Model Mathematics

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Component: Environment

FonRT = \frac{F}{R T}

Component: cell

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} Itotal = - (xina + xik1 + xikr + xiks + xito + xiNaCa + xica + xiNaK + i_Stim) \frac{d}{d time} (V) = Itotal

Component: INa

am = \{\begin{matrix} \frac{0.32 1 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} if | V + 47.13 | > 0.001 \\ 3.2 otherwise \end{matrix} bm = 0.08 ⅇ^{\frac{- V}{11}} ah = \{\begin{matrix} 0.135 ⅇ^{\frac{80 + V}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} bh = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} aj = \{\begin{matrix} \frac{((- 127140) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(- 0.04391) V})) 1 (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} bj = \{\begin{matrix} \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.3 ⅇ^{(- 0.0000002535) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} \frac{d}{d time} (xh) = ah (1 - xh) - (bh xh) \frac{d}{d time} (xj) = aj (1 - xj) - (bj xj) \frac{d}{d time} (xm) = am (1 - xm) - (bm xm) xina = gna xh xj xm xm xm (V - ena)

Component: ICaL

za = V 2 FonRT rxa = \{\begin{matrix} \frac{4 pca F FonRT (csm ⅇ^{za} - (0.341 Ca_o))}{2 FonRT} if | za | < 0.001 \\ \frac{4 pca V F FonRT (csm ⅇ^{za} - (0.341 Ca_o))}{ⅇ^{za} - 1} otherwise \end{matrix} poinf = \frac{1}{1 + ⅇ^{\frac{- (V - vth)}{s6}}} alpha = \frac{poinf}{taupo} beta = \frac{1 - poinf}{taupo} fca = \frac{1}{1 + {(\frac{cat}{Ca_dyad})}^{3}} s1 = 0.0182688 fca k1 = 0.024168 fca s2 = \frac{\frac{s1 r1}{r2} k2}{k1} s2t = \frac{\frac{s1t r1}{r2} k2t}{k1t} poi = \frac{1}{1 + ⅇ^{\frac{- (V - vx)}{sx}}} k3 = \frac{1 - poi}{tau3} k3t = k3 \Pr = 1 - (\frac{1}{1 + ⅇ^{\frac{- (V - vy)}{sy}}}) recov = 10 + 4954 ⅇ^{\frac{V}{15.6}} tau_ca = \frac{tca}{1 + {(\frac{Ca_dyad}{cpt})}^{4}} + 0.1 tauca = (recov - tau_ca) \Pr + tau_ca tauba = (recov - 450) \Pr + 450 Ps = \frac{1}{1 + ⅇ^{\frac{- (V - vyr)}{syr}}} k6 = \frac{fca Ps}{tauca} k5 = \frac{1 - Ps}{tauca} k6t = \frac{Ps}{tauba} k5t = \frac{1 - Ps}{tauba} k4 = \frac{\frac{\frac{k3 alpha}{beta} k1}{k2} k5}{k6} k4t = \frac{\frac{\frac{k3t alpha}{beta} k1t}{k2t} k5t}{k6t} po = 1 - xi1ca - xi2ca - xi1ba - xi2ba - c1 - c2 \frac{d}{d time} (c1) = alpha c2 + k2 xi1ca + k2t xi1ba + r2 po - ((beta + r1 + k1t + k1) c1) \frac{d}{d time} (c2) = beta c1 + k5 xi2ca + k5t xi2ba - ((k6 + k6t + alpha) c2) \frac{d}{d time} (xi1ca) = k1 c1 + k4 xi2ca + s1 po - ((k3 + k2 + s2) xi1ca) \frac{d}{d time} (xi1ba) = k1t c1 + k4t xi2ba + s1t po - ((k3t + k2t + s2t) xi1ba) \frac{d}{d time} (xi2ca) = k3 xi1ca + k6 c2 - ((k5 + k4) xi2ca) \frac{d}{d time} (xi2ba) = k3t xi1ba + k6t c2 - ((k5t + k4t) xi2ba) jca = gca po rxa xica = 2 wca jca

