# Model Mathematics

### Component: membrane

$ddtimeV=-1Cm⁢i_Na+i_Ca_L+i_Ca_T+i_Kv+i_BK+i_NCX+i_NaK+i_NsK+i_NsNa+I_stim$

### Component: I_stim

$stim_start=period⁢0iftime≥period⁢0∧time≦period⁢1period⁢1iftime≥period⁢1∧time≦period⁢2period⁢2iftime≥period⁢2∧time

### Component: L_type_Ca_channel

$E_Ca=R⁢T2⁢F⁢ln⁡CaoCaii_Ca_L=g_CaL⁢O_CaL⁢V-E_Ca$

### Component: L_type_Ca_channel_states

$T_correction_Ca=Q10CaT-31010alpha=T_correction_Ca⁢0.731⁢ⅇV30beta=T_correction_Ca⁢0.2149⁢ⅇ-V40alpha_0=4⁢alphaalpha_1=3⁢alphaalpha_2=2⁢alphaalpha_3=1⁢alphabeta_0=1⁢betabeta_1=2⁢betabeta_2=3⁢betabeta_3=4⁢betaphi_f=T_correction_Ca⁢0.4742⁢ⅇV10phi_s=T_correction_Ca⁢0.05956⁢ⅇ-V40xi_f=T_correction_Ca⁢0.01407⁢ⅇ-V300xi_s=T_correction_Ca⁢0.01213⁢ⅇV500psi_f=T_correction_Ca⁢0.02197⁢ⅇV500psi_s=T_correction_Ca⁢0.00232⁢ⅇ-V280omega_f=beta_3⁢xi_f⁢phi_falpha_3⁢psi_fomega_s=beta_3⁢xi_s⁢phi_salpha_3⁢psi_somega_sf=xi_s⁢psi_fxi_fomega_fs=psi_sdelta=T_correction_Ca⁢0.01theta=T_correction_Ca⁢41+1Cainorm=C3+C2+C1+C0+C3Ca+C2Ca+C1Ca+C0Ca+O_CaL+ICa+IVs+IVf+IVsCa+IVfCaddtimeC3=alpha_2⁢C2norm-alpha_3+beta_2+phi_s+phi_f+theta⁢C3norm+beta_3⁢O_CaLnorm+omega_f⁢IVfnorm+omega_s⁢IVsnorm+delta⁢C3CanormddtimeC2=alpha_1⁢C1norm-alpha_2+beta_1+theta⁢C2norm+beta_2⁢C3norm+delta⁢C2CanormddtimeC1=alpha_0⁢C0norm-alpha_1+beta_0+theta⁢C1norm+beta_1⁢C2norm+delta⁢C1CanormddtimeC0=-alpha_0+theta⁢C0norm+beta_0⁢C1norm+delta⁢C0CanormddtimeC3Ca=theta⁢C3norm+alpha_2⁢C2Canorm-alpha_3+beta_2+phi_f+phi_s+delta⁢C3Canorm+beta_3⁢ICanorm+omega_f⁢IVfCanorm+omega_s⁢IVsCanormddtimeC2Ca=theta⁢C2norm+alpha_1⁢C1Canorm-alpha_2+beta_1+delta⁢C2Canorm+beta_2⁢C3CanormddtimeC1Ca=theta⁢C1norm+alpha_0⁢C0Canorm-alpha_1+beta_0+delta⁢C1Canorm+beta_1⁢C2CanormddtimeC0Ca=theta⁢C0norm-alpha_0+delta⁢C0Canorm+beta_0⁢C1CanormddtimeO_CaL=alpha_3⁢C3norm-beta_3+psi_f+psi_s+theta⁢O_CaLnorm+xi_f⁢IVfnorm+xi_s⁢IVsnorm+delta⁢ICanormddtimeICa=theta⁢O_CaLnorm+alpha_3⁢C3Canorm-beta_3+psi_f+psi_s+delta⁢ICanorm+xi_f⁢IVfCanorm+xi_s⁢IVsCanormddtimeIVs=phi_s⁢C3norm+psi_s⁢O_CaLnorm+omega_fs⁢IVfnorm-omega_s+xi_s+omega_sf+theta⁢IVsnorm+delta⁢IVsCanormddtimeIVf=phi_f⁢C3norm+psi_f⁢O_CaLnorm-omega_f+xi_f+omega_fs+theta⁢IVfnorm+omega_sf⁢IVsnorm+delta⁢IVfCanormddtimeIVsCa=theta⁢IVsnorm+phi_s⁢C3Canorm+psi_s⁢ICanorm+omega_fs⁢IVfCanorm-omega_s+xi_s+omega_sf+delta⁢IVsCanormddtimeIVfCa=theta⁢IVfnorm+phi_f⁢C3Canorm+psi_f⁢ICanorm-omega_f+xi_f+omega_fs+delta⁢IVfCanorm+omega_sf⁢IVsCanorm$

