Model Mathematics

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Component: environment

Component: model_parameters

Vc = \frac{Smtot}{7850} Vmyo = Vc 0.585 VSRup = Vc 0.0315 VSRrel = Vc 0.0035 Vd = Vc 0.000075 Sms = Smtot - (0.44 e 4 Vc) Smt = Smtot - Sms pt = ptcm Sms Lt = \frac{Smt}{2 3.14159 radiust pt} Vt = 3.14159 {radiust}^{2} Lt pt Cmt = Smt 1 Cms = Sms 1 Rst = \frac{\frac{Rot Lt}{2}}{3.14159 {radiust}^{2} pt}

Component: i_circ

i_circ = 1000 (\frac{Vmt}{Rst} - (\frac{Vms}{Rst}))

Component: i_Stim

past = ⌊ \frac{time}{stim_period} ⌋ stim_period i_Stim = \{\begin{matrix} stim_amplitude if time - past \geq stim_offset \land time - past ≦ stim_offset + stim_duration \\ 0 otherwise \end{matrix}

Component: membrane_permeabilities

gNas = gNa (Sms + Smt) (1 - fNat) gNat = gNa (Sms + Smt) fNat gKfs = gKf (Sms + Smt) (1 - fKft) gKft = gKf (Sms + Smt) fKft gKsts = gKst (Sms + Smt) (1 - fKstt) gKstt = gKst (Sms + Smt) fKstt gK1s = gK1 (Sms + Smt) (1 - fK1t) gK1t = gK1 (Sms + Smt) fK1t gKbs = gKb (Sms + Smt) (1 - fKbt) gKbt = gKb (Sms + Smt) fKbt gKtos = gKto (Sms + Smt) (1 - fKtot) gKtot = gKto (Sms + Smt) fKtot gNabs = gNab (Sms + Smt) (1 - fNabt) gNabt = gNab (Sms + Smt) fNabt gCabs = gCab (Sms + Smt) (1 - fCabt) gCabt = gCab (Sms + Smt) fCabt gCas = gCa (Sms + Smt) (1 - fCat) gCat = gCa (Sms + Smt) fCat i_NaCa_maxs = i_NaCa_max (Sms + Smt) (1 - fNaCat) i_NaCa_maxt = i_NaCa_max (Sms + Smt) fNaCat i_NaK_maxs = i_NaK_max (Sms + Smt) (1 - fNaKt) i_NaK_maxt = i_NaK_max (Sms + Smt) fNaKt i_pCa_maxs = i_pCa_max (Sms + Smt) (1 - fpCat) i_pCa_maxt = i_pCa_max (Sms + Smt) fpCat

Component: i_Nas

E_Nas = \frac{\ln (\frac{Nae}{Nai}) R T}{F} i_Nas = gNas m^{3} h (Vms - (\frac{\ln (\frac{Nae + 0.12 Ke}{Nai + 0.12 Ki}) R T}{F}))

Component: i_Nas_m_gate

alpha_m = \frac{117.26 (Vms + 59.3)}{1 - (ⅇ^{(- 0.55) (Vms + 59.3)})} beta_m = 3800 ⅇ^{(- 0.072) (Vms + 61)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: i_Nas_h_gate

alpha_h = \frac{284.4}{1 + ⅇ^{0.0812 (Vms + 115.9)}} beta_h = \frac{18.77 (Vms + 64.4)}{1 - (ⅇ^{(- 0.22) (Vms + 64.4)})} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: i_Nat

E_Nat = \frac{\ln (\frac{Nat}{Nai}) R T}{F} i_Nat = gNat m^{3} h (Vmt - (\frac{\ln (\frac{Nat + 0.12 Kt}{Nai + 0.12 Ki}) R T}{F}))

Component: i_Nat_m_gate

alpha_m = \frac{117.26 (Vmt + 59.3)}{1 - (ⅇ^{(- 0.55) (Vmt + 59.3)})} beta_m = 3800 ⅇ^{(- 0.072) (Vmt + 61)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: i_Nat_h_gate

alpha_h = \frac{284.4}{1 + ⅇ^{0.0812 (Vmt + 115.9)}} beta_h = \frac{18.77 (Vmt + 64.4)}{1 - (ⅇ^{(- 0.22) (Vmt + 64.4)})} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: i_Cas