Component: IK1

aki = \frac{1.02}{1 + ⅇ^{0.2385 (V - ek - 59.215)}} bki = \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)}} xkin = \frac{aki}{aki + bki} xik1 = gkix \sqrt{\frac{K_o}{5.4}} xkin (V - ek)

Component: IKr

xkrv1 = \{\begin{matrix} \frac{0.00138 1 (V + 7)}{1 - (ⅇ^{(- 0.123) (V + 7)})} if | V + 7 | > 0.001 \\ \frac{0.00138}{0.123} otherwise \end{matrix} xkrv2 = \{\begin{matrix} \frac{0.00061 1 (V + 10)}{ⅇ^{0.145 (V + 10)} - 1} if | V + 10 | > 0.001 \\ \frac{0.00061}{0.145} otherwise \end{matrix} taukr = \frac{1}{xkrv1 + xkrv2} xkrinf = \frac{1}{1 + ⅇ^{\frac{- (V + 50)}{7.5}}} rg = \frac{1}{1 + ⅇ^{\frac{V + 33}{22.4}}} xikr = gkr \sqrt{\frac{K_o}{5.4}} xr rg (V - ek) \frac{d}{d time} (xr) = \frac{xkrinf - xr}{taukr}

Component: IKs

xs1ss = \frac{1}{1 + ⅇ^{\frac{- (V - 1.5)}{16.7}}} xs2ss = xs1ss tauxs1 = \{\begin{matrix} \frac{1}{\frac{0.0000719}{0.148} + \frac{0.000131}{0.0687}} if | V + 30 | < \frac{0.001}{0.0687} \\ \frac{1}{\frac{0.0000719 (V + 30)}{1 - (ⅇ^{(- 0.148) (V + 30)})} + \frac{0.000131 (V + 30)}{ⅇ^{0.0687 (V + 30)} - 1}} otherwise \end{matrix} tauxs2 = 4 tauxs1 gksx = 1 + \frac{0.8}{1 + {(\frac{0.5}{Ca_i})}^{3}} xiks = gks gksx xs1 xs2 (V - eks) \frac{d}{d time} (xs1) = \frac{xs1ss - xs1}{tauxs1} \frac{d}{d time} (xs2) = \frac{xs2ss - xs2}{tauxs2}

Component: Ito

rt1 = \frac{- (V + 3)}{15} rt2 = \frac{V + 33.5}{10} rt3 = \frac{V + 60}{10} rt4 = \frac{\frac{- V}{30} V}{30} rt5 = \frac{V + 33.5}{10} xtos_inf = \frac{1}{1 + ⅇ^{rt1}} ytos_inf = \frac{1}{1 + ⅇ^{rt2}} xtof_inf = xtos_inf ytof_inf = ytos_inf rs_inf = \frac{1}{1 + ⅇ^{rt2}} txs = \frac{9}{1 + ⅇ^{- rt1}} + 0.5 tys = \frac{3000}{1 + ⅇ^{rt3}} + 30 txf = 3.5 ⅇ^{rt4} + 1.5 tyf = \frac{20}{1 + ⅇ^{rt5}} + 20 \frac{d}{d time} (xtos) = \frac{xtos_inf - xtos}{txs} \frac{d}{d time} (ytos) = \frac{ytos_inf - ytos}{tys} \frac{d}{d time} (xtof) = \frac{xtof_inf - xtof}{txf} \frac{d}{d time} (ytof) = \frac{ytof_inf - ytof}{tyf} xitos = gtos xtos (ytos + 0.5 rs_inf) (V - ek) xitof = gtof xtof ytof (V - ek) xito = xitos + xitof

Component: INaK

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}} xiNaK = \frac{\frac{gNaK fNaK Na_i}{Na_i + xkmnai} K_o}{K_o + xkmko}

Component: INaCa

zw3 = {Na_i}^{3} Ca_o ⅇ^{V 0.35 FonRT} - ({Na_o}^{3} csm ⅇ^{V (0.35 - 1) FonRT}) zw4 = 1 + 0.2 ⅇ^{V (0.35 - 1) FonRT} aloss = \frac{1}{1 + {(\frac{xkdna}{Ca_submem})}^{3}} yz1 = xmcao {Na_i}^{3} + {xmnao}^{3} csm yz2 = {xmnai}^{3} Ca_o (1 + \frac{csm}{xmcai}) yz3 = xmcai {Na_o}^{3} (1 + {(\frac{Na_i}{xmnai})}^{3}) yz4 = {Na_i}^{3} Ca_o + {Na_o}^{3} csm zw8 = yz1 + yz2 + yz3 + yz4 jNaCa = \frac{gNaCa aloss zw3}{zw4 zw8} xiNaCa = wca jNaCa

