Model Mathematics

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 = \frac{i_Na + i_Ca_L + i_t + i_Kur + i_K1 + i_Kr + i_Ks + i_B_Na + i_B_Ca + i_NaK + i_CaP + i_NaCa + i_KACh}{Cm} + i_Stim \frac{d}{d time} (V) = (- I) 1000 Q_tot = 0.05 V

i_Na = \frac{\frac{P_Na m m m (0.9 h1 + 0.1 h2) Na_c V F F}{R T} (ⅇ^{\frac{(V - E_Na) F}{R T}} - 1)}{ⅇ^{\frac{V F}{R T}} - 1} E_Na = \frac{R T}{F} \ln (\frac{Na_c}{Na_i})

\frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m} m_factor = \frac{V + 25.57}{28.8} m_infinity = \frac{1}{1 + ⅇ^{\frac{V + 27.12}{- 8.21}}} tau_m = 0.000042 ⅇ^{(- m_factor) m_factor} + 0.000024

\frac{d}{d time} (h1) = \frac{h_infinity - h1}{tau_h1} h_infinity = \frac{1}{1 + ⅇ^{\frac{V + 63.6}{5.3}}} h_factor = \frac{1}{1 + ⅇ^{\frac{V + 35.1}{3.2}}} tau_h1 = 0.03 h_factor + 0.0003

\frac{d}{d time} (h2) = \frac{h_infinity - h2}{tau_h2} tau_h2 = 0.12 h_factor + 0.003

i_Ca_L = g_Ca_L d_L (f_Ca f_L1 + (1 - f_Ca) f_L2) (V - E_Ca_app) f_Ca = \frac{Ca_d}{Ca_d + k_Ca}

\frac{d}{d time} (d_L) = \frac{d_L_infinity - d_L}{tau_d_L} d_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 9}{- 5.8}}} d_L_factor = \frac{V + 35}{30} tau_d_L = 0.0027 ⅇ^{(- d_L_factor) d_L_factor} + 0.002

\frac{d}{d time} (f_L1) = \frac{f_L_infinity - f_L1}{tau_f_L1} f_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 27.4}{7.1}}} f_L_factor = V + 40 tau_f_L1 = 0.161 ⅇ^{\frac{\frac{(- f_L_factor) f_L_factor}{14.4}}{14.4}} + 0.01

\frac{d}{d time} (f_L2) = \frac{f_L_infinity - f_L2}{tau_f_L2} tau_f_L2 = 1.3323 ⅇ^{\frac{\frac{(- f_L_factor) f_L_factor}{14.2}}{14.2}} + 0.0626

i_t = g_t r s (V - E_K) E_K = \frac{R T}{F} \ln (\frac{K_c}{K_i})

\frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r} r_infinity = \frac{1}{1 + ⅇ^{\frac{V - 1}{- 11}}} tau_r = 0.0035 ⅇ^{\frac{\frac{(- V) V}{30}}{30}} + 0.0015

\frac{d}{d time} (s) = \frac{s_infinity - s}{tau_s} s_infinity = \frac{1}{1 + ⅇ^{\frac{V + 40.5}{11.5}}} s_factor = \frac{V + 52.45}{15.8827} tau_s = 0.025635 ⅇ^{(- s_factor) s_factor} + 0.01414

i_Kur = g_kur a_ur i_ur (V - E_K)

\frac{d}{d time} (a_ur) = \frac{a_ur_infinity - a_ur}{tau_a_ur} a_ur_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 6)}{8.6}}} tau_a_ur = \frac{0.009}{1 + ⅇ^{\frac{V + 5}{12}}} + 0.0005

\frac{d}{d time} (i_ur) = \frac{i_ur_infinity - i_ur}{tau_i_ur} i_ur_infinity = \frac{1}{1 + ⅇ^{\frac{V + 7.5}{10}}} tau_i_ur = \frac{0.59}{1 + ⅇ^{\frac{V + 60}{10}}} + 3.05

i_K1 = \frac{g_K1 {(\frac{K_c}{1})}^{0.4457} (V - E_K)}{1 + ⅇ^{\frac{1.5 ((V - E_K) + 3.6) F}{R T}}}

i_Ks = g_Ks n (V - E_K) i_Kr = g_Kr pa pip (V - E_K)

\frac{d}{d time} (n) = \frac{n_infinity - n}{tau_n} n_infinity = \frac{1}{1 + ⅇ^{\frac{V - 19.9}{- 12.7}}} n_factor = \frac{V - 20}{20} tau_n = 0.7 + 0.4 ⅇ^{(- n_factor) n_factor}

