Model Mathematics

i_Stim = \{\begin{matrix} - stim_amplitude if time - (⌊ \frac{time}{stim_period} ⌋ stim_period) \geq stim_start \land time - (⌊ \frac{time}{stim_period} ⌋ stim_period) ≦ stim_start + stim_duration \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = - (i_K1 + i_to + i_Kr + i_Ks + i_CaL + i_NaK + i_Na + i_b_Na + i_NaCa + i_b_Ca + i_p_K + i_p_Ca + i_Stim)

E_Na = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) E_Ks = \frac{R T}{F} \ln (\frac{K_o + P_kna Na_o}{K_i + P_kna Na_i}) E_Ca = \frac{0.5 R T}{F} \ln (\frac{Ca_o}{Ca_i})

alpha_K1 = \frac{0.1}{1 + ⅇ^{0.06 (V - E_K - 200)}} beta_K1 = \frac{3 ⅇ^{0.0002 ((V - E_K) + 100)} + ⅇ^{0.1 (V - E_K - 10)}}{1 + ⅇ^{(- 0.5) (V - E_K)}} xK1_inf = \frac{alpha_K1}{alpha_K1 + beta_K1} i_K1 = g_K1 xK1_inf \sqrt{\frac{K_o}{5.4}} (V - E_K)

i_Kr = g_Kr \sqrt{\frac{K_o}{5.4}} Xr1 Xr2 (V - E_K)

xr1_inf = \frac{1}{1 + ⅇ^{\frac{- 26 - V}{7}}} alpha_xr1 = \frac{450}{1 + ⅇ^{\frac{- 45 - V}{10}}} beta_xr1 = \frac{6}{1 + ⅇ^{\frac{V + 30}{11.5}}} tau_xr1 = 1 alpha_xr1 beta_xr1 \frac{d}{d time} (Xr1) = \frac{xr1_inf - Xr1}{tau_xr1}

xr2_inf = \frac{1}{1 + ⅇ^{\frac{V + 88}{24}}} alpha_xr2 = \frac{3}{1 + ⅇ^{\frac{- 60 - V}{20}}} beta_xr2 = \frac{1.12}{1 + ⅇ^{\frac{V - 60}{20}}} tau_xr2 = 1 alpha_xr2 beta_xr2 \frac{d}{d time} (Xr2) = \frac{xr2_inf - Xr2}{tau_xr2}

i_Ks = g_Ks {Xs}^{2} (V - E_Ks)

xs_inf = \frac{1}{1 + ⅇ^{\frac{- 5 - V}{14}}} alpha_xs = \frac{1400}{\sqrt{1 + ⅇ^{\frac{5 - V}{6}}}} beta_xs = \frac{1}{1 + ⅇ^{\frac{V - 35}{15}}} tau_xs = 1 alpha_xs beta_xs + 80 \frac{d}{d time} (Xs) = \frac{xs_inf - Xs}{tau_xs}

i_Na = g_Na m^{3} h j (V - E_Na)

m_inf = \frac{1}{{(1 + ⅇ^{\frac{- 56.86 - V}{9.03}})}^{2}} alpha_m = \frac{1}{1 + ⅇ^{\frac{- 60 - V}{5}}} beta_m = \frac{0.1}{1 + ⅇ^{\frac{V + 35}{5}}} + \frac{0.1}{1 + ⅇ^{\frac{V - 50}{200}}} tau_m = 1 alpha_m beta_m \frac{d}{d time} (m) = \frac{m_inf - m}{tau_m}

h_inf = \frac{1}{{(1 + ⅇ^{\frac{V + 71.55}{7.43}})}^{2}} alpha_h = \{\begin{matrix} 0.057 ⅇ^{\frac{- (V + 80)}{6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 2.7 ⅇ^{0.079 V} + 310000 ⅇ^{0.3485 V} if V < - 40 \\ \frac{0.77}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} tau_h = \frac{1}{alpha_h + beta_h} \frac{d}{d time} (h) = \frac{h_inf - h}{tau_h}

j_inf = \frac{1}{{(1 + ⅇ^{\frac{V + 71.55}{7.43}})}^{2}} alpha_j = \{\begin{matrix} \frac{\frac{((- 25428) ⅇ^{0.2444 V} - (6.948 e -6 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1}}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.02424 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.6 ⅇ^{0.057 V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} tau_j = \frac{1}{alpha_j + beta_j} \frac{d}{d time} (j) = \frac{j_inf - j}{tau_j}

i_b_Na = g_bna (V - E_Na)

i_CaL = \frac{\frac{g_CaL d f f2 fCass 4 (V - 15) F^{2}}{R T} (0.25 Ca_ss ⅇ^{\frac{2 (V - 15) F}{R T}} - Ca_o)}{ⅇ^{\frac{2 (V - 15) F}{R T}} - 1}

d_inf = \frac{1}{1 + ⅇ^{\frac{- 8 - V}{7.5}}} alpha_d = \frac{1.4}{1 + ⅇ^{\frac{- 35 - V}{13}}} + 0.25 beta_d = \frac{1.4}{1 + ⅇ^{\frac{V + 5}{5}}} gamma_d = \frac{1}{1 + ⅇ^{\frac{50 - V}{20}}} tau_d = 1 alpha_d beta_d + gamma_d \frac{d}{d time} (d) = \frac{d_inf - d}{tau_d}

