Model Mathematics

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Component: environment

Component: membrane

RTONF = \frac{R T}{F} \frac{d}{d time} (V) = \frac{- (i_Na + i_CaL + i_to + i_Kr + i_f + i_st + i_K1 + i_NaCa + i_p + i_b + i_ACh)}{C}

Component: hyperpolarising_activated_current

i_f = y g_f (V - (- 30))

Component: hyperpolarising_activated_current_y_gate

y_inf = \frac{1}{1 + ⅇ^{\frac{V + 83.19 - (\frac{(- 7.2) {ACh}^{0.69}}{{1.26 e -5}^{0.69} + {ACh}^{0.69}})}{13.56}}} tau_y = 0.25 + 2 ⅇ^{\frac{- ({(V + 70)}^{2})}{500}} \frac{d}{d time} (y) = \frac{y_inf - y}{tau_y}

Component: rapid_delayed_rectifier_potassium_current

E_K = RTONF \ln (\frac{Kc}{Ki}) i_Kr = g_Kr (0.9 paf + 0.1 pas) pik (V - E_K)

Component: rapid_delayed_rectifier_potassium_current_paf_gate

paf_infinity = \frac{1}{1 + ⅇ^{\frac{V + 10.22}{- 8.5}}} tau_paf = \frac{1}{17 ⅇ^{0.0398 V} + 0.211 ⅇ^{(- 0.051) V}} \frac{d}{d time} (paf) = \frac{paf_infinity - paf}{tau_paf}

Component: rapid_delayed_rectifier_potassium_current_pas_gate

pas_infinity = \frac{1}{1 + ⅇ^{\frac{V + 10.22}{- 8.5}}} tau_pas = 0.33581 + 0.90673 ⅇ^{\frac{- ({(V + 10)}^{2})}{988.05}} \frac{d}{d time} (pas) = \frac{pas_infinity - pas}{tau_pas}

Component: rapid_delayed_rectifier_potassium_current_pik_gate

pik_infinity = \frac{1}{1 + ⅇ^{\frac{V + 4.9}{15.14}}} (1 - (0.3 ⅇ^{\frac{- (V^{2})}{500}})) alpha_pik = 92.01 ⅇ^{(- 0.0183) V} beta_pik = 603.6 ⅇ^{0.00942 V} tau_pik = \frac{1}{alpha_pik + beta_pik} \frac{d}{d time} (pik) = \frac{pik_infinity - pik}{tau_pik}

Component: time_independent_potassium_current

g_K1_prime = g_K1 (0.5 + \frac{0.5}{1 + ⅇ^{\frac{V + 30}{5}}}) i_K1 = \frac{g_K1_prime {(\frac{Kc}{Kc + 0.59})}^{3} (V + 81.9)}{1 + ⅇ^{\frac{1.393 (V + 81.9 + 3.6)}{RTONF}}}

Component: background_current

i_b = g_b (V - E_b)

Component: sodium_potassium_pump

i_p = \frac{I_p {(\frac{Nai}{5.64 + Nai})}^{3} {(\frac{Kc}{0.621 + Kc})}^{2} 1.6}{1.5 + ⅇ^{\frac{- (V + 60)}{40}}}