### Component: T_type_Ca_channel

$E_Ca=R⁢T2⁢F⁢ln⁡CaoCaii_Ca_T=g_CaT⁢d_CaT⁢f_CaT⁢V-E_Ca$

### Component: T_type_Ca_channel_d_gate

$T_correction_Ca=Q10CaT-29710d_CaT_inf=11+ⅇ-V+60.55.3tau_d_CaT=1.9058T_correction_Caddtimed_CaT=d_CaT_inf-d_CaTtau_d_CaT$

### Component: T_type_Ca_channel_f_gate

$T_correction_Ca=Q10CaT-29710f_CaT_inf=11+ⅇV+75.54tau_f_CaT=0.38117⁢8.6+14.7⁢ⅇ-V+50⁢V+50900T_correction_Caddtimef_CaT=f_CaT_inf-f_CaTtau_f_CaT$

### Component: voltage_dep_K_channel

$i_Kv=g_Kv⁢x_Kv⁢y_Kv⁢V-E_KE_K=R⁢TF⁢ln⁡KoKi$

### Component: voltage_dep_K_channel_x_gate

$T_correction_K=Q10KT-29710x_Kv_inf=11+ⅇ-V+4317.36tau_x_Kv=4.7803T_correction_Kddtimex_Kv=x_Kv_inf-x_Kvtau_x_Kv$

### Component: voltage_dep_K_channel_y_gate

$T_correction_K=Q10KT-29710y_Kv_inf=11+ⅇV-44.912.0096tau_y_Kv=763.7564T_correction_Kddtimey_Kv=y_Kv_inf-y_Kvtau_y_Kv$

### Component: Calcium_voltage_activated_potassium_channel

$E_K=R⁢TF⁢ln⁡KoKii_BK=g_BK⁢O4⁢V-E_K$

### Component: Calcium_voltage_activated_potassium_channel_states

$alpha=1⁢ⅇ8.47188⁢V1⁢Tbeta=1⁢ⅇ-7.77556⁢V1⁢Tk_on=40633k_off_C=11k_off_O=1.1k_C0O0=0.02162⁢alphak_C1O1=0.000869⁢alphak_C2O2=0.0000281⁢alphak_C3O3=0.000781⁢alphak_C4O4=0.044324⁢alphak_O0C0=318.1084⁢betak_O1C1=144.1736⁢betak_O2C2=32.6594⁢betak_O3C3=0.095312⁢betak_O4C4=0.000106⁢betak_C0C1=4⁢k_on⁢Caik_C1C2=3⁢k_on⁢Caik_C2C3=2⁢k_on⁢Caik_C3C4=1⁢k_on⁢Caik_C4C3=4⁢k_off_Ck_C3C2=3⁢k_off_Ck_C2C1=2⁢k_off_Ck_C1C0=1⁢k_off_Ck_O0O1=4⁢k_on⁢Caik_O1O2=3⁢k_on⁢Caik_O2O3=2⁢k_on⁢Caik_O3O4=1⁢k_on⁢Caik_O4O3=4⁢k_off_Ok_O3O2=3⁢k_off_Ok_O2O1=2⁢k_off_Ok_O1O0=1⁢k_off_Onorm=C0+C1+C2+C3+C4+O0+O1+O2+O3+O4ddtimeC4=-k_C4C3+k_C4O4⁢C4norm+k_C3C4⁢C3norm+k_O4C4⁢O4normddtimeC3=-k_C3C2+k_C3O3+k_C3C4⁢C3norm+k_C2C3⁢C2norm+k_O3C3⁢O3norm+k_C4C3⁢C4normddtimeC2=-k_C2C1+k_C2O2+k_C2C3⁢C2norm+k_C1C2⁢C1norm+k_O2C2⁢O2norm+k_C3C2⁢C3normddtimeC1=-k_C1C0+k_C1O1+k_C1C2⁢C1norm+k_C0C1⁢C0norm+k_O1C1⁢O1norm+k_C2C1⁢C2normddtimeC0=-k_C0C1+k_C0O0⁢C0norm+k_C1C0⁢C1norm+k_O0C0⁢O0normddtimeO4=-k_O4O3+k_O4C4⁢O4norm+k_O3O4⁢O3norm+k_C4O4⁢C4normddtimeO3=-k_O3O2+k_O3C3+k_O3O4⁢O3norm+k_O2O3⁢O2norm+k_C3O3⁢C3norm+k_O4O3⁢O4normddtimeO2=-k_O2O1+k_O2C2+k_O2O3⁢O2norm+k_O1O2⁢O1norm+k_C2O2⁢C2norm+k_O3O2⁢O3normddtimeO1=-k_O1O0+k_O1C1+k_O1O2⁢O1norm+k_O0O1⁢O0norm+k_C1O1⁢C1norm+k_O2O1⁢O2normddtimeO0=-k_O0O1+k_O0C0⁢O0norm+k_O1O0⁢O1norm+k_C0O0⁢C0norm$

