Model Mathematics

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Component: environment

Component: membrane

i_ext = \{\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} (Vm) = \frac{- (i_tot + i_ext)}{Cm} i_tot = i_Na + i_Ca_L + i_Ca_T + i_K1 + i_Kr + i_Ks + i_to + i_I + i_NaK + i_NaCa i_I = i_bNSC + i_Cab + i_Kpl + i_lCa + i_KATP

Component: external_ion_concentrations

Component: internal_ion_concentrations

b1 = (CMDN_max - Ca_Total) + K_mCMDN c1 = K_mCMDN Ca_Total Cai = \frac{\sqrt{{b1}^{2} + 4 c1} - b1}{2} \frac{d}{d time} (Nai) = \frac{- i_net_Na}{F Vi} \frac{d}{d time} (Ki) = \frac{- (i_net_K + i_ext)}{F Vi} \frac{d}{d time} (Ca_Total) = \frac{- (i_net_Ca - i_SR_U - i_RyR - i_SR_L)}{2 F Vi} + dCaidt_NL i_net_Na = i_Na_Na + i_Ks_Na + i_to_Na + i_CaL_Na + i_bNSC_Na + i_lCa_Na + 3 i_NaK + 3 i_NaCa i_net_K = i_K1 + i_Kr + i_to_K + i_KATP + i_Ks_K + i_Na_K + i_CaL_K + i_bNSC_K + i_lCa_K + i_Kpl - (2 i_NaK) i_net_Ca = i_CaL_Ca + i_Ca_T + i_Cab - (2 i_NaCa)

Component: constant_field_equations

CF_Na = \{\begin{matrix} - Nao if Vm = 0 \\ \frac{\frac{F Vm}{R T} (Nai - (Nao ⅇ^{\frac{(- F) Vm}{R T}}))}{1 - (ⅇ^{\frac{(- F) Vm}{R T}})} otherwise \end{matrix} CF_Ca = \{\begin{matrix} - Cao if Vm = 0 \\ \frac{\frac{2 F Vm}{R T} (Cai - (Cao ⅇ^{\frac{(- 2) F Vm}{R T}}))}{1 - (ⅇ^{\frac{(- 2) F Vm}{R T}})} otherwise \end{matrix} CF_K = \{\begin{matrix} Ki if Vm = 0 \\ \frac{\frac{F Vm}{R T} (Ki - (Ko ⅇ^{\frac{(- F) Vm}{R T}}))}{1 - (ⅇ^{\frac{(- F) Vm}{R T}})} otherwise \end{matrix}

Component: ATP_production

\frac{d}{d time} (ATPi) = (ProducingRate_Max (Adenosine_Total - ATPi) + dATPdt_NL - (\frac{i_NaK}{F Vi})) + \frac{i_SR_U}{4 F Vi}

Component: sodium_current

i_Na = i_Na_Na + i_Na_K i_Na_Na = P_Na CF_Na p_AP_Na y i_Na_K = 0.1 P_Na CF_K p_AP_Na y

Component: sodium_current_voltage_dependent_gate

\frac{d}{d time} (p_RP_Na) = p_AP_Na k_AP_RP + p_RI_Na k_RI_RP - (p_RP_Na (k_RP_RI + k_RP_AP)) \frac{d}{d time} (p_AP_Na) = p_RP_Na k_RP_AP + p_AI_Na k_AI_AP - (p_AP_Na (k_AP_RP + k_AP_AI)) \frac{d}{d time} (p_AI_Na) = p_RI_Na k_RI_AI + p_AP_Na k_AP_AI - (p_AI_Na (k_AI_RI + k_AI_AP)) p_RI_Na = 1 - p_RP_Na - p_AP_Na - p_AI_Na k_RP_AP = \frac{1}{0.1027 ⅇ^{\frac{- Vm}{8}} + 0.25 ⅇ^{\frac{- Vm}{50}}} k_AP_RP = \frac{1}{26 ⅇ^{\frac{Vm}{17}} + 0.02 ⅇ^{\frac{Vm}{800}}} k_AP_AI = \frac{1}{0.8 ⅇ^{\frac{- Vm}{400}}} k_AI_RI = \frac{1}{1300 ⅇ^{\frac{Vm}{20}} + 0.04 ⅇ^{\frac{Vm}{800}}} k_RI_AI = \frac{1}{0.0001027 ⅇ^{\frac{- Vm}{8}} + 5 ⅇ^{\frac{- Vm}{400}}} k_RP_RI = \frac{0.01}{1 + \frac{k_AI_AP k_AP_RP k_RI_AI}{k_AP_AI k_RP_AP k_AI_RI}} k_RI_RP = 0.01 - k_RP_RI

