# Model Mathematics

### Component: membrane

$V_cell=π⁢Radius_cell2⁢Length_cellV_myo=0.65⁢V_cellV_sr=0.035⁢V_cellV_sl=0.02⁢V_cellV_junc=5.39 e -4⁢V_cellFoRT=FrdyR⁢TF_sl=1-F_juncF_sl_Ca_L=1-F_junc_Ca_LI_Stim=I_Stim_Amplitudeift≥I_Stim_Start∧t≦I_Stim_End∧t-I_Stim_Start-⌊t-I_Stim_StartI_Stim_Period⌋⁢I_Stim_Period≦I_Stim_PulseDuration0otherwiseI_Na_tot=I_Na_tot_junc+I_Na_tot_slI_Cl_tot=I_ClCa+I_Cl_BkI_Ca_tot=I_Ca_tot_junc+I_Ca_tot_slI_tot=I_Na_tot+I_Cl_tot+I_Ca_tot+I_K_totddtV=-I_tot-I_Stim$

### Component: Fast_Na_Current

$m_ss=11+ⅇ-56.86+V9.032tau_m=0.1292⁢ⅇ-V+45.7915.542+0.06487⁢ⅇ-V-4.82351.122a_h=0ifV≥-400.057⁢ⅇ-V+806.8otherwiseb_h=5.92311+ⅇ-V+10.6611.1ifV≥-402.7⁢ⅇ0.079⁢V+3.1 e 5⁢ⅇ0.3485⁢Votherwisea_j=0ifV≥-40-2.5428 e 4⁢ⅇ0.2444⁢V-6.948 e -6⁢ⅇ-0.04391⁢V⁢V+37.781+ⅇ0.311⁢V+79.23otherwiseb_j=0.6⁢ⅇ0.057⁢V1+ⅇ-0.1⁢V+32ifV≥-400.02424⁢ⅇ-0.01052⁢V1+ⅇ-0.1378⁢V+40.14otherwisetau_h=1a_h+b_hh_ss=11+ⅇV+71.557.432tau_j=1a_j+b_jj_ss=11+ⅇV+71.557.432ddtm=m_ss-mtau_mddth=h_ss-htau_hddtj=j_ss-jtau_jI_Na_junc=F_junc⁢G_Na⁢m3⁢h⁢j⁢V-E_Na_juncI_Na_sl=F_sl⁢G_Na⁢m3⁢h⁢j⁢V-E_Na_slI_Na=I_Na_junc+I_Na_sl$

### Component: Background_Na_Current

$I_Na_Bk_junc=F_junc⁢G_Na_B⁢V-E_Na_juncI_Na_Bk_sl=F_sl⁢G_Na_B⁢V-E_Na_slI_Na_Bk=I_Na_Bk_junc+I_Na_Bk_sl$

### Component: Na_K_Pump_Current

$sigma=ⅇNa_o67.3-17f_NaK=11+0.1245⁢ⅇ-0.1⁢V⁢FoRT+0.0365⁢sigma⁢ⅇ-V⁢FoRTI_NaK_junc=F_junc⁢Ibar_NaK⁢f_NaK⁢K_o1+Km_NaipNa_j4⁢K_o+Km_KoI_NaK_sl=F_sl⁢Ibar_NaK⁢f_NaK⁢K_o1+Km_NaipNa_sl4⁢K_o+Km_KoI_NaK=I_NaK_junc+I_NaK_sl$

### Component: Rapidly_Activating_K_Current

$x_r_ss=11+ⅇ-V+105tau_xr=33001+ⅇ-22-V9⁢1+ⅇV+119+2301+ⅇV+4020ddtx_Kr=x_r_ss-x_Krtau_xrr_Kr=11+ⅇV+7424I_Kr=G_Kr⁢K_o5.4⁢x_Kr⁢r_Kr⁢V-E_K$