E_Cas = \frac{\ln (\frac{Cae}{Cai}) R T}{2 F} Coinf = \frac{1}{1 + ⅇ^{\frac{- (Vms + 15.3)}{5}}} tauCo = 0.00025 + 0.00305 ⅇ^{(- 0.0045) {(Vms + 7)}^{2}} + 0.00105 ⅇ^{(- 0.002) {(Vms - 18)}^{2}} C1inf = \frac{1}{1 + ⅇ^{\frac{Vms + 26.7}{5.4}}} tauC1 = 0.105 ⅇ^{- ({(\frac{Vms + 45}{12})}^{2})} + \frac{0.04}{1 + ⅇ^{\frac{(- Vms) + 25}{25}}} + \frac{0.015}{1 + ⅇ^{\frac{Vms + 75}{25}}} C2inf = C1inf tauC2 = 0.041 ⅇ^{- ({(\frac{Vms + 47}{12})}^{2})} + \frac{0.08}{1 + ⅇ^{\frac{- (Vms + 55)}{5}}} + \frac{0.015}{1 + ⅇ^{\frac{Vms + 75}{25}}} Ccastinf = \frac{1}{1 + \frac{Cass}{0.01}} \frac{d}{d time} (C1) = \frac{C1inf - C1}{tauC1} \frac{d}{d time} (C2) = \frac{C2inf - C2}{tauC2} \frac{d}{d time} (Co) = \frac{Coinf - Co}{tauCo} \frac{d}{d time} (Ccast) = \frac{Ccastinf - Ccast}{tauCcast} i_Cas = \frac{\frac{gCas 4 Vms F^{2}}{R T} (Cass ⅇ^{\frac{2 Vms F}{R T}} - (0.341 Cae))}{ⅇ^{\frac{2 Vms F}{R T}} - 1} Co ((0.9 + \frac{Ccast}{10}) C1 + (0.1 - (\frac{Ccast}{10})) C2)

Component: i_Cat

E_Cat = \frac{\ln (\frac{Cat}{Cai}) R T}{2 F} Coinf = \frac{1}{1 + ⅇ^{\frac{- (Vmt + 15.3)}{5}}} tauCo = 0.00025 + 0.00305 ⅇ^{(- 0.0045) {(Vmt + 7)}^{2}} + 0.00105 ⅇ^{(- 0.002) {(Vmt - 18)}^{2}} C1inf = \frac{1}{1 + ⅇ^{\frac{Vmt + 26.7}{5.4}}} tauC1 = 0.105 ⅇ^{- ({(\frac{Vmt + 45}{12})}^{2})} + \frac{0.04}{1 + ⅇ^{\frac{(- Vmt) + 25}{25}}} + \frac{0.015}{1 + ⅇ^{\frac{Vmt + 75}{25}}} C2inf = C1inf tauC2 = 0.041 ⅇ^{- ({(\frac{Vmt + 47}{12})}^{2})} + \frac{0.08}{1 + ⅇ^{\frac{- (Vmt + 55)}{5}}} + \frac{0.015}{1 + ⅇ^{\frac{Vmt + 75}{25}}} Ccastinf = \frac{1}{1 + \frac{Cass}{0.01}} \frac{d}{d time} (C1) = \frac{C1inf - C1}{tauC1} \frac{d}{d time} (C2) = \frac{C2inf - C2}{tauC2} \frac{d}{d time} (Co) = \frac{Coinf - Co}{tauCo} \frac{d}{d time} (Ccast) = \frac{Ccastinf - Ccast}{tauCcast} i_Cat = \frac{\frac{gCat 4 Vmt F^{2}}{R T} (Cass ⅇ^{\frac{2 Vmt F}{R T}} - (0.341 Cat))}{ⅇ^{\frac{2 Vmt F}{R T}} - 1} Co ((0.9 + \frac{Ccast}{10}) C1 + (0.1 - (\frac{Ccast}{10})) C2)

Component: i_Ktos

i_Ktos = gKtos rs (0.886 ss + 0.114 sss) (Vms - E_Ks) E_Ks = \frac{\ln (\frac{Ke}{Ki}) R T}{F}

