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) = - (i_CaL + i_CaT + i_pCa + i_NaCa + i_Cab + i_Na + i_Nab + i_NaK + i_Kto_f + i_K1 + i_Ks + i_Kur + i_Kss + i_Kr + i_ClCa + i_stim)

Component: calcium_concentration

\frac{d}{d time} (Cai) = Bi (J_leak + J_xfer - (J_up + J_trpn)) \frac{d}{d time} (Cass) = Bss (\frac{J_rel VJSR}{Vss} - (\frac{J_xfer Vmyo}{Vss} + \frac{(i_CaL + i_Cab + i_pCa + i_CaT - (2 i_NaCa)) Acap Cm}{2 Vss F})) \frac{d}{d time} (CaJSR) = BJSR (J_xfer - J_rel) \frac{d}{d time} (CaNSR) = \frac{(J_up - J_leak) Vmyo}{VNSR} - (\frac{J_tr VJSR}{VNSR}) Bi = {(1 + \frac{CMDN_tot Km_CMDN}{{(Km_CMDN + Cai)}^{2}})}^{- 1} Bss = {(1 + \frac{CMDN_tot Km_CMDN}{{(Km_CMDN + Cass)}^{2}})}^{- 1} BJSR = {(1 + \frac{CSQN_tot Km_CSQN}{{(Km_CSQN + CaJSR)}^{2}})}^{- 1}

Component: calcium_fluxes

J_rel = v1 (P_O1 + P_O2) (CaJSR - Cass) P_RyR J_tr = \frac{CaNSR - CaJSR}{tau_tr} J_xfer = \frac{Cass - Cai}{tau_xfer} J_leak = v2 (CaNSR - Cai) J_up = \frac{v3 {Cai}^{2}}{{Km_up}^{2} + {Cai}^{2}} J_trpn = k_plus_htrpn Cai (HTRPN_tot - HTRPN_Ca) + k_plus_ltrpn Cai (LTRPN_tot - LTRPN_Ca) - (k_minus_htrpn HTRPN_Ca + k_minus_ltrpn LTRPN_Ca) \frac{d}{d time} (P_RyR) = (- 0.04) P_RyR - (\frac{0.1 i_CaL}{i_CaL_max} ⅇ^{\frac{- ({(V - 5)}^{2})}{648}})

Component: calcium_buffering

\frac{d}{d time} (LTRPN_Ca) = k_plus_ltrpn Cai (LTRPN_tot - LTRPN_Ca) - (k_minus_ltrpn LTRPN_Ca) \frac{d}{d time} (HTRPN_Ca) = k_plus_htrpn Cai (HTRPN_tot - HTRPN_Ca) - (k_minus_htrpn HTRPN_Ca)

Component: ryanodine_receptors

\frac{d}{d time} (P_O1) = k_plus_a {Cass}^{n} P_C1 + k_minus_b P_O2 + k_minus_c P_C2 - (k_minus_a P_O1 + k_plus_b {Cass}^{m} P_O1 + k_plus_c P_O1) P_C1 = 1 - (P_C2 + P_O1 + P_O2) \frac{d}{d time} (P_O2) = k_plus_b {Cass}^{m} P_O1 - (k_minus_b P_O2) \frac{d}{d time} (P_C2) = k_plus_c P_O1 - (k_minus_c P_C2)