### Component: Slowly_Activating_K_Current

$x_s_ss=11+ⅇ-V+3.814.25tau_xs=990.11+ⅇ-V+2.43614.12ddtx_Ks=x_s_ss-x_Kstau_xsI_Ks_junc=F_junc⁢G_Ks_junc⁢x_Ks2⁢V-E_KsI_Ks_sl=F_sl⁢G_Ks_sl⁢x_Ks2⁢V-E_KsI_Ks=I_Ks_junc+I_Ks_sl$

### Component: Plateau_K_Current

$kp_Kp=11+ⅇ7.488-V5.98I_Kp=G_Kp⁢kp_Kp⁢V-E_K$

### Component: Transient_Outward_K_Current

$x_to_ss=11+ⅇ-V-1913y_to_ss=11+ⅇV+19.55tau_x_tos=91+ⅇV+315+0.5tau_y_tos=8001+ⅇV+6010+30ddtx_to_s=x_to_ss-x_to_stau_x_tosddty_to_s=y_to_ss-y_to_stau_y_tostau_x_tof=8.5⁢ⅇ-V+45502+0.5tau_y_tof=85⁢ⅇ-V+402220+7ddtx_to_f=x_to_ss-x_to_ftau_x_tofddty_to_f=y_to_ss-y_to_ftau_y_tofI_to_s=G_to_s⁢x_to_s⁢y_to_s⁢V-E_KI_to_f=G_to_f⁢x_to_f⁢y_to_f⁢V-E_KI_to=I_to_s+I_to_f$

### Component: Inward_Rectifier_K_Current

$a_K1=4.09381+ⅇ0.12165⁢V-E_K-49.9344b_K1=15.7197⁢ⅇ0.06739⁢V-E_K-3.2571+ⅇ0.06175⁢V-E_K-594.311+ⅇ-0.16285⁢V-E_K+14.2067K1_ss=a_K1a_K1+b_K1I_K1=G_K1⁢K_o5.4⁢K1_ss⁢V-E_K$

### Component: Ca_Activated_Cl_Current

$I_ClCa_junc=F_junc⁢G_ClCa1+Kd_ClCaCa_j⁢V-E_ClI_ClCa_sl=F_sl⁢G_ClCa1+Kd_ClCaCa_sl⁢V-E_ClI_ClCa=I_ClCa_junc+I_ClCa_sl$

