# Model Mathematics

### Component: cell_geom

$rho=A_capV_myo$

### Component: membrane

$Cm=CmFA_capddtimeV=-I_H+I_Cl+i_Na+I_CaL+i_t+i_ss+i_f+i_K1+I_Cab+I_SAC_Na+i_B_K+i_B_Na+I_Ko+I_SAC_K+i_NaK+I_NaCa+I_pCa+i_StimCm$

### Component: I_stimulus

$i_Stim=stim_amplitudeA_capiftime-⌊timestim_period⌋⁢stim_period≥0∧time-⌊timestim_period⌋⁢stim_period≦stim_duration0otherwise$

### Component: SAC_current

$gamma_SLSAC=ExtensionRatio-1⁢10ifSAC_on=10otherwiser=-E_R+85E_R-65I_SAC_Na=g_SAC⁢gamma_SLSAC⁢V-E_NaA_cap⁢rI_SAC_K=g_SAC⁢gamma_SLSAC⁢V-E_KA_capI_SAC=I_SAC_K+I_SAC_Na$

### Component: KSA_current

$gamma_SLKO=0.7+ExtensionRatio-1⁢3ifSAC_on=10.7otherwiseI_Ko=g_Ko1+ⅇ-10+V45⁢V-E_K⁢gamma_SLKOA_cap$

### Component: sodium_current

$g_Na_endo=1.33⁢g_NaE_Na=R⁢TF⁢ln⁡Na_oNa_ii_Na=g_Na_endo⁢m3⁢h⁢j⁢V-E_NaA_cap$

### Component: sodium_current_m_gate

$m_infinity=11+ⅇV+45-6.5tau_m=0.00136e30.32⁢V+47.131-ⅇ-0.1⁢V+47.13+0.08⁢ⅇ-V11ddtimem=m_infinity-mtau_m$

### Component: sodium_current_h_gate

$h_infinity=11+ⅇV+76.16.07tau_h=0.0004537e3⁢1+ⅇ-V+10.6611.1ifV≥-400.00349e30.135⁢ⅇ-V+806.8+3.56⁢ⅇ0.079⁢V+310000⁢ⅇ0.35⁢Votherwiseddtimeh=h_infinity-htau_h$

### Component: sodium_current_j_gate

$j_infinity=11+ⅇV+76.16.07tau_j=0.01163e3⁢1+ⅇ-0.1⁢V+32ⅇ-0.0000002535⁢VifV≥-400.00349e3V+37.781+ⅇ0.311⁢V+79.23⁢-127140⁢ⅇ0.2444⁢V-0.00003474⁢ⅇ-0.04391⁢V+0.1212⁢ⅇ-0.01052⁢V1+ⅇ-0.1378⁢V+40.14otherwiseddtimej=j_infinity-jtau_j$

### Component: Ca_independent_transient_outward_K_current

$g_t_endo=0.4647⁢g_tE_K=R⁢TF⁢ln⁡K_oK_ii_t=g_t_endo⁢r⁢a_endo⁢s+b_endo⁢s_slow⁢V-E_KA_cap$

### Component: Ca_independent_transient_outward_K_current_r_gate

$r_infinity=11+ⅇV+10.6-11.42tau_r=1e345.16⁢ⅇ0.03577⁢V+50+98.9⁢ⅇ-0.1⁢V+38ddtimer=r_infinity-rtau_r$

### Component: Ca_independent_transient_outward_K_current_s_gate

$s_infinity=11+ⅇV+45.36.8841tau_s_endo=0.55e3⁢ⅇ-V+70252+0.049e3ddtimes=s_infinity-stau_s_endo$

### Component: Ca_independent_transient_outward_K_current_s_slow_gate

$s_slow_infinity=11+ⅇV+45.36.8841tau_s_slow_endo=3.3e3⁢ⅇ-V+70302+0.049e3ddtimes_slow=s_slow_infinity-s_slowtau_s_slow_endo$

$i_ss=g_ss⁢r_ss⁢s_ss⁢V-E_KA_cap$

$r_ss_infinity=11+ⅇV+11.5-11.82tau_r_ss=10e345.16⁢ⅇ0.03577⁢V+50+98.9⁢ⅇ-0.1⁢V+38ddtimer_ss=r_ss_infinity-r_sstau_r_ss$

$s_ss_infinity=11+ⅇV+87.510.3tau_s_ss=2.1e3ddtimes_ss=s_ss_infinity-s_sstau_s_ss$