Component: L_type_calcium_current

i_CaL = g_CaL O (V - E_CaL) \frac{d}{d time} (O) = alpha C4 + Kpcb I1 + 0.001 (alpha I2 - (Kpcf O)) - (4 beta O + gamma O) C1 = 1 - (O + C2 + C3 + C4 + I1 + I2 + I3) \frac{d}{d time} (C2) = 4 alpha C1 + 2 beta C3 - (beta C2 + 3 alpha C2) \frac{d}{d time} (C3) = 3 alpha C2 + 3 beta C4 - (2 beta C3 + 2 alpha C3) \frac{d}{d time} (C4) = 2 alpha C3 + 4 beta O + 0.01 (4 Kpcb beta I1 - (alpha gamma C4)) + 0.002 (4 beta I2 - (Kpcf C4)) + 4 beta Kpcb I3 - (3 beta C4 + alpha C4 + 1 gamma Kpcf C4) \frac{d}{d time} (I1) = gamma O + 0.001 (alpha I3 - (Kpcf I1)) + 0.01 (alpha gamma C4 - (4 beta Kpcf I1)) - (Kpcb I1) \frac{d}{d time} (I2) = 0.001 (Kpcf O - (alpha I2)) + Kpcb I3 + 0.002 (Kpcf C4 - (4 beta I2)) - (gamma I2) \frac{d}{d time} (I3) = 0.001 (Kpcf I1 - (alpha I3)) + gamma I2 + 1 gamma Kpcf C4 - (4 beta Kpcb I3 + Kpcb I3) alpha = \frac{0.4 ⅇ^{\frac{V + 12}{10}} (1 + 0.7 ⅇ^{\frac{- ({(V + 40)}^{2})}{10}} - (0.75 ⅇ^{\frac{- ({(V + 20)}^{2})}{400}}))}{1 + 0.12 ⅇ^{\frac{V + 12}{10}}} beta = 0.05 ⅇ^{\frac{- (V + 12)}{13}} gamma = \frac{Kpc_max Cass}{Kpc_half + Cass} Kpcf = 13 (1 - (ⅇ^{\frac{- ({(V + 14.5)}^{2})}{100}}))

Component: T_type_calcium_current

i_CaT = g_CaT b g (V - E_CaT)

Component: T_type_calcium_current_b_gate

b_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 48)}{6.1}}} tau_b = 0.1 + \frac{5.4}{1 + ⅇ^{\frac{V + 100}{6.6}}} \frac{d}{d time} (b) = \frac{b_infinity - b}{tau_b}

Component: T_type_calcium_current_g_gate

g_infinity = \frac{1}{1 + ⅇ^{\frac{V + 66}{6.6}}} tau_g = 8 + \frac{32}{1 + ⅇ^{\frac{V + 65}{5}}} \frac{d}{d time} (g) = \frac{g_infinity - g}{tau_g}

Component: calcium_pump_current

i_pCa = \frac{i_pCa_max {Cai}^{2}}{{Km_pCa}^{2} + {Cai}^{2}}

Component: sodium_calcium_exchange_current

i_NaCa = \frac{\frac{\frac{k_NaCa 1}{{K_mNa}^{3} + {Nao}^{3}} 1}{K_mCa + Cao} 1}{1 + k_sat ⅇ^{\frac{(eta - 1) V F}{R T}}} (ⅇ^{\frac{eta V F}{R T}} {Nai}^{3} Cao - (ⅇ^{\frac{(eta - 1) V F}{R T}} {Nao}^{3} Cai))

Component: calcium_background_current

i_Cab = g_Cab (V - E_CaN) E_CaN = \frac{R T}{2 F} \ln (\frac{Cao}{Cai})

Component: sodium_concentration

\frac{d}{d time} (Nai) = \frac{(- (i_Na + i_Nab + 3 i_NaK + 3 i_NaCa)) Acap Cm}{Vmyo F}

Component: fast_sodium_current

i_Na = g_Na m^{3} 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 - (ⅇ^{(- 0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (m) = alpha_m - ((alpha_m + beta_m) m)

Component: fast_sodium_current_h_gate

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{80 + V}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} \frac{d}{d time} (h) = alpha_h - ((alpha_h + beta_h) h)

Component: fast_sodium_current_j_gate

alpha_j = \{\begin{matrix} \frac{(- (127140 ⅇ^{0.2444 V} + 3.474 e -5 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.3 ⅇ^{(- 2.535 e -7) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} \frac{d}{d time} (j) = alpha_j - ((alpha_j + beta_j) j)

Component: sodium_background_current

i_Nab = g_Nab (V - E_Na)

Component: potassium_concentration

\frac{d}{d time} (Ki) = \frac{(- (i_Kto_f + i_K1 + i_Ks + i_Kss + i_Kur + i_Kr - (2 i_NaK))) Acap Cm}{Vmyo F}