### Component: Background_Cl_Current

$I_Cl_Bk=G_Cl_B⁢V-E_Cl$

### Component: L_Type_Calcium_Current

$d_ss=11+ⅇ-V+56alpha_d=1.41+ⅇ-35-V13+0.25beta_d=1.41+ⅇV+55gamma_d=11+ⅇ50-V20tau_d=alpha_d⁢beta_d+gamma_df_ss=11+ⅇV+207alpha_f=1102.5⁢ⅇ-V+27152beta_f=2001+ⅇ13-V10gamma_f=1801+ⅇV+3010+20tau_f=alpha_f+beta_f+gamma_ff_2_ss=0.671+ⅇV+357+0.33alpha_f_2=300⁢ⅇ-V+252170beta_f_2=311+ⅇ25-V10gamma_f_2=161+ⅇV+3010tau_f_2=alpha_f_2+beta_f_2+gamma_f_2ddtd=d_ss-dtau_dddtf=f_ss-ftau_fddtf_2=f_2_ss-f_2tau_f_2ddtf_Ca_B_j=1.7⁢Ca_j⁢1-f_Ca_B_j-11.9 e -3⁢f_Ca_B_jddtf_Ca_B_sl=1.7⁢Ca_sl⁢1-f_Ca_B_sl-11.9 e -3⁢f_Ca_B_slIbar_Ca_j=p_Ca⁢V⁢Frdy⁢FoRT⁢Ca_j⁢ⅇ2⁢V⁢FoRT-Ca_oⅇ2⁢V⁢FoRT-1Ibar_Ca_sl=p_Ca⁢V⁢Frdy⁢FoRT⁢Ca_sl⁢ⅇ2⁢V⁢FoRT-Ca_oⅇ2⁢V⁢FoRT-1Ibar_K=p_K⁢V⁢Frdy⁢FoRT⁢K_i⁢ⅇV⁢FoRT-K_oⅇV⁢FoRT-1Ibar_Na_j=p_Na⁢V⁢Frdy⁢FoRT⁢Na_j⁢ⅇV⁢FoRT-Na_oⅇV⁢FoRT-1Ibar_Na_sl=p_Na⁢V⁢Frdy⁢FoRT⁢Na_sl⁢ⅇV⁢FoRT-Na_oⅇV⁢FoRT-1I_Ca_junc=F_junc_Ca_L⁢Ibar_Ca_j⁢d⁢f⁢f_2⁢1-f_Ca_B_jI_Ca_sl=F_sl_Ca_L⁢Ibar_Ca_sl⁢d⁢f⁢f_2⁢1-f_Ca_B_slI_Ca=I_Ca_junc+I_Ca_slI_Ca_K=Ibar_K⁢d⁢f⁢f_2⁢F_junc_Ca_L⁢1-f_Ca_B_j+F_sl_Ca_L⁢1-f_Ca_B_slI_Ca_Na_junc=F_junc_Ca_L⁢Ibar_Na_j⁢d⁢f⁢f_2⁢1-f_Ca_B_jI_Ca_Na_sl=F_sl_Ca_L⁢Ibar_Na_sl⁢d⁢f⁢f_2⁢1-f_Ca_B_slI_Ca_Na=I_Ca_Na_junc+I_Ca_Na_slI_Ca_L=I_Ca+I_Ca_K+I_Ca_Na$

### Component: Na_Ca_Exchanger_Current

$Ka_junc=11+Kd_actCa_j2Ka_sl=11+Kd_actCa_sl2s1_junc=ⅇnu⁢V⁢FoRT⁢Na_j3⁢Ca_os1_sl=ⅇnu⁢V⁢FoRT⁢Na_sl3⁢Ca_os2_junc=ⅇnu-1⁢V⁢FoRT⁢Na_o3⁢Ca_js2_sl=ⅇnu-1⁢V⁢FoRT⁢Na_o3⁢Ca_sls3_junc=Km_Ca_i⁢Na_o3⁢1+Na_jKm_Na_i3+Km_Na_o3⁢Ca_j⁢1+Ca_jKm_Ca_i+Km_Ca_o⁢Na_j3+Na_j3⁢Ca_o+Na_o3⁢Ca_js3_sl=Km_Ca_i⁢Na_o3⁢1+Na_slKm_Na_i3+Km_Na_o3⁢Ca_sl⁢1+Ca_slKm_Ca_i+Km_Ca_o⁢Na_sl3+Na_sl3⁢Ca_o+Na_o3⁢Ca_slI_ncx_junc=F_junc⁢Ibar_NCX⁢Ka_junc⁢s1_junc-s2_juncs3_junc⁢1+k_sat⁢ⅇnu-1⁢V⁢FoRTI_ncx_sl=F_sl⁢Ibar_NCX⁢Ka_sl⁢s1_sl-s2_sls3_sl⁢1+k_sat⁢ⅇnu-1⁢V⁢FoRTI_ncx=I_ncx_junc+I_ncx_sl$

### Component: Sarcolemmal_Ca_Pump_Current

$I_pCa_junc=F_junc⁢Ibar_PMCA⁢Ca_j1.6Km_P_Ca1.6+Ca_j1.6I_pCa_sl=F_sl⁢Ibar_PMCA⁢Ca_sl1.6Km_P_Ca1.6+Ca_sl1.6I_pCa=I_pCa_junc+I_pCa_sl$