f_inf = \frac{1}{1 + ⅇ^{\frac{V + 20}{7}}} tau_f = 1102.5 ⅇ^{\frac{- ({(V + 27)}^{2})}{225}} + \frac{200}{1 + ⅇ^{\frac{13 - V}{10}}} + \frac{180}{1 + ⅇ^{\frac{V + 30}{10}}} + 20 \frac{d}{d time} (f) = \frac{f_inf - f}{tau_f}

f2_inf = \frac{0.67}{1 + ⅇ^{\frac{V + 35}{7}}} + 0.33 tau_f2 = 562 ⅇ^{\frac{- ({(V + 27)}^{2})}{240}} + \frac{31}{1 + ⅇ^{\frac{25 - V}{10}}} + \frac{80}{1 + ⅇ^{\frac{V + 30}{10}}} \frac{d}{d time} (f2) = \frac{f2_inf - f2}{tau_f2}

fCass_inf = \frac{0.6}{1 + {(\frac{Ca_ss}{0.05})}^{2}} + 0.4 tau_fCass = \frac{80}{1 + {(\frac{Ca_ss}{0.05})}^{2}} + 2 \frac{d}{d time} (fCass) = \frac{fCass_inf - fCass}{tau_fCass}

i_b_Ca = g_bca (V - E_Ca)

i_to = g_to r s (V - E_K)

s_inf = \frac{1}{1 + ⅇ^{\frac{V + 20}{5}}} tau_s = 85 ⅇ^{\frac{- ({(V + 45)}^{2})}{320}} + \frac{5}{1 + ⅇ^{\frac{V - 20}{5}}} + 3 \frac{d}{d time} (s) = \frac{s_inf - s}{tau_s}

r_inf = \frac{1}{1 + ⅇ^{\frac{20 - V}{6}}} tau_r = 9.5 ⅇ^{\frac{- ({(V + 40)}^{2})}{1800}} + 0.8 \frac{d}{d time} (r) = \frac{r_inf - r}{tau_r}

i_NaK = \frac{\frac{\frac{P_NaK K_o}{K_o + K_mk} Na_i}{Na_i + K_mNa}}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0353 ⅇ^{\frac{(- V) F}{R T}}}

i_NaCa = \frac{K_NaCa (ⅇ^{\frac{gamma V F}{R T}} {Na_i}^{3} Ca_o - (ⅇ^{\frac{(gamma - 1) V F}{R T}} {Na_o}^{3} Ca_i alpha))}{({Km_Nai}^{3} + {Na_o}^{3}) (Km_Ca + Ca_o) (1 + K_sat ⅇ^{\frac{(gamma - 1) V F}{R T}})}

i_p_Ca = \frac{g_pCa Ca_i}{Ca_i + K_pCa}

i_p_K = \frac{g_pK (V - E_K)}{1 + ⅇ^{\frac{25 - V}{5.98}}}

i_rel = V_rel O (Ca_SR - Ca_ss) i_up = \frac{Vmax_up}{1 + \frac{{K_up}^{2}}{{Ca_i}^{2}}} i_leak = V_leak (Ca_SR - Ca_i) i_xfer = V_xfer (Ca_ss - Ca_i) O = \frac{k1 {Ca_ss}^{2} R_prime}{k3 + k1 {Ca_ss}^{2}} \frac{d}{d time} (R_prime) = (- k2) Ca_ss R_prime + k4 (1 - R_prime) k1 = \frac{k1_prime}{kcasr} k2 = k2_prime kcasr kcasr = max_sr - (\frac{max_sr - min_sr}{1 + {(\frac{EC}{Ca_SR})}^{2}}) Ca_i_bufc = \frac{1}{1 + \frac{Buf_c K_buf_c}{{(Ca_i + K_buf_c)}^{2}}} Ca_sr_bufsr = \frac{1}{1 + \frac{Buf_sr K_buf_sr}{{(Ca_SR + K_buf_sr)}^{2}}} Ca_ss_bufss = \frac{1}{1 + \frac{Buf_ss K_buf_ss}{{(Ca_ss + K_buf_ss)}^{2}}} \frac{d}{d time} (Ca_i) = Ca_i_bufc (\frac{(i_leak - i_up) V_sr}{V_c} + i_xfer - (\frac{1 (i_b_Ca + i_p_Ca - (2 i_NaCa)) Cm}{2 1 V_c F})) \frac{d}{d time} (Ca_SR) = Ca_sr_bufsr (i_up - (i_rel + i_leak)) \frac{d}{d time} (Ca_ss) = Ca_ss_bufss (\frac{(- 1) i_CaL Cm}{2 1 V_ss F} + \frac{i_rel V_sr}{V_ss} - (\frac{i_xfer V_c}{V_ss}))

\frac{d}{d time} (Na_i) = \frac{(- 1) (i_Na + i_b_Na + 3 i_NaK + 3 i_NaCa)}{1 V_c F} Cm

\frac{d}{d time} (K_i) = \frac{(- 1) (i_K1 + i_to + i_Kr + i_Ks + i_p_K + i_Stim - (2 i_NaK))}{1 V_c F} Cm