}{norm} - (\frac{(alpha_1 + beta_0 + theta) C1}{norm})) + \frac{beta_1 C2}{norm} + \frac{delta C1Ca}{norm} \frac{d}{d time} (C0) = \frac{(- (alpha_0 + theta)) C0}{norm} + \frac{beta_0 C1}{norm} + \frac{delta C0Ca}{norm} \frac{d}{d time} (C3Ca) = (\frac{theta C3}{norm} + \frac{alpha_2 C2Ca}{norm} - (\frac{(alpha_3 + beta_2 + phi_f + phi_s + delta) C3Ca}{norm})) + \frac{beta_3 ICa}{norm} + \frac{omega_f IVfCa}{norm} + \frac{omega_s IVsCa}{norm} \frac{d}{d time} (C2Ca) = (\frac{theta C2}{norm} + \frac{alpha_1 C1Ca}{norm} - (\frac{(alpha_2 + beta_1 + delta) C2Ca}{norm})) + \frac{beta_2 C3Ca}{norm} \frac{d}{d time} (C1Ca) = (\frac{theta C1}{norm} + \frac{alpha_0 C0Ca}{norm} - (\frac{(alpha_1 + beta_0 + delta) C1Ca}{norm})) + \frac{beta_1 C2Ca}{norm} \frac{d}{d time} (C0Ca) = (\frac{theta C0}{norm} - (\frac{(alpha_0 + delta) C0Ca}{norm})) + \frac{beta_0 C1Ca}{norm} \frac{d}{d time} (O_CaL) = (\frac{alpha_3 C3}{norm} - (\frac{(beta_3 + psi_f + psi_s + theta) O_CaL}{norm})) + \frac{xi_f IVf}{norm} + \frac{xi_s IVs}{norm} + \frac{delta ICa}{norm} \frac{d}{d time} (ICa) = (\frac{theta O_CaL}{norm} + \frac{alpha_3 C3Ca}{norm} - (\frac{(beta_3 + psi_f + psi_s + delta) ICa}{norm})) + \frac{xi_f IVfCa}{norm} + \frac{xi_s IVsCa}{norm} \frac{d}{d time} (IVs) = (\frac{phi_s C3}{norm} + \frac{psi_s O_CaL}{norm} + \frac{omega_fs IVf}{norm} - (\frac{(omega_s + xi_s + omega_sf + theta) IVs}{norm})) + \frac{delta IVsCa}{norm} \frac{d}{d time} (IVf) = (\frac{phi_f C3}{norm} + \frac{psi_f O_CaL}{norm} - (\frac{(omega_f + xi_f + omega_fs + theta) IVf}{norm})) + \frac{omega_sf IVs}{norm} + \frac{delta IVfCa}{norm} \frac{d}{d time} (IVsCa) = \frac{theta IVs}{norm} + \frac{phi_s C3Ca}{norm} + \frac{psi_s ICa}{norm} + \frac{omega_fs IVfCa}{norm} - (\frac{(omega_s + xi_s + omega_sf + delta) IVsCa}{norm}) \frac{d}{d time} (IVfCa) = (\frac{theta IVf}{norm} + \frac{phi_f C3Ca}{norm} + \frac{psi_f ICa}{norm} - (\frac{(omega_f + xi_f + omega_fs + delta) IVfCa}{norm})) + \frac{omega_sf IVsCa}{norm}

Component: T_type_Ca_channel

E_Ca = \frac{R T}{2 F} \ln (\frac{Cao}{Cai}) i_Ca_T = g_CaT d_CaT f_CaT (V - E_Ca)

Component: T_type_Ca_channel_d_gate

T_correction_Ca = {Q10Ca}^{\frac{T - 297}{10}} d_CaT_inf = \frac{1}{1 + ⅇ^{\frac{- (V + 60.5)}{5.3}}} tau_d_CaT = \frac{1.9058}{T_correction_Ca} \frac{d}{d time} (d_CaT) = \frac{d_CaT_inf - d_CaT}{tau_d_CaT}

Component: T_type_Ca_channel_f_gate

T_correction_Ca = {Q10Ca}^{\frac{T - 297}{10}} f_CaT_inf = \frac{1}{1 + ⅇ^{\frac{V + 75.5}{4}}} tau_f_CaT = \frac{0.38117 (8.6 + 14.7 ⅇ^{\frac{(- (V + 50)) (V + 50)}{900}})}{T_correction_Ca} \frac{d}{d time} (f_CaT) = \frac{f_CaT_inf - f_CaT}{tau_f_CaT}

Component: voltage_dep_K_channel

i_Kv = g_Kv x_Kv y_Kv (V - E_K) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki})