Component: sodium_current_ultra_slow_gate

alpha_y = \frac{1}{9000000000 ⅇ^{\frac{Vm}{5}} + 8000 ⅇ^{\frac{Vm}{100}}} beta_y = \frac{1}{0.014 ⅇ^{\frac{- Vm}{5}} + 4000 ⅇ^{\frac{- Vm}{100}}} \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

Component: L_type_Ca_channel

i_Ca_L = i_CaL_Na + i_CaL_Ca + i_CaL_K i_CaL_Ca = P_CaL CF_Ca p_open_CaL i_CaL_Na = 0.0000185 P_CaL CF_Na p_open_CaL i_CaL_K = 0.000365 P_CaL CF_K p_open_CaL p_open_CaL = \frac{p_AP_CaL (p_U + p_UCa) y}{1 + {(\frac{1.4}{ATPi})}^{3}}

Component: L_type_Ca_channel_voltage_dependent_gate

\frac{d}{d time} (p_RP_CaL) = p_AP_CaL k_AP_RP + p_RI_CaL k_RI_RP - (p_RP_CaL (k_RP_RI + k_RP_AP)) \frac{d}{d time} (p_AP_CaL) = p_RP_CaL k_RP_AP + p_AI_CaL k_AI_AP - (p_AP_CaL (k_AP_RP + k_AP_AI)) \frac{d}{d time} (p_AI_CaL) = p_RI_CaL k_RI_AI + p_AP_CaL k_AP_AI - (p_AI_CaL (k_AI_RI + k_AI_AP)) p_RI_CaL = 1 - p_AP_CaL - p_RP_CaL - p_AI_CaL k_RP_AP = \frac{1}{0.27 ⅇ^{\frac{- Vm}{5.9}} + 1.5 ⅇ^{\frac{- Vm}{65}}} k_AP_RP = \frac{1}{480 ⅇ^{\frac{Vm}{7}} + 2.2 ⅇ^{\frac{Vm}{65}}} k_RI_AI = \frac{1}{0.0018 ⅇ^{\frac{- Vm}{7.4}} + 2 ⅇ^{\frac{- Vm}{100}}} k_AI_RI = \frac{1}{2200000 ⅇ^{\frac{Vm}{7.4}} + 11 ⅇ^{\frac{Vm}{100}}} k_RP_RI = \frac{0.04}{1 + \frac{k_AI_AP k_AP_RP k_RI_AI}{k_AP_AI k_RP_AP k_AI_RI}} k_RI_RP = 0.04 - k_RP_RI

Component: L_type_Ca_channel_Ca_dependent_gate

iCaL = 0.0676 CF_Ca CaDiadic = iCaL p_open_CaL Cacm = Cai - (0.3 iCaL) CaEffC = Cacm p_AP_CaL CaEffU = CaEffC + Cai (1 - p_AP_CaL) k_UUCa_Ca = k_U_UCa CaEffU k_CCCa_Ca = k_C_CCa CaEffC \frac{d}{d time} (p_U) = p_C k_C_U + p_UCa k_UCa_U - (p_U (k_UUCa_Ca + k_U_C)) \frac{d}{d time} (p_UCa) = p_U k_UUCa_Ca + p_CCa k_CCa_UCa - (p_UCa (k_UCa_CCa + k_UCa_U)) \frac{d}{d time} (p_C) = p_CCa k_CCa_C + p_U k_U_C - (p_C (k_C_U + k_C_CCa Cacm p_AP_CaL)) p_CCa = 1 - p_C - p_U - p_UCa k_UCa_U = \frac{k_CCa_C k_C_U k_U_UCa k_UCa_CCa}{k_U_C k_C_CCa k_CCa_UCa}

