Model Mathematics

i_Stim = \{\begin{matrix} stim_amplitude if time \geq stim_start \land time ≦ stim_end \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Stim + i_Na + i_Ca_L + i_t + i_sus + i_K1 + i_Kr + i_Ks + i_B_Na + i_B_Ca + i_NaK + i_CaP + i_NaCa)

i_Na = \frac{\frac{P_Na m^{3} (0.9 h1 + 0.1 h2) Na_c V F^{2}}{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_infinity = \frac{1}{1 + ⅇ^{\frac{V + 27.12}{- 8.21}}} tau_m = (4.2e-5) ⅇ^{- ({(\frac{V + 25.57}{28.8})}^{2})} + (2.4e-5)

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

\frac{d}{d time} (h2) = \frac{h_infinity - h2}{tau_h2} tau_h2 = \frac{0.12}{1 + ⅇ^{\frac{V + 35.1}{3.2}}} + 0.003

i_Ca_L = g_Ca_L d_L (f_Ca f_L_1 + (1 - f_Ca) f_L_2) (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}}} tau_d_L = 0.0027 ⅇ^{- ({(\frac{V + 35}{30})}^{2})} + 0.002

\frac{d}{d time} (f_L_1) = \frac{f_L_infinity - f_L_1}{tau_f_L1} f_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 27.4}{7.1}}} tau_f_L1 = 0.161 ⅇ^{- ({(\frac{V + 40}{14.4})}^{2})} + 0.01

\frac{d}{d time} (f_L_2) = \frac{f_L_infinity - f_L_2}{tau_f_L2} tau_f_L2 = 1.3323 ⅇ^{- ({(\frac{V + 40}{14.2})}^{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{V}{30})}^{2})} + 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}}} tau_s = 0.4812 ⅇ^{- ({(\frac{V + 52.45}{14.97})}^{2})} + 0.01414

i_sus = g_sus r_sus s_sus (V - E_K)

\frac{d}{d time} (r_sus) = \frac{r_sus_infinity - r_sus}{tau_r_sus} r_sus_infinity = \frac{1}{1 + ⅇ^{\frac{V + 4.3}{- 8}}} tau_r_sus = \frac{0.009}{1 + ⅇ^{\frac{V + 5}{12}}} + 0.0005

\frac{d}{d time} (s_sus) = \frac{s_sus_infinity - s_sus}{tau_s_sus} s_sus_infinity = \frac{0.4}{1 + ⅇ^{\frac{V + 20}{10}}} + 0.6 tau_s_sus = \frac{0.047}{1 + ⅇ^{\frac{V + 60}{10}}} + 0.3

i_Ks = g_Ks n (V - E_K) i_Kr = g_Kr p_a p_i (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}}} tau_n = 0.7 + 0.4 ⅇ^{- ({(\frac{V - 20}{20})}^{2})}

\frac{d}{d time} (p_a) = \frac{p_a_infinity - p_a}{tau_p_a} p_a_infinity = \frac{1}{1 + ⅇ^{\frac{V + 15}{- 6}}} tau_p_a = 0.03118 + 0.21718 ⅇ^{- ({(\frac{V + 20.1376}{22.1996})}^{2})}

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

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_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})

i_NaK = \frac{\frac{\frac{i_NaK_max K_c}{K_c + k_NaK_K} {Na_i}^{1.5}}{{Na_i}^{1.5} + {k_NaK_Na}^{1.5}} (V + 150)}{V + 200}

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

i_NaCa = \frac{k_NaCa ({Na_i}^{3} Ca_c ⅇ^{\frac{gamma F V}{R T}} - ({Na_c}^{3} Ca_i ⅇ^{\frac{(gamma - 1) V F}{R T}}))}{1 + d_NaCa ({Na_c}^{3} Ca_i + {Na_i}^{3} Ca_c)}

Vol_d = 0.02 Vol_i \frac{d}{d time} (Na_i) = \frac{- (i_Na + i_B_Na + 3 i_NaK + 3 i_NaCa + phi_Na_en)}{Vol_i F} \frac{d}{d time} (K_i) = \frac{- (i_t + i_sus + i_K1 + i_Kr + i_Ks - (2 i_NaK))}{Vol_i F} \frac{d}{d time} (Ca_i) = \frac{- (((- i_di) + i_B_Ca + i_CaP - (2 i_NaCa)) + i_up - i_rel)}{2 Vol_i F} - (0.08 dOTCdt + 0.16 dOTMgCdt + 0.045 dOCdt) \frac{d}{d time} (Ca_d) = \frac{- (i_Ca_L + i_di)}{2 Vol_d F} i_di = \frac{(Ca_d - Ca_i) 2 F Vol_d}{tau_di}

\frac{d}{d time} (O_C) = 200000 Ca_i (1 - O_C) - (476 O_C) dOCdt = \frac{d}{d time} (O_C) \frac{d}{d time} (O_TC) = 78400 Ca_i (1 - O_TC) - (392 O_TC) dOTCdt = \frac{d}{d time} (O_TC) \frac{d}{d time} (O_TMgC) = 200000 Ca_i (1 - O_TMgC - O_TMgMg) - (6.6 O_TMgC) dOTMgCdt = \frac{d}{d time} (O_TMgC) \frac{d}{d time} (O_TMgMg) = 2000 Mg_i (1 - O_TMgC - O_TMgMg) - (666 O_TMgMg)

Vol_c = 0.136 Vol_i \frac{d}{d time} (Na_c) = \frac{Na_b - Na_c}{tau_Na} + \frac{i_Na + i_B_Na + 3 i_NaK + 3 i_NaCa + phi_Na_en}{Vol_c F} \frac{d}{d time} (K_c) = \frac{K_b - K_c}{tau_K} + \frac{i_t + i_sus + i_K1 + i_Kr + i_Ks - (2 i_NaK)}{Vol_c F} \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}

i_up = \frac{I_up_max (\frac{Ca_i}{k_cyca} - (\frac{{k_xcs}^{2} 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 F Vol_rel}{tau_tr} i_rel = alpha_rel {(\frac{F2}{F2 + 0.25})}^{2} (Ca_rel - Ca_i) \frac{d}{d time} (O_Calse) = 480 Ca_rel (1 - O_Calse) - (400 O_Calse) \frac{d}{d time} (Ca_rel) = \frac{i_tr - i_rel}{2 Vol_rel F} - (31 \frac{d}{d time} (O_Calse)) \frac{d}{d time} (Ca_up) = \frac{i_up - i_tr}{2 Vol_up F} \frac{d}{d time} (F1) = r_recov (1 - F1 - F2) - (r_act F1) \frac{d}{d time} (F2) = r_act F1 - (r_inact F2) r_act = 203.8 ({(\frac{Ca_i}{Ca_i + k_rel_i})}^{4} + {(\frac{Ca_d}{Ca_d + k_rel_d})}^{4}) r_inact = 33.96 + 339.6 {(\frac{Ca_i}{Ca_i + k_rel_i})}^{4}