### Component: sodium_current

$i_Na=g_Na⁢O_Na⁢V-E_NaE_Na=R⁢TF⁢ln⁡NaoNai$

### Component: Na_channel_states

$T_correction_Na=1⁢Q10NaT-29710k_I2I1=T_correction_Na⁢0.0039239⁢ⅇ2.6793+0.0061468⁢Vk_I1O=T_correction_Na⁢0.12052⁢ⅇ-9.6028+0.083025⁢Vk_OC1=T_correction_Na⁢2.391⁢ⅇ-13.335-0.25289⁢Vk_C1C2=T_correction_Na⁢3.1566⁢ⅇ0.36352+0.077193⁢Vk_C2C3=T_correction_Na⁢0.55432⁢ⅇ-0.099074+0.036441⁢Vk_C3C2=T_correction_Na⁢0.00052548⁢ⅇ-0.069102+0.0031945⁢Vk_C2C1=T_correction_Na⁢1.4496⁢ⅇ-0.1566+0.058353⁢Vk_C1O=T_correction_Na⁢1.5329⁢ⅇ0.0093193+0.041075⁢Vk_OI1=T_correction_Na⁢1.6164⁢ⅇ0.30763+0.0060535⁢Vk_I1I2=T_correction_Na⁢0.027735⁢ⅇ0.05149-0.046865⁢Vk_I1C1=T_correction_Na⁢1.9046⁢ⅇ-2.484+0.020406⁢Vk_C1I1=T_correction_Na⁢0.00021688⁢ⅇ-0.063438+0.0046683⁢Vnorm=O_Na+C1+C2+C3+I1+I2ddtimeC3=-k_C3C2⁢C3norm+k_C2C3⁢C2normddtimeC2=-k_C2C1+k_C2C3⁢C2norm+k_C1C2⁢C1norm+k_C3C2⁢C3normddtimeC1=-k_C1C2+k_C1O+k_C1I1⁢C1norm+k_OC1⁢O_Nanorm+k_C2C1⁢C2norm+k_I1C1⁢I1normddtimeO_Na=-k_OC1+k_OI1⁢O_Nanorm+k_C1O⁢C1norm+k_I1O⁢I1normddtimeI2=-k_I2I1⁢I2norm+k_I1I2⁢I1normddtimeI1=-k_I1O+k_I1I2+k_I1C1⁢I1norm+k_I2I1⁢I2norm+k_C1I1⁢C1norm+k_OI1⁢O_Nanorm$

### Component: Na_Ca_exchanger

$i_NCX=P_NCX⁢ⅇgamma⁢V⁢FR⁢T⁢Nai3⁢Cao-2.5⁢ⅇgamma-1⁢V⁢FR⁢T⁢Nao3⁢CaiK_mNai3+Nao3⁢K_mCa+Cao⁢1+k_sat⁢ⅇgamma-1⁢V⁢FR⁢T$

### Component: sodium_potassium_pump

$i_NaK=P_NaK⁢Ko⁢NaiKo+K_mK⁢K_mNa+Nai⁢1+0.1245⁢ⅇ-0.1⁢V⁢FR⁢T+0.0353⁢ⅇ-V⁢FR⁢T$

### Component: non_specific_Na_current

$E_Na=R⁢TF⁢ln⁡NaoNaii_NsNa=g_NsNa⁢V-E_Na$

### Component: non_specific_K_current

$E_K=R⁢TF⁢ln⁡KoKii_NsK=g_NsK⁢V-E_K$

### Component: ionic_concentrations

$V_myo=3.5 e -12ddtimeNai=-11 e 15⁢i_Na+i_NsNa+i_NCX⁢3+i_NaK⁢3V_myo⁢FddtimeKi=-1 e -15⁢i_Kv+i_BK+i_NsK+I_stim-i_NaK⁢2V_myo⁢FddtimeCai=-1 e -15⁢i_Ca_L+i_Ca_T-i_NCX⁢22⁢V_myo⁢F1+n_CRT⁢CRT_total⁢K_D_CRT⁢Cain_CRT-1Cain_CRT+K_D_CRT2+n_CaM⁢CaM_total⁢K_D_CaM⁢Cain_CaM-1Cain_CaM+K_D_CaM2$
Source
Derived from workspace Poh, 2012 at changeset 713cd2940e5d.
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