# Model Mathematics

### Component: membrane

$i_Stim=stim_amplitudeiftime≥stim_start∧time≦stim_end∧time-stim_start-⌊time-stim_startstim_period⌋⁢stim_period≦stim_duration0otherwiseddtimeV=-1Cm⁢i_Stim+i_K1+i_to+i_Kr+i_Ks+i_K_ATP+i_K_ACh+i_NaK+i_Na+i_b_Na+i_p_Na+i_Ca_L_Na_cyt+i_Ca_L_Na_ds+i_NaCa_cyt+i_NaCa_ds+i_Ca_L_Ca_cyt+i_Ca_L_Ca_ds+i_Ca_L_K_cyt+i_Ca_L_K_ds+i_b_Ca+i_stretch$

### Component: reversal_potentials

$E_Na=R⁢TF⁢ln⁡Na_oNa_iE_K=R⁢TF⁢ln⁡K_oK_iE_Ks=R⁢TF⁢ln⁡K_o+P_kna⁢Na_oK_i+P_kna⁢Na_iE_Ca=0.5⁢R⁢TF⁢ln⁡Ca_oCa_iE_mh=R⁢TF⁢ln⁡Na_o+0.12⁢K_oNa_i+0.12⁢K_i$

### Component: time_independent_potassium_current

$i_K1=g_K1⁢K_oK_o+K_mk1⁢V-E_K1+ⅇV-E_K-10⁢F⁢1.25R⁢T$

### Component: rapid_delayed_rectifier_potassium_current

$i_Kr=g_Kr1⁢xr1+g_Kr2⁢xr2⁢11+ⅇV+922.4⁢V-E_K$

### Component: rapid_delayed_rectifier_potassium_current_xr1_gate

$alpha_xr1=501+ⅇ-V-59beta_xr1=0.05⁢ⅇ-V-2015ddtimexr1=alpha_xr1⁢1-xr1-beta_xr1⁢xr1$

### Component: rapid_delayed_rectifier_potassium_current_xr2_gate

$alpha_xr2=501+ⅇ-V-59beta_xr2=0.4⁢ⅇ-V+30303ddtimexr2=alpha_xr2⁢1-xr2-beta_xr2⁢xr2$

### Component: slow_delayed_rectifier_potassium_current

$i_Ks=g_Ks⁢xs2⁢V-E_Ks$

### Component: slow_delayed_rectifier_potassium_current_xs_gate

$alpha_xs=141+ⅇ-V-409beta_xs=1⁢ⅇ-V45ddtimexs=alpha_xs⁢1-xs-beta_xs⁢xs$

### Component: ATP_dependent_potassium_current

$i_K_ATP=g_K_ATP⁢V+801+ATPK_ATP2$

### Component: sodium_activated_potassium_current

$i_KNa=g_K_Na⁢Na_iNa_i+K_kna⁢V-E_K$

### Component: fast_sodium_current

$i_Na=g_Na⁢m3⁢h⁢V-E_mh$

### Component: fast_sodium_current_m_gate

$E0_m=V+41alpha_m=2000if|E0_m|

### Component: fast_sodium_current_h_gate

$alpha_h=20⁢ⅇ-0.125⁢V+75beta_h=20001+320⁢ⅇ-0.1⁢V+75ddtimeh=alpha_h⁢1-h-beta_h⁢h$

### Component: persistent_sodium_current

$i_p_Na=g_pna⁢11+ⅇ-V+528⁢V-E_Na$

### Component: sodium_background_current

$i_b_Na=g_bna⁢V-E_Na$

### Component: L_type_Ca_channel

$i_Ca_L_Ca_cyt=1-FrICa⁢4⁢P_Ca_L⁢d⁢f⁢f2⁢V-50⁢FR⁢T1-ⅇ-V-50⁢F⁢2R⁢T⁢Ca_i⁢ⅇ100⁢FR⁢T-Ca_o⁢ⅇ-V-50⁢F⁢2R⁢Ti_Ca_L_K_cyt=1-FrICa⁢P_CaK⁢P_Ca_L⁢d⁢f⁢f2⁢V-50⁢FR⁢T1-ⅇ-V-50⁢FR⁢T⁢K_i⁢ⅇ50⁢FR⁢T-K_o⁢ⅇ-V-50⁢FR⁢Ti_Ca_L_Na_cyt=1-FrICa⁢P_CaNa⁢P_Ca_L⁢d⁢f⁢f2⁢V-50⁢FR⁢T1-ⅇ-V-50⁢FR⁢T⁢Na_i⁢ⅇ50⁢FR⁢T-Na_o⁢ⅇ-V-50⁢FR⁢Ti_Ca_L_Ca_ds=FrICa⁢4⁢P_Ca_L⁢d⁢f⁢f2ds⁢V-50⁢FR⁢T1-ⅇ-V-50⁢F⁢2R⁢T⁢Ca_i⁢ⅇ100⁢FR⁢T-Ca_o⁢ⅇ-V-50⁢F⁢2R⁢Ti_Ca_L_K_ds=FrICa⁢P_CaK⁢P_Ca_L⁢d⁢f⁢f2ds⁢V-50⁢FR⁢T1-ⅇ-V-50⁢FR⁢T⁢K_i⁢ⅇ50⁢FR⁢T-K_o⁢ⅇ-V-50⁢FR⁢Ti_Ca_L_Na_ds=FrICa⁢P_CaNa⁢P_Ca_L⁢d⁢f⁢f2ds⁢V-50⁢FR⁢T1-ⅇ-V-50⁢FR⁢T⁢Na_i⁢ⅇ50⁢FR⁢T-Na_o⁢ⅇ-V-50⁢FR⁢Ti_Ca_L=i_Ca_L_Ca_cyt+i_Ca_L_K_cyt+i_Ca_L_Na_cyt+i_Ca_L_Ca_ds+i_Ca_L_K_ds+i_Ca_L_Na_ds$