Component: sodium_calcium_exchange_current

do = 1 + \frac{Cao}{Kco} (1 + ⅇ^{\frac{Qco V}{RTONF}}) + \frac{Nao}{K1no} + \frac{{Nao}^{2}}{K1no K2no} + \frac{{Nao}^{3}}{K1no K2no K3no} k32 = ⅇ^{\frac{Qn V}{2 RTONF}} k23 = \frac{(\frac{{Nao}^{2}}{K1no K2no} + \frac{{Nao}^{3}}{K1no K2no K3no}) ⅇ^{\frac{(- Qn) V}{2 RTONF}}}{do} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{(- Qco) V}{RTONF}}}{do} k34 = \frac{Nao}{K3no + Nao} di = 1 + \frac{Casub}{Kci} (1 + ⅇ^{\frac{(- Qci) V}{RTONF}} + \frac{Nai}{Kcni}) + \frac{Nai}{K1ni} + \frac{{Nai}^{2}}{K1ni K2ni} + \frac{{Nai}^{3}}{K1ni K2ni K3ni} k41 = ⅇ^{\frac{(- Qn) V}{2 RTONF}} k14 = \frac{(\frac{{Nai}^{2}}{K1ni K2ni} + \frac{{Nai}^{3}}{K1ni K2ni K3ni}) ⅇ^{\frac{Qn V}{2 RTONF}}}{di} k12 = \frac{\frac{Casub}{Kci} ⅇ^{\frac{(- Qci) V}{RTONF}}}{di} k43 = \frac{Nai}{K3ni + Nai} x1 = k41 k34 (k23 + k21) + k21 k32 (k43 + k41) x2 = k32 k43 (k14 + k12) + k41 k12 (k34 + k32) x3 = k14 k43 (k23 + k21) + k12 k23 (k43 + k41) x4 = k23 k34 (k14 + k12) + k14 k21 (k34 + k32) i_NaCa = \frac{kNaCa (x2 k21 - (x1 k12))}{x1 + x2 + x3 + x4}

Component: fast_sodium_current

E_Na = RTONF \ln (\frac{Nao}{Nai}) i_Na = \frac{\frac{g_Na m^{3} (0.635 h1 + 0.365 h2) Nao V F}{RTONF} (ⅇ^{\frac{V - E_Na}{RTONF}} - 1)}{ⅇ^{\frac{V}{RTONF}} - 1}

Component: fast_sodium_current_m_gate

E0_m = V + 44.4 alpha_m = \{\begin{matrix} \frac{(- 460) (- 12.673)}{ⅇ^{\frac{E0_m}{- 12.673}}} if | E0_m | < delta_m \\ \frac{(- 460) E0_m}{ⅇ^{\frac{E0_m}{- 12.673}} - 1} otherwise \end{matrix} beta_m = 18400 ⅇ^{\frac{E0_m}{- 12.673}} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: fast_sodium_current_h1_gate

alpha_h1 = 44.9 ⅇ^{\frac{V + 66.9}{- 5.57}} beta_h1 = \frac{1491}{1 + 323.3 ⅇ^{\frac{V + 94.6}{- 12.9}}} h1_inf = \frac{alpha_h1}{alpha_h1 + beta_h1} tau_h1 = \frac{0.03}{1 + ⅇ^{\frac{V + 40}{6}}} + 0.00035 \frac{d}{d time} (h1) = \frac{h1_inf - h1}{tau_h1}

Component: fast_sodium_current_h2_gate

alpha_h2 = 44.9 ⅇ^{\frac{V + 66.9}{- 5.57}} beta_h2 = \frac{1491}{1 + 323.3 ⅇ^{\frac{V + 94.6}{- 12.9}}} h2_inf = \frac{alpha_h2}{alpha_h2 + beta_h2} tau_h2 = \frac{0.12}{1 + ⅇ^{\frac{V + 60}{2}}} + 0.00295 \frac{d}{d time} (h2) = \frac{h2_inf - h2}{tau_h2}

Component: L_type_calcium_current

i_CaL = g_CaL d (0.675 f + 0.325 f2) (V - E_CaL) (1 - (\frac{\frac{i_ACh ACh}{0.9 e -4 + ACh}}{1}))

Component: L_type_calcium_current_d_gate

alpha_d = \frac{(- 26.12) (V + 35)}{ⅇ^{\frac{V + 35}{- 2.5}} - 1} + \frac{(- 78.11) V}{ⅇ^{(- 0.208) V} - 1} beta_d = \frac{10.52 (V - 5)}{ⅇ^{0.4 (V - 5)} - 1} d_inf = \frac{1}{1 + ⅇ^{\frac{V - ((- 3.2) + act_shift)}{slope_factor_act}}} tau_d = \frac{1}{alpha_d + beta_d} \frac{d}{d time} (d) = \frac{d_inf - d}{tau_d}

Component: L_type_calcium_current_f_gate

f_inf = \frac{1}{1 + ⅇ^{\frac{V - ((- 24) + inact_shift)}{6.31}}} tau_f = 0.01 + 0.1539 ⅇ^{\frac{- ({(V + 40)}^{2})}{185.67}} \frac{d}{d time} (f) = \frac{f_inf - f}{tau_f}