Component: L_type_Ca_channel_ultra_slow_gate

alpha_y = \frac{1}{250000 ⅇ^{\frac{Vm}{9}} + 58 ⅇ^{\frac{Vm}{65}}} beta_y = \frac{1}{1800 ⅇ^{\frac{- Vm}{14}} + 66 ⅇ^{\frac{- Vm}{65}}} \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

Component: T_type_Ca_channel

i_Ca_T = P_CaT CF_Ca y1 y2

Component: T_type_Ca_channel_y1_gate

alpha_y1 = \frac{1}{0.019 ⅇ^{\frac{- Vm}{5.6}} + 0.82 ⅇ^{\frac{- Vm}{250}}} beta_y1 = \frac{1}{40 ⅇ^{\frac{Vm}{6.3}} + 1.5 ⅇ^{\frac{Vm}{10000}}} \frac{d}{d time} (y1) = alpha_y1 (1 - y1) - (beta_y1 y1)

Component: T_type_Ca_channel_y2_gate

alpha_y2 = \frac{1}{62000 ⅇ^{\frac{Vm}{10.1}} + 30 ⅇ^{\frac{Vm}{3000}}} beta_y2 = \frac{1}{0.0006 ⅇ^{\frac{- Vm}{6.7}} + 1.2 ⅇ^{\frac{- Vm}{25}}} \frac{d}{d time} (y2) = alpha_y2 (1 - y2) - (beta_y2 y2)

Component: time_independent_potassium_current

i_K1 = g_K1 (Vm - E_K) (fO4 + fO3 + fO2) y E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) g_K1 = P_K1_0 Cm {(\frac{Ko}{5.4})}^{0.4} fB = \frac{mu}{mu + lambda} fO = \frac{lambda}{mu + lambda} fO2 = 2 {fO}^{2} {fB}^{2} fO3 = \frac{8}{3} {fO}^{3} fB fO4 = {fO}^{4} mu = \frac{0.75 ⅇ^{0.035 (Vm - E_K - 10)}}{1 + ⅇ^{0.015 (Vm - E_K - 140)}} lambda = \frac{3 ⅇ^{(- 0.048) (Vm - E_K - 10)} (1 + ⅇ^{0.064 (Vm - E_K - 38)})}{1 + ⅇ^{0.03 (Vm - E_K - 70)}}

Component: time_independent_potassium_current_y_gate

alpha_y = \frac{1}{8000 ⅇ^{\frac{Vm - E_K - 97}{8.5}} + 7 ⅇ^{\frac{Vm - E_K - 97}{300}}} beta_y = \frac{{fO}^{4} 1}{0.00014 ⅇ^{\frac{- (Vm - E_K - 97)}{9.1}} + 0.2 ⅇ^{\frac{- (Vm - E_K - 97)}{500}}} \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

Component: rapid_time_dependent_potassium_current

i_Kr = g_Kr (Vm - E_K) (0.6 y1 + 0.4 y2) y3 g_Kr = P_Kr Cm {(\frac{Ko}{5.4})}^{0.2}

Component: rapid_time_dependent_potassium_current_y1_gate

alpha_y1 = \frac{1}{20 ⅇ^{\frac{- Vm}{11.5}} + 5 ⅇ^{\frac{- Vm}{300}}} beta_y1 = \frac{1}{160 ⅇ^{\frac{Vm}{28}} + 200 ⅇ^{\frac{Vm}{1000}}} + \frac{1}{2500 ⅇ^{\frac{Vm}{20}}} \frac{d}{d time} (y1) = alpha_y1 (1 - y1) - (beta_y1 y1)

