Model Mathematics

Component based math viewer is available

Component: environment

Component: cell

FonRT = \frac{F}{R T} 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} I_tot = i_Na + i_Ca + i_to + i_Kr + i_Ks + i_K1 + i_NaCa + i_NaK + i_b_Na + i_b_Ca + i_Stim \frac{d}{d time} (V) = dVdt dVdt = \{\begin{matrix} \frac{V_hold - V}{1} if V_clamp = 1 \land (time ≦ 500 \lor time > 800) \\ \frac{V_step - V}{1} if V_clamp = 1 \land time > 500 \land time ≦ 800 \\ (- 1) I_tot otherwise \end{matrix}

Component: INa

i_Na = g_Na m^{3} h j (V - E_Na) E_Na = \frac{\ln (\frac{Nao}{Nai})}{FonRT}

Component: INa_m_gate

alpha_m = \{\begin{matrix} \frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} if | V + 47.13 | > 0.001 \\ 3.2 otherwise \end{matrix} beta_m = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: INa_h_gate

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{80 + V}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 3.1 e 5 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{0.13 (1 + ⅇ^{\frac{- (V + 10.66)}{11.1}})} otherwise \end{matrix} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: INa_j_gate

alpha_j = \{\begin{matrix} \frac{((- 127140) ⅇ^{0.244 V} - (3.474 e -5 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.3 ⅇ^{(- 2.535 e -7) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} \frac{d}{d time} (j) = alpha_j (1 - j) - (beta_j j)

Component: ICa

i_Ca = g_Ca_max d f f_Ca (V - E_Ca) E_Ca = \frac{\ln (\frac{Cao}{Cai})}{2 FonRT}

Component: ICa_d_gate

alpha_d = \frac{14.9859}{16.6813 \sqrt{2 π}} ⅇ^{\frac{- ({(\frac{V - 22.36}{16.6813})}^{2})}{2}} beta_d = 0.1471 - (\frac{5.3}{14.93 \sqrt{2 π}} ⅇ^{\frac{- ({(\frac{V - 6.2744}{14.93})}^{2})}{2}}) \frac{d}{d time} (d) = alpha_d (1 - d) - (beta_d d)

Component: ICa_f_gate

alpha_f = \frac{6.872 e -3}{1 + ⅇ^{\frac{V - 6.1546}{6.1223}}} beta_f = \frac{0.0687 ⅇ^{(- 0.1081) (V + 9.8255)} + 0.0112}{1 + ⅇ^{(- 0.2779) (V + 9.8255)}} + 5.474 e -4 \frac{d}{d time} (f) = alpha_f (1 - f) - (beta_f f)

Component: ICa_f_Ca_gate

f_Ca = \frac{Km_Ca}{Km_Ca + Cai}

Component: Ito

g_to_max = \{\begin{matrix} 0.3 if failing = 0 \\ 0.191 otherwise \end{matrix} i_to = g_to_max r t (V - E_to) E_to = \frac{\ln (\frac{0.043 Nao + Ko}{0.043 Nai + Ki})}{FonRT}

Component: Ito_r_gate

\frac{d}{d time} (r) = alpha_r (1 - r) - (beta_r r) alpha_r = \frac{0.5266 ⅇ^{(- 0.0166) (V - 42.2912)}}{1 + ⅇ^{(- 0.0943) (V - 42.2912)}} beta_r = \frac{5.186 e -5 V + 0.5149 ⅇ^{(- 0.1344) (V - 5.0027)}}{1 + ⅇ^{(- 0.1348) (V - 5.186 e -5)}}

Component: Ito_t_gate

\frac{d}{d time} (t) = alpha_t (1 - t) - (beta_t t) alpha_t = \frac{5.612 e -5 V + 0.0721 ⅇ^{(- 0.173) (V + 34.2531)}}{1 + ⅇ^{(- 0.1732) (V + 34.2531)}} beta_t = \frac{1.215 e -4 V + 0.0767 ⅇ^{(- 1.66 e -9) (V + 34.0235)}}{1 + ⅇ^{(- 0.1604) (V + 34.0235)}}

Component: IKs

i_Ks = g_Ks_max {Xs}^{2} (V - E_Ks) E_Ks = \frac{\ln (\frac{0.01833 Nao + Ko}{0.01833 Nai + Ki})}{FonRT}

