# Model Mathematics

### Component: Membrane

$i_tot=i_f+i_Kr+i_Ks+i_to+i_NaK+i_NaCa+i_Na+i_CaL+i_CaT+i_KACh+i_KurddtimeV_ode=-i_totCRTONF=R⁢TFV=V_clampifclamp_mode≥1V_odeotherwise$

### Component: Voltage_clamp

$V_clamp=V_testiftime>t_holding∧time

### Component: Ionic_values

$E_Na=RTONF⁢ln⁡NaoNaiE_K=RTONF⁢ln⁡KoKiE_Ca=0.5⁢RTONF⁢ln⁡CaoCa_sub$

### Component: Nai_concentration

$ddtimeNai_=1-Nai_clamp⁢-1⁢i_Na+i_fNa+i_siNa+3⁢i_NaK+3⁢i_NaCa1⁢V_i+V_sub⁢FNai=Nai_$

### Component: i_f

$G_f=g_fKoKo+Km_fG_f_K=G_falpha+1G_f_Na=alpha⁢G_f_Kg_f_Na=G_f_Na⁢KoKo+Km_fg_f_K=G_f_K⁢KoKo+Km_fi_fNa=y⁢g_f_Na⁢V-E_Na⁢1-blockadei_fK=y⁢g_f_K⁢V-E_K⁢1-blockadei_f=i_fNa+i_fK$

### Component: i_f_y_gate

$ACh_shift=-1-9.898⁢1⁢ACh0.6181⁢ACh0.618+0.00122423ifACh>00otherwiseIso_shift=7.5ifIso_1_uM>00otherwisetau_y=10.36⁢V+148.8-ACh_shift-Iso_shiftⅇ0.066⁢V+148.8-ACh_shift-Iso_shift-1+0.1⁢V+87.3-ACh_shift-Iso_shift1-ⅇ-0.2⁢V+87.3-ACh_shift-Iso_shift-0.054y_infinity=0.01329+0.999211+ⅇV+97.134-ACh_shift-Iso_shift-y_shift8.1752ifV<-80-ACh_shift-Iso_shift-y_shift0.0002501⁢ⅇ-V-ACh_shift-Iso_shift-y_shift12.861otherwiseddtimey=y_infinity-ytau_y$

### Component: i_NaK

$Iso_increase=1.2ifIso_1_uM>01otherwisei_NaK=Iso_increase⁢i_NaK_max⁢1+Km_KpKo1.2-1⁢1+Km_NapNai1.3-1⁢1+ⅇ-V-E_Na+11020-1$

### Component: i_NaCa

$i_NaCa=1-blockade_NaCa⁢K_NaCa⁢x2⁢k21-x1⁢k12x1+x2+x3+x4x1=k41⁢k34⁢k23+k21+k21⁢k32⁢k43+k41x2=k32⁢k43⁢k14+k12+k41⁢k12⁢k34+k32x3=k14⁢k43⁢k23+k21+k12⁢k23⁢k43+k41x4=k23⁢k34⁢k14+k12+k14⁢k21⁢k34+k32k43=NaiK3ni+Naik12=Ca_subKci⁢ⅇ-Qci⁢VRTONFdik14=NaiK1ni⁢NaiK2ni⁢1+NaiK3ni⁢ⅇQn⁢V2⁢RTONFdik41=ⅇ-Qn⁢V2⁢RTONFdi=1+Ca_subKci⁢1+ⅇ-Qci⁢VRTONF+NaiKcni+NaiK1ni⁢1+NaiK2ni⁢1+NaiK3nik34=NaoK3no+Naok21=CaoKco⁢ⅇQco⁢VRTONFdok23=NaoK1no⁢NaoK2no⁢1+NaoK3no⁢ⅇ-Qn⁢V2⁢RTONFdok32=ⅇQn⁢V2⁢RTONFdo=1+CaoKco⁢1+ⅇQco⁢VRTONF+NaoK1no⁢1+NaoK2no⁢1+NaoK3no$

### Component: i_Na

$E_mh=RTONF⁢ln⁡Nao+0.12⁢KoNai+0.12⁢Kii_Na_=g_Na⁢m3⁢h⁢V-E_mhi_Na_L=g_Na_L⁢m3⁢V-E_mhi_Na=i_Na_+i_Na_L$

### Component: i_Na_m_gate

$m_infinity=11+ⅇ-V+42.05048.3106E0_m=V+41alpha_m=2000if|E0_m|

### Component: i_Na_h_gate

$h_infinity=11+ⅇV+69.8044.4565alpha_h=20⁢ⅇ-0.125⁢V+75beta_h=2000320⁢ⅇ-0.1⁢V+75+1tau_h=1alpha_h+beta_hddtimeh=h_infinity-htau_h$

