# Model Mathematics

### Component: Environment

$FoRT=FR⁢TRToF=R⁢TF$

### Component: membrane

$i_stim=stim_amplitudeiftime≥stim_start∧time≦stim_end∧time-stim_start-⌊time-stim_startstim_period⌋⁢stim_period≦stim_duration0otherwisebulk_volume=bulk_fraction⁢cell_volumeperipheral_volume=periphery_fraction⁢cell_volumesr_volume=sr_fraction⁢cell_volumediffusable_volume=bulk_fraction+periphery_fraction⁢cell_volumeddtimeVm=-1⁢1Cm⁢i_k1+i_to_fast+i_to_sustained+i_kr+i_ks+i_kb+i_nak+i_cal+i_cat+i_na_fast+i_na_late+i_pmca+i_cab+i_f_na+i_f_k+i_naca+i_nab+i_stimddtimeCa_i_peripheral=-1⁢i_cal+-1⁢i_cat+-1⁢i_cab+1⁢i_rel_per+-1⁢i_serca_per+-1⁢i_diff+2⁢i_naca+-1⁢i_pmca⁢10002⁢F⁢peripheral_volumeddtimeCa_i_bulk=-1⁢i_serca_bulk+i_diff+i_leak⁢10002⁢F⁢bulk_volumeddtimeCa_sr=-1⁢i_rel_per+-1⁢i_leak+i_serca_bulk+i_serca_per⁢10002⁢F⁢sr_volumeddtimeK_i=-1⁢i_to_fast+-1⁢i_to_sustained+-1⁢i_kr+-1⁢i_ks+-1⁢i_k1+-1⁢i_kb+-1⁢i_f_k+-1⁢i_stim+2⁢i_nak⁢1000F⁢diffusable_volumeddtimeNa_i=-1⁢i_na_fast+-1⁢i_na_late+-3⁢i_nak+-3⁢i_naca+-1⁢i_f_na+-1⁢i_nab⁢1000F⁢diffusable_volume$

### Component: x_Ttype

$x_inf_Ttype=11+ⅇVm+47.8-5.5ddtimex_Ttype=x_inf_Ttype-x_Ttypetau_x_Ttype$

### Component: y_Ttype

$y_inf_Ttype=11+ⅇVm+67.93.87tau_y_Ttype=1.42271⁢ⅇ-0.05119⁢Vmddtimey_Ttype=y_inf_Ttype-y_Ttypetau_y_Ttype$

### Component: i_cat

$E_Ca=0.5⁢RToF⁢ln⁡Ca_oCa_i_peripherali_cat=G_max_Ttype⁢x_Ttype⁢y_Ttype⁢Vm-E_Ca$

### Component: x_Ltype

$x_inf_Ltype=11+ⅇVm+14.6-5.5ddtimex_Ltype=x_inf_Ltype-x_Ltypetau_x_Ltype$

### Component: y_Ltype

$y_inf_Ltype=11+ⅇVm+31.05.54tau_y_Ltype=25.10.04+0.7⁢ⅇ-1⁢0.028⁢Vm+14.52.0ddtimey_Ltype=y_inf_Ltype-y_Ltypetau_y_Ltype$

### Component: y_ca_Ltype

$y_ca_inf_Ltype=0.4+0.61+Ca_i_peripheral0.00012.0tau_y_ca_Ltype=2.0+80.01+Ca_i_peripheral0.00012.0ddtimey_ca_Ltype=y_ca_inf_Ltype-y_ca_Ltypetau_y_ca_Ltype$

### Component: i_cal

$E_Ca=0.5⁢RToF⁢ln⁡Ca_oCa_i_peripherali_cal=G_max_Ltype⁢x_Ltype⁢y_Ltype⁢y_ca_Ltype⁢Vm-E_Ca$

### Component: x_to_fast

$x_inf_to_fast=11+ⅇVm+-7.0-9.0ddtimex_to_fast=x_inf_to_fast-x_to_fasttau_x_to_fast$

### Component: y_to_fast

$y_inf_to_fast=11+ⅇVm+27.58.0ddtimey_to_fast=y_inf_to_fast-y_to_fasttau_y_to_fast$

### Component: i_to_fast

$E_k=RToF⁢ln⁡K_oK_ii_to_fast=G_max_to_fast⁢x_to_fast⁢y_to_fast⁢Vm-E_k$

### Component: x_to_sustained

$x_to_sustained=11+ⅇ5.0-Vm17.0$

### Component: i_to_sustained

$E_k=RToF⁢ln⁡K_oK_ii_to_sustained=G_max_to_sustained⁢x_to_sustained⁢Vm-E_k$

### Component: x_na_fast

$x_inf_na_fast=11+ⅇVm+25.0-5.0ddtimex_na_fast=x_inf_na_fast-x_na_fasttau_x_na_fast$

### Component: y_na_fast

$y_inf_na_fast=11+ⅇVm+69.03.96ddtimey_na_fast=y_inf_na_fast-y_na_fasttau_y_na_fast$

### Component: i_na_fast

$E_na=RToF⁢ln⁡Na_oNa_ii_na_fast=G_max_na_fast⁢x_na_fast⁢y_na_fast⁢Vm-E_na$

### Component: x_na_late

$x_inf_na_late=11+ⅇVm+30.0-5.0ddtimex_na_late=x_inf_na_late-x_na_latetau_x_na_late$

