Model Mathematics

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Component: environment

Component: membrane

i_Stim = \{\begin{matrix} stim_amplitude if time \geq stim_start \land time ≦ stim_end \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Kr + i_Ks + i_Na + i_Ca_L + i_Ca_T + i_to + i_sus + i_K1 + i_B_Na + i_B_Ca + i_p + i_CaP + i_NaCa + i_Stim)

Component: sodium_current

E_Na = \frac{R T}{F} \ln (\frac{Na_c}{Na_i}) i_Na = \frac{\frac{P_Na m^{3} (0.635 h1 + 0.365 h2) Na_c V F^{2}}{R T} (ⅇ^{\frac{(V - E_Na) F}{R T}} - 1)}{ⅇ^{\frac{V F}{R T}} - 1}

Component: sodium_current_m_gate

E0_m = V + 44.4 alpha_m = \frac{(- 460) E0_m}{ⅇ^{\frac{E0_m}{- 12.673}} - 1} beta_m = 18400 ⅇ^{\frac{E0_m}{- 12.673}} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: sodium_current_h1_gate

alpha_h = 44.9 ⅇ^{\frac{V + 66.9}{- 5.57}} beta_h = \frac{1491}{1 + 323.3 ⅇ^{\frac{V + 94.6}{- 12.9}}} h_infinity = \frac{alpha_h}{alpha_h + beta_h} tau_h1 = \frac{0.03}{1 + ⅇ^{\frac{V + 40}{6}}} + 0.00015 \frac{d}{d time} (h1) = \frac{h_infinity - h1}{tau_h1}

Component: sodium_current_h2_gate

tau_h2 = \frac{0.12}{1 + ⅇ^{\frac{V + 60}{2}}} + 0.00045 \frac{d}{d time} (h2) = \frac{h_infinity - h2}{tau_h2}

Component: L_type_Ca_channel

d_prime = \frac{1}{1 + ⅇ^{\frac{V - 23}{- 12}}} i_Ca_L = \{\begin{matrix} 1.8 g_Ca_L (d_L f_L + d_prime) (V - E_Ca_app) if CT = 1 \land PM = 0 \\ 2.1 g_Ca_L (d_L f_L + d_prime) (V - E_Ca_app) if CT = 0 \land PM = 1 \\ g_Ca_L (d_L f_L + d_prime) (V - E_Ca_app) otherwise \end{matrix}

Component: L_type_Ca_channel_d_L_gate

E0_alpha_d_L = V + 45 E0_beta_d_L = V + 5 E10 = V + 10 alpha_d_L = \frac{(- 16.72) E0_alpha_d_L}{ⅇ^{\frac{E0_alpha_d_L}{- 2.5}} - 1} + \frac{(- 50) E10}{ⅇ^{\frac{E10}{- 4.808}} - 1} beta_d_L = \frac{4.48 E0_beta_d_L}{ⅇ^{\frac{E0_beta_d_L}{2.5}} - 1} d_L_infinity = \frac{1}{1 + ⅇ^{\frac{E10 + 0.95}{- 6.6}}} tau_d_L = \frac{1}{alpha_d_L + beta_d_L} \frac{d}{d time} (d_L) = \frac{d_L_infinity - d_L}{tau_d_L}

Component: L_type_Ca_channel_f_L_gate

E0_f_L = V + 18 alpha_f_L = \frac{8.49 E0_f_L}{ⅇ^{\frac{E0_f_L}{4}} - 1} beta_f_L = \frac{67.922}{1 + ⅇ^{\frac{E0_f_L}{- 4}}} f_L_infinity = \frac{alpha_f_L}{alpha_f_L + beta_f_L} tau_f_L = \frac{1}{alpha_f_L + beta_f_L} \frac{d}{d time} (f_L) = \frac{f_L_infinity - f_L}{tau_f_L}

Component: T_type_Ca_channel

i_Ca_T = g_Ca_T d_T f_T (V - E_Ca_T)

Component: T_type_Ca_channel_d_T_gate

E0_d_T = V + 23.3 alpha_d_T = 674.173 ⅇ^{\frac{E0_d_T}{30}} beta_d_T = 674.173 ⅇ^{\frac{E0_d_T}{- 30}} d_T_infinity = \frac{1}{1 + ⅇ^{\frac{E0_d_T - 0.3}{- 6.1}}} tau_d_T = \frac{1}{alpha_d_T + beta_d_T} \frac{d}{d time} (d_T) = \frac{d_T_infinity - d_T}{tau_d_T}

Component: T_type_Ca_channel_f_T_gate

E0_f_T = V + 75 alpha_f_T = 9.637 ⅇ^{\frac{E0_f_T}{- 83.3}} beta_f_T = 9.637 ⅇ^{\frac{E0_f_T}{15.38}} f_T_infinity = \frac{alpha_f_T}{alpha_f_T + beta_f_T} tau_f_T = \frac{1}{alpha_f_T + beta_f_T} \frac{d}{d time} (f_T) = \frac{f_T_infinity - f_T}{tau_f_T}