Component: rapid_time_dependent_potassium_current_y2_gate

alpha_y2 = \frac{1}{200 ⅇ^{\frac{- Vm}{13}} + 20 ⅇ^{\frac{- Vm}{300}}} beta_y2 = \frac{1}{1600 ⅇ^{\frac{Vm}{28}} + 2000 ⅇ^{\frac{Vm}{1000}}} + \frac{1}{10000 ⅇ^{\frac{Vm}{20}}} \frac{d}{d time} (y2) = alpha_y2 (1 - y2) - (beta_y2 y2)

Component: rapid_time_dependent_potassium_current_y3_gate

alpha_y3 = \frac{1}{10 ⅇ^{\frac{Vm}{17}} + 2.5 ⅇ^{\frac{Vm}{300}}} beta_y3 = \frac{1}{0.35 ⅇ^{\frac{- Vm}{17}} + 2 ⅇ^{\frac{- Vm}{150}}} \frac{d}{d time} (y3) = alpha_y3 (1 - y3) - (beta_y3 y3)

Component: slow_time_dependent_potassium_current

i_Ks = i_Ks_Na + i_Ks_K i_Ks_K = P_Ks_K CF_K {y1}^{2} (0.9 y2 + 0.1) i_Ks_Na = P_Ks_Na CF_Na {y1}^{2} (0.9 y2 + 0.1)

Component: slow_time_dependent_potassium_current_y1_gate

alpha_y1 = \frac{1}{85 ⅇ^{\frac{- Vm}{10.5}} + 370 ⅇ^{\frac{- Vm}{62}}} beta_y1 = \frac{1}{1450 ⅇ^{\frac{Vm}{20}} + 260 ⅇ^{\frac{Vm}{100}}} \frac{d}{d time} (y1) = alpha_y1 (1 - y1) - (beta_y1 y1)

Component: slow_time_dependent_potassium_current_y2_gate

alpha_y2 = 3.7 Cai \frac{d}{d time} (y2) = alpha_y2 (1 - y2) - (beta_y2 y2)

Component: transient_outward_current

i_to = i_to_Na + i_to_K i_to_K = P_to_K CF_K {y1}^{3} y2 i_to_Na = P_to_Na CF_Na {y1}^{3} y2

Component: transient_outward_current_y1_gate

alpha_y1 = \frac{1}{11 ⅇ^{\frac{- Vm}{28}} + 0.2 ⅇ^{\frac{- Vm}{400}}} beta_y1 = \frac{1}{4.4 ⅇ^{\frac{Vm}{16}} + 0.2 ⅇ^{\frac{Vm}{500}}} \frac{d}{d time} (y1) = alpha_y1 (1 - y1) - (beta_y1 y1)

Component: transient_outward_current_y2_gate

alpha_y2 = \frac{0.0038 ⅇ^{\frac{- (Vm + 13.5)}{11.3}}}{1 + 0.051335 ⅇ^{\frac{- (Vm + 13.5)}{11.3}}} beta_y2 = \frac{0.0038 ⅇ^{\frac{Vm + 13.5}{11.3}}}{1 + 0.067083 ⅇ^{\frac{Vm + 13.5}{11.3}}} \frac{d}{d time} (y2) = alpha_y2 (1 - y2) - (beta_y2 y2)

Component: background_NSC_current

i_bNSC = i_bNSC_K + i_bNSC_Na i_bNSC_K = 0.4 P_bNSC CF_K i_bNSC_Na = P_bNSC CF_Na

Component: background_Kpl_current

i_Kpl = \{\begin{matrix} P_Kpl CF_K 13.0077 if Vm = - 3 \\ \frac{P_Kpl CF_K (Vm + 3)}{1 - (ⅇ^{\frac{- (Vm + 3)}{13}})} otherwise \end{matrix} P_Kpl = 0.00011 {(\frac{Ko}{5.4})}^{0.16}

