Model Mathematics

Component: membrane

$i_ext=stim_amplitudeiftime≥stim_start∧time≦stim_end∧time-stim_start-⌊time-stim_startstim_period⌋⁢stim_period≦stim_duration0otherwiseddtimeVm=-i_tot+i_extCmi_tot=i_Na+i_Ca_L+i_Ca_T+i_K1+i_Kr+i_Ks+i_to+i_I+i_NaK+i_NaCai_I=i_bNSC+i_Cab+i_Kpl+i_lCa+i_KATP$

Component: internal_ion_concentrations

$b1=CMDN_max-Ca_Total+K_mCMDNc1=K_mCMDN⁢Ca_TotalCai=b12+4⁢c1-b12ddtimeNai=-i_net_NaF⁢ViddtimeKi=-i_net_K+i_extF⁢ViddtimeCa_Total=-i_net_Ca-i_SR_U-i_RyR-i_SR_L2⁢F⁢Vi+dCaidt_NLi_net_Na=i_Na_Na+i_Ks_Na+i_to_Na+i_CaL_Na+i_bNSC_Na+i_lCa_Na+3⁢i_NaK+3⁢i_NaCai_net_K=i_K1+i_Kr+i_to_K+i_KATP+i_Ks_K+i_Na_K+i_CaL_K+i_bNSC_K+i_lCa_K+i_Kpl-2⁢i_NaKi_net_Ca=i_CaL_Ca+i_Ca_T+i_Cab-2⁢i_NaCa$

Component: constant_field_equations

$CF_Na=-NaoifVm=0F⁢VmR⁢T⁢Nai-Nao⁢ⅇ-F⁢VmR⁢T1-ⅇ-F⁢VmR⁢TotherwiseCF_Ca=-CaoifVm=02⁢F⁢VmR⁢T⁢Cai-Cao⁢ⅇ-2⁢F⁢VmR⁢T1-ⅇ-2⁢F⁢VmR⁢TotherwiseCF_K=KiifVm=0F⁢VmR⁢T⁢Ki-Ko⁢ⅇ-F⁢VmR⁢T1-ⅇ-F⁢VmR⁢Totherwise$

Component: ATP_production

$ddtimeATPi=ProducingRate_Max⁢Adenosine_Total-ATPi+dATPdt_NL-i_NaKF⁢Vi+i_SR_U4⁢F⁢Vi$

Component: sodium_current

$i_Na=i_Na_Na+i_Na_Ki_Na_Na=P_Na⁢CF_Na⁢p_AP_Na⁢yi_Na_K=0.1⁢P_Na⁢CF_K⁢p_AP_Na⁢y$

Component: sodium_current_voltage_dependent_gate

$ddtimep_RP_Na=p_AP_Na⁢k_AP_RP+p_RI_Na⁢k_RI_RP-p_RP_Na⁢k_RP_RI+k_RP_APddtimep_AP_Na=p_RP_Na⁢k_RP_AP+p_AI_Na⁢k_AI_AP-p_AP_Na⁢k_AP_RP+k_AP_AIddtimep_AI_Na=p_RI_Na⁢k_RI_AI+p_AP_Na⁢k_AP_AI-p_AI_Na⁢k_AI_RI+k_AI_APp_RI_Na=1-p_RP_Na-p_AP_Na-p_AI_Nak_RP_AP=10.1027⁢ⅇ-Vm8+0.25⁢ⅇ-Vm50k_AP_RP=126⁢ⅇVm17+0.02⁢ⅇVm800k_AP_AI=10.8⁢ⅇ-Vm400k_AI_RI=11300⁢ⅇVm20+0.04⁢ⅇVm800k_RI_AI=10.0001027⁢ⅇ-Vm8+5⁢ⅇ-Vm400k_RP_RI=0.011+k_AI_AP⁢k_AP_RP⁢k_RI_AIk_AP_AI⁢k_RP_AP⁢k_AI_RIk_RI_RP=0.01-k_RP_RI$

Component: sodium_current_ultra_slow_gate

$alpha_y=19000000000⁢ⅇVm5+8000⁢ⅇVm100beta_y=10.014⁢ⅇ-Vm5+4000⁢ⅇ-Vm100ddtimey=alpha_y⁢1-y-beta_y⁢y$

Component: L_type_Ca_channel

$i_Ca_L=i_CaL_Na+i_CaL_Ca+i_CaL_Ki_CaL_Ca=P_CaL⁢CF_Ca⁢p_open_CaLi_CaL_Na=0.0000185⁢P_CaL⁢CF_Na⁢p_open_CaLi_CaL_K=0.000365⁢P_CaL⁢CF_K⁢p_open_CaLp_open_CaL=p_AP_CaL⁢p_U+p_UCa⁢y1+1.4ATPi3$