### Component: L_type_Ca_channel_d_gate

$E0_d=V+24-5alpha_d=120if|E0_d|<0.000130⁢E0_d1-ⅇ-E0_d4otherwisebeta_d=120if|E0_d|<0.000112⁢E0_dⅇE0_d10-1otherwiseddtimed=speed_d⁢alpha_d⁢1-d-beta_d⁢d$

### Component: L_type_Ca_channel_f_gate

$E0_f=V+34alpha_f=25if|E0_f|

### Component: L_type_Ca_channel_f2_gate

$ddtimef2=1-1⁢Ca_iKm_f2+Ca_i+f2$

### Component: L_type_Ca_channel_f2ds_gate

$ddtimef2ds=R_decay⁢1-Ca_dsKm_f2ds+Ca_ds+f2ds$

### Component: calcium_background_current

$i_b_Ca=g_bca⁢V-E_Ca$

### Component: transient_outward_current

$i_to=g_to⁢g_tos+s⁢1-g_tos⁢r⁢V-E_K$

### Component: transient_outward_current_s_gate

$alpha_s=0.033⁢ⅇ-V17beta_s=331+ⅇ-0.125⁢V+10ddtimes=alpha_s⁢1-s-beta_s⁢s$

### Component: transient_outward_current_r_gate

$ddtimer=333⁢11+ⅇ-V+45-r$

### Component: ACh_dependent_potassium_current

$i_K_ACh=g_KACh⁢K_oK_o+K_mk1⁢x_ACh⁢11+K_DACh2⁢V-E_K1+ⅇ2⁢F⁢V-E_K+10R⁢T$

### Component: ACh_dependent_potassium_current_xACh_gate

$ddtimex_ACh=alpha_ACh⁢1-x_ACh-beta_ACh⁢x_ACh$

### Component: sodium_potassium_pump

$i_NaK=i_NaK_max⁢K_oK_mK+K_o⁢Na_iK_mNa+Na_i$

### Component: sodium_calcium_exchanger

$i_NaCa_cyt=1-FRiNaCa⁢k_NaCa⁢ⅇgamma⁢n_NaCa-2⁢V⁢FR⁢T⁢Na_in_NaCa⁢Ca_o-ⅇgamma-1⁢n_NaCa-2⁢V⁢FR⁢T⁢Na_on_NaCa⁢Ca_i1+d_NaCa⁢Ca_i⁢Na_on_NaCa+Ca_o⁢Na_in_NaCa⁢1+Ca_i0.0069i_NaCa_ds=FRiNaCa⁢k_NaCa⁢ⅇgamma⁢n_NaCa-2⁢V⁢FR⁢T⁢Na_in_NaCa⁢Ca_o-ⅇgamma-1⁢n_NaCa-2⁢V⁢FR⁢T⁢Na_on_NaCa⁢Ca_ds1+d_NaCa⁢Ca_ds⁢Na_on_NaCa+Ca_o⁢Na_in_NaCa⁢1+Ca_ds0.0069i_NaCa=i_NaCa_cyt+i_NaCa_ds$

### Component: sarcoplasmic_reticulum_calcium_pump

$K_1=K_cyca⁢K_xcsK_srcaK_2=Ca_i+Ca_up⁢K_1+K_cyca⁢K_xcs+K_cycai_up=Ca_iK_2⁢alpha_up-Ca_up⁢K_1K_2⁢beta_up$