Component: i_Ktos_rs_gate

rinfs = \frac{1}{1 + ⅇ^{\frac{Vms + 10.6}{- 11.42}}} taurs = \frac{1}{45.16 ⅇ^{0.03577 (Vms + 50)} + 98.9 ⅇ^{(- 0.1) (Vms + 38)}} \frac{d}{d time} (rs) = \frac{rinfs - rs}{taurs}

Component: i_Ktos_ss_gate

sinfs = \frac{1}{1 + ⅇ^{\frac{Vms + 45.3}{6.8841}}} tauss = 0.35 ⅇ^{- ({(0.0666 (Vms + 70))}^{2})} + 0.035 \frac{d}{d time} (ss) = \frac{sinfs - ss}{tauss}

Component: i_Ktos_sss_gate

ssinfs = \frac{1}{1 + ⅇ^{\frac{Vms + 45.3}{6.8841}}} tausss = 3.7 ⅇ^{- ({(0.0333 (Vms + 70))}^{2})} + 0.035 \frac{d}{d time} (sss) = \frac{ssinfs - sss}{tausss}

Component: i_Ktot

E_Kt = \frac{\ln (\frac{Kt}{Ki}) R T}{F} i_Ktot = gKtot rt (0.886 st + 0.114 sst) (Vmt - E_Kt)

Component: i_Ktot_rt_gate

rinft = \frac{1}{1 + ⅇ^{\frac{Vmt + 10.6}{- 11.42}}} taurt = \frac{1}{45.16 ⅇ^{0.03577 (Vmt + 50)} + 98.9 ⅇ^{(- 0.1) (Vmt + 38)}} \frac{d}{d time} (rt) = \frac{rinft - rt}{taurt}

Component: i_Ktot_st_gate

sinft = \frac{1}{1 + ⅇ^{\frac{Vmt + 45.3}{6.8841}}} taust = 0.35 ⅇ^{- ({(0.0666 (Vmt + 70))}^{2})} + 0.035 \frac{d}{d time} (st) = \frac{sinft - st}{taust}

Component: i_Ktot_sst_gate

ssinft = \frac{1}{1 + ⅇ^{\frac{Vmt + 45.3}{6.8841}}} tausst = 3.7 ⅇ^{- ({(0.0333 (Vmt + 70))}^{2})} + 0.035 \frac{d}{d time} (sst) = \frac{ssinft - sst}{tausst}

Component: i_Ksts

i_Ksts = gKsts rsts ssts (Vms - E_Ks)

Component: i_Ksts_rsts_gate

rstinfs = \frac{1}{1 + ⅇ^{\frac{- (Vms + 11.5)}{11.82}}} taursts = \frac{10}{45.16 ⅇ^{0.03577 (Vms + 50)} + 98.9 ⅇ^{(- 0.1) (Vms + 38)}} \frac{d}{d time} (rsts) = \frac{rstinfs - rsts}{taursts}

Component: i_Ksts_ssts_gate

sstinfs = \frac{1}{1 + ⅇ^{\frac{Vms + 87.5}{10.3}}} \frac{d}{d time} (ssts) = \frac{sstinfs - ssts}{taussts}

Component: i_Kstt

i_Kstt = gKstt rstt sstt (Vmt - E_Kt)

Component: i_Kstt_rstt_gate

rstinft = \frac{1}{1 + ⅇ^{\frac{- (Vmt + 11.5)}{11.82}}} taurstt = \frac{10}{45.16 ⅇ^{0.03577 (Vmt + 50)} + 98.9 ⅇ^{(- 0.1) (Vmt + 38)}} \frac{d}{d time} (rstt) = \frac{rstinft - rstt}{taurstt}

Component: i_Kstt_sstt_gate

sstinft = \frac{1}{1 + ⅇ^{\frac{Vmt + 87.5}{10.3}}} \frac{d}{d time} (sstt) = \frac{sstinft - sstt}{tausstt}

Component: i_Kfs

i_Kfs = gKfs qs (0.2 (Vms - E_Nas) + 0.8 (Vms - E_Ks))

