# Model Mathematics

### Component: membrane

$i_Stim=stim_amplitudeiftime-⌊timestim_period⌋⁢stim_period≥stim_start∧time-⌊timestim_period⌋⁢stim_period≦stim_start+stim_duration0otherwiseddtimeV=-i_K1+i_to+i_Kr+i_Ks+i_CaL+i_NaK+i_Na+i_b_Na+i_NaCa+i_b_Ca+i_p_K+i_p_Ca+i_Stim$

### Component: reversal_potentials

$E_Na=R⁢TF⁢ln⁡Na_oNa_iE_K=R⁢TF⁢ln⁡K_oK_iE_Ks=R⁢TF⁢ln⁡K_o+P_kna⁢Na_oK_i+P_kna⁢Na_iE_Ca=0.5⁢R⁢TF⁢ln⁡Ca_oCa_i$

### Component: inward_rectifier_potassium_current

$alpha_K1=0.11+ⅇ0.06⁢V-E_K-200beta_K1=3⁢ⅇ0.0002⁢V-E_K+100+1⁢ⅇ0.1⁢V-E_K-101+ⅇ-0.5⁢V-E_KxK1_inf=alpha_K1alpha_K1+beta_K1i_K1=g_K1⁢xK1_inf⁢K_o5.4⁢V-E_K$

### Component: rapid_time_dependent_potassium_current

$i_Kr=g_Kr⁢K_o5.4⁢Xr1⁢Xr2⁢V-E_K$

### Component: rapid_time_dependent_potassium_current_Xr1_gate

$xr1_inf=11+ⅇ-26-V7alpha_xr1=4501+ⅇ-45-V10beta_xr1=61+ⅇV+3011.5tau_xr1=1⁢alpha_xr1⁢beta_xr1ddtimeXr1=xr1_inf-Xr1tau_xr1$

### Component: rapid_time_dependent_potassium_current_Xr2_gate

$xr2_inf=11+ⅇV+8824alpha_xr2=31+ⅇ-60-V20beta_xr2=1.121+ⅇV-6020tau_xr2=1⁢alpha_xr2⁢beta_xr2ddtimeXr2=xr2_inf-Xr2tau_xr2$

### Component: slow_time_dependent_potassium_current

$i_Ks=g_Ks⁢Xs2⁢V-E_Ks$

### Component: slow_time_dependent_potassium_current_Xs_gate

$xs_inf=11+ⅇ-5-V14alpha_xs=11001+ⅇ-10-V6beta_xs=11+ⅇV-6020tau_xs=1⁢alpha_xs⁢beta_xsddtimeXs=xs_inf-Xstau_xs$

### Component: fast_sodium_current

$i_Na=g_Na⁢m3⁢h⁢j⁢V-E_Na$

### Component: fast_sodium_current_m_gate

$m_inf=11+ⅇ-56.86-V9.032alpha_m=11+ⅇ-60-V5beta_m=0.11+ⅇV+355+0.11+ⅇV-50200tau_m=1⁢alpha_m⁢beta_mddtimem=m_inf-mtau_m$

### Component: fast_sodium_current_h_gate

$h_inf=11+ⅇV+71.557.432alpha_h=0.057⁢ⅇ-V+806.8ifV<-400otherwisebeta_h=2.7⁢ⅇ0.079⁢V+310000⁢ⅇ0.3485⁢VifV<-400.770.13⁢1+ⅇV+10.66-11.1otherwisetau_h=1alpha_h+beta_hddtimeh=h_inf-htau_h$

### Component: fast_sodium_current_j_gate

$j_inf=11+ⅇV+71.557.432alpha_j=-25428⁢ⅇ0.2444⁢V-6.948e-6⁢ⅇ-0.04391⁢V⁢V+37.7811+ⅇ0.311⁢V+79.23ifV<-400otherwisebeta_j=0.02424⁢ⅇ-0.01052⁢V1+ⅇ-0.1378⁢V+40.14ifV<-400.6⁢ⅇ0.057⁢V1+ⅇ-0.1⁢V+32otherwisetau_j=1alpha_j+beta_jddtimej=j_inf-jtau_j$

