Model Mathematics

FonRT = \frac{\frac{F}{Temp}}{R}

vcell = 1000 π ra ra l ageo = 2 π ra ra + 2 π ra l Acap = ageo 2 vmyo = vcell 0.68 vmito = vcell 0.24 vsr = vcell 0.06 vnsr = vcell 0.0552 vjsr = vcell 0.0048 vss = vcell 0.02 AF = \frac{Acap}{F} 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} caiont = (ilca + icab + ipca - (2 inaca)) + icat naiont = ina + inab + 3 inaca + ilcana + 3 inak kiont = (ikr + iks + IK1 + ikp + ilcak - (2 inak)) + i_Stim \frac{d}{d time} (V) = - (naiont + kiont + caiont)

ina = GNa m m m H J (V - ENa) a = 1 - (\frac{1}{1 + ⅇ^{\frac{- (V + 40)}{0.024}}}) aH = a 0.135 ⅇ^{\frac{80 + V}{- 6.8}} bH = \frac{1 - a}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} + a (3.56 ⅇ^{0.079 V} + 3.1 1e5 ⅇ^{0.35 V}) aj = \frac{a ((- 1.2714e5) ⅇ^{0.2444 V} - ((3.474e-5) ⅇ^{(- 0.04391) V})) 1 (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} bj = \frac{(1 - a) 0.3 ⅇ^{(- (2.535e-7)) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} + \frac{a 0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} am = \frac{0.32 1 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} bm = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (H) = aH (1 - H) - (bH H) \frac{d}{d time} (m) = am (1 - m) - (bm m) \frac{d}{d time} (J) = aj (1 - J) - (bj J)

dss0 = \frac{1}{1 + ⅇ^{\frac{- (V + 10)}{6.24}}} taud = \frac{dss0 1 (1 - (ⅇ^{\frac{- (V + 10)}{6.24}}))}{0.035 (V + 10)} dss1 = \frac{1}{1 + ⅇ^{\frac{- (V + 60)}{0.024}}} dss = dss0 dss1 fss = \frac{1}{1 + ⅇ^{\frac{V + 32}{8}}} + \frac{0.6}{1 + ⅇ^{\frac{50 - V}{20}}} tauf = \frac{1}{0.0197 ⅇ^{- ({(0.0337 (V + 10))}^{2})} + 0.02} ibarca = \frac{pca 4 V F FonRT (gacai Ca_i ⅇ^{2 V FonRT} - (gacao Ca_o))}{ⅇ^{2 V FonRT} - 1} ibarna = \frac{pna V F FonRT (ganai Na_i ⅇ^{V FonRT} - (ganao Na_o))}{ⅇ^{V FonRT} - 1} ibark = \frac{pk V F FonRT (gaki K_i ⅇ^{V FonRT} - (gako K_o))}{ⅇ^{V FonRT} - 1} fca = \frac{1}{1 + \frac{Ca_i}{kmca}} ilca = d f fca ibarca ilcana = d f fca ibarna ilcak = d f fca ibark \frac{d}{d time} (d) = \frac{dss - d}{taud} \frac{d}{d time} (f) = \frac{fss - f}{tauf}

GK1_= GK1max \sqrt{\frac{K_o}{5.4}} ak1 = \frac{1.02}{1 + ⅇ^{0.2385 (V - EK - 59.215)}} bk1 = \frac{0.49124 ⅇ^{0.08032 ((V - EK) + 5.476)} + 1 ⅇ^{0.06175 (V - EK - 594.31)}}{1 + ⅇ^{(- 0.5143) ((V - EK) + 4.753)}} gK1 = \frac{GK1_ak1}{ak1 + bk1} IK1 = gK1 (V - EK)

r = \frac{1}{1 + ⅇ^{\frac{V + 9}{22.4}}} ikr = gkrmax \sqrt{\frac{K_o}{5.4}} xr r (V - EK) xrss = \frac{1}{1 + ⅇ^{\frac{- (V + 10.085)}{4.25}}} tauxr = \frac{1}{\frac{0.00138 (V + 14.2)}{1 - (ⅇ^{(- 0.123) (V + 14.2)})} + \frac{0.00061 (V + 38.9)}{ⅇ^{0.145 (V + 38.9)} - 1}} \frac{d}{d time} (xr) = \frac{xrss - xr}{tauxr}

gks = GKsmax (1 + \frac{0.6}{1 + {(\frac{3.8e-5}{Ca_i})}^{1.4}}) xss = \frac{1}{1 + ⅇ^{\frac{- (V - 1.5)}{16.7}}} tauxs = \frac{1}{\frac{0.0000719 (V + 30)}{1 - (ⅇ^{(- 0.148) (V + 30)})} + \frac{0.000131 (V + 30)}{ⅇ^{0.0687 (V + 30)} - 1}} iks = gks xs1 xs2 (V - EKs) \frac{d}{d time} (xs1) = \frac{xss - xs1}{tauxs} \frac{d}{d time} (xs2) = \frac{\frac{xss - xs2}{tauxs}}{4}