### Component: calcium_translocation

$i_trans=50⁢Ca_up-Ca_rel$

### Component: calcium_release

$VoltDep=ⅇ0.08⁢V-40CaiReg=Ca_iCa_i+K_m_Ca_cytCadsReg=Ca_dsCa_ds+K_m_Ca_dsRegBindSite=CaiReg+1-CaiReg⁢CadsRegActRate=0⁢VoltDep+500⁢RegBindSite2InactRate=60+500⁢RegBindSite2SpeedRel=5ifV<-501otherwiseSRLeak=K_leak_rate⁢ⅇgama_SR_IT⁢isometric_tensionifisometric_tension>0K_leak_rate⁢ⅇgama_SR_SL⁢sarcomere_lengthotherwisePrecFrac=1-ActFrac-ProdFracddtimeActFrac=PrecFrac⁢SpeedRel⁢ActRate-ActFrac⁢SpeedRel⁢InactRateddtimeProdFrac=ActFrac⁢SpeedRel⁢InactRate-SpeedRel⁢1⁢ProdFraci_rel=ActFracActFrac+0.252⁢K_m_rel+SRLeak⁢Ca_rel$

### Component: intracellular_sodium_concentration

$ddtimeNa_i=-11⁢V_i⁢F⁢i_Na+i_p_Na+i_b_Na+3⁢i_NaK+3⁢i_NaCa_cyt+i_Ca_L_Na_cyt+i_Ca_L_Na_ds$

### Component: intracellular_potassium_concentration

$ddtimeK_i=-11⁢V_i⁢F⁢i_K1+i_Kr+i_Ks+i_Ca_L_K_cyt+i_Ca_L_K_ds+i_to-2⁢i_NaK$

### Component: intracellular_calcium_concentration

$V_Cell=3.141592654⁢radius10002⁢length1000V_i_ratio=1-V_e_ratio-V_up_ratio-V_rel_ratioV_i=V_Cell⁢V_i_ratioddtimeCa_i=-12⁢1⁢V_i⁢F⁢i_Ca_L_Ca_cyt+i_b_Ca-2⁢i_NaCa_cyt-2⁢i_NaCa_ds+Ca_ds⁢V_ds_ratio⁢Kdecay+i_rel⁢V_rel_ratioV_i_ratio-ddtimeCa_Calmod-ddtimeCa_Trop-i_upddtimeCa_ds=-1⁢i_Ca_L_Ca_ds2⁢1⁢V_ds_ratio⁢V_i⁢F-Ca_ds⁢KdecayddtimeCa_up=V_i_ratioV_up_ratio⁢i_up-i_transddtimeCa_rel=V_up_ratioV_rel_ratio⁢i_trans-i_relddtimeCa_Calmod=alpha_Calmod⁢Ca_i⁢Calmod-Ca_Calmod-beta_Calmod⁢Ca_Calmodalpha_Trop=KTrop⁢ⅇgama_Trop_SL⁢sarcomere_lengthddtimeCa_Trop=alpha_Trop⁢Ca_i⁢Trop-Ca_Trop-beta_Trop⁢Ca_Trop$

### Component: contraction

$XCont2=Ca_CalmodCalmodXCont1=Ca_TropTropddtimelight_chain=KCont1⁢XCont12⁢XCont2⁢1-light_chain-KCont2⁢light_chainddtimecross_bridge=KCont3⁢light_chain⁢1-cross_bridge-KCont4⁢cross_bridgetension_rest=2e-4⁢ⅇ2⁢sarcomere_lengthtension_active=1-ⅇ-3⁢sarcomere_length-1ifsarcomere_length>10otherwiseoverlap=1-0.625⁢sarcomere_length-2ifsarcomere_length>21otherwisecross_bridge_availability=tension_active⁢overlap⁢cross_bridge_densityisometric_tension=cross_bridge⁢cross_bridge_availability+tension_rest$

### Component: stretch_current

$f_stretch=11+ⅇ-2⁢gama_SAC_IT⁢isometric_tension-ITHSTifisometric_tension>011+ⅇ-2⁢gama_SAC_SL⁢sarcomere_length-SLHSTotherwisei_Ca_stretch=g_Ca_stretch⁢f_stretch⁢V-E_Cai_Na_stretch=g_Na_stretch⁢f_stretch⁢V-E_Nai_K_stretch=g_K_stretch⁢f_stretch⁢V-E_Ki_An_stretch=g_An_stretch⁢f_stretch⁢V-E_An_stretchi_stretch=i_Ca_stretch+i_Na_stretch+i_K_stretch+i_An_stretch$

### Component: fibroblast

$i_fibro=g_fibro⁢V_fibro+20+g_fibro_stretch⁢V_fibro-E_fibro_stretchi_fibro_junct=-g_fibro_junct⁢V-V_fibroddtimeV_fibro=-i_fibro+i_fibro_junctc_fibro$
Source
Derived from workspace Noble, Varghese, Kohl, Noble, 1998 at changeset 995dda261096.
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