Component: L_type_calcium_current_f2_gate

f2_inf = \frac{1}{1 + ⅇ^{\frac{V - ((- 24) + inact_shift)}{6.31}}} tau_f2 = 0.06 + 0.48076 2.25 ⅇ^{\frac{- ({(V - (- 40))}^{2})}{138.04}} \frac{d}{d time} (f2) = \frac{f2_inf - f2}{tau_f2}

Component: transient_outward_potassium_current

i_to = g_to r (0.45 q_fast + 0.55 q_slow) (V - E_K) E_K = RTONF \ln (\frac{Kc}{Ki})

Component: transient_outward_potassium_current_r_gate

\frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r} r_infinity = \frac{1}{1 + ⅇ^{\frac{V - 7.44}{- 16.4}}} tau_r = 0.596 e -3 + \frac{3.118 e -3}{1.037 ⅇ^{0.09 (V + 30.61)} + 0.396 ⅇ^{(- 0.12) (V + 23.84)}}

Component: transient_outward_potassium_current_qfast_gate

\frac{d}{d time} (q_fast) = \frac{qfast_infinity - q_fast}{tau_qfast} qfast_infinity = \frac{1}{1 + ⅇ^{\frac{V + 33.8}{6.12}}} tau_qfast = 12.66 e -3 + \frac{4.72716}{1 + ⅇ^{\frac{V + 154.5}{23.96}}}

Component: transient_outward_potassium_current_qslow_gate

\frac{d}{d time} (q_slow) = \frac{qslow_infinity - q_slow}{tau_qslow} qslow_infinity = \frac{1}{1 + ⅇ^{\frac{V + 33.8}{6.12}}} tau_qslow = 0.1 + 4 ⅇ^{\frac{- ({(V + 65)}^{2})}{500}}

Component: sustained_outward_potassium_current

i_st = g_st qa qi (V - E_st)

Component: sustained_outward_potassium_current_qa_gate

\frac{d}{d time} (qa) = \frac{qa_infinity - qa}{tau_qa} qa_infinity = \frac{1}{1 + ⅇ^{\frac{V - (- 49.1)}{- 8.98}}} alpha_qa = \frac{1}{0.15 ⅇ^{\frac{- V}{11}} + 0.2 ⅇ^{\frac{- V}{700}}} beta_qa = \frac{1}{16 ⅇ^{\frac{V}{8}} + 15 ⅇ^{\frac{V}{50}}} tau_qa = \frac{0.001}{alpha_qa + beta_qa}

Component: sustained_outward_potassium_current_qi_gate

\frac{d}{d time} (qi) = \frac{qi_infinity - qi}{tau_qi} alpha_qi = \frac{0.1504}{3100 ⅇ^{\frac{V}{13}} + 700 ⅇ^{\frac{V}{70}}} beta_qi = \frac{0.1504}{95 ⅇ^{\frac{- V}{10}} + 50 ⅇ^{\frac{- V}{700}}} + \frac{0.000229}{1 + ⅇ^{\frac{- V}{5}}} qi_infinity = \frac{alpha_qi}{alpha_qi + beta_qi} tau_qi = \frac{0.001}{alpha_qi + beta_qi}

Component: acetylcholine_sensitive_current

g_ACh = \frac{g_ACh_max achf achs {ACh}^{1.5}}{{K_ACh}^{1.5} + {ACh}^{1.5}} i_ACh = \frac{\frac{g_ACh Kc}{10 + Kc} (V - E_K)}{1 + ⅇ^{\frac{V - E_K - 140}{2.5 RTONF}}}

Component: acetylcholine_sensitive_current_achf_gate

beta_achf = \frac{120}{1 + ⅇ^{\frac{- (V + 50)}{15}}} \frac{d}{d time} (achf) = alpha_achf (1 - achf) - (beta_achf achf)

Component: acetylcholine_sensitive_current_achs_gate

beta_achs = \frac{5.82}{1 + ⅇ^{\frac{- (V + 50)}{15}}} \frac{d}{d time} (achs) = alpha_achs (1 - achs) - (beta_achs achs)

