Model Mathematics

\frac{d}{d time} (V) = \frac{- (i_Na + i_Ca_T + i_Ca_L + i_K + i_f + i_B + i_NaK + i_NaCa + i_Ca_P + i_K_ACh)}{Cm}

F_ACh_Na = 1 - (\frac{ACh}{ACh + 1 e -3}) i_Na = \frac{\frac{F_ACh_Na P_Na m^{3} h1 h2 Na_c V F^{2}}{R T} (ⅇ^{\frac{(V - E_Na) F}{R T}} - 1)}{ⅇ^{\frac{V F}{R T}} - 1}

alpha_m = \frac{(- 824) (V + 51.9)}{ⅇ^{\frac{V + 51.9}{- 8.9}} - 1} beta_m = 32960 ⅇ^{\frac{V + 51.9}{- 8.9}} m_infinity = \frac{alpha_m}{alpha_m + beta_m} tau_m = \frac{1}{alpha_m + beta_m} + 0.000015 \frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m}

alpha_h1 = 165 ⅇ^{\frac{V + 101.3}{- 12.6}} beta_h1 = \frac{12360}{320 ⅇ^{\frac{V + 101.3}{- 12.6}} + 1} h1_infinity = \frac{alpha_h1}{alpha_h1 + beta_h1} tau_h1 = \frac{1}{alpha_h1 + beta_h1} h2_infinity = h1_infinity tau_h2 = 20 tau_h1 \frac{d}{d time} (h1) = \frac{h1_infinity - h1}{tau_h1} \frac{d}{d time} (h2) = \frac{h2_infinity - h2}{tau_h2}

F_cAMP_CaL = 0.4 (1 + \frac{4.5 cAMP}{cAMP + 6.5 e -3}) + 0.03157 g_Ca_L = g_Ca_L_cont F_cAMP_CaL i_Ca_L = g_Ca_L (f_L d_L + 0.095 d_L_infinity) (V - E_Ca_L)

alpha_d_L = \frac{(- 28.39) (V + 35)}{ⅇ^{\frac{V + 35}{- 2.5}} - 1} + \frac{(- 84.9) V}{ⅇ^{(- 0.208) V} - 1} beta_d_L = \frac{11.43 (V - 5)}{ⅇ^{0.4 (V - 5)} - 1} tau_d_L = \frac{1}{alpha_d_L + beta_d_L} d_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 14.1}{- 6}}} \frac{d}{d time} (d_L) = \frac{d_L_infinity - d_L}{tau_d_L}

alpha_f_L = \frac{3.75 (V + 28)}{ⅇ^{\frac{V + 28}{4}} - 1} beta_f_L = \frac{30}{1 + ⅇ^{\frac{V + 28}{- 4}}} tau_f_L = \frac{1}{alpha_f_L + beta_f_L} f_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 30}{5}}} \frac{d}{d time} (f_L) = \frac{f_L_infinity - f_L}{tau_f_L}

i_Ca_T = g_Ca_T d_T f_T (V - E_Ca_T)

alpha_d_T = 1068 ⅇ^{\frac{V + 26.3}{30}} beta_d_T = 1068 ⅇ^{\frac{V + 26.3}{- 30}} tau_d_T = \frac{1}{alpha_d_T + beta_d_T} d_T_infinity = \frac{1}{1 + ⅇ^{\frac{V + 26.3}{- 6}}} \frac{d}{d time} (d_T) = \frac{d_T_infinity - d_T}{tau_d_T}

alpha_f_T = 15.3 ⅇ^{\frac{V + 61.7}{- 83.3}} beta_f_T = 15 ⅇ^{\frac{V + 61.7}{15.38}} tau_f_T = \frac{1}{alpha_f_T + beta_f_T} f_T_infinity = \frac{1}{1 + ⅇ^{\frac{V + 61.7}{5.6}}} \frac{d}{d time} (f_T) = \frac{f_T_infinity - f_T}{tau_f_T}

F_cAMP_K = 0.62 (1 + \frac{2.6129 cAMP}{cAMP + 9 e -3}) - 0.025 g_K = F_cAMP_K 0.00693 {(\frac{K_b}{1})}^{0.59} i_K = g_K P_a P_i (V - E_K)