Component: Irel

\frac{d}{d time} (Ca_JSR) = \frac{Ca_NSR - Ca_JSR}{taua} bv = (1 - av) cstar - 50 Qr0 = \{\begin{matrix} \frac{Ca_JSR - 50}{1} if Ca_JSR > 50 \land Ca_JSR < cstar \\ av Ca_JSR + bv if Ca_JSR \geq cstar \\ 0 otherwise \end{matrix} Qr = \frac{Ca_NSR Qr0}{cstar} sparkV = \frac{ⅇ^{(- ay) (V + 30)}}{1 + ⅇ^{(- ay) (V + 30)}} spark_rate = \frac{gryr}{1} po | rxa | sparkV \frac{d}{d time} (xir) = spark_rate Qr - (\frac{xir (1 - (\frac{taur dCa_JSR}{Ca_NSR}))}{taur}) xirp = \frac{\frac{po Qr | rxa | gbarsr}{1} ⅇ^{(- ax) (V + 30)}}{1 + ⅇ^{(- ax) (V + 30)}} xicap = po gdyad | rxa | xiryr = xirp + xicap

Component: Ileak_Iup_Ixfer

jup = \frac{vup Ca_i Ca_i}{Ca_i Ca_i + cup cup} jleak = \frac{gleak Ca_NSR Ca_NSR}{Ca_NSR Ca_NSR + kj kj} (Ca_NSR 16.667 - Ca_i)

Component: Na

\frac{d}{d time} (Na_i) = \frac{- (xina + 3 xiNaK + 3 xiNaCa)}{wca 1000}

Component: Ca

bpxs = \frac{bcal xkcal}{(xkcal + Ca_submem) (xkcal + Ca_submem)} spxs = \frac{srmax srkd}{(srkd + Ca_submem) (srkd + Ca_submem)} mempxs = \frac{bmem kmem}{(kmem + Ca_submem) (kmem + Ca_submem)} sarpxs = \frac{bsar ksar}{(ksar + Ca_submem) (ksar + Ca_submem)} dcsib = \frac{1}{1 + bpxs + spxs + mempxs + sarpxs} bpxi = \frac{bcal xkcal}{(xkcal + Ca_i) (xkcal + Ca_i)} spxi = \frac{srmax srkd}{(srkd + Ca_i) (srkd + Ca_i)} mempxi = \frac{bmem kmem}{(kmem + Ca_i) (kmem + Ca_i)} sarpxi = \frac{bsar ksar}{(ksar + Ca_i) (ksar + Ca_i)} dciib = \frac{1}{1 + bpxi + spxi + mempxi + sarpxi} jd = \frac{Ca_submem - Ca_i}{taud} xbi = xkon Ca_i (btrop - tropi) - (xkoff tropi) xbs = xkon Ca_submem (btrop - trops) - (xkoff trops) dCa_JSR = (- xir) + jup - jleak \frac{d}{d time} (Ca_dyad) = xiryr - (\frac{Ca_dyad - Ca_submem}{taups}) \frac{d}{d time} (Ca_submem) = dcsib (50 ((xir - jd - jca) + jNaCa) - xbs) \frac{d}{d time} (Ca_i) = dciib ((jd - jup) + jleak - xbi) \frac{d}{d time} (Ca_NSR) = dCa_JSR \frac{d}{d time} (tropi) = xbi \frac{d}{d time} (trops) = xbs csm = \frac{Ca_submem}{1000}

Component: reversal_potentials

ek = \frac{1}{FonRT} \ln (\frac{K_o}{K_i}) eks = \frac{1}{FonRT} \ln (\frac{K_o + prNaK Na_o}{K_i + prNaK Na_i}) ena = \frac{1}{FonRT} \ln (\frac{Na_o}{Na_i})