Component: background_lCa_current

i_lCa = i_lCa_K + i_lCa_Na i_lCa_K = P_lCa CF_K p_open i_lCa_Na = P_lCa CF_Na p_open p_open = \frac{1}{1 + {(\frac{0.0012}{Cai})}^{3}}

Component: background_KATP_current

i_KATP = gamma (Vm - E_K) p_open gamma = P_KATP N {(\frac{Ko}{1})}^{0.24} p_open = \frac{0.8}{1 + {(\frac{ATPi}{0.1})}^{2}}

Component: background_Cab_current

i_Cab = P_Cab CF_Ca

Component: sodium_calcium_exchanger

i_NaCa = P_NaCa Cm 1 (k1 p_E1Na y - (k2 p_E2Na (1 - y))) p_E1Na = \frac{1}{1 + {(\frac{Km_Nai}{Nai})}^{3} (1 + \frac{Cai}{Km_Cai})} p_E2Na = \frac{1}{1 + {(\frac{Km_Nao}{Nao})}^{3} (1 + \frac{Cao}{Km_Cao})} p_E1Ca = \frac{1}{1 + \frac{Km_Cai}{Cai} (1 + {(\frac{Nai}{Km_Nai})}^{3})} p_E2Ca = \frac{1}{1 + \frac{Km_Cao}{Cao} (1 + {(\frac{Nao}{Km_Nao})}^{3})} k1 = 1 ⅇ^{\frac{Partition F Vm}{R T}} k2 = 1 ⅇ^{\frac{(Partition - 1) F Vm}{R T}}

Component: sodium_calcium_exchanger_y_gate

alpha_y = k2 p_E2Na + k4 p_E2Ca beta_y = k1 p_E1Na + k3 p_E1Ca \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

Component: sodium_potassium_pump

i_NaK = P_NaK Cm 1 (k1 p_E1Na y - (k2 p_E2Na (1 - y))) p_E1Na = \frac{1}{1 + {(\frac{Km_Nai}{Nai})}^{1.06} (1 + {(\frac{Ki}{Km_Ki})}^{1.12})} p_E2Na = \frac{1}{1 + {(\frac{Km_Nao}{Nao_Eff})}^{1.06} (1 + {(\frac{Ko}{Km_Ko})}^{1.12})} p_E1K = \frac{1}{1 + {(\frac{Km_Ki}{Ki})}^{1.12} (1 + {(\frac{Nai}{Km_Nai})}^{1.06})} p_E2K = \frac{1}{1 + {(\frac{Km_Ko}{Ko})}^{1.12} (1 + {(\frac{Nao_Eff}{Km_Nao})}^{1.06})} k1 = \frac{0.37}{1 + \frac{Km_ATP}{ATPi}} Nao_Eff = Nao ⅇ^{\frac{(- 0.82) F Vm}{R T}}

Component: sodium_potassium_pump_y_gate

alpha_y = k2 p_E2Na + k4 p_E2K beta_y = k1 p_E1Na + k3 p_E1K \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

Component: SR_calcium_pump

i_SR_U = i_max 1 (k1 p_E1Ca y - (k2 p_E2Ca (1 - y))) p_E1Ca = \frac{1}{1 + \frac{Km_CaSR}{Caup}} p_E2Ca = \frac{1}{1 + \frac{Km_CaCyto}{Cai}} p_E1 = 1 - p_E1Ca p_E2 = 1 - p_E2Ca k2 = \frac{1}{1 + \frac{Km_ATP}{ATPi}}

Component: SR_calcium_pump_y_gate

alpha_y = k2 p_E2Ca + k4 p_E2 beta_y = k1 p_E1Ca + k3 p_E1 \frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y)

Component: RyR_channel

i_RyR = P_RyR (Carel - Cai) p_open_RyR \frac{d}{d time} (p_open_RyR) = p_close_RyR k1 - (p_open_RyR k2) \frac{d}{d time} (p_close_RyR) = k3 (1 - (p_open_RyR + p_close_RyR)) - ((k1 + k4) p_close_RyR) k1 = 280000 {(\frac{Cai}{1})}^{2} + Diadid_Factor CaDiadic k2 = \frac{0.08}{1 + \frac{0.36}{Carel}} k3 = 0.000377 {(\frac{Carel}{1})}^{2}