### Component: inward_rectifier

$i_K1=48e-3ⅇV+3725+ⅇV+37-25+10e-3⁢0.0011+ⅇV-E_K+76.77-17+g_K1⁢V-E_K+1.731+ⅇ1.613⁢F⁢V-E_K+1.73R⁢T⁢1+ⅇK_o-0.9988-0.124A_cap$

### Component: hyperpolarisation_activated_current

$f_K=1-f_Nai_f_Na=g_f⁢y⁢f_Na⁢V-E_NaA_capi_f_K=g_f⁢y⁢f_K⁢V-E_KA_capi_f=i_f_Na+i_f_K$

### Component: hyperpolarisation_activated_current_y_gate

$y_infinity=11+ⅇV+138.610.48tau_y=1e30.11885⁢ⅇV+8028.37+0.5623⁢ⅇV+80-14.19ddtimey=y_infinity-ytau_y$

### Component: background_currents

$i_B_Na=scale_Na⁢g_B_Na⁢V-E_NaA_capi_B_K=scale_K⁢g_B_K⁢V-E_KA_cap$

### Component: sodium_potassium_pump

$p_nai=11+K_m_NaNa_i4sigma=ⅇNa_o67.3-17p_v=11+0.1245⁢ⅇ-0.1⁢V⁢FR⁢T+0.0365⁢sigma⁢ⅇ-V⁢FR⁢Ti_NaK=i_NaK_maxA_cap⁢p_v⁢K_oK_o+K_m_K⁢p_nai$

### Component: J_CO2

$jco2=V_myo⁢rho⁢Pco2⁢CO2e-CO2i$

### Component: comp_v_nhe_exchanger

$gamma_NHE=1-2.804⁢ExtensionRatio-1ifpH_on=11otherwiseK_Hs=K_Hi⁢gamma_NHEkm2=kp1⁢kp2km1Be=10-pH_e⁢1e3Bi=10-pH_i⁢1e3ap1=KB⁢Ae⁢kp1KA⁢KB+KB⁢Ae+Ae⁢Be+KA⁢Beam1=KB⁢Ai⁢km1KA⁢KB+KB⁢Ai+Ai⁢Bi+KA⁢Biap2=KA⁢Bi⁢kp2KA⁢KB+KB⁢Ai+Ai⁢Bi+KA⁢Biam2=KA⁢Be⁢km2KA⁢KB+KB⁢Ae+Ae⁢Be+KA⁢Bereg=Bin_HiBin_Hi+K_Hsn_Hiflux_nhe=reg⁢ap1⁢ap2-am1⁢am2ap1+ap2+am1+am2v_nhe=flux_nhe⁢V_myo⁢Q_10Scaler$

### Component: comp_v_che_exchanger

$km2=kp2⁢km1kp1OHe=1000⁢10-14+pH_eOHi=1000⁢10-14+pH_ia=1+K_OHOHe+K_OH⁢CleOHe⁢K_Clb=1+K_ClCle+K_Cl⁢OHeCle⁢K_OHc=1+K_ClCli+K_Cl⁢OHiCli⁢K_OHd=1+K_OHOHi+K_OH⁢CliOHi⁢K_Cls1=1a+d⁢kp1+kp2⁢abkm1+km2⁢dcs6=1d+a⁢km1+km2⁢dckp1+kp2⁢abv_che=V_myo⁢km1⁢s6-kp1⁢s160⁢1000⁢Q_10Scaler$

### Component: comp_v_nbc

$km2=kp2⁢km1kp1a=1+NaeK_Na+Nae⁢HCO3eK_Na⁢K_HCO3b=1+K_HCO3HCO3e+K_Na⁢K_HCO3HCO3e⁢Naec=1+K_HCO3HCO3i+K_Na⁢K_HCO3HCO3i⁢Naid=1+NaiK_Na+Nai⁢HCO3iK_Na⁢K_HCO3He=1000⁢10-pH_eHi=1000⁢10-pH_is1=1a+d⁢kp1+kp2⁢abkm1+km2⁢dcs6=1d+a⁢km1+km2⁢dckp1+kp2⁢abreg=Hin_HiHin_Hi+K_Hin_Hi⁢1-Hen_HeHen_He+K_Hen_Hev_nbc=V_myo⁢reg⁢km1⁢s6-kp1⁢s160⁢1000⁢Q_10Scaler$