Component: Ca_independent_transient_outward_K_current

E_K = \frac{R T}{F} \ln (\frac{K_c}{K_i}) i_to = \{\begin{matrix} 0.2 g_to r (0.59 {s1}^{3} + 0.41 {s2}^{3}) (0.6 {s3}^{6} + 0.4) (V - E_K) if CT = 1 \land PM = 0 \\ 0.35 g_to r (0.59 {s1}^{3} + 0.41 {s2}^{3}) (0.6 {s3}^{6} + 0.4) (V - E_K) if CT = 0 \land PM = 1 \\ g_to r (0.59 {s1}^{3} + 0.41 {s2}^{3}) (0.6 {s3}^{6} + 0.4) (V - E_K) otherwise \end{matrix} i_sus = \{\begin{matrix} 0.0014 (V + 70) if CT = 1 \land PM = 0 \\ 0.0024 (V + 70) if CT = 0 \land PM = 1 \\ 0.001 (V + 70) otherwise \end{matrix}

Component: Ca_independent_transient_outward_K_current_r_gate

alpha_r = 386.6 ⅇ^{\frac{V}{12}} beta_r = 8.011 ⅇ^{\frac{V}{- 7.2}} r_infinity = \frac{1}{1 + ⅇ^{\frac{V + 15}{- 5.633}}} tau_r = \frac{1}{alpha_r + beta_r} + 4 e -4 \frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r}

Component: Ca_independent_transient_outward_K_current_s1_gate

s1_infinity = \frac{1}{1 + ⅇ^{\frac{V + 28.29}{7.06}}} tau_s1 = \frac{0.5466}{1 + ⅇ^{\frac{V + 32.8}{0.1}}} + 0.0204 \frac{d}{d time} (s1) = \frac{s1_infinity - s1}{tau_s1}

Component: Ca_independent_transient_outward_K_current_s2_gate

s2_infinity = \frac{1}{1 + ⅇ^{\frac{V + 28.29}{7.06}}} tau_s2 = \frac{5.75}{1 + ⅇ^{\frac{V + 32.8}{0.1}}} + \frac{0.45}{1 + ⅇ^{\frac{V - 13.54}{- 13.97}}} \frac{d}{d time} (s2) = \frac{s2_infinity - s2}{tau_s2}

Component: Ca_independent_transient_outward_K_current_s3_gate

s3_infinity = \frac{\frac{1}{1 + ⅇ^{\frac{V + 50.67}{27.38}}} + 0.666}{1.666} tau_s3 = \frac{7.5}{1 + ⅇ^{\frac{V + 23}{0.5}}} + 0.5 \frac{d}{d time} (s3) = \frac{s3_infinity - s3}{tau_s3}

Component: delayed_rectifier_K_current

i_Ks = g_Ks z (V - E_K) i_Kr = g_Kr p_a p_i (V - E_K)

Component: delayed_rectifier_K_current_z_gate

alpha_z = 1.66 ⅇ^{\frac{V}{69.452}} beta_z = 0.3 ⅇ^{\frac{V}{- 21.826}} z_infinity = \frac{1}{1 + ⅇ^{\frac{V - 0.9}{- 13.8}}} tau_z = \frac{1}{alpha_z + beta_z} + 0.06 \frac{d}{d time} (z) = \frac{z_infinity - z}{tau_z}

Component: delayed_rectifier_K_current_pa_gate

alpha_p_a = 9 ⅇ^{\frac{V}{25.371}} beta_p_a = 1.3 ⅇ^{\frac{V}{- 13.026}} p_a_infinity = \frac{1}{1 + ⅇ^{\frac{V + 5.1}{- 7.4}}} tau_p_a = \frac{1}{alpha_p_a + beta_p_a} \frac{d}{d time} (p_a) = \frac{p_a_infinity - p_a}{tau_p_a}

Component: delayed_rectifier_K_current_pi_gate

alpha_p_i = 100 ⅇ^{\frac{V}{- 54.645}} beta_p_i = 656 ⅇ^{\frac{V}{106.157}} p_i_infinity = \frac{1}{1 + ⅇ^{\frac{V + 47.3921}{18.6603}}} tau_p_i = \frac{1}{alpha_p_i + beta_p_i} \frac{d}{d time} (p_i) = \frac{p_i_infinity - p_i}{tau_p_i}

Component: inward_rectifier

i_K1 = \{\begin{matrix} \frac{2 g_K1 (V - E_K) {(\frac{K_c}{K_c + KmK1})}^{3} 1}{1 + ⅇ^{\frac{steepK1 F (V - E_K - shiftK1)}{R T}}} if CT = 1 \land PM = 0 \\ \frac{2.5 g_K1 (V - E_K) {(\frac{K_c}{K_c + KmK1})}^{3} 1}{1 + ⅇ^{\frac{steepK1 F (V - E_K - shiftK1)}{R T}}} if CT = 0 \land PM = 1 \\ \frac{g_K1 (V - E_K) {(\frac{K_c}{K_c + KmK1})}^{3} 1}{1 + ⅇ^{\frac{steepK1 F (V - E_K - shiftK1)}{R T}}} otherwise \end{matrix}