Component: voltage_dep_K_channel_x_gate

T_correction_K = {Q10K}^{\frac{T - 297}{10}} x_Kv_inf = \frac{1}{1 + ⅇ^{\frac{- (V + 43)}{17.36}}} tau_x_Kv = \frac{4.7803}{T_correction_K} \frac{d}{d time} (x_Kv) = \frac{x_Kv_inf - x_Kv}{tau_x_Kv}

Component: voltage_dep_K_channel_y_gate

T_correction_K = {Q10K}^{\frac{T - 297}{10}} y_Kv_inf = \frac{1}{1 + ⅇ^{\frac{V - 44.9}{12.0096}}} tau_y_Kv = \frac{763.7564}{T_correction_K} \frac{d}{d time} (y_Kv) = \frac{y_Kv_inf - y_Kv}{tau_y_Kv}

Component: Calcium_voltage_activated_potassium_channel

E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_BK = g_BK O4 (V - E_K)

Component: Calcium_voltage_activated_potassium_channel_states

alpha = 1 ⅇ^{\frac{8.47188 V}{1 T}} beta = 1 ⅇ^{\frac{(- 7.77556) V}{1 T}} k_on = 40633 k_off_C = 11 k_off_O = 1.1 k_C0O0 = 0.02162 alpha k_C1O1 = 0.000869 alpha k_C2O2 = 0.0000281 alpha k_C3O3 = 0.000781 alpha k_C4O4 = 0.044324 alpha k_O0C0 = 318.1084 beta k_O1C1 = 144.1736 beta k_O2C2 = 32.6594 beta k_O3C3 = 0.095312 beta k_O4C4 = 0.000106 beta k_C0C1 = 4 k_on Cai k_C1C2 = 3 k_on Cai k_C2C3 = 2 k_on Cai k_C3C4 = 1 k_on Cai k_C4C3 = 4 k_off_C k_C3C2 = 3 k_off_C k_C2C1 = 2 k_off_C k_C1C0 = 1 k_off_C k_O0O1 = 4 k_on Cai k_O1O2 = 3 k_on Cai k_O2O3 = 2 k_on Cai k_O3O4 = 1 k_on Cai k_O4O3 = 4 k_off_O k_O3O2 = 3 k_off_O k_O2O1 = 2 k_off_O k_O1O0 = 1 k_off_O norm = C0 + C1 + C2 + C3 + C4 + O0 + O1 + O2 + O3 + O4 \frac{d}{d time} (C4) = \frac{(- (k_C4C3 + k_C4O4)) C4}{norm} + \frac{k_C3C4 C3}{norm} + \frac{k_O4C4 O4}{norm} \frac{d}{d time} (C3) = \frac{(- (k_C3C2 + k_C3O3 + k_C3C4)) C3}{norm} + \frac{k_C2C3 C2}{norm} + \frac{k_O3C3 O3}{norm} + \frac{k_C4C3 C4}{norm} \frac{d}{d time} (C2) = \frac{(- (k_C2C1 + k_C2O2 + k_C2C3)) C2}{norm} + \frac{k_C1C2 C1}{norm} + \frac{k_O2C2 O2}{norm} + \frac{k_C3C2 C3}{norm} \frac{d}{d time} (C1) = \frac{(- (k_C1C0 + k_C1O1 + k_C1C2)) C1}{norm} + \frac{k_C0C1 C0}{norm} + \frac{k_O1C1 O1}{norm} + \frac{k_C2C1 C2}{norm} \frac{d}{d time} (C0) = \frac{(- (k_C0C1 + k_C0O0)) C0}{norm} + \frac{k_C1C0 C1}{norm} + \frac{k_O0C0 O0}{norm} \frac{d}{d time} (O4) = \frac{(- (k_O4O3 + k_O4C4)) O4}{norm} + \frac{k_O3O4 O3}{norm} + \frac{k_C4O4 C4}{norm} \frac{d}{d time} (O3) = \frac{(- (k_O3O2 + k_O3C3 + k_O3O4)) O3}{norm} + \frac{k_O2O3 O2}{norm} + \frac{k_C3O3 C3}{norm} + \frac{k_O4O3 O4}{norm} \frac{d}{d time} (O2) = \frac{(- (k_O2O1 + k_O2C2 + k_O2O3)) O2}{norm} + \frac{k_O1O2 O1}{norm} + \frac{k_C2O2 C2}{norm} + \frac{k_O3O2 O3}{norm} \frac{d}{d time} (O1) = \frac{(- (k_O1O0 + k_O1C1 + k_O1O2)) O1}{norm} + \frac{k_O0O1 O0}{norm} + \frac{k_C1O1 C1}{norm} + \frac{k_O2O1 O2}{norm} \frac{d}{d time} (O0) = \frac{(- (k_O0O1 + k_O0C0)) O0}{norm} + \frac{k_O1O0 O1}{norm} + \frac{k_C0O0 C0}{norm}