Component: fast_transient_outward_potassium_current

i_Kto_f = g_Kto_f {ato_f}^{6.5} ito_f (V - E_K) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) \frac{d}{d time} (ato_f) = alpha_a (1 - ato_f) - (beta_a ato_f) \frac{d}{d time} (ito_f) = \frac{ito_f_infinity - ito_f}{tau_ito_f} ito_f_infinity = \frac{alpha_i}{alpha_i + beta_i} tau_ito_f = \frac{1}{\frac{0.000152 ⅇ^{\frac{- (V + 13.5)}{7}}}{0.067083 ⅇ^{\frac{- (V + 33.5)}{7}} + 1} + \frac{0.00095 ⅇ^{\frac{V + 33.5}{7}}}{0.051335 ⅇ^{\frac{V + 33.5}{7}} + 1}} alpha_a = 0.18064 ⅇ^{0.03577 (V + 30)} beta_a = 0.3956 ⅇ^{(- 0.06237) (V + 30)} alpha_i = \frac{0.000152 ⅇ^{\frac{- (V - 3.81)}{15.75}}}{0.0067083 ⅇ^{\frac{- (V + 132.05)}{15.75}} + 1} beta_i = \frac{0.00095 ⅇ^{\frac{V + 132.05}{15.75}}}{0.051335 ⅇ^{\frac{V + 132.05}{15.75}} + 1}

Component: ultra_rapidly_activating_delayed_rectifier_potassium_current

i_Kur = g_Kur aur iur (V - E_K) \frac{d}{d time} (aur) = \frac{ass - aur}{tau_aur} \frac{d}{d time} (iur) = \frac{iss - iur}{tau_iur} ass = \frac{1}{1 + ⅇ^{\frac{- (V + 22.5)}{7.7}}} tau_aur = 0.493 ⅇ^{(- 0.0629) V} + 2.058 iss = \frac{1}{1 + ⅇ^{\frac{V + 45.2}{5.7}}} tau_iur = 1200 - (\frac{170}{1 + ⅇ^{\frac{V + 45.2}{5.7}}})

Component: non_inactivating_steady_state_potassium_current

i_Kss = g_Kss aKss (V - E_K) \frac{d}{d time} (aKss) = \frac{ass - aKss}{tau_Kss} tau_Kss = 39.3 ⅇ^{(- 0.0862) V} + 13.17

Component: time_independent_potassium_current

i_K1 = \frac{\frac{g_K1 Ko}{Ko + 210} (V - E_K)}{1 + ⅇ^{0.0896 (V - E_K)}}

Component: slow_delayed_rectifier_potassium_current

i_Ks = g_Ks {nKs}^{2} (V - E_K) \frac{d}{d time} (nKs) = alpha_n (1 - nKs) - (beta_n nKs) alpha_n = \frac{0.00000481333 (V + 26.5)}{1 - (ⅇ^{(- 0.128) (V + 26.5)})} beta_n = 0.0000953333 ⅇ^{(- 0.038) (V + 26.5)}

Component: rapid_delayed_rectifier_potassium_current

i_Kr = g_Kr O_K (V - E_K) C_K0 = 1 - (C_K1 + C_K2 + O_K + I_K) \frac{d}{d time} (C_K2) = kf C_K1 + beta_a1 O_K - (kb C_K2 + alpha_a1 C_K2) \frac{d}{d time} (C_K1) = alpha_a0 C_K0 + kb C_K2 - (beta_a0 C_K1 + kf C_K1) \frac{d}{d time} (O_K) = alpha_a1 C_K2 + beta_i I_K - (beta_a1 O_K + alpha_i O_K) \frac{d}{d time} (I_K) = alpha_i O_K - (beta_i I_K) alpha_a0 = 0.022348 ⅇ^{0.01176 V} beta_a0 = 0.047002 ⅇ^{(- 0.0631) V} alpha_a1 = 0.013733 ⅇ^{0.038198 V} beta_a1 = 0.0000689 ⅇ^{(- 0.04178) V} alpha_i = 0.090821 ⅇ^{0.023391 (V + 5)} beta_i = 0.006497 ⅇ^{(- 0.03268) (V + 5)}

Component: sodium_potassium_pump_current

i_NaK = \frac{\frac{i_NaK_max f_NaK 1}{1 + {(\frac{Km_Nai}{Nai})}^{1.5}} Ko}{Ko + Km_Ko} f_NaK = \frac{1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- V) F}{R T}}} sigma = \frac{1}{7} (ⅇ^{\frac{Nao}{67300}} - 1)

Component: calcium_activated_chloride_current

f_ClCa = {(\frac{Cass}{Km_Cl + Cass})}^{n_ClCa} i_ClCa = \frac{\frac{1 P_ClCa f_ClCa V F^{2}}{R T} (Clo ⅇ^{\frac{V F}{R T}} - Cli)}{ⅇ^{\frac{V F}{R T}} - 1}