sigma = \frac{ⅇ^{\frac{Na_o}{67.3}} - 1}{7} fnak = \frac{1}{1 + 0.1245 ⅇ^{(- 0.1) V FonRT} + 0.0365 sigma ⅇ^{(- V) FonRT}} inak = \frac{\frac{ibarnak fnak}{1 + {(\frac{kmnai}{Na_i})}^{2}}}{1 + \frac{kmko}{K_o}}

inaca = \frac{c1 ⅇ^{(gammas - 1) V FonRT} (ⅇ^{V FonRT} {Na_i}^{3} Ca_o - ({Na_o}^{3} Ca_i))}{1 + c2 ⅇ^{(gammas - 1) V FonRT} (ⅇ^{V FonRT} {Na_i}^{3} Ca_o + {Na_o}^{3} Ca_i)}

ikp = \frac{GKpmax (V - EK)}{1 + ⅇ^{\frac{7.488 - V}{5.98}}}

ipca = \frac{ibarpca Ca_i}{kmpca + Ca_i}

icab = gcab (V - (\frac{\ln (\frac{Ca_o}{Ca_i})}{2 FonRT}))

inab = GNab (V - ENa)

icat = gcat b b g (V - ECa) bss = \frac{1}{1 + ⅇ^{\frac{- (V + 14)}{10.8}}} taub = 3.7 + \frac{6.1}{1 + ⅇ^{\frac{V + 25}{4.5}}} gss = \frac{1}{1 + ⅇ^{\frac{V + 60}{5.6}}} aa = 1 - (\frac{1}{1 + ⅇ^{\frac{- V}{0.0024}}}) taug = aa 1 ((- 0.875) V + 12) + 12 (1 - aa) \frac{d}{d time} (b) = \frac{bss - b}{taub} \frac{d}{d time} (g) = \frac{gss - g}{taug}

ENa = \frac{\ln (\frac{Na_o}{Na_i})}{FonRT} ECa = \frac{\frac{\ln (\frac{Ca_o}{Ca_i})}{2}}{FonRT} EK = \frac{\ln (\frac{K_o}{K_i})}{FonRT} EKs = \frac{\ln (\frac{K_o + prnak Na_o}{K_i + prnak Na_i})}{FonRT}

alpha_Rel = tau kappa bbb = csqnbar + kmcsqn - Ca_JSR_T c = Ca_JSR_T kmcsqn Ca_JSR_free = \frac{- bbb}{2} + \frac{\sqrt{{bbb}^{2} + 4 c}}{2} Rel_ss = \frac{ilca alpha_Rel}{1 + {(\frac{K_Relss}{Ca_JSR_free})}^{qn}} tau_Rel = \frac{tau}{1 + \frac{0.0123}{Ca_JSR_free}} \frac{d}{d time} (Rel) = \frac{- (Rel_ss + Rel)}{tau_Rel}

iup = \frac{iupbar Ca_i}{Ca_i + kmup} ileak = \frac{iupbar Ca_NSR}{nsrbar}

itr = \frac{Ca_NSR - Ca_JSR_free}{tautr}

\frac{d}{d time} (Na_i) = \frac{(- naiont) AF}{vmyo}

\frac{d}{d time} (K_i) = \frac{(- kiont) AF}{vmyo}

bmyo = (cmdnbar + trpnbar - Ca_T) + kmtrpn + kmcmdn cmyo = (kmcmdn kmtrpn - (Ca_T (kmtrpn + kmcmdn))) + trpnbar kmcmdn + cmdnbar kmtrpn dmyo = (- kmtrpn) kmcmdn Ca_T Ca_i = \frac{2 \sqrt{bmyo bmyo - (3 cmyo)}}{3} \cos (\frac{\arccos (\frac{9 bmyo cmyo - (2 bmyo bmyo bmyo) - (27 dmyo)}{2 {(bmyo bmyo - (3 cmyo))}^{1.5}})}{3}) - (\frac{bmyo}{3}) \frac{d}{d time} (Over) = 0 \frac{d}{d time} (Ca_T) = \frac{(- caiont) AF}{vmyo 2} + \frac{(ileak - iup) vnsr}{vmyo} + \frac{Rel vjsr}{vmyo} \frac{d}{d time} (Ca_NSR) = iup - (\frac{itr vjsr}{vnsr}) - ileak \frac{d}{d time} (Ca_JSR_T) = itr - Rel