### Component: comp_v_ae

$gamma_AE=1+2.5⁢ExtensionRatio-1ifpH_on=11otherwiseK_Hs=K_Hi⁢gamma_AEkm2=kp2⁢km1kp1He=1000⁢10-pH_eHi=1000⁢10-pH_ia=1+K_HCO3HCO3e+K_HCO3⁢CleHCO3e⁢K_Clb=1+K_ClCle+K_Cl⁢HCO3eCle⁢K_HCO3c=1+K_ClCli+K_Cl⁢HCO3iCli⁢K_HCO3d=1+K_HCO3HCO3i+K_HCO3⁢CliHCO3i⁢K_Cls1=1a+d⁢kp1+kp2⁢abkm1+km2⁢dcs6=1d+a⁢km1+km2⁢dckp1+kp2⁢abreg=K_Hsn_HiHin_Hi+K_Hsn_Hi⁢Hen_HeHen_He+K_Hen_Hev_ae=V_myo⁢reg⁢km1⁢s6-kp1⁢s160⁢1000⁢Q_10Scaler$

### Component: intracellular_ion_concentrations

$E_Cl=R⁢TF⁢ln⁡CliCleI_Cl=g_Cl⁢V-E_ClH_o=1000⁢10-pH_eH_i=1000⁢10-pH_iE_H=R⁢TF⁢ln⁡H_oH_iI_H=g_H⁢V-E_HK_Nak=i_NaK⁢-2⁢A_capV_myo⁢FK_flux=-I_SAC_K+i_ss+I_Ko+i_t+i_K1+i_f_K+i_NaK⁢-2+i_B_K⁢A_capV_myo⁢Fnai_flux=-i_B_Na+I_SAC_Na+i_Na+I_NaCa⁢3+i_NaK⁢3+i_f_Na⁢A_capV_myo⁢Fnai_total=nai_flux+nai_NHE+nai_NBCnai_NHE=v_nheV_myonai_NBC=v_nbcV_myonai_Nak=i_NaK⁢3⁢A_capV_myo⁢Fnai_NaCa=I_NaCa⁢3⁢A_capV_myo⁢Fnai_na=i_Na⁢A_capV_myo⁢Fnai_bg=i_f_Na⁢A_capV_myo⁢Fbeta_intr=ln⁡10⁢10-pH_i+10pH_i+pKa_ib1⁢ib110pH_i+10pKa_ib12+10pH_i+pKa_ib2⁢ib210pH_i+10pKa_ib22CO2e=PP_co2e⁢CO_2sol⁢P_atmHCO3e=kf_co2hydkr_co2hyd⁢CO2e10-pH_e⁢1e3v_co2hyd=V_myo+V_SR⁢kf_co2hyd⁢CO2i-kr_co2hyd⁢HCO3i⁢10-pH_i⁢1e3ddtimeHCO3i=v_co2hydV_myo+V_SR+v_nbc-v_aeV_myoddtimeCO2i=jco2V_myo-v_co2hydV_myo+V_SR+J_CO2ddtimepH_i=1-beta_intr⁢-v_nhe+v_cheV_myo+v_co2hydV_myo+V_SR-I_H⁢A_capV_myo⁢FddtimeNa_i=-i_B_Na+I_SAC_Na+i_Na+I_NaCa⁢3+i_NaK⁢3+i_f_Na⁢A_capV_myo⁢F+v_nhe+v_nbcV_myoddtimeK_i=-i_Stim+I_SAC_K+i_ss+I_Ko+i_t+i_K1+i_f_K-2⁢i_NaK+i_B_K⁢A_capV_myo⁢FddtimeCli=I_Cl⁢A_capV_myo⁢F+pH_scale⁢v_che+v_aeV_myo$

### Component: SL_pump

$I_pCa=g_pCa⁢Ca_iK_mpCa+Ca_i⁢2⁢V_myo⁢FA_cap$

### Component: Cab

$E_Ca=log10⁡Ca_eCa_ideltaI_Cab=g_Cab⁢V-E_Ca⁢2⁢V_myo⁢FA_cap$

### Component: I_Ca_L

$I_CaL=-J_LC⁢2⁢V_myo⁢FA_cap$

### Component: NCX

$edv=ⅇdelta⁢0.5⁢V⁢etaedv2=ⅇdelta⁢0.5⁢V⁢eta-1Nai3=Na_i3Nae3=Na_e3I_NaCa=g_NCXNae3+K_mNa3⁢Ca_e+K_mCa⁢edv⁢Nai3⁢Ca_e-edv2⁢Nae3⁢Ca_i1+k_sat⁢edv2⁢V_myo⁢FA_cap$