### Component: i_CaL

$Iso_increase=1.23ifIso_1_uM>01otherwisei_siCa=2⁢P_CaL⁢V-0RTONF⁢1-ⅇ-1⁢V-0⁢2RTONF⁢Ca_sub-Cao⁢ⅇ-2⁢V-0RTONF⁢dL⁢fL⁢fCai_siK=0.000365⁢P_CaL⁢V-0RTONF⁢1-ⅇ-1⁢V-0RTONF⁢Ki-Ko⁢ⅇ-1⁢V-0RTONF⁢dL⁢fL⁢fCai_siNa=0.0000185⁢P_CaL⁢V-0RTONF⁢1-ⅇ-1⁢V-0RTONF⁢Nai-Nao⁢ⅇ-1⁢V-0RTONF⁢dL⁢fL⁢fCaACh_block=0.31⁢AChACh+0.00009i_CaL=i_siCa+i_siK+i_siNa⁢1-ACh_block⁢1⁢Iso_increase$

### Component: i_CaL_dL_gate

$Iso_shift_dL=-8ifIso_1_uM>00otherwiseIso_slope_dL=-27ifIso_1_uM>00otherwisedL_infinity=11+ⅇ-V-V_dL-Iso_shift_dLk_dL⁢1+Iso_slope_dL100tau_dL=0.001alpha_dL+beta_dLalpha_dL=-0.02839⁢adVm+41.8ⅇ-adVm+41.82.5-1-0.0849⁢adVm+6.8ⅇ-adVm+6.84.8-1adVm=-41.80001ifV=-41.80ifV=0-6.80001ifV=-6.8Votherwisebeta_dL=0.01143⁢bdVm+1.8ⅇbdVm+1.82.5-1bdVm=-1.80001ifV=-1.8VotherwiseddtimedL=dL_infinity-dLtau_dL$

### Component: i_CaL_fL_gate

$fL_infinity=11+ⅇV+37.4+shift_fL5.3+k_fLtau_fL=0.001⁢44.3+230⁢ⅇ-V+36102ddtimefL=fL_infinity-fLtau_fL$

### Component: i_CaL_fCa_gate

$fCa_infinity=Km_fCaKm_fCa+Ca_subtau_fCa=0.001⁢fCa_infinityalpha_fCaddtimefCa=fCa_infinity-fCatau_fCa$

### Component: i_CaT

$i_CaT=2⁢P_CaT⁢VRTONF⁢1-ⅇ-1⁢V⁢2RTONF⁢Ca_sub-Cao⁢ⅇ-2⁢VRTONF⁢dT⁢fT$

### Component: i_CaT_dT_gate

$dT_infinity=11+ⅇ-V+38.35.5tau_dT=0.0011.068⁢ⅇV+38.330+1.068⁢ⅇ-V+38.330ddtimedT=dT_infinity-dTtau_dT$

### Component: i_CaT_fT_gate

$fT_infinity=11+ⅇV+58.73.8tau_fT=116.67⁢ⅇ-V+7583.3+16.67⁢ⅇV+7515.38+offset_fTddtimefT=fT_infinity-fTtau_fT$

### Component: Ca_SR_release

$j_SRCarel=ks⁢O⁢Ca_jsr-Ca_subdiff=Ca_jsr-Ca_subkCaSR=MaxSR-MaxSR-MinSR1+EC50_SRCa_jsrHSRkoSRCa=koCakCaSRkiSRCa=kiCa⁢kCaSRddtimeR=kim⁢RI-kiSRCa⁢Ca_sub⁢R-koSRCa⁢Ca_sub2⁢R-kom⁢OddtimeO=koSRCa⁢Ca_sub2⁢R-kom⁢O-kiSRCa⁢Ca_sub⁢O-kim⁢IddtimeI=kiSRCa⁢Ca_sub⁢O-kim⁢I-kom⁢I-koSRCa⁢Ca_sub2⁢RIddtimeRI=kom⁢I-koSRCa⁢Ca_sub2⁢RI-kim⁢RI-kiSRCa⁢Ca_sub⁢RP_tot=R+O+I+RI$

### Component: Ca_intracellular_fluxes

$b_up=-0.25ifIso_1_uM>00.7⁢ACh0.00009+AChifACh>00otherwiseP_up=P_up_basal⁢1-b_upj_Ca_dif=Ca_sub-Caitau_dif_Caj_up=P_up1+ⅇ-Cai+K_upslope_upj_tr=Ca_nsr-Ca_jsrtau_tr$

### Component: Ca_buffering

$ddtimefTC=delta_fTCdelta_fTC=kf_TC⁢Cai⁢1-fTC-kb_TC⁢fTCddtimefTMC=delta_fTMCdelta_fTMC=kf_TMC⁢Cai⁢1-fTMC+fTMM-kb_TMC⁢fTMCddtimefTMM=delta_fTMMdelta_fTMM=kf_TMM⁢Mgi⁢1-fTMC+fTMM-kb_TMM⁢fTMMddtimefCMi=delta_fCMidelta_fCMi=kf_CM⁢Cai⁢1-fCMi-kb_CM⁢fCMiddtimefCMs=delta_fCMsdelta_fCMs=kf_CM⁢Ca_sub⁢1-fCMs-kb_CM⁢fCMsddtimefCQ=delta_fCQdelta_fCQ=kf_CQ⁢Ca_jsr⁢1-fCQ-kb_CQ⁢fCQ$