Component: i_Kfs_qs_gate

qinfs = \frac{1}{1 + ⅇ^{\frac{Vms + 138.6}{10.48}}} tauqs = \frac{1}{0.11885 ⅇ^{\frac{Vms + 80}{28.37}} + 0.5623 ⅇ^{\frac{- (Vms + 80)}{14.19}}} \frac{d}{d time} (qs) = \frac{qinfs - qs}{tauqs}

Component: i_Kft

i_Kft = gKft qt (0.2 (Vmt - E_Nat) + 0.8 (Vmt - E_Kt))

Component: i_Kft_qt_gate

qinft = \frac{1}{1 + ⅇ^{\frac{Vmt + 138.6}{10.48}}} tauqt = \frac{1}{0.11885 ⅇ^{\frac{Vmt + 80}{28.37}} + 0.5623 ⅇ^{\frac{- (Vmt + 80)}{14.19}}} \frac{d}{d time} (qt) = \frac{qinft - qt}{tauqt}

Component: i_K1s

i_K1s = \frac{gK1s (Vms - E_Ks - 1.73)}{1 + ⅇ^{\frac{1.613 F}{R T} (Vms - E_Ks - 1.73)} (1 + ⅇ^{\frac{Ke - 0.9988}{- 0.124}})}

Component: i_K1t

i_K1t = \frac{gK1t (Vmt - E_Kt - 1.73)}{1 + ⅇ^{\frac{1.613 F}{R T} (Vmt - E_Kt - 1.73)} (1 + ⅇ^{\frac{Kt - 0.9988}{- 0.124}})}

Component: i_Nabs

i_Nabs = gNabs (Vms - E_Nas)

Component: i_Nabt

i_Nabt = gNabt (Vmt - E_Nat)

Component: i_Cabs

i_Cabs = gCabs (Vms - E_Cas)

Component: i_Cabt

i_Cabt = gCabt (Vmt - E_Cat)

Component: i_Kbs

i_Kbs = gKbs (Vms - E_Ks)

Component: i_Kbt

i_Kbt = gKbt (Vmt - E_Kt)

Component: i_NaCas

i_NaCas = \frac{i_NaCa_maxs (ⅇ^{0.0187 Vms} {Nai}^{3} Cae - (ⅇ^{(- 0.0187) Vms} {Nae}^{3} Cai))}{1 + 0.0001 ({Nae}^{3} Cai + {Nai}^{3} Cae)}

Component: i_NaCat

i_NaCat = \frac{i_NaCa_maxt (ⅇ^{0.0187 Vmt} {Nai}^{3} Cat - (ⅇ^{(- 0.0187) Vmt} {Nat}^{3} Cai))}{1 + 0.0001 ({Nat}^{3} Cai + {Nai}^{3} Cat)}

Component: i_NaKs

i_NaKs = \frac{\frac{\frac{i_NaK_maxs}{1 + 0.1245 ⅇ^{\frac{(- 0.1) Vms F}{R T}} + \frac{0.0365 1}{7} (ⅇ^{\frac{Nae}{67.3}} - 1) ⅇ^{\frac{(- Vms) F}{R T}}} 1}{1 + {(\frac{10}{Nai})}^{1.5}} Ke}{Ke + 1.5}

Component: i_NaKt

i_NaKt = \frac{\frac{\frac{i_NaK_maxt}{1 + 0.1245 ⅇ^{\frac{(- 0.1) Vmt F}{R T}} + \frac{0.0365 1}{7} (ⅇ^{\frac{Nat}{67.3}} - 1) ⅇ^{\frac{(- Vmt) F}{R T}}} 1}{1 + {(\frac{10}{Nai})}^{1.5}} Kt}{Kt + 1.5}

Component: i_pCas

i_pCas = \frac{i_pCa_maxs Cai}{0.0004 + Cai}

Component: i_pCat

i_pCat = \frac{i_pCa_maxt Cai}{0.0004 + Cai}

Component: t_tubular_ion_fluxes

JteNa = \frac{1 Vt}{tauNa} (Nat - Nae) JteCa = \frac{1 Vt}{tauCa} (Cat - Cae) JteK = \frac{1 Vt}{tauK} (Kt - Ke)

Component: JCaSRup

JCaSRup = \frac{1 Vmyo (0.04 10 {(\frac{Cai}{0.000168})}^{1.2} - (\frac{0.9 CaSRup}{3.29}))}{1 + {(\frac{Cai}{0.000168})}^{1.2} + \frac{CaSRup}{3.29}}