### Component: SERCA

$I_SERCA=g_SERCA⁢Ca_i2K_SERCA2+Ca_i2$

### Component: ionic_concentrations

$Ca_b=B_TRPN-TRPNJ_SR=-J_RY+I_SERCA-g_SRl⁢Ca_SR-Ca_iddtimeCa_SR=V_myoV_SR⁢J_SRddtimeCa_i=11+B_CMDN⁢K_CMDNCa_i+K_CMDN⁢Ca_i+K_CMDN⁢J_TPRN-J_SR+2⁢I_NaCa-I_pCa-I_Cab-I_CaL⁢A_cap2⁢V_myo⁢FddtimeTRPN=J_TPRN$

### Component: Ca_voltage

$dV=delta⁢VexpmdV=ⅇ-dVexpVL=ⅇV-V_L0delta_VL$

### Component: C_ij

$C_cc=Ca_iC_co=Ca_i+J_Rg_D⁢Ca_SR1+J_Rg_DC_oc=Ca_i+J_Lg_D⁢Ca_e⁢dV⁢expmdV1-expmdV1+J_Lg_D⁢dV1-expmdVif|dV|>0.000000001Ca_i+J_Lg_D⁢Ca_e1+J_Lg_DotherwiseC_oo=Ca_i+J_Rg_D⁢Ca_SR+J_Lg_D⁢Ca_e⁢dV⁢expmdV1-expmdV1+J_Rg_D+J_Lg_D⁢dV1-expmdVif|dV|>0.000000001Ca_i+J_Rg_D⁢Ca_SR+J_Lg_D⁢Ca_e1+J_Rg_D+J_Lg_Dotherwise$

### Component: J_ij

$J_Rco=J_R⁢Ca_SR-Ca_i1+J_Rg_DJ_Roo=J_R⁢Ca_SR-Ca_i+J_Lg_D⁢dV1-expmdV⁢Ca_SR-Ca_e⁢expmdV1+J_Rg_D+J_Lg_D⁢dV1-expmdVif|dV|>0.00001J_R⁢Ca_SR-Ca_i+J_Lg_D⁢0.000011-ⅇ-0.00001⁢Ca_SR-Ca_e⁢ⅇ-0.000011+J_Rg_D+J_Lg_D⁢0.000011-ⅇ-0.00001otherwiseJ_Loc=J_L⁢dV1-expmdV⁢Ca_e⁢expmdV-Ca_i1+J_Lg_D⁢dV1-expmdVif|dV|>0.00001J_L⁢0.000011-ⅇ-0.00001⁢Ca_e⁢ⅇ-0.00001-Ca_i1+J_Lg_D⁢0.000011-ⅇ-0.00001otherwiseJ_Loo=J_L⁢dV1-expmdV⁢Ca_e⁢expmdV-Ca_i+J_Rg_D⁢Ca_e⁢expmdV-Ca_SR1+J_Rg_D+J_Lg_D⁢dV1-expmdVif|dV|>0.00001J_L⁢0.000011-ⅇ-0.00001⁢Ca_e⁢ⅇ-0.00001-Ca_i+J_Rg_D⁢Ca_e⁢ⅇ-0.00001-Ca_SR1+J_Rg_D+J_Lg_D⁢0.000011-ⅇ-0.00001otherwise$

### Component: Ca_tau

$t_R=1.17⁢t_L$

### Component: epsilon

$epsilon_pco=1tau_L⁢C_coK_L⁢expVL+aexpVL+1epsilon_pcc=1tau_L⁢Ca_iK_L⁢expVL+aexpVL+1epsilon_m=1tau_L⁢b⁢expVL+ab⁢expVL+a$

### Component: alpha

$alpha_p=expVLt_L⁢expVL+1alpha_m=phi_Lt_L$

### Component: RyR_param

$gamma_NO=1+22.41⁢ExtensionRatio-1ifNO_on=11otherwisephi_R=phi_R_base⁢gamma_NO$

### Component: beta

$beta_poc=1t_R⁢C_oc2C_oc2+K_RyR2beta_pcc=1t_R⁢Ca_i2Ca_i2+K_RyR2beta_m=phi_Rt_R$

### Component: mu_ij

$mu_poc=1tau_R⁢C_oc2+c⁢K_RyR2C_oc2+K_RyR2mu_pcc=1tau_R⁢Ca_i2+c⁢K_RyR2Ca_i2+K_RyR2mu_moc=theta_Rtau_R⁢d⁢C_oc2+c⁢K_RyR2d⁢C_oc2+c⁢K_RyR2mu_mcc=theta_Rtau_R⁢d⁢Ca_i2+c⁢K_RyR2d⁢Ca_i2+c⁢K_RyR2$