### Component: Background_Ca_Current

$I_Ca_Bk_junc=F_junc⁢G_Ca_B⁢V-E_Ca_juncI_Ca_Bk_sl=F_sl⁢G_Ca_B⁢V-E_Ca_slI_Ca_Bk=I_Ca_Bk_junc+I_Ca_Bk_sl$

### Component: SR_Fluxes

$k_Ca_SR=Max_SR-Max_SR-Min_SR1+ec50_SRCa_SR2.5ko_SR_Ca=ko_Cak_Ca_SRki_SR_Ca=ki_Ca⁢k_Ca_SRRI=1-Ry_Rr-Ry_Ro-Ry_RiddtRy_Rr=ki_m⁢RI-ki_SR_Ca⁢Ca_j⁢Ry_Rr-ko_SR_Ca⁢Ca_j2⁢Ry_Rr-ko_m⁢Ry_RoddtRy_Ro=ko_SR_Ca⁢Ca_j2⁢Ry_Rr-ko_m⁢Ry_Ro-ki_SR_Ca⁢Ca_j⁢Ry_Ro-ki_m⁢Ry_RiddtRy_Ri=ki_SR_Ca⁢Ca_j⁢Ry_Ro-ki_m⁢Ry_Ri-ko_m⁢Ry_Ri-ko_SR_Ca⁢Ca_j2⁢RIJ_SR_Ca_rel=ks⁢Ry_Ro⁢Ca_SR-Ca_jJ_ser_Ca=V_max_SR_CaP⁢Ca_iKm_fhill_SR_CaP-Ca_SRKm_rhill_SR_CaP1+Ca_iKm_fhill_SR_CaP+Ca_SRKm_rhill_SR_CaPJ_SR_leak=5.348 e -6⁢Ca_SR-Ca_j$

### Component: Cytosolic_Ca_Buffers

$ddtTnC_l=k_on_TnC_l⁢Ca_i⁢B_max_TnC_low-TnC_l-k_off_TnC_l⁢TnC_lddtTnC_h_c=k_on_TnC_h_Ca⁢Ca_i⁢B_max_TnC_high-TnC_h_c-TnC_h_m-k_off_TnC_h_Ca⁢TnC_h_cddtTnC_h_m=k_on_TnC_h_Mg⁢Mg_i⁢B_max_TnC_high-TnC_h_c-TnC_h_m-k_off_TnC_h_Mg⁢TnC_h_mddtCaM=k_on_CaM⁢Ca_i⁢B_max_CaM-CaM-k_off_CaM⁢CaMddtMyo_c=k_on_myo_Ca⁢Ca_i⁢B_max_myosin-Myo_c-Myo_m-k_off_myo_Ca⁢Myo_cddtMyo_m=k_on_myo_Mg⁢Mg_i⁢B_max_myosin-Myo_c-Myo_m-k_off_myo_Mg⁢Myo_mddtSRB=k_on_SR⁢Ca_i⁢B_max_SR-SRB-k_off_SR⁢SRBJ_Ca_B_cytosol=ddtTnC_l+ddtTnC_h_c+ddtTnC_h_m+ddtCaM+ddtMyo_c+ddtMyo_m+ddtSRB$

### Component: Junctional_and_SL_Ca_Buffers

$B_max_SL_low_sl=37.4 e -3⁢V_myoV_slB_max_SL_low_j=4.6 e -4⁢V_myoV_juncB_max_SL_high_sl=13.4 e -3⁢V_myoV_slB_max_SL_high_j=1.65 e -4⁢V_myoV_juncddtSLL_j=k_on_sl_l⁢Ca_j⁢B_max_SL_low_j-SLL_j-k_off_sl_l⁢SLL_jddtSLL_sl=k_on_sl_l⁢Ca_sl⁢B_max_SL_low_sl-SLL_sl-k_off_sl_l⁢SLL_slddtSLH_j=k_on_sl_h⁢Ca_j⁢B_max_SL_high_j-SLH_j-k_off_sl_h⁢SLH_jddtSLH_sl=k_on_sl_h⁢Ca_sl⁢B_max_SL_high_sl-SLH_sl-k_off_sl_h⁢SLH_slJ_Ca_B_junction=ddtSLL_j+ddtSLH_jJ_Ca_B_sl=ddtSLL_sl+ddtSLH_sl$