Component: IKs_Xs_gate

\frac{d}{d time} (Xs) = alpha_Xs (1 - Xs) - (beta_Xs Xs) alpha_Xs = \frac{3.013 e -3}{1 + ⅇ^{\frac{7.4454 - (V + 10)}{14.3171}}} beta_Xs = \frac{5.87 e -3}{1 + ⅇ^{\frac{5.95 + V + 10}{15.82}}}

Component: IKr

i_Kr = g_Kr_max Xr rik (V - E_K) E_K = \frac{\ln (\frac{Ko}{Ki})}{FonRT} rik = \frac{1}{1 + ⅇ^{\frac{V + 26}{23}}}

Component: IKr_Xr_gate

\frac{d}{d time} (Xr) = alpha_Xr (1 - Xr) - (beta_Xr Xr) alpha_Xr = \frac{0.005 ⅇ^{5.266 e -4 (V + 4.067)}}{1 + ⅇ^{(- 0.1262) (V + 4.067)}} beta_Xr = \frac{0.016 ⅇ^{1.6 e -3 (V + 65.66)}}{1 + ⅇ^{0.0783 (V + 65.66)}}

Component: IK1

g_K1_max = \{\begin{matrix} 2.5 if failing = 0 \\ 2 otherwise \end{matrix} i_K1 = g_K1_max K1_infinity (V - E_K1) E_K1 = \frac{\ln (\frac{Ko}{Ki})}{FonRT}

Component: IK1_K1_gate

K1_infinity = \frac{alpha_K1}{alpha_K1 + beta_K1} alpha_K1 = \frac{0.1}{1 + ⅇ^{0.06 (V - (E_K1 + 200))}} beta_K1 = \frac{3 ⅇ^{2 e -4 (V + 100 + (- E_K1))} + 1 ⅇ^{0.1 (V - (10 + E_K1))}}{1 + ⅇ^{(- 0.5) (V - E_K1)}}

Component: ICab

g_b_Ca_max = \{\begin{matrix} 0.00085 if failing = 0 \\ 0.0013 otherwise \end{matrix} i_b_Ca = g_b_Ca_max (V - E_Ca)

Component: INab

g_b_Na_max = \{\begin{matrix} 0.001 if failing = 0 \\ 0 otherwise \end{matrix} i_b_Na = g_b_Na_max (V - E_Na)

Component: INaK

I_NaK_max = \{\begin{matrix} 1.3 if failing = 0 \\ 0.75 otherwise \end{matrix} f_NaK = \frac{1}{1 + 0.1245 ⅇ^{(- 0.1) V FonRT} + 0.0365 sigma ⅇ^{(- V) FonRT}} sigma = \frac{1}{7} (ⅇ^{\frac{Nao}{67.3}} - 1) i_NaK = \frac{\frac{I_NaK_max f_NaK 1}{1 + {(\frac{K_mNai}{Nai})}^{1.5}} Ko}{Ko + K_mKo}

Component: INaCa

K_NaCa = \{\begin{matrix} 1000 if failing = 0 \\ 1650 otherwise \end{matrix} i_NaCa = \frac{\frac{\frac{K_NaCa 1}{{K_mNa}^{3} + {Nao}^{3}} 1}{K_mCa + Cao} 1}{1 + K_sat ⅇ^{(eta - 1) V FonRT}} (ⅇ^{eta V FonRT} {Nai}^{3} Cao - (ⅇ^{(eta - 1) V FonRT} {Nao}^{3} Cai))