Component: background_currents

E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_c}{Ca_i}) i_B_Na = \{\begin{matrix} 0.00002 (V - E_Na) if CT = 1 \land PM = 0 \\ 0.00003 (V - E_Na) if CT = 0 \land PM = 1 \\ g_B_Na (V - E_Na) otherwise \end{matrix} i_B_Ca = \{\begin{matrix} 0.00002 (V - E_Ca) if CT = 1 \land PM = 0 \\ 0.00003 (V - E_Ca) if CT = 0 \land PM = 1 \\ g_B_Ca (V - E_Ca) otherwise \end{matrix}

Component: sodium_potassium_pump

i_p = \frac{\frac{\frac{i_NaK_max K_c}{K_c + k_NaK_K} {Na_i}^{1.5}}{{Na_i}^{1.5} + {k_NaK_Na}^{1.5}} 1.6}{1.5 + ⅇ^{\frac{V + 60}{- 40}}}

Component: sarcolemmal_calcium_pump_current

i_CaP = \frac{i_CaP_max Ca_i}{Ca_i + k_CaP}

Component: Na_Ca_ion_exchanger_current

i_NaCa = \frac{k_NaCa ({Na_i}^{3} Ca_c ⅇ^{\frac{gamma F V}{R T}} - ({Na_c}^{3} Ca_i ⅇ^{\frac{(gamma - 1) V F}{R T}}))}{1 + d_NaCa ({Na_c}^{3} Ca_i + {Na_i}^{3} Ca_c)}

Component: intracellular_ion_concentrations

\frac{d}{d time} (Na_i) = \frac{- (i_Na + i_B_Na + 3 i_p + 3 i_NaCa)}{Vol_i F} \frac{d}{d time} (K_i) = \frac{- (i_to + i_sus + i_K1 + i_Kr + i_Ks - (2 i_p))}{Vol_i F} \frac{d}{d time} (Ca_i) = \frac{- ((i_Ca_L + i_Ca_T + i_B_Ca + i_CaP - (2 i_NaCa)) + i_up - i_rel)}{2 Vol_Ca F} - (0.08 dOTCdt + 0.16 dOTMgCdt + 0.045 dOCdt)

Component: intracellular_Ca_buffering

dOCdt = 200000 Ca_i (1 - O_C) - (476 O_C) \frac{d}{d time} (O_C) = dOCdt dOTCdt = 78400 Ca_i (1 - O_TC) - (392 O_TC) \frac{d}{d time} (O_TC) = dOTCdt dOTMgCdt = 200000 Ca_i (1 - O_TMgC - O_TMgMg) - (6.6 O_TMgC) \frac{d}{d time} (O_TMgC) = dOTMgCdt \frac{d}{d time} (O_TMgMg) = 2000 Mg_i (1 - O_TMgC - O_TMgMg) - (666 O_TMgMg)

Component: cleft_space_ion_concentrations

\frac{d}{d time} (K_c) = \frac{i_to + i_sus + i_K1 + i_Kr + i_Ks - (2 i_p)}{Vol_c F}

Component: Ca_handling_by_the_SR

\frac{d}{d time} (O_Calse) = 480 Ca_rel (1 - O_Calse) - (400 O_Calse) \frac{d}{d time} (Ca_rel) = \frac{i_tr - i_rel}{2 Vol_rel F} - (31 \frac{d}{d time} (O_Calse)) \frac{d}{d time} (Ca_up) = \frac{i_up - i_tr}{2 Vol_up F} \frac{d}{d time} (F1) = k_F3 F3 - (r_act F1) \frac{d}{d time} (F2) = r_act F1 - (r_inact F2) \frac{d}{d time} (F3) = F2 r_inact - (k_F3 F3) r_act = 240 ⅇ^{0.08 (V - 20)} + 203.8 {(\frac{Ca_i}{Ca_i + k_rel})}^{4} r_inact = 33.96 + 339.6 {(\frac{Ca_i}{Ca_i + k_rel})}^{4} i_up = \frac{I_up_max (\frac{Ca_i}{k_cyca} - (\frac{{k_xcs}^{2} Ca_up}{k_srca}))}{\frac{Ca_i + k_cyca}{k_cyca} + \frac{k_xcs (Ca_up + k_srca)}{k_srca}} i_tr = \frac{(Ca_up - Ca_rel) 2 F Vol_rel}{tau_tr} i_rel = alpha_rel {(\frac{F2}{F2 + 0.25})}^{2} (Ca_rel - Ca_i)