Component: Jtr

Jtr = \frac{1 VSRrel}{tautr} (CaSRup - CaSRrel)

Component: JCaSRrel

JCaSRrel = 1800 VSRrel (F2 + F3) (CaSRrel - Cass) kap = 12.15 e +12 kbp = 4.05 e +9 \frac{d}{d time} (F1) = F2 kam - (F1 kap {Cass}^{4}) \frac{d}{d time} (F2) = F1 kap {Cass}^{4} + F3 kbm + F4 kcm - (F2 (kam + kbp {Cass}^{3} + kcp)) \frac{d}{d time} (F3) = F2 kbp {Cass}^{3} - (F3 kbm) \frac{d}{d time} (F4) = F2 kcp - (F4 kcm)

Component: JCad

JCad = \frac{1 (Cass - Cai) Vmyo}{taud}

Component: CaSRup

\frac{d}{d time} (CaSRup) = \frac{JCaSRup - Jtr}{1 VSRup}

Component: CaSRrel

\frac{d}{d time} (CaSRrel) = \frac{\frac{1}{1 + \frac{15 0.8}{{(0.8 + CaSRrel)}^{2}}} (Jtr - JCaSRrel)}{VSRrel}

Component: ion_concentrations

dBTRH = 20000 Cai (1 - BTRH) - (0.07 BTRH) \frac{d}{d time} (BTRH) = dBTRH dBTRL = 40000 Cai (1 - BTRL) - (BTRL 40) \frac{d}{d time} (BTRL) = dBTRL \frac{d}{d time} (Cass) = \frac{1}{1 + \frac{0.05 0.00238}{{(0.00238 + Cass)}^{2}}} (\frac{- (i_Cas + i_Cat)}{2 F Vd} + \frac{1 JCaSRrel}{Vd} - (\frac{1 JCad}{Vd})) \frac{d}{d time} (Cai) = \frac{1}{1 + \frac{0.05 0.00238}{{(0.00238 + Cai)}^{2}}} (\frac{(- 2) (i_NaCas + i_NaCat) + i_Cabs + i_Cabt + i_pCas + i_pCat}{(- 2) F Vmyo} + \frac{1 (JCad - JCaSRup)}{Vmyo} - (dBTRH 0.14) - (dBTRL 0.07)) \frac{d}{d time} (Nai) = \frac{- (i_Nas + i_Nat + i_Nabs + i_Nabt + 3 i_NaCas + 3 i_NaCat + 3 i_NaKs + 3 i_NaKt)}{F Vmyo} \frac{d}{d time} (Ki) = \frac{- (((- i_Stim) + i_Kfs + i_Kft + i_Ksts + i_Kstt + i_K1s + i_K1t + i_Kbs + i_Kbt - (2 i_NaKs) - (2 i_NaKt)) + i_Ktos + i_Ktot)}{F Vmyo} \frac{d}{d time} (Nat) = \frac{\frac{i_Nat + i_Nabt + 3 i_NaCat + 3 i_NaKt}{F} - (1 JteNa)}{Vt} \frac{d}{d time} (Cat) = \frac{\frac{(- 2) i_NaCat + i_Cat + i_Cabt + i_pCat}{2 F} - (1 JteCa)}{Vt} \frac{d}{d time} (Kt) = \frac{\frac{(i_Kft + i_Kstt + i_K1t + i_Kbt - (2 i_NaKt)) + i_Ktot}{F} - (1 JteK)}{Vt}

Component: Vms

\frac{d}{d time} (Vms) = \frac{1000 (i_Stim + i_circ - (i_Nas + i_Cas + i_Ktos + i_Ksts + i_Kfs + i_K1s + i_Nabs + i_Cabs + i_Kbs + i_NaKs + i_NaCas + i_pCas))}{Cms}

Component: Vmt

\frac{d}{d time} (Vmt) = \frac{1000 (- (i_circ + i_Nat + i_Cat + i_Ktot + i_Kstt + i_Kft + i_K1t + i_Nabt + i_Cabt + i_Kbt + i_NaKt + i_NaCat + i_pCat))}{Cmt}