### Component: y_na_late

$y_inf_na_late=0.1+0.91+ⅇVm+75.66.3tau_y_na_late=120.0+1.0⁢ⅇVm+100.025.0ddtimey_na_late=y_inf_na_late-y_na_latetau_y_na_late$

### Component: i_na_late

$E_na=RToF⁢ln⁡Na_oNa_ii_na_late=G_max_na_late⁢x_na_late⁢y_na_late⁢Vm-E_na$

### Component: x_k1

$x_k1=11+ⅇ92.0+Vm10.0$

### Component: i_k1

$E_k=RToF⁢ln⁡K_oK_ii_k1=G_max_k1⁢K_o5.40.8⁢x_k1⁢Vm-E_k$

### Component: x_kr

$x_kr=11+ⅇ33.0+Vm22.4$

### Component: y_kr

$y_inf_kr=11+ⅇVm+50.0-7.5ykrv1=0.00138⁢1⁢Vm+71-ⅇ-0.123⁢Vm+7if|Vm+7|>0.0010.001380.123otherwiseykrv2=0.000061⁢1⁢Vm+10ⅇ0.145⁢Vm+10-1if|Vm+10|>0.0010.000610.145otherwisetau_y_kr=1ykrv1+ykrv2ddtimey_kr=y_inf_kr-y_krtau_y_kr$

### Component: i_kr

$E_k=RToF⁢ln⁡K_oK_ii_kr=G_max_kr⁢K_o5.41.0⁢x_kr⁢y_kr⁢Vm-E_k$

### Component: x_ks

$x_inf_ks=11+ⅇVm-1.5-16.7tau_x_ks=417.9462if|Vm+30|<0.014510.0000719⁢Vm+301-ⅇ-0.148⁢Vm+30+0.000131⁢Vm+30ⅇ0.0687⁢Vm+30-1otherwiseddtimex_ks=x_inf_ks-x_kstau_x_ks$

### Component: y_ks

$y_inf_ks=x_inf_kstau_y_ks=4⁢tau_x_ksddtimey_ks=y_inf_ks-y_kstau_y_ks$

### Component: i_ks

$E_k=RToF⁢ln⁡K_oK_ii_ks=G_max_ks⁢x_ks⁢y_ks⁢Vm-E_k$

### Component: i_naca

$i_naca=512.009113204⁢g_NaCa⁢ⅇgamma⁢n_NaCa-2⁢VmRToF⁢Nain_NaCa⁢Cao-Cai⁢ⅇgamma-1⁢n_NaCa-2⁢VmRToF⁢Naon_NaCa1+d_NaCa⁢Cai⁢Naon_NaCa+Cao⁢Nain_NaCa⁢1+Cai0.0069$

### Component: x_nak

$x_nak=11+ⅇVm+80.0-45.0$

### Component: y_nak

$y_nak=11+ⅇVm+0.0125.0$

### Component: i_nak

$i_nak=g_nak⁢x_nak⁢y_nak⁢1.01.0+1.9K_o1.45⁢1.01.0+31.98Na_i1.0$

### Component: i_kb

$E_k=RToF⁢ln⁡K_oK_ii_kb=G_max_kb⁢Vm-E_k$

### Component: i_nab

$E_na=RToF⁢ln⁡Na_oNa_ii_nab=G_max_nab⁢Vm-E_na$

### Component: i_cab

$E_ca=0.5⁢RToF⁢ln⁡Ca_oCa_i_peripherali_cab=G_max_cab⁢Vm-E_ca$

### Component: i_pmca

$i_pmca=PMCA_max⁢1.01.0+KpmcaCa_i_peripheralHpmca$

### Component: i_f_k

$E_k=RToF⁢ln⁡K_oK_ii_f_k=G_f_k⁢y_gate_f_k⁢Vm-E_k$

### Component: i_f_na

$E_na=RToF⁢ln⁡Na_oNa_ii_f_na=G_f_na⁢y_gate_f_na⁢Vm-E_na$

### Component: y_gate_f_k

$y_inf_f_gate=1.01.0+ⅇVm+109.010.0tau_y_f_gate=6000.0ⅇ-1.0⁢2.9+0.04⁢Vm+ⅇ1.0⁢3.6+0.11⁢Vmddtimey_gate_f_k=y_inf_f_gate-y_gate_f_ktau_y_f_gate$

### Component: y_gate_f_na

$y_inf_f_gate=1.01.0+ⅇVm+109.010.0tau_y_f_gate=6000.0ⅇ-1.0⁢2.9+0.04⁢Vm+ⅇ1.0⁢3.6+0.11⁢Vmddtimey_gate_f_na=y_inf_f_gate-y_gate_f_natau_y_f_gate$

### Component: i_rel_per

$i_rel_per=REL_max⁢Ca_sr1.0+KrelCa_i_peripheral2$

### Component: i_serca_per

$i_serca_per=SERCA_max⁢Ca_i_peripheralKmfH-Ca_srKmrH1.0+Ca_i_peripheralKmfH+Ca_srKmrH$

### Component: i_serca_bulk

$i_serca_bulk=SERCA_max⁢Ca_i_bulkKmfH-Ca_srKmrH1.0+Ca_i_bulkKmfH+Ca_srKmrH$

### Component: i_leak

$i_leak=LEAK_max⁢Ca_sr-Ca_i_bulk$

### Component: i_diff

$i_diff=DIFF_max⁢Ca_i_peripheral-Ca_i_bulk$
Source
Derived from workspace Model of a rabbit Purkinje cell at changeset 72bceeb3ab3e.
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