Component: L_type_Ca_channel_voltage_dependent_gate

$ddtimep_RP_CaL=p_AP_CaL⁢k_AP_RP+p_RI_CaL⁢k_RI_RP-p_RP_CaL⁢k_RP_RI+k_RP_APddtimep_AP_CaL=p_RP_CaL⁢k_RP_AP+p_AI_CaL⁢k_AI_AP-p_AP_CaL⁢k_AP_RP+k_AP_AIddtimep_AI_CaL=p_RI_CaL⁢k_RI_AI+p_AP_CaL⁢k_AP_AI-p_AI_CaL⁢k_AI_RI+k_AI_APp_RI_CaL=1-p_AP_CaL-p_RP_CaL-p_AI_CaLk_RP_AP=10.27⁢ⅇ-Vm5.9+1.5⁢ⅇ-Vm65k_AP_RP=1480⁢ⅇVm7+2.2⁢ⅇVm65k_RI_AI=10.0018⁢ⅇ-Vm7.4+2⁢ⅇ-Vm100k_AI_RI=12200000⁢ⅇVm7.4+11⁢ⅇVm100k_RP_RI=0.041+k_AI_AP⁢k_AP_RP⁢k_RI_AIk_AP_AI⁢k_RP_AP⁢k_AI_RIk_RI_RP=0.04-k_RP_RI$

Component: L_type_Ca_channel_Ca_dependent_gate

$iCaL=0.0676⁢CF_CaCaDiadic=iCaL⁢p_open_CaLCacm=Cai-0.3⁢iCaLCaEffC=Cacm⁢p_AP_CaLCaEffU=CaEffC+Cai⁢1-p_AP_CaLk_UUCa_Ca=k_U_UCa⁢CaEffUk_CCCa_Ca=k_C_CCa⁢CaEffCddtimep_U=p_C⁢k_C_U+p_UCa⁢k_UCa_U-p_U⁢k_UUCa_Ca+k_U_Cddtimep_UCa=p_U⁢k_UUCa_Ca+p_CCa⁢k_CCa_UCa-p_UCa⁢k_UCa_CCa+k_UCa_Uddtimep_C=p_CCa⁢k_CCa_C+p_U⁢k_U_C-p_C⁢k_C_U+k_C_CCa⁢Cacm⁢p_AP_CaLp_CCa=1-p_C-p_U-p_UCak_UCa_U=k_CCa_C⁢k_C_U⁢k_U_UCa⁢k_UCa_CCak_U_C⁢k_C_CCa⁢k_CCa_UCa$

Component: L_type_Ca_channel_ultra_slow_gate

$alpha_y=1250000⁢ⅇVm9+58⁢ⅇVm65beta_y=11800⁢ⅇ-Vm14+66⁢ⅇ-Vm65ddtimey=alpha_y⁢1-y-beta_y⁢y$

Component: T_type_Ca_channel

$i_Ca_T=P_CaT⁢CF_Ca⁢y1⁢y2$

Component: T_type_Ca_channel_y1_gate

$alpha_y1=10.019⁢ⅇ-Vm5.6+0.82⁢ⅇ-Vm250beta_y1=140⁢ⅇVm6.3+1.5⁢ⅇVm10000ddtimey1=alpha_y1⁢1-y1-beta_y1⁢y1$

Component: T_type_Ca_channel_y2_gate

$alpha_y2=162000⁢ⅇVm10.1+30⁢ⅇVm3000beta_y2=10.0006⁢ⅇ-Vm6.7+1.2⁢ⅇ-Vm25ddtimey2=alpha_y2⁢1-y2-beta_y2⁢y2$

Component: time_independent_potassium_current

$i_K1=g_K1⁢Vm-E_K⁢fO4+fO3+fO2⁢yE_K=R⁢TF⁢ln⁡KoKig_K1=P_K1_0⁢Cm⁢Ko5.40.4fB=mumu+lambdafO=lambdamu+lambdafO2=2⁢fO2⁢fB2fO3=83⁢fO3⁢fBfO4=fO4mu=0.75⁢ⅇ0.035⁢Vm-E_K-101+ⅇ0.015⁢Vm-E_K-140lambda=3⁢ⅇ-0.048⁢Vm-E_K-10⁢1+ⅇ0.064⁢Vm-E_K-381+ⅇ0.03⁢Vm-E_K-70$

