# Model Mathematics

### Component: membrane

$i_stim=stim_amplitudeiftime≥stim_start∧time≦stim_end0otherwiseddtimeV=-i_tot+i_stimi_tot=i_Na+i_Na_L+i_Ca_L+i_Ca_T+i_to_1+i_to_2+i_Kr+i_Ks+i_K1+i_NaCa+i_NaK+i_Na_b+i_K_b+i_Ca_b+i_Cl_b+i_Ca_p+i_K_p$

### Component: equilibrium_potentials

$E_Na=R⁢TF⁢ln⁡Na_oNa_iE_K=R⁢TF⁢ln⁡K_oK_iE_Ca=R⁢T2⁢F⁢ln⁡Ca_oCa_iE_Cl=-R⁢TF⁢ln⁡Cl_oCl_iE_Ks=R⁢TF⁢ln⁡K_o+r_NaK⁢Na_oK_i+r_NaK⁢Na_i$

### Component: i_Na

$i_Na=g_Na⁢m3⁢0.8⁢h+0.2⁢j⁢V-E_Na$

### Component: i_Na_m_gate

$alpha_m=0.32⁢V+47.131-ⅇ-0.1⁢V+47.13beta_m=0.08⁢ⅇ-V11m_infinity=alpha_malpha_m+beta_mtau_m=1alpha_m+beta_mddtimem=m_infinity-mtau_m$

### Component: i_Na_h_gate

$alpha_h=0.135⁢ⅇV+80-6.8ifV<-400otherwisebeta_h=3.56⁢ⅇ0.079⁢V+310000⁢ⅇ0.35⁢VifV<-4010.13⁢1+ⅇV+10.66-11.1otherwiseh_infinity=alpha_halpha_h+beta_htau_h=1alpha_h+beta_hddtimeh=h_infinity-htau_h$

### Component: i_Na_j_gate

$alpha_j=-127140⁢ⅇ0.2444⁢V-0.00003474⁢ⅇ-0.04391⁢V⁢V+37.781+ⅇ0.311⁢V+79.23ifV<-400otherwisebeta_j=0.1212⁢ⅇ-0.01052⁢V1+ⅇ-0.1378⁢V+40.14ifV<-400.3⁢ⅇ-0.0000002535⁢V1+ⅇ-0.1⁢V+32otherwisej_infinity=0.1⁢alpha_jalpha_j+beta_jtau_j=0.1alpha_j+beta_jddtimej=j_infinity-jtau_j$

### Component: i_Na_L

$i_Na_L=g_Na_L⁢m_L3⁢h_L⁢V-E_Na$

### Component: i_Na_L_m_L_gate

$alpha_m_L=0.32⁢V+47.131-ⅇ-0.1⁢V+47.13beta_m_L=0.08⁢ⅇ-V11m_L_infinity=alpha_m_Lalpha_m_L+beta_m_Ltau_m_L=1alpha_m_L+beta_m_Lddtimem_L=m_L_infinity-m_Ltau_m_L$

### Component: i_Na_L_h_L_gate

$h_L_infinity=11+ⅇV+696.1tau_h_L=175+1251+ⅇ-V+256ddtimeh_L=h_L_infinity-h_Ltau_h_L$

### Component: i_Ca_L

$i_Ca_L=g_Ca_L⁢d⁢f⁢f2⁢f_Ca⁢f_Ca2⁢i_Ca_L_maxi_Ca_L_max=1⁢p_CaCm⁢z_Ca2⁢V-15⁢F2R⁢T⁢gamma_Cai⁢Ca_r⁢ⅇz_Ca⁢F⁢V-15R⁢T-gamma_Cao⁢Ca_oⅇz_Ca⁢F⁢V-15R⁢T-1$

### Component: i_Ca_L_d_gate

$ddtimed=d_infinity-dtau_dd_infinity=11+ⅇ-V-46.74tau_d=0.59+0.8⁢ⅇ0.052⁢V+131+ⅇ0.132⁢V+13$