\frac{d}{d time} (pa) = \frac{p_a_infinity - pa}{tau_pa} p_a_infinity = \frac{1}{1 + ⅇ^{\frac{V + 15}{- 6}}} pa_factor = \frac{V + 20.1376}{22.1996} tau_pa = 0.03118 + 0.21718 ⅇ^{(- pa_factor) pa_factor}

pip = \frac{1}{1 + ⅇ^{\frac{V + 55}{24}}}

i_B_Na = g_B_Na (V - E_Na) i_B_Ca = g_B_Ca (V - E_Ca) E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_c}{Ca_i})

pow_Na_i_15 = {Na_i}^{1.5} i_NaK = \frac{\frac{\frac{i_NaK_max K_c}{K_c + K_NaK_K} pow_Na_i_15}{pow_Na_i_15 + pow_K_NaK_Na_15} (V + 150)}{V + 200}

i_CaP = \frac{i_CaP_max Ca_i}{Ca_i + k_CaP}

i_NaCa = \frac{K_NaCa (Na_i Na_i Na_i Ca_c ⅇ^{\frac{F V gamma_Na}{R T}} - (Na_c Na_c Na_c Ca_i ⅇ^{\frac{(gamma_Na - 1) V F}{R T}}))}{1 + d_NaCa (Na_c Na_c Na_c Ca_i + Na_i Na_i Na_i Ca_c)}

i_KACh = \frac{10}{1 + \frac{9.13652 1^{0.477811}}{{ACh}^{0.477811}}} (0.0517 + \frac{0.4516}{1 + ⅇ^{\frac{V + 59.53}{17.18}}}) (V - E_K) Cm

\frac{d}{d time} (K_i) = \frac{- ((i_t + i_Kur + i_K1 + i_Ks + i_Kr - (2 i_NaK)) + i_Stim Cm)}{Vol_i F} \frac{d}{d time} (Na_i) = \frac{- (i_Na + i_B_Na + 3 i_NaCa + 3 i_NaK + phi_Na_en)}{Vol_i F} \frac{d}{d time} (Ca_i) = \frac{- (i_B_Ca + i_CaP + i_up - (i_di + i_rel + 2 i_NaCa))}{2 Vol_i F} - (1 J_O) \frac{d}{d time} (Ca_d) = \frac{- (i_Ca_L + i_di)}{2 Vol_d F} i_di = \frac{(Ca_d - Ca_i) 2 Vol_d F}{tau_di}

J_O_C = 200000 Ca_i (1 - O_C) - (476 O_C) J_O_TC = 78400 Ca_i (1 - O_TC) - (392 O_TC) J_O_TMgC = 200000 Ca_i (1 - O_TMgC - O_TMgMg) - (6.6 O_TMgC) J_O_TMgMg = 2000 Mg_i (1 - O_TMgC - O_TMgMg) - (666 O_TMgMg) \frac{d}{d time} (O_C) = J_O_C \frac{d}{d time} (O_TC) = J_O_TC \frac{d}{d time} (O_TMgC) = J_O_TMgC \frac{d}{d time} (O_TMgMg) = J_O_TMgMg J_O = 0.08 J_O_TC + 0.16 J_O_TMgC + 0.045 J_O_C \frac{d}{d time} (O) = J_O

\frac{d}{d time} (Ca_c) = \frac{Ca_b - Ca_c}{tau_Ca} + \frac{i_Ca_L + i_B_Ca + i_CaP - (2 i_NaCa)}{2 Vol_c F} \frac{d}{d time} (K_c) = \frac{K_b - K_c}{tau_K} + \frac{i_t + i_Kur + i_K1 + i_Ks + i_Kr - (2 i_NaK)}{Vol_c F} \frac{d}{d time} (Na_c) = \frac{Na_b - Na_c}{tau_Na} + \frac{i_Na + i_B_Na + 3 i_NaCa + 3 i_NaK + phi_Na_en}{Vol_c F}

r_Ca_d_term = \frac{Ca_d}{Ca_d + k_rel_d} r_Ca_i_term = \frac{Ca_i}{Ca_i + k_rel_i} r_Ca_d_factor = r_Ca_d_term r_Ca_d_term r_Ca_d_term r_Ca_d_term r_Ca_i_factor = r_Ca_i_term r_Ca_i_term r_Ca_i_term r_Ca_i_term r_act = 203.8 (r_Ca_i_factor + r_Ca_d_factor) r_inact = 33.96 + 339.6 r_Ca_i_factor \frac{d}{d time} (F1) = r_recov (1 - F1 - F2) - (r_act F1) \frac{d}{d time} (F2) = r_act F1 - (r_inact F2) i_rel_f2 = \frac{F2}{F2 + 0.25} i_rel_factor = i_rel_f2 i_rel_f2 i_rel = alpha_rel i_rel_factor (Ca_rel - Ca_i) i_up = \frac{I_up_max (\frac{Ca_i}{k_cyca} - (\frac{k_xcs k_xcs Ca_up}{k_srca}))}{\frac{Ca_i + k_cyca}{k_cyca} + \frac{k_xcs (Ca_up + k_srca)}{k_srca}} i_tr = \frac{(Ca_up - Ca_rel) 2 Vol_rel F}{tau_tr} J_O_Calse = 480 Ca_rel (1 - O_Calse) - (400 O_Calse) \frac{d}{d time} (O_Calse) = J_O_Calse \frac{d}{d time} (Ca_up) = \frac{i_up - i_tr}{2 Vol_up F} \frac{d}{d time} (Ca_rel) = \frac{i_tr - i_rel}{2 Vol_rel F} - (31 J_O_Calse)