Component: time_independent_potassium_current_y_gate

$alpha_y=18000⁢ⅇVm-E_K-978.5+7⁢ⅇVm-E_K-97300beta_y=fO4⁢10.00014⁢ⅇ-Vm-E_K-979.1+0.2⁢ⅇ-Vm-E_K-97500ddtimey=alpha_y⁢1-y-beta_y⁢y$

Component: rapid_time_dependent_potassium_current

$i_Kr=g_Kr⁢Vm-E_K⁢0.6⁢y1+0.4⁢y2⁢y3g_Kr=P_Kr⁢Cm⁢Ko5.40.2$

Component: rapid_time_dependent_potassium_current_y1_gate

$alpha_y1=120⁢ⅇ-Vm11.5+5⁢ⅇ-Vm300beta_y1=1160⁢ⅇVm28+200⁢ⅇVm1000+12500⁢ⅇVm20ddtimey1=alpha_y1⁢1-y1-beta_y1⁢y1$

Component: rapid_time_dependent_potassium_current_y2_gate

$alpha_y2=1200⁢ⅇ-Vm13+20⁢ⅇ-Vm300beta_y2=11600⁢ⅇVm28+2000⁢ⅇVm1000+110000⁢ⅇVm20ddtimey2=alpha_y2⁢1-y2-beta_y2⁢y2$

Component: rapid_time_dependent_potassium_current_y3_gate

$alpha_y3=110⁢ⅇVm17+2.5⁢ⅇVm300beta_y3=10.35⁢ⅇ-Vm17+2⁢ⅇ-Vm150ddtimey3=alpha_y3⁢1-y3-beta_y3⁢y3$

Component: slow_time_dependent_potassium_current

$i_Ks=i_Ks_Na+i_Ks_Ki_Ks_K=P_Ks_K⁢CF_K⁢y12⁢0.9⁢y2+0.1i_Ks_Na=P_Ks_Na⁢CF_Na⁢y12⁢0.9⁢y2+0.1$

Component: slow_time_dependent_potassium_current_y1_gate

$alpha_y1=185⁢ⅇ-Vm10.5+370⁢ⅇ-Vm62beta_y1=11450⁢ⅇVm20+260⁢ⅇVm100ddtimey1=alpha_y1⁢1-y1-beta_y1⁢y1$

Component: slow_time_dependent_potassium_current_y2_gate

$alpha_y2=3.7⁢Caiddtimey2=alpha_y2⁢1-y2-beta_y2⁢y2$

Component: transient_outward_current

$i_to=i_to_Na+i_to_Ki_to_K=P_to_K⁢CF_K⁢y13⁢y2i_to_Na=P_to_Na⁢CF_Na⁢y13⁢y2$

Component: transient_outward_current_y1_gate

$alpha_y1=111⁢ⅇ-Vm28+0.2⁢ⅇ-Vm400beta_y1=14.4⁢ⅇVm16+0.2⁢ⅇVm500ddtimey1=alpha_y1⁢1-y1-beta_y1⁢y1$

Component: transient_outward_current_y2_gate

$alpha_y2=0.0038⁢ⅇ-Vm+13.511.31+0.051335⁢ⅇ-Vm+13.511.3beta_y2=0.0038⁢ⅇVm+13.511.31+0.067083⁢ⅇVm+13.511.3ddtimey2=alpha_y2⁢1-y2-beta_y2⁢y2$

Component: background_NSC_current

$i_bNSC=i_bNSC_K+i_bNSC_Nai_bNSC_K=0.4⁢P_bNSC⁢CF_Ki_bNSC_Na=P_bNSC⁢CF_Na$

Component: background_Kpl_current

$i_Kpl=P_Kpl⁢CF_K⁢13.0077ifVm=-3P_Kpl⁢CF_K⁢Vm+31-ⅇ-Vm+313otherwiseP_Kpl=0.00011⁢Ko5.40.16$

Component: background_lCa_current

$i_lCa=i_lCa_K+i_lCa_Nai_lCa_K=P_lCa⁢CF_K⁢p_openi_lCa_Na=P_lCa⁢CF_Na⁢p_openp_open=11+0.0012Cai3$

