# Model Mathematics

### Component: cell

$I_st=stim_amplitudeiftime≥stim_start∧time≦stim_end∧time-stim_start-⌊time-stim_startstim_period⌋⁢stim_period≦stim_duration0otherwisedVdt=-i_Na+i_Ca_L+i_Ca_T+i_Kr+i_Ks+i_K_Na+i_K_ATP+i_to+i_K1+i_Kp+i_NaCa+i_p_Ca+i_Na_b+i_Ca_b+i_NaK+i_ns_Ca+I_stddtimeV=dVdt$

### Component: fast_sodium_current

$E_Na=R⁢TF⁢ln⁡NaoNaii_Na=g_Na⁢m3⁢h⁢j⁢V-E_Na$

### Component: fast_sodium_current_m_gate

$E0_m=V+47.13alpha_m=0.32⁢E0_m1-ⅇ-0.1⁢E0_mif|E0_m|≥delta_m3.2otherwisebeta_m=0.08⁢ⅇ-V11ddtimem=alpha_m⁢1-m-beta_m⁢m$

### Component: fast_sodium_current_h_gate

$alpha_h=0.135⁢ⅇ80+V-6.8ifV<-400otherwisebeta_h=3.56⁢ⅇ0.079⁢V+310000⁢ⅇ0.35⁢VifV<-4010.13⁢1+ⅇV+10.66-11.1otherwiseddtimeh=alpha_h⁢1-h-beta_h⁢h$

### Component: fast_sodium_current_j_gate

$alpha_j=-127140⁢ⅇ0.2444⁢V+3.474 e -5⁢ⅇ-0.04391⁢V⁢V+37.781+ⅇ0.311⁢V+79.23ifV<-400otherwisebeta_j=0.1212⁢ⅇ-0.01052⁢V1+ⅇ-0.1378⁢V+40.14ifV<-400.3⁢ⅇ-2.535 e -7⁢V1+ⅇ-0.1⁢V+32otherwiseddtimej=alpha_j⁢1-j-beta_j⁢j$

### Component: L_type_Ca_channel

$I_CaCa=P_Ca⁢22⁢V⁢F2R⁢T⁢gamma_Cai⁢Cai⁢ⅇ2⁢V⁢FR⁢T-gamma_Cao⁢Caoⅇ2⁢V⁢FR⁢T-1I_CaNa=P_Na⁢12⁢V⁢F2R⁢T⁢gamma_Nai⁢Nai⁢ⅇ1⁢V⁢FR⁢T-gamma_Nao⁢Naoⅇ1⁢V⁢FR⁢T-1I_CaK=P_K⁢12⁢V⁢F2R⁢T⁢gamma_Ki⁢Ki⁢ⅇ1⁢V⁢FR⁢T-gamma_Ko⁢Koⅇ1⁢V⁢FR⁢T-1i_CaCa=d⁢f⁢f_Ca⁢I_CaCai_CaNa=d⁢f⁢f_Ca⁢I_CaNai_CaK=d⁢f⁢f_Ca⁢I_CaKi_Ca_L=i_CaCa+i_CaK+i_CaNa$

### Component: L_type_Ca_channel_d_gate

$E0_d=V+10d_infinity=11+ⅇ-E0_d6.24tau_d=10.035⁢6.24⁢2if|E0_d|<1 e -51⁢d_infinity⁢1-ⅇ-E0_d6.240.035⁢E0_dotherwisealpha_d=d_infinitytau_dbeta_d=1-d_infinitytau_dddtimed=alpha_d⁢1-d-beta_d⁢d$

### Component: L_type_Ca_channel_f_gate

$f_infinity=11+ⅇV+35.068.6+0.61+ⅇ50-V20tau_f=10.0197⁢ⅇ-0.0337⁢V+102+0.02alpha_f=f_infinitytau_fbeta_f=1-f_infinitytau_fddtimef=alpha_f⁢1-f-beta_f⁢f$

### Component: L_type_Ca_channel_f_Ca_gate

$f_Ca=11+CaiKm_Ca$

### Component: T_type_Ca_channel

$i_Ca_T=g_CaT⁢b⁢b⁢g⁢V-E_Ca$

### Component: T_type_Ca_channel_b_gate

$b_inf=1.1.+ⅇ-V+50.5.tau_b=5.1.068⁢ⅇV+16.330.+1.068⁢ⅇ-V+16.330.ddtimeb=b_inf-btau_b$

### Component: T_type_Ca_channel_g_gate

$g_inf=1.1.+ⅇV+61.5.tau_g=5.0.015⁢ⅇ-V+71.783.3+0.015⁢ⅇV+71.715.4ddtimeg=g_inf-gtau_g$

### Component: rapid_delayed_rectifier_potassium_current

$g_Kr=G_Kr⁢Ko5.4Rect=11+ⅇV+922.4i_Kr=g_Kr⁢xr⁢Rect⁢V-E_K$

### Component: rapid_delayed_rectifier_potassium_current_xr_gate

$xr_infinity=11+ⅇ-V+21.57.5tau_xr=10.00138⁢V+14.21-ⅇ-0.123⁢V+14.2+0.00061⁢V+38.9ⅇ0.145⁢V+38.9-1ddtimexr=xr_infinity-xrtau_xr$

### Component: slow_delayed_rectifier_potassium_current

$E_Ks=R⁢TF⁢ln⁡Ko+PNaK⁢NaoKi+PNaK⁢Naig_Ks=G_Ks⁢1+0.61+3.8 e -5Cai1.4i_Ks=g_Ks⁢xs1⁢xs2⁢V-E_Ks$

### Component: slow_delayed_rectifier_potassium_current_xs1_gate

$xs1_infinity=11+ⅇ-V-1.516.7tau_xs1=17.19 e -5⁢V+301-ⅇ-0.148⁢V+30+0.000131⁢V+30ⅇ0.0687⁢V+30-1ddtimexs1=xs1_infinity-xs1tau_xs1$

### Component: slow_delayed_rectifier_potassium_current_xs2_gate

$xs2_infinity=11+ⅇ-V-1.516.7tau_xs2=47.19 e -5⁢V+301-ⅇ-0.148⁢V+30+0.000131⁢V+30ⅇ0.0687⁢V+30-1ddtimexs2=xs2_infinity-xs2tau_xs2$

### Component: time_independent_potassium_current

$g_K1=G_K1⁢Ko5.4E_K=R⁢TF⁢ln⁡KoKii_K1=g_K1⁢K1_infinity⁢V-E_K$

### Component: time_independent_potassium_current_K1_gate

$alpha_K1=1.021+ⅇ0.2385⁢V-E_K-59.215beta_K1=1⁢0.49124⁢ⅇ0.08032⁢V-E_K+5.476+ⅇ0.06175⁢V-E_K-594.311+ⅇ-0.5143⁢V-E_K+4.753K1_infinity=alpha_K1alpha_K1+beta_K1$

### Component: plateau_potassium_current

$Kp=11+ⅇ7.488-V5.98i_Kp=g_Kp⁢Kp⁢V-E_K$

### Component: sodium_activated_potassium_current

$pona=0.851+kdKNaNainKNapov=0.8-0.651+ⅇV+12515i_K_Na=g_K_Na⁢pona⁢pov⁢V-E_K$

### Component: ATP_sensitive_potassium_current

$g_K_ATP=i_K_ATP_on⁢0.000193nicholsareapATP=11+ATPikATPhATPGKbaraATP=g_K_ATP⁢pATP⁢Ko4nATPi_K_ATP=GKbaraATP⁢V-E_K$

### Component: transient_outward_current

$g_to=0⁢0.5rvdv=ⅇV100i_to=g_to⁢zdv3⁢ydv⁢rvdv⁢V-E_K$

### Component: transient_outward_current_zdv_gate

$alpha_zdv=10⁢ⅇV-40251+ⅇV-4025beta_zdv=10⁢ⅇ-V+90251+ⅇ-V+9025tau_zdv=1alpha_zdv+beta_zdvzdv_ss=alpha_zdvalpha_zdv+beta_zdvddtimezdv=zdv_ss-zdvtau_zdv$

### Component: transient_outward_current_ydv_gate

$alpha_ydv=0.0151+ⅇV+605beta_ydv=0.1⁢ⅇV+2551+ⅇV+255tau_ydv=1alpha_ydv+beta_ydvydv_ss=alpha_ydvalpha_ydv+beta_ydvddtimeydv=ydv_ss-ydvtau_ydv$

### Component: sarcolemmal_calcium_pump

$i_p_Ca=I_pCa⁢CaiK_mpCa+Cai$

### Component: sodium_background_current

$i_Na_b=g_Nab⁢V-E_Na$

### Component: calcium_background_current

$E_Ca=R⁢T2⁢F⁢ln⁡CaoCaii_Ca_b=g_Cab⁢V-E_Ca$

### Component: sodium_potassium_pump

$sigma=17⁢ⅇNao67.3-1f_NaK=11+0.1245⁢ⅇ-0.1⁢V⁢FR⁢T+0.0365⁢sigma⁢ⅇ-V⁢FR⁢Ti_NaK=I_NaK⁢f_NaK⁢11+K_mNaiNai2⁢KoKo+K_mKo$

### Component: non_specific_calcium_activated_current

$P_ns_Ca=1.75 e -7I_ns_Na=P_ns_Ca⁢12⁢V⁢F2R⁢T⁢gamma_Nai⁢Nai⁢ⅇ1⁢V⁢FR⁢T-gamma_Nao⁢Naoⅇ1⁢V⁢FR⁢T-1I_ns_K=P_ns_Ca⁢12⁢V⁢F2R⁢T⁢gamma_Ki⁢Ki⁢ⅇ1⁢V⁢FR⁢T-gamma_Ko⁢Koⅇ1⁢V⁢FR⁢T-1i_ns_Na=I_ns_Na⁢11+K_m_ns_CaCai3i_ns_K=I_ns_K⁢11+K_m_ns_CaCai3i_ns_Ca=i_ns_Na+i_ns_K$

### Component: Na_Ca_exchanger

$i_NaCa=c1⁢ⅇgamma-1⁢V⁢FR⁢T⁢ⅇV⁢FR⁢T⁢Nai3⁢Cao-Nao3⁢Cai1+c2⁢ⅇgamma-1⁢V⁢FR⁢T⁢ⅇV⁢FR⁢T⁢Nai3⁢Cao+Nao3⁢Cai$

### Component: calcium_dynamics

$ddtimeAPtrack=100⁢1-APtrack-0.5⁢APtrackifdVdt>150-0.5⁢APtrackotherwiseddtimeAPtrack2=100⁢1-APtrack2-0.5⁢APtrack2ifAPtrack<0.2∧APtrack>0.18-0.5⁢APtrack2otherwiseddtimeAPtrack3=100⁢1-APtrack3-0.5⁢APtrack3ifAPtrack<0.2∧APtrack>0.18-0.01⁢APtrack3otherwiseddtimeCainfluxtrack=-1⁢A_cap⁢i_CaCa+i_Ca_T-2⁢i_NaCa+i_p_Ca+i_Ca_b2⁢V_myo⁢FifAPtrack>0.20ifAPtrack2>0.01∧APtrack≦0.2-0.5⁢CainfluxtrackotherwiseddtimeOVRLDtrack=50⁢1-OVRLDtrackif11+K_mCSQNCa_JSR>CSQNthresh∧OVRLDtrack3<0.37∧APtrack3<0.37-0.5⁢OVRLDtrackotherwiseddtimeOVRLDtrack2=50⁢1-OVRLDtrack2ifOVRLDtrack>Logicthresh∧OVRLDtrack2Logicthresh∧OVRLDtrack3delta_Ca_ithG_rel_overload⁢1-OVRLDtrack2⁢OVRLDtrack2ifCainfluxtrack≦delta_Ca_ith∧OVRLDtrack2>00otherwiseRyRopen=11+ⅇ2⁢APtrack2RyRclose=1-RyRopenG_rel=RyRopen⁢RyRclose⁢1501+ⅇi_CaCa+i_Ca_T-2⁢i_NaCa+i_p_Ca+i_Ca_b+50.9i_rel=G_rel⁢Ca_JSR-Caii_up=I_up⁢CaiCai+K_mupK_leak=I_upCa_NSR_maxi_leak=K_leak⁢Ca_NSRi_tr=Ca_NSR-Ca_JSRtau_trddtimeCa_JSR=11+CSQN_max⁢K_mCSQNK_mCSQN+Ca_JSR2⁢i_tr-i_relddtimeCa_NSR=-i_tr⁢V_JSRV_NSR-i_leak+i_upddtimeCai=11+CMDN_max⁢K_mCMDNK_mCMDN+Cai2+Tn_max⁢K_mTnK_mTn+Cai2⁢-1⁢A_cap⁢i_CaCa+i_Ca_T-2⁢i_NaCa+i_p_Ca+i_Ca_b2⁢V_myo⁢F+i_rel⁢V_JSRV_myo+i_leak-i_up⁢V_NSRV_myo$

### Component: ionic_concentrations

$ddtimeNai=-1⁢i_Na+i_CaNa+i_Na_b+i_ns_Na+i_NaCa⁢3+i_NaK⁢3⁢A_capV_myo⁢FddtimeKi=-1⁢I_st+i_CaK+i_Kr+i_Ks+i_K1+i_Kp+i_K_Na+i_K_ATP+i_to+i_ns_K+-i_NaK⁢2⁢A_capV_myo⁢F$

### Component: geometry

$volume=π⁢preplength⁢radius2V_myo=0.68⁢volumeV_JSR=0.0048⁢volumeV_NSR=0.0552⁢volume$

### Component: Ions_n_reversal_potentials

Source
Derived from workspace Faber, Rudy, 2000 and Tong, Ghouri, Taggart, 2014 at changeset 976a11364041.
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