### Component: y_ij

$denom=alpha_p+alpha_m⁢beta_m+beta_poc+alpha_m⁢beta_m+beta_pcc+alpha_p⁢beta_m+beta_pocy_oc=alpha_p⁢beta_m⁢alpha_p+alpha_m+beta_m+beta_pccdenomy_co=alpha_m⁢beta_pcc⁢alpha_m+beta_m+beta_poc+beta_poc⁢alpha_pdenomy_oo=alpha_p⁢beta_poc⁢alpha_p+beta_m+beta_pcc+beta_pcc⁢alpha_mdenomy_cc=alpha_m⁢beta_m⁢alpha_m+alpha_p+beta_m+beta_pocdenom$

### Component: r_i

$r_1=y_oc⁢mu_poc+y_cc⁢mu_pccr_2=alpha_p⁢mu_moc+alpha_m⁢mu_mccalpha_p+alpha_mr_3=beta_m⁢mu_pccbeta_m+beta_pccr_4=mu_mccr_5=y_co⁢epsilon_pco+y_cc⁢epsilon_pccr_6=epsilon_mr_7=alpha_m⁢epsilon_pccalpha_p+alpha_mr_8=epsilon_m$

### Component: z_i

$z_4=1-z_1-z_2-z_3ddtimez_1=-r_1+r_5⁢z_1+r_2⁢z_2+r_6⁢z_3ddtimez_2=r_1⁢z_1-r_2+r_7⁢z_2+r_8⁢z_4ddtimez_3=r_5⁢z_1-r_6+r_3⁢z_3+r_4⁢z_4$

### Component: J_values

$J_R1=y_oo⁢J_Roo+J_Rco⁢y_coJ_R3=J_Rco⁢beta_pccbeta_m+beta_pccJ_L1=J_Loo⁢y_oo+J_Loc⁢y_ocJ_L2=J_Loc⁢alpha_palpha_p+alpha_m$

### Component: L_flux

$J_LC=z_1⁢J_L1+z_2⁢J_L2⁢NV_myo$

### Component: R_flux

$J_RY=z_1⁢J_R1+z_3⁢J_R3⁢NV_myo$

### Component: troponin

$betaCab=k_off⁢1-Tensiongamma_trpn⁢T_refif1-Tensiongamma_trpn⁢T_ref>0.1k_off⁢0.1otherwiseJ_TPRN=B_TRPN-TRPN⁢betaCab-Ca_i⁢TRPN⁢k_on$

### Component: Myofilaments

$lambda_prev=ExtensionRatiodExtensionRatiodt=0lamda=ExtensionRatioifExtensionRatio>0.8∧ExtensionRatio≦1.151.15ifExtensionRatio>1.150.8otherwise$

### Component: tropomyosin

$K_2=alpha_r2⁢z_pn_Relz_pn_Rel+K_zn_Rel⁢1-n_Rel⁢K_zn_Relz_pn_Rel+K_zn_RelK_1=alpha_r2⁢z_pn_Rel-1⁢n_Rel⁢K_zn_Relz_pn_Rel+K_zn_Rel2z_max=alpha_0Ca_TRPN_50TRPN_totn_Hill-K_2alpha_r1+K_1+alpha_0Ca_TRPN_50TRPN_totn_HillCa_50=Ca_50ref⁢1+beta_1⁢lamda-1Ca_TRPN_50=Ca_50⁢TRPN_totCa_50+k_offk_on⁢1-1+beta_0⁢lamda-1⁢0.5gamma_trpnalpha_Tm=alpha_0⁢Ca_bCa_TRPN_50n_Hillbeta_Tm=alpha_r1+alpha_r2⁢zn_Rel-1zn_Rel+K_zn_Relddtimez=alpha_Tm⁢1-z-beta_Tm⁢z$

### Component: filament_overlap

$overlap=1+beta_0⁢lamda-1$

### Component: length_independent_tension

$T_Base=T_ref⁢zz_max$

### Component: isometric_tension

$T_0=T_Base⁢overlap$

### Component: Cross_Bridges

$Q=Q_1+Q_2+Q_3Tension=T_0⁢a⁢Q+11-QifQ<0T_0⁢1+a+2⁢Q1+QotherwiseddtimeQ_1=A_1⁢dExtensionRatiodt-alpha_1⁢Q_1ddtimeQ_2=A_2⁢dExtensionRatiodt-alpha_2⁢Q_2ddtimeQ_3=A_3⁢dExtensionRatiodt-alpha_3⁢Q_3$

### Component: pH_regulation

Source
Derived from workspace Niederer, Smith, 2007 at changeset 1e96989c8bcd.
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