Component: Irel

\frac{d}{d time} (APtrack) = \{\begin{matrix} 100 (1 - APtrack) - (0.5 APtrack) if dVdt > 150 \\ (- 0.5) APtrack otherwise \end{matrix} \frac{d}{d time} (APtrack2) = \{\begin{matrix} 100 (1 - APtrack2) - (0.5 APtrack2) if APtrack < 0.2 \land APtrack > 0.18 \\ (- 0.5) APtrack2 otherwise \end{matrix} \frac{d}{d time} (APtrack3) = \{\begin{matrix} 100 (1 - APtrack3) - (0.5 APtrack3) if APtrack < 0.2 \land APtrack > 0.18 \\ (- 0.01) APtrack3 otherwise \end{matrix} \frac{d}{d time} (Cainfluxtrack) = \{\begin{matrix} \frac{(- Cm) ((i_Ca - i_NaCa) + i_b_Ca)}{2 V_myo F} if APtrack > 0.2 \\ 0 if APtrack2 > 0.01 \land APtrack ≦ 0.2 \\ (- 0.5) Cainfluxtrack otherwise \end{matrix} \frac{d}{d time} (OVRLDtrack) = \{\begin{matrix} 0 50 (1 - OVRLDtrack) if \frac{1}{1 + \frac{K_mCSQN}{Ca_JSR}} > CSQNthresh \land OVRLDtrack3 < 0.37 \land APtrack3 < 0.37 \\ (- 0) 0.5 OVRLDtrack otherwise \end{matrix} \frac{d}{d time} (OVRLDtrack2) = \{\begin{matrix} 0 50 (1 - OVRLDtrack2) if OVRLDtrack > Logicthresh \land OVRLDtrack2 < Logicthresh \\ (- 0) 0.5 OVRLDtrack2 otherwise \end{matrix} \frac{d}{d time} (OVRLDtrack3) = \{\begin{matrix} 0 50 (1 - OVRLDtrack3) if OVRLDtrack > Logicthresh \land OVRLDtrack3 < Logicthresh \\ (- 0) 0.01 OVRLDtrack3 otherwise \end{matrix} G_rel = \{\begin{matrix} \frac{1 G_rel_max (Cainfluxtrack - delta_Ca_ith)}{K_mrel + Cainfluxtrack - delta_Ca_ith} (1 - APtrack2) APtrack2 if Cainfluxtrack > delta_Ca_ith \\ 0 G_rel_overload (1 - OVRLDtrack2) OVRLDtrack2 if Cainfluxtrack ≦ delta_Ca_ith \land OVRLDtrack2 > 0 \\ 0 otherwise \end{matrix} i_rel = G_rel (Ca_JSR - Cai)

Component: Iup

I_up_max = \{\begin{matrix} 0.0045 if failing = 0 \\ 0.0015 otherwise \end{matrix} i_up = \frac{I_up_max Cai}{Cai + K_mup}

Component: Ileak

K_leak = \{\begin{matrix} 0.00026 if failing = 0 \\ 0.00017 otherwise \end{matrix} i_leak = K_leak Ca_NSR

Component: Itr

i_tr = \frac{1 (Ca_NSR - Ca_JSR)}{tau_tr}

Component: calcium_buffers_in_the_myoplasm

Cai_bufc = \frac{1}{1 + buffon (\frac{CMDN_max K_mCMDN}{{(K_mCMDN + Cai)}^{2}} + \frac{Tn_max K_mTn}{{(K_mTn + Cai)}^{2}})}

Component: calcium_buffers_in_the_JSR

Ca_JSR_bufc = \frac{1}{1 + \frac{CSQN_max K_mCSQN}{{(K_mCSQN + Ca_JSR)}^{2}}}

Component: Ionic_concentrations

volume = π preplength {radius}^{2} V_myo = 0.68 volume V_NSR = 0.0552 volume V_JSR = 0.0048 volume \frac{d}{d time} (Nai) = \frac{(- 1) Cm (i_Na + i_b_Na + i_NaCa 3 + i_NaK 3)}{V_myo F} \frac{d}{d time} (Ki) = \frac{(- 1) Cm (i_to + i_Kr + i_K1 + i_Stim + i_Ks - (2 i_NaK))}{V_myo F} \frac{d}{d time} (Cai) = Cai_bufc (\frac{(- Cm) ((i_Ca - (2 i_NaCa)) + i_b_Ca)}{2 V_myo F} + \frac{i_rel V_JSR}{V_myo} + \frac{(i_leak - i_up) V_NSR}{V_myo}) \frac{d}{d time} (Ca_JSR) = Ca_JSR_bufc (i_tr - i_rel) \frac{d}{d time} (Ca_NSR) = (- 1) (i_leak + \frac{V_JSR}{V_NSR} i_tr - i_up)

Model Mathematics

Component: environment

Component: cell

Component: INa

Component: INa_m_gate

Component: INa_h_gate

Component: INa_j_gate

Component: ICa

Component: ICa_d_gate

Component: ICa_f_gate

Component: ICa_f_Ca_gate

Component: Ito

Component: Ito_r_gate

Component: Ito_t_gate

Component: IKs

Component: IKs_Xs_gate

Component: IKr

Component: IKr_Xr_gate

Component: IK1

Component: IK1_K1_gate

Component: ICab

Component: INab

Component: INaK

Component: INaCa

Component: Irel

Component: Iup

Component: Ileak

Component: Itr

Component: calcium_buffers_in_the_myoplasm

Component: calcium_buffers_in_the_JSR

Component: Ionic_concentrations

Component: transmembrane_currents

Component: intracellular_currents

Component: Ions_n_reversal_potentials