### Component: i_Ca_L_f_gate

$ddtimef=f_infinity-ftau_ff_infinity=11+ⅇV+1810tau_f=4+0.005⁢V-2.52$

### Component: i_Ca_L_f2_gate

$ddtimef2=f2_infinity-f2tau_f2f2_infinity=11+ⅇV+1810tau_f2=38+0.07⁢V-18.62$

### Component: i_Ca_L_f_Ca_gate

$ddtimef_Ca=f_Ca_infinity-f_Catau_f_Caf_Ca_infinity=0.31-i_Ca_L0.05+0.551+Ca_r0.003+0.15tau_f_Ca=0.5+10⁢1⁢Ca_MK_act1⁢Ca_MK_act+km_Ca_MK+11+Ca_r0.003$

### Component: i_Ca_L_f_Ca2_gate

$ddtimef_Ca2=f_Ca2_infinity-f_Ca2tau_f_Ca2f_Ca2_infinity=11-i_Ca_L0.01tau_f_Ca2=125+3001+ⅇ-i_Ca_L-0.1750.04$

### Component: i_Ca_T

$i_Ca_T=g_Ca_T⁢b⁢g⁢V-50$

### Component: i_Ca_T_b_gate

$alpha_b=1.068⁢ⅇV+16.330beta_b=1.068⁢ⅇ-V+16.330b_infinity=11+ⅇ-V+336.1tau_b=1alpha_b+beta_bddtimeb=b_infinity-btau_b$

### Component: i_Ca_T_g_gate

$alpha_g=0.015⁢ⅇ-V+71.783.3beta_g=0.015⁢ⅇV+71.715.4g_infinity=11+ⅇV+606.6tau_g=1alpha_g+beta_gddtimeg=g_infinity-gtau_g$

### Component: i_to_1

$i_to_1=g_to_1⁢a⁢0.8⁢i+0.2⁢i2⁢V-E_K$

### Component: i_to_1_a_gate

$ddtimea=a_infinity-atau_aalpha_a=25⁢ⅇV-76201+ⅇV-7620beta_a=25⁢ⅇ-V+54201+ⅇ-V+5420tau_a=1alpha_a+beta_aa_infinity=alpha_aalpha_a+beta_a$

### Component: i_to_1_i_gate

$ddtimei=i_infinity-itau_ialpha_i=0.031+ⅇV+2515beta_i=0.1⁢ⅇV-40151+ⅇV-4015tau_i=6+51+ⅇV-16.510i_infinity=alpha_ialpha_i+beta_i$

### Component: i_to_1_i2_gate

$ddtimei2=i2_infinity-i2tau_i2alpha_i2=0.004421+ⅇV+2615beta_i2=0.05⁢ⅇV-10151+ⅇV-1015tau_i2=21.5+301+ⅇV-2510i2_infinity=alpha_i2alpha_i2+beta_i2$

### Component: i_Kr

$i_Kr=g_Kr⁢xr⁢rr_infinity⁢V-E_Kg_Kr=0.040008488⁢K_o5.4rr_infinity=11+ⅇV-5.420.4$

### Component: i_Kr_xr_gate

$ddtimexr=xr_infinity-xrtau_xrtau_xr=9001+ⅇV5+100xr_infinity=11+ⅇ-V+0.08512.25$

### Component: i_Ks

$i_Ks=g_Ks⁢xs1⁢xs2⁢V-E_Ksg_Ks=0.052581329⁢1+0.61+0.000038Ca_i1.4$

### Component: i_Ks_xs1_gate

$ddtimexs1=xs1_infinity-xs1tau_xs1tau_xs1=10.0000761⁢V+44.61-ⅇ-9.97⁢V+44.6+0.00036⁢V-0.55ⅇ0.128⁢V-0.55-1xs1_infinity=11+ⅇ-V-913.7$

### Component: i_Ks_xs2_gate

$ddtimexs2=xs2_infinity-xs2tau_xs2tau_xs2=2⁢10.0000761⁢V+44.61-ⅇ-9.97⁢V+44.6+0.00036⁢V-0.55ⅇ0.128⁢V-0.55-1xs2_infinity=11+ⅇ-V-913.7$