Component: extracellular_sodium_concentration

Component: intracellular_sodium_concentration

Component: extracellular_calcium_concentration

Component: intracellular_calcium_concentration

i_diff = \frac{Casub - Cai}{0.04 e -3} i_up = \frac{5}{1 + \frac{K_up}{Cai}} i_tr = \frac{Ca_up - Ca_rel}{tau_tr} i_rel = \frac{P_rel (Ca_rel - Casub)}{1 + {(\frac{0.0012}{Casub})}^{2}} \frac{d}{d time} (Ca_up) = i_up - (\frac{i_tr V_rel}{V_up}) V_up = 0.0116 V_cell \frac{d}{d time} (Ca_rel) = i_tr - i_rel - (10 diff_f_CQ) V_rel = 0.0012 V_cell \frac{d}{d time} (Cai) = \frac{i_diff V_sub - (i_up V_up)}{Vi} - (0.045 diff_f_CMi + 0.031 diff_f_TC + 0.062 diff_f_TMC) \frac{d}{d time} (Casub) = \frac{\frac{- (i_CaL - (2 i_NaCa))}{2 F} + i_rel V_rel}{V_sub} - i_diff - (0.045 diff_f_CMs) - (\frac{0.031}{1.2} diff_f_CSL) Vi = 0.46 V_cell - V_sub V_sub = 0.01 V_cell diff_f_TC = 88.8 e 3 Cai (1 - f_TC) - (0.446 e 3 f_TC) \frac{d}{d time} (f_TC) = diff_f_TC diff_f_TMC = 227.7 e 3 Cai (1 - f_TMC - f_TMM) - (0.00751 e 3 f_TMC) \frac{d}{d time} (f_TMC) = diff_f_TMC diff_f_TMM = 2.277 e 3 2.5 (1 - f_TMC - f_TMM) - (0.751 e 3 f_TMM) \frac{d}{d time} (f_TMM) = diff_f_TMM diff_f_CMi = 227.7 e 3 Cai (1 - f_CMi) - (0.542 e 3 f_CMi) \frac{d}{d time} (f_CMi) = diff_f_CMi diff_f_CMs = 227.7 e 3 Casub (1 - f_CMs) - (0.542 e 3 f_CMs) \frac{d}{d time} (f_CMs) = diff_f_CMs diff_f_CQ = 0.534 e 3 Ca_rel (1 - f_CQ) - (0.445 e 3 f_CQ) \frac{d}{d time} (f_CQ) = diff_f_CQ diff_f_CSL = 0.001 (115 Casub (1 - f_CSL) - (1000 f_CSL)) \frac{d}{d time} (f_CSL) = diff_f_CSL

Model Mathematics

Component: environment

Component: membrane

Component: hyperpolarising_activated_current

Component: hyperpolarising_activated_current_y_gate

Component: rapid_delayed_rectifier_potassium_current

Component: rapid_delayed_rectifier_potassium_current_paf_gate

Component: rapid_delayed_rectifier_potassium_current_pas_gate

Component: rapid_delayed_rectifier_potassium_current_pik_gate

Component: time_independent_potassium_current

Component: background_current

Component: sodium_potassium_pump

Component: sodium_calcium_exchange_current

Component: fast_sodium_current

Component: fast_sodium_current_m_gate

Component: fast_sodium_current_h1_gate

Component: fast_sodium_current_h2_gate

Component: L_type_calcium_current

Component: L_type_calcium_current_d_gate

Component: L_type_calcium_current_f_gate

Component: L_type_calcium_current_f2_gate

Component: transient_outward_potassium_current

Component: transient_outward_potassium_current_r_gate

Component: transient_outward_potassium_current_qfast_gate

Component: transient_outward_potassium_current_qslow_gate

Component: sustained_outward_potassium_current

Component: sustained_outward_potassium_current_qa_gate

Component: sustained_outward_potassium_current_qi_gate

Component: acetylcholine_sensitive_current

Component: acetylcholine_sensitive_current_achf_gate

Component: acetylcholine_sensitive_current_achs_gate

Component: extracellular_sodium_concentration

Component: intracellular_sodium_concentration

Component: extracellular_calcium_concentration

Component: intracellular_calcium_concentration

Component: extracellular_potassium_concentration

Component: intracellular_potassium_concentration