P_a_infinity = \frac{1}{1 + ⅇ^{\frac{V + 5.1}{- 7.4}}} tau_P_a = \frac{1}{17 ⅇ^{0.0398 V} + 2.11 ⅇ^{(- 0.051) V}} \frac{d}{d time} (P_a) = \frac{P_a_infinity - P_a}{tau_P_a}

alpha_P_i = 100 ⅇ^{(- 0.0183) V} beta_P_i = 656 ⅇ^{0.00942 V} \frac{d}{d time} (P_i) = alpha_P_i (1 - P_i) - (beta_P_i P_i)

F_ACh_bNa = 1 - (\frac{ACh}{ACh + 5 e -1}) i_B_Na = F_ACh_bNa g_B_Na (V - E_Na) i_B_Ca = g_B_Ca (V - E_Ca) i_B_K = g_B_K (V - E_K) i_B = i_B_Na + i_B_Ca + i_B_K

i_f_Na = g_f_Na y^{2} (V - 75) i_f_K = g_f_K y^{2} (V + 85) i_f = i_f_Na + i_f_K

V_half = \frac{20.5}{1 + ⅇ^{\frac{cAMP - 3.4 e -3}{- 5 e -4}}} - 78.56 y_infinity = \frac{1}{1 + ⅇ^{\frac{V - V_half}{9}}} tau_y = \frac{1}{1.6483 ⅇ^{\frac{V + 54.06}{- 24.33}} + \frac{14.01055}{0.7 + ⅇ^{\frac{V + 60}{- 5.5}}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

F_cAMP_NaK = \frac{1.6}{1 + ⅇ^{\frac{cAMP - 3.75 e -3}{- 1.5 e -4}}} + 0.99 i_NaK = \{\begin{matrix} \frac{(2 - F_cAMP_NaK) i_NaK_max {(\frac{Na_i}{K_m_Na + Na_i})}^{3} {(\frac{K_c}{K_m_K + K_c})}^{2} 1.6}{1.5 + ⅇ^{\frac{V + 60}{- 40}}} if Ca_i < 0.00015 \\ \frac{F_cAMP_NaK i_NaK_max {(\frac{Na_i}{K_m_Na + Na_i})}^{3} {(\frac{K_c}{K_m_K + K_c})}^{2} 1.6}{1.5 + ⅇ^{\frac{V + 60}{- 40}}} otherwise \end{matrix}

i_Ca_P = \frac{i_Ca_P_max Ca_i}{Ca_i + 0.0004}

i_NaCa = \frac{K_NaCa ({Na_i}^{3} Ca_c ⅇ^{0.03743 V gamma} - ({Na_c}^{3} Ca_i ⅇ^{0.03743 V (gamma - 1)}))}{1 + d_NaCa (Ca_i {Na_c}^{3} + Ca_c {Na_i}^{3})}

i_K_ACh = a I_K_ACh \frac{d}{d time} (a) = beta_a (1 - a) - (alpha_a a) beta_a = \frac{12.32}{1 + \frac{0.0042}{ACh}} alpha_a = 17 ⅇ^{0.0133 (V + 40)} I_K_ACh = g_K_ACh (V - E_K) g_K_ACh = P_M2_KACh g_K_ACh_base P_M2_KACh = \{\begin{matrix} \frac{1.026}{1 + ⅇ^{\frac{f_Vagal + 11.05}{- 7.5095}}} - 0.99 if f_Vagal < 100 \land f_Vagal > 25 \\ 0.0006 otherwise \end{matrix}

V_i = 0.465 Vol phi_C = 129000 Ca_i (1 - Ca_Calmod) - (307 Ca_Calmod) \frac{d}{d time} (Ca_Calmod) = phi_C phi_TC = 50500 Ca_i (1 - Ca_Trop) - (252 Ca_Trop) \frac{d}{d time} (Ca_Trop) = phi_TC phi_TMgC = 129000 Ca_i (1 - (Ca_Mg_Trop + Mg_Mg_Trop)) - (4.25 Ca_Mg_Trop) \frac{d}{d time} (Ca_Mg_Trop) = phi_TMgC phi_TMgM = 1290 Mg_i (1 - (Ca_Mg_Trop + Mg_Mg_Trop)) - (429 Mg_Mg_Trop) \frac{d}{d time} (Mg_Mg_Trop) = phi_TMgM F_C = 0.09 phi_C F_TC = 0.031 phi_TC F_TMgC = 0.062 phi_TMgC phi_B = F_C + F_TC + F_TMgC \frac{d}{d time} (Na_i) = \frac{- (3 i_NaK + 3 i_NaCa + i_B_Na + i_f_Na + i_Na)}{