### Component: i_K1

$i_K1=g_K1⁢xK1+0.004⁢V-E_Kg_K1=0.25⁢K_o5.4$

### Component: i_K1_xK1_gate

$xK1=alpha_xK1alpha_xK1+beta_xK1beta_xK1=0.49124⁢ⅇ0.08032⁢V+5.476-E_K+ⅇ0.06175⁢V-594.31+E_K1+ⅇ-0.5143⁢V+4.753-E_Kalpha_xK1=1.021+ⅇ0.2385⁢V-E_K+59.215$

### Component: i_K_p

$i_K_p=g_K_p⁢kp⁢V-E_Kkp=11+ⅇ7.488-V5.98$

### Component: i_to_2

$i_to_2=20⁢i_to_2_max⁢ai_to_2_max=1⁢p_ClCm⁢z_Cl2⁢V⁢F2R⁢T⁢Cl_i-Cl_o⁢ⅇ-z_Cl⁢V⁢FR⁢T1-ⅇ-z_Cl⁢V⁢FR⁢T$

### Component: i_to_2_a_gate

$ddtimea=a_infinity-atau_aa_infinity=11+km_to_2Ca_rtau_a=1$

### Component: i_NaCa

$i_NaCa=X_NaCa⁢i_NaCa_max⁢Na_i3⁢Ca_o⁢ⅇ0.35⁢F⁢VR⁢T-1.5⁢Na_o3⁢Ca_i⁢ⅇ-0.65⁢F⁢VR⁢T1+km_Ca_act1.5⁢Ca_i2⁢1+k_sat⁢ⅇ-0.65⁢V⁢FR⁢T⁢dNaCa_1+dNaCa_2dNaCa_1=km_Ca_o⁢Na_i3+1.5⁢km_Na_o3⁢Ca_i+km_Na_i_13⁢Ca_o⁢1+1.5⁢Ca_ikm_Ca_idNaCa_2=km_Ca_i⁢Na_o3⁢1+Na_ikm_Na_i_1+Na_i3⁢Ca_o+1.5⁢Na_o3⁢Ca_i$

### Component: i_NaK

$i_NaK=g_NaK⁢f_NaK⁢11+km_Na_i_2Na_i2⁢K_oK_o+km_K_of_NaK=11+0.1245⁢ⅇ-0.1⁢F⁢VR⁢T+0.0365⁢sigma⁢ⅇ-F⁢VR⁢Tsigma=17⁢ⅇNa_o67.3-1$

### Component: i_Ca_p

$i_Ca_p=i_Ca_p_max⁢Ca_ikm_Ca_p+Ca_i$

### Component: CT_K_Cl

$CT_K_Cl=CT_K_Cl_max⁢E_K-E_ClE_K+87.8251-E_Cl$

### Component: CT_Na_Cl

$CT_Na_Cl=CT_Na_Cl_max⁢E_Na-E_Cl4E_Na-E_Cl4+87.82514$

### Component: background_currents

$i_Na_b=g_Na_b⁢V-E_Nai_K_b=g_K_b⁢V-E_Ki_Cl_b=g_Cl_b⁢V-E_Cli_Ca_b=1⁢p_Ca_bCm⁢z_Ca2⁢V⁢F2R⁢T⁢gamma_Cai⁢Ca_i⁢ⅇz_Ca⁢V⁢FR⁢T-gamma_Cao⁢Ca_oⅇz_Ca⁢V⁢FR⁢T-1$

### Component: intracellular_ion_concentrations

$ddtimeNa_i=-Cm⁢i_Na+i_Na_L+i_Na_b+3⁢i_NaK+3⁢i_NaCa⁢a_capVol_myo⁢F+CT_Na_ClddtimeK_i=-Cm⁢i_to_1+i_K1+i_Kr+i_Ks+i_K_p+i_K_b-2⁢i_NaK⁢a_capVol_myo⁢F+CT_K_ClddtimeCl_i=-Cm⁢i_to_2+i_Cl_b⁢a_capVol_myo⁢F+CT_Na_Cl+CT_K_Cl$