Component: sodium_current

i_Na = g_Na O_Na (V - E_Na) E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai})

Component: Na_channel_states

T_correction_Na = 1 {Q10Na}^{\frac{T - 297}{10}} k_I2I1 = T_correction_Na 0.0039239 ⅇ^{2.6793 + 0.0061468 V} k_I1O = T_correction_Na 0.12052 ⅇ^{(- 9.6028) + 0.083025 V} k_OC1 = T_correction_Na 2.391 ⅇ^{- 13.335 - (0.25289 V)} k_C1C2 = T_correction_Na 3.1566 ⅇ^{0.36352 + 0.077193 V} k_C2C3 = T_correction_Na 0.55432 ⅇ^{(- 0.099074) + 0.036441 V} k_C3C2 = T_correction_Na 0.00052548 ⅇ^{(- 0.069102) + 0.0031945 V} k_C2C1 = T_correction_Na 1.4496 ⅇ^{(- 0.1566) + 0.058353 V} k_C1O = T_correction_Na 1.5329 ⅇ^{0.0093193 + 0.041075 V} k_OI1 = T_correction_Na 1.6164 ⅇ^{0.30763 + 0.0060535 V} k_I1I2 = T_correction_Na 0.027735 ⅇ^{0.05149 - (0.046865 V)} k_I1C1 = T_correction_Na 1.9046 ⅇ^{(- 2.484) + 0.020406 V} k_C1I1 = T_correction_Na 0.00021688 ⅇ^{(- 0.063438) + 0.0046683 V} norm = O_Na + C1 + C2 + C3 + I1 + I2 \frac{d}{d time} (C3) = \frac{(- k_C3C2) C3}{norm} + \frac{k_C2C3 C2}{norm} \frac{d}{d time} (C2) = \frac{(- (k_C2C1 + k_C2C3)) C2}{norm} + \frac{k_C1C2 C1}{norm} + \frac{k_C3C2 C3}{norm} \frac{d}{d time} (C1) = \frac{(- (k_C1C2 + k_C1O + k_C1I1)) C1}{norm} + \frac{k_OC1 O_Na}{norm} + \frac{k_C2C1 C2}{norm} + \frac{k_I1C1 I1}{norm} \frac{d}{d time} (O_Na) = \frac{(- (k_OC1 + k_OI1)) O_Na}{norm} + \frac{k_C1O C1}{norm} + \frac{k_I1O I1}{norm} \frac{d}{d time} (I2) = \frac{(- k_I2I1) I2}{norm} + \frac{k_I1I2 I1}{norm} \frac{d}{d time} (I1) = \frac{(- (k_I1O + k_I1I2 + k_I1C1)) I1}{norm} + \frac{k_I2I1 I2}{norm} + \frac{k_C1I1 C1}{norm} + \frac{k_OI1 O_Na}{norm}

Component: Na_Ca_exchanger

i_NCX = \frac{P_NCX (ⅇ^{\frac{gamma V F}{R T}} {Nai}^{3} Cao - (2.5 ⅇ^{\frac{(gamma - 1) V F}{R T}} {Nao}^{3} Cai))}{({K_mNai}^{3} + {Nao}^{3}) (K_mCa + Cao) (1 + k_sat ⅇ^{\frac{(gamma - 1) V F}{R T}})}

Component: sodium_potassium_pump

i_NaK = \frac{P_NaK Ko Nai}{(Ko + K_mK) (K_mNa + Nai) (1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0353 ⅇ^{\frac{(- V) F}{R T}})}

Component: non_specific_Na_current

E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai}) i_NsNa = g_NsNa (V - E_Na)

Component: non_specific_K_current

E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_NsK = g_NsK (V - E_K)

Component: ionic_concentrations

V_myo = 3.5 e -12 \frac{d}{d time} (Nai) = \frac{\frac{- 1}{1 e 15} (i_Na + i_NsNa + i_NCX 3 + i_NaK 3)}{V_myo F} \frac{d}{d time} (Ki) = \frac{(- 1 e -15) (i_Kv + i_BK + i_NsK + I_stim - (i_NaK 2))}{V_myo F} \frac{d}{d time} (Cai) = \frac{\frac{(- 1 e -15) (i_Ca_L + i_Ca_T - (i_NCX 2))}{2 V_myo F}}{1 + \frac{n_CRT CRT_total K_D_CRT {Cai}^{n_CRT - 1}}{{({Cai}^{n_CRT} + K_D_CRT)}^{2}} + \frac{n_CaM CaM_total K_D_CaM {Cai}^{n_CaM - 1}}{{({Cai}^{n_CaM} + K_D_CaM)}^{2}}}