### Component: Sodium_Concentrations

$I_Na_tot_junc=I_Na_junc+I_Na_Bk_junc+3⁢I_ncx_junc+3⁢I_NaK_junc+I_Ca_Na_juncI_Na_tot_sl=I_Na_sl+I_Na_Bk_sl+3⁢I_ncx_sl+3⁢I_NaK_sl+I_Ca_Na_slddtNa_j=-I_Na_tot_junc⁢C_memV_junc⁢Frdy+J_Na_juncslV_junc⁢Na_sl-Na_j-ddtNa_B_jddtNa_sl=-I_Na_tot_sl⁢C_memV_sl⁢Frdy+J_Na_juncslV_sl⁢Na_j-Na_sl+J_Na_slmyoV_sl⁢Na_i-Na_sl-ddtNa_B_slddtNa_i=J_Na_slmyoV_myo⁢Na_sl-Na_iddtNa_B_j=k_on_Na⁢Na_j⁢B_max_Na_j-Na_B_j-k_off_Na⁢Na_B_jddtNa_B_sl=k_on_Na⁢Na_sl⁢B_max_Na_sl-Na_B_sl-k_off_Na⁢Na_B_sl$

### Component: Potassium_Concentrations

$I_K_tot=I_to+I_Kr+I_Ks+I_K1-2⁢I_NaK+I_Ca_K+I_Kp$

### Component: Calcium_Concentrations

$I_Ca_tot_junc=I_Ca_junc+I_Ca_Bk_junc+I_pCa_junc-2⁢I_ncx_juncI_Ca_tot_sl=I_Ca_sl+I_Ca_Bk_sl+I_pCa_sl-2⁢I_ncx_slB_max_csqn=140 e -3⁢V_myoV_srddtCsqn_b=k_on_csqn⁢Ca_SR⁢B_max_csqn-Csqn_b-k_off_csqn⁢Csqn_bddtCa_j=-I_Ca_tot_junc⁢C_memV_junc⁢2⁢Frdy+J_Ca_juncslV_junc⁢Ca_sl-Ca_j-J_Ca_B_junction+J_SR_Ca_rel⁢V_srV_junc+J_SR_leak⁢V_myoV_juncddtCa_sl=-I_Ca_tot_sl⁢C_memV_sl⁢2⁢Frdy+J_Ca_juncslV_sl⁢Ca_j-Ca_sl+J_Ca_slmyoV_sl⁢Ca_i-Ca_sl-J_Ca_B_slddtCa_i=-J_ser_Ca⁢V_srV_myo-J_Ca_B_cytosol+J_Ca_slmyoV_myo⁢Ca_sl-Ca_iddtCa_SR=J_ser_Ca-J_SR_leak⁢V_myoV_sr+J_SR_Ca_rel-ddtCsqn_b$

### Component: Nerst_Potentials

$E_Na_junc=1FoRT⁢ln⁡Na_oNa_jE_Na_sl=1FoRT⁢ln⁡Na_oNa_slE_Ks=1FoRT⁢ln⁡K_o+p_Na_K⁢Na_oK_i+p_Na_K⁢Na_iE_K=1FoRT⁢ln⁡K_oK_iE_Ca_junc=12⁢FoRT⁢ln⁡Ca_oCa_jE_Ca_sl=12⁢FoRT⁢ln⁡Ca_oCa_slE_Cl=1FoRT⁢ln⁡Cl_iCl_o$
Source
Derived from workspace Carro, Rodriguez, Laguna, Pueyo (2011) at changeset 6ae1765d034d.
Collaboration
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