### Component: Ca_i

$ddtimeCa_i=-b_myo⁢Cm⁢i_Ca_b+i_Ca_p-2⁢i_NaCa⁢a_cap2⁢Vol_myo⁢F+q_up-q_leak⁢Vol_nsrVol_myo-q_diff⁢Vol_ssVol_myoCMDN=2⁢CMDN_max⁢Ca_iCa_i+km_CMDN2TRPN=2⁢TRPN_max⁢Ca_iCa_i+km_TRPN2b_myo=11+TRPN+CMDN$

### Component: Ca_MK_act

$Ca_MK_act=Ca_MK_bound+Ca_MK_trapddtimeCa_MK_trap=alpha_Ca_MK⁢Ca_MK_bound⁢Ca_MK_bound+Ca_MK_trap-beta_Ca_MK⁢Ca_MK_trapCa_MK_bound=Ca_MK_0⁢1-Ca_MK_trap1+km_Ca_MKCa_r$

### Component: Ca_NSR

$ddtimeCa_NSR=q_up-q_leak+q_tr⁢Vol_jsrVol_nsr$

### Component: Ca_JSR

$ddtimeCa_JSR=q_tr-q_rel1+CSQN_max⁢km_CSQNkm_CSQN+Ca_JSR2$

### Component: Ca_r

$ddtimeCa_r=Ca_r_tot⁢-Cm⁢i_Ca_L⁢a_capVol_ss⁢z_Ca⁢F+q_rel⁢Vol_jsrVol_ss-q_diffq_diff=Ca_r-Ca_itau_ssb_SL=2⁢b_SL_max⁢Ca_rCa_r+km_b_SL2b_SR=2⁢b_SR_max⁢Ca_rCa_r+km_b_SR2Ca_r_tot=11+b_SR+b_SL$

### Component: q_rel

$q_rel=g_rel⁢ro⁢ri⁢Ca_JSR-Ca_rg_rel=3000⁢vgvg=11+ⅇg_Ca_L⁢i_Ca_L_max+135$

### Component: q_rel_ri_gate

$ddtimeri=ri_infinity-ritau_ritau_ri=350-tau_Ca_MK1+ⅇCa_r-0.003+0.003⁢Ca_fac0.0002+3+tau_Ca_MKri_infinity=11+ⅇCa_r-0.0004+0.002⁢Ca_fac0.000025Ca_fac=11+ⅇi_Ca_L+0.050.015tau_Ca_MK=tau_Ca_MK_max⁢1⁢Ca_MK_actkm_Ca_MK+1⁢Ca_MK_act$

### Component: q_rel_ro_gate

$ddtimero=ro_infinity-rotau_roro_infinity=ro_infinity_JSR⁢i_Ca_L2i_Ca_L2+1ro_infinity_JSR=Ca_JSR1.9Ca_JSR1.9+49.28⁢Ca_rCa_r+0.00281.9$

### Component: q_leak

$q_leak=q_leak_max⁢Ca_NSRNSR_max$

### Component: q_up

$q_up=X_q_up⁢dq_up_Ca_MK+1⁢q_up_max⁢Ca_iCa_i+km_up-dkm_plbdq_up_Ca_MK=dq_up_Ca_MK_max⁢Ca_MK_act⁢1km_Ca_MK+Ca_MK_act⁢1dkm_plb=dkm_plb_max⁢Ca_MK_act⁢1km_Ca_MK+Ca_MK_act⁢1$

### Component: q_tr

$q_tr=Ca_NSR-Ca_JSRtau_tr$

### Component: model_parameters

$a_geo=2⁢3.14⁢radius2+2⁢3.14⁢radius⁢lengtha_cap=rcg⁢a_geoVol_myo=Vol_cell⁢0.68Vol_nsr=Vol_cell⁢0.0552Vol_jsr=Vol_cell⁢0.0048Vol_ss=Vol_cell⁢0.02$
Source
Derived from workspace Aslanidi 2009 at changeset 794b3900e6b2.
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