Component: SR_T_current

i_SR_T = P_SR_T (Caup - Carel)

Component: SR_L_current

i_SR_L = P_SR_L (Caup - Cai)

Component: Ca_concentrations_in_SR

b1 = (CSQN_max - Ca_Total) + K_mCSQN c1 = K_mCSQN Ca_Total Carel = \frac{\sqrt{{b1}^{2} + 4 c1} - b1}{2} \frac{d}{d time} (Ca_Total) = \frac{i_SR_T - i_RyR}{2 F V_rel} \frac{d}{d time} (Caup) = \frac{- i_SR_U - i_SR_T - i_SR_L}{2 F V_up}

Component: NL_model

h = L - X p = 1 - pCa - pCaCB - pCB Q_b = Y_1 Cai p - (Z_1 pCa) EffFraction = ⅇ^{(- 20) {(L - L_a)}^{2}} Q_a = Y_2 pCa EffFraction - (Z_2 pCaCB) Q_r = Y_3 pCaCB - (Z_3 pCB Cai) Q_d = Y_4 pCB Q_d1 = Y_d {(\frac{d}{d time} (X))}^{2} pCB Q_d2 = Y_d {(\frac{d}{d time} (X))}^{2} pCaCB \frac{d}{d time} (pCa) = Q_b - Q_a \frac{d}{d time} (pCaCB) = Q_a - Q_r - Q_d2 \frac{d}{d time} (pCB) = Q_r - Q_d - Q_d1 dCaidt = T_t (Q_d2 + Q_r - Q_b) dATPdt = (- 0.4) pCaCB T_t CBBound = T_t (pCaCB + pCB) NewCBF = ForceFactor CBBound ForceEcomp = KForceEC {(ZeroForceEL - L)}^{5} + KForceLinearEc (ZeroForceEL - L) ForceCB = NewCBF h ForceExt = (- ForceEcomp) + ForceCB \frac{d}{d time} (X) = B (h - h_c)

Model Mathematics

Component: environment

Component: membrane

Component: external_ion_concentrations

Component: internal_ion_concentrations

Component: constant_field_equations

Component: ATP_production

Component: sodium_current

Component: sodium_current_voltage_dependent_gate

Component: sodium_current_ultra_slow_gate

Component: L_type_Ca_channel

Component: L_type_Ca_channel_voltage_dependent_gate

Component: L_type_Ca_channel_Ca_dependent_gate

Component: L_type_Ca_channel_ultra_slow_gate

Component: T_type_Ca_channel

Component: T_type_Ca_channel_y1_gate

Component: T_type_Ca_channel_y2_gate

Component: time_independent_potassium_current

Component: time_independent_potassium_current_y_gate

Component: rapid_time_dependent_potassium_current

Component: rapid_time_dependent_potassium_current_y1_gate

Component: rapid_time_dependent_potassium_current_y2_gate

Component: rapid_time_dependent_potassium_current_y3_gate

Component: slow_time_dependent_potassium_current

Component: slow_time_dependent_potassium_current_y1_gate

Component: slow_time_dependent_potassium_current_y2_gate

Component: transient_outward_current

Component: transient_outward_current_y1_gate

Component: transient_outward_current_y2_gate

Component: background_NSC_current

Component: background_Kpl_current

Component: background_lCa_current

Component: background_KATP_current

Component: background_Cab_current

Component: sodium_calcium_exchanger

Component: sodium_calcium_exchanger_y_gate

Component: sodium_potassium_pump

Component: sodium_potassium_pump_y_gate

Component: SR_calcium_pump

Component: SR_calcium_pump_y_gate

Component: RyR_channel

Component: SR_T_current

Component: SR_L_current

Component: Ca_concentrations_in_SR

Component: NL_model

Component: internal_concentrations

Component: intracellular_currents

Component: transmembrane_currents