### Component: Ca_dynamics

$ddtimeCai=1⁢j_Ca_dif⁢V_sub-j_up⁢V_nsrV_i-CM_tot⁢delta_fCMi+TC_tot⁢delta_fTC+TMC_tot⁢delta_fTMCddtimeCa_sub=j_SRCarel⁢V_jsrV_sub-i_siCa+i_CaT-2⁢i_NaCa2⁢F⁢V_sub+j_Ca_dif+CM_tot⁢delta_fCMsddtimeCa_nsr=j_up-j_tr⁢V_jsrV_nsrddtimeCa_jsr=j_tr-j_SRCarel+CQ_tot⁢delta_fCQ$

### Component: Cell_parameters

$V_cell=0.000000001⁢π⁢R_cell2⁢L_cellV_sub=0.000000001⁢2⁢π⁢L_sub⁢R_cell-L_sub2⁢L_cellV_jsr=V_jsr_part⁢V_cellV_i=V_i_part⁢V_cell-V_subV_nsr=V_nsr_part⁢V_cell$

### Component: i_Kur

$i_Kur=g_Kur⁢r_Kur⁢s_Kur⁢V-E_K$

### Component: i_Kur_rKur_gate

$ddtimer_Kur=r_Kur_infinity-r_Kurtau_r_Kurr_Kur_infinity=11+ⅇV+6-8.6tau_r_Kur=0.0091+ⅇV+512+0.0005$

### Component: i_Kur_sKur_gate

$ddtimes_Kur=s_Kur_infinity-s_Kurtau_s_Kurs_Kur_infinity=11+ⅇV+7.510tau_s_Kur=0.591+ⅇV+6010+3.05$

### Component: i_to

$i_to=g_to⁢V-E_K⁢q⁢r$

### Component: i_to_q_gate

$q_infinity=11+ⅇV+4913tau_q=0.001⁢0.6⁢65.170.57⁢ⅇ-0.08⁢V+44+0.065⁢ⅇ0.1⁢V+45.93+10.1ddtimeq=q_infinity-qtau_q$

### Component: i_to_r_gate

$r_infinity=11+ⅇ-V-19.315tau_r=0.001⁢0.66⁢1.4⁢15.591.037⁢ⅇ0.09⁢V+30.61+0.369⁢ⅇ-0.12⁢V+23.84+2.98ddtimer=r_infinity-rtau_r$

### Component: i_Kr

$i_Kr=g_Kr⁢V-E_K⁢0.9⁢paF+0.1⁢paS⁢piy$

### Component: i_Kr_pa_gate

$alfapaF=11+ⅇ-V+23.26.60.8465535437.2⁢ⅇV11.9+0.96⁢ⅇ-V18.5betapaF=4⁢37.2⁢ⅇV15.9+0.96⁢ⅇ-V22.50.84655354-11+ⅇ-V+23.210.60.8465535437.2⁢ⅇV15.9+0.96⁢ⅇ-V22.5pa_infinity=11+ⅇ-V+10.01447.6607tau_paS=0.846553544.2⁢ⅇV17+0.15⁢ⅇ-V21.6tau_paF=130⁢ⅇV10+ⅇ-V12ddtimepaS=pa_infinity-paStau_paSddtimepaF=pa_infinity-paFtau_paF$

### Component: i_Kr_pi_gate

$tau_pi=1100⁢ⅇ-V54.645+656⁢ⅇV106.157pi_infinity=11+ⅇV+28.617.1ddtimepiy=pi_infinity-piytau_pi$

### Component: i_Ks

$g_Ks=1.2⁢g_Ks_ifIso_1_uM>0g_Ks_otherwiseE_Ks=RTONF⁢ln⁡Ko+0.12⁢NaoKi+0.12⁢Naii_Ks=g_Ks⁢V-E_Ks⁢n2$

### Component: i_Ks_n_gate

$Iso_shift=-14ifIso_1_uM>00otherwisen_infinity=11+ⅇ-V+0.6383-Iso_shift10.7071tau_n=1alpha_n+beta_nalpha_n=281+ⅇ-V-40-Iso_shift3beta_n=1⁢ⅇ-V-Iso_shift-525ddtimen=n_infinity-ntau_n$

### Component: i_KACh

$i_KACh=ACh_on⁢g_KACh⁢V-E_K⁢1+ⅇV+2020⁢aifACh>00otherwise$

### Component: i_KACh_a_gate

$alpha_a=3.5988-0.0256411+0.00000121551⁢ACh1.6951+0.025641beta_a=10⁢ⅇ0.0133⁢V+40a_infinity=alpha_aalpha_a+beta_atau_a=1alpha_a+beta_addtimea=a_infinity-atau_a$
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