Component: background_KATP_current

$i_KATP=gamma⁢Vm-E_K⁢p_opengamma=P_KATP⁢N⁢Ko10.24p_open=0.81+ATPi0.12$

Component: background_Cab_current

$i_Cab=P_Cab⁢CF_Ca$

Component: sodium_calcium_exchanger

$i_NaCa=P_NaCa⁢Cm⁢1⁢k1⁢p_E1Na⁢y-k2⁢p_E2Na⁢1-yp_E1Na=11+Km_NaiNai3⁢1+CaiKm_Caip_E2Na=11+Km_NaoNao3⁢1+CaoKm_Caop_E1Ca=11+Km_CaiCai⁢1+NaiKm_Nai3p_E2Ca=11+Km_CaoCao⁢1+NaoKm_Nao3k1=1⁢ⅇPartition⁢F⁢VmR⁢Tk2=1⁢ⅇPartition-1⁢F⁢VmR⁢T$

Component: sodium_calcium_exchanger_y_gate

$alpha_y=k2⁢p_E2Na+k4⁢p_E2Cabeta_y=k1⁢p_E1Na+k3⁢p_E1Caddtimey=alpha_y⁢1-y-beta_y⁢y$

Component: sodium_potassium_pump

$i_NaK=P_NaK⁢Cm⁢1⁢k1⁢p_E1Na⁢y-k2⁢p_E2Na⁢1-yp_E1Na=11+Km_NaiNai1.06⁢1+KiKm_Ki1.12p_E2Na=11+Km_NaoNao_Eff1.06⁢1+KoKm_Ko1.12p_E1K=11+Km_KiKi1.12⁢1+NaiKm_Nai1.06p_E2K=11+Km_KoKo1.12⁢1+Nao_EffKm_Nao1.06k1=0.371+Km_ATPATPiNao_Eff=Nao⁢ⅇ-0.82⁢F⁢VmR⁢T$

Component: sodium_potassium_pump_y_gate

$alpha_y=k2⁢p_E2Na+k4⁢p_E2Kbeta_y=k1⁢p_E1Na+k3⁢p_E1Kddtimey=alpha_y⁢1-y-beta_y⁢y$

Component: SR_calcium_pump

$i_SR_U=i_max⁢1⁢k1⁢p_E1Ca⁢y-k2⁢p_E2Ca⁢1-yp_E1Ca=11+Km_CaSRCaupp_E2Ca=11+Km_CaCytoCaip_E1=1-p_E1Cap_E2=1-p_E2Cak2=11+Km_ATPATPi$

Component: SR_calcium_pump_y_gate

$alpha_y=k2⁢p_E2Ca+k4⁢p_E2beta_y=k1⁢p_E1Ca+k3⁢p_E1ddtimey=alpha_y⁢1-y-beta_y⁢y$

Component: RyR_channel

$i_RyR=P_RyR⁢Carel-Cai⁢p_open_RyRddtimep_open_RyR=p_close_RyR⁢k1-p_open_RyR⁢k2ddtimep_close_RyR=k3⁢1-p_open_RyR+p_close_RyR-k1+k4⁢p_close_RyRk1=280000⁢Cai12+Diadid_Factor⁢CaDiadick2=0.081+0.36Carelk3=0.000377⁢Carel12$

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

$b1=CSQN_max-Ca_Total+K_mCSQNc1=K_mCSQN⁢Ca_TotalCarel=b12+4⁢c1-b12ddtimeCa_Total=i_SR_T-i_RyR2⁢F⁢V_relddtimeCaup=-i_SR_U-i_SR_T-i_SR_L2⁢F⁢V_up$

Component: NL_model

$h=L-Xp=1-pCa-pCaCB-pCBQ_b=Y_1⁢Cai⁢p-Z_1⁢pCaEffFraction=ⅇ-20⁢L-L_a2Q_a=Y_2⁢pCa⁢EffFraction-Z_2⁢pCaCBQ_r=Y_3⁢pCaCB-Z_3⁢pCB⁢CaiQ_d=Y_4⁢pCBQ_d1=Y_d⁢ddtimeX2⁢pCBQ_d2=Y_d⁢ddtimeX2⁢pCaCBddtimepCa=Q_b-Q_addtimepCaCB=Q_a-Q_r-Q_d2ddtimepCB=Q_r-Q_d-Q_d1dCaidt=T_t⁢Q_d2+Q_r-Q_bdATPdt=-0.4⁢pCaCB⁢T_tCBBound=T_t⁢pCaCB+pCBNewCBF=ForceFactor⁢CBBoundForceEcomp=KForceEC⁢ZeroForceEL-L5+KForceLinearEc⁢ZeroForceEL-LForceCB=NewCBF⁢hForceExt=-ForceEcomp+ForceCBddtimeX=B⁢h-h_c$

Component: transmembrane_currents

Source
Derived from workspace Matsuoka, Sarai, Kuratomi, Ono, Noma, 2003 at changeset dd1892d2a56a.
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