### Component: sodium_background_current

$i_b_Na=g_bna⁢V-E_Na$

### Component: L_type_Ca_current

$i_CaL=g_CaL⁢d⁢f⁢fCa⁢4⁢V⁢F2R⁢T⁢Ca_i⁢ⅇ2⁢V⁢FR⁢T-0.341⁢Ca_oⅇ2⁢V⁢FR⁢T-1$

### Component: L_type_Ca_current_d_gate

$d_inf=11+ⅇ-5-V7.5alpha_d=1.41+ⅇ-35-V13+0.25beta_d=1.41+ⅇV+55gamma_d=11+ⅇ50-V20tau_d=1⁢alpha_d⁢beta_d+gamma_dddtimed=d_inf-dtau_d$

### Component: L_type_Ca_current_f_gate

$f_inf=11+ⅇV+207tau_f=1125⁢ⅇ-V+272240+80+1651+ⅇ25-V10ddtimef=f_inf-ftau_f$

### Component: L_type_Ca_current_fCa_gate

$alpha_fCa=11+Ca_i0.0003258beta_fCa=0.11+ⅇCa_i-0.00050.0001gama_fCa=0.21+ⅇCa_i-0.000750.0008fCa_inf=alpha_fCa+beta_fCa+gama_fCa+0.231.46tau_fCa=2d_fCa=fCa_inf-fCatau_fCaddtimefCa=0iffCa_inf>fCa∧V>-60d_fCaotherwise$

### Component: calcium_background_current

$i_b_Ca=g_bca⁢V-E_Ca$

### Component: transient_outward_current

$i_to=g_to⁢r⁢s⁢V-E_K$

### Component: transient_outward_current_s_gate

$s_inf=11+ⅇV+205tau_s=85⁢ⅇ-V+452320+51+ⅇV-205+3ddtimes=s_inf-stau_s$

### Component: transient_outward_current_r_gate

$r_inf=11+ⅇ20-V6tau_r=9.5⁢ⅇ-V+4021800+0.8ddtimer=r_inf-rtau_r$

### Component: sodium_potassium_pump_current

$i_NaK=P_NaK⁢K_oK_o+K_mk⁢Na_iNa_i+K_mNa1+0.1245⁢ⅇ-0.1⁢V⁢FR⁢T+0.0353⁢ⅇ-V⁢FR⁢T$

### Component: sodium_calcium_exchanger_current

$i_NaCa=K_NaCa⁢ⅇgamma⁢V⁢FR⁢T⁢Na_i3⁢Ca_o-ⅇgamma-1⁢V⁢FR⁢T⁢Na_o3⁢Ca_i⁢alphaKm_Nai3+Na_o3⁢Km_Ca+Ca_o⁢1+K_sat⁢ⅇgamma-1⁢V⁢FR⁢T$

### Component: calcium_pump_current

$i_p_Ca=g_pCa⁢Ca_iCa_i+K_pCa$

### Component: potassium_pump_current

$i_p_K=g_pK⁢V-E_K1+ⅇ25-V5.98$

### Component: calcium_dynamics

$i_rel=a_rel⁢Ca_SR2b_rel2+Ca_SR2+c_rel⁢d⁢gi_up=Vmax_up1+K_up2Ca_i2i_leak=V_leak⁢Ca_SR-Ca_ig_inf=11+Ca_i0.000356ifCa_i<0.0003511+Ca_i0.0003516otherwised_g=g_inf-gtau_gddtimeg=0ifg_inf>g∧V>-60d_gotherwiseddt_Ca_i_total=-i_CaL+i_b_Ca+i_p_Ca-2⁢i_NaCa2⁢V_c⁢F⁢Cm+i_leak-i_up+i_relddt_Ca_sr_total=V_cV_sr⁢i_up-i_rel+i_leakf_JCa_i_free=11+Buf_c⁢K_buf_cCa_i+K_buf_c2f_JCa_sr_free=11+Buf_sr⁢K_buf_srCa_SR+K_buf_sr2ddtimeCa_i=ddt_Ca_i_total⁢f_JCa_i_freeddtimeCa_SR=ddt_Ca_sr_total⁢f_JCa_sr_free$

### Component: sodium_dynamics

$ddtimeNa_i=-i_Na+i_b_Na+3⁢i_NaK+3⁢i_NaCa⁢CmV_c⁢F$

### Component: potassium_dynamics

$ddtimeK_i=-i_K1+i_to+i_Kr+i_Ks+i_p_K+i_Stim-2⁢i_NaK⁢CmV_c⁢F$
Source
Derived from workspace Tentusscher, Noble, Noble, Panfilov, 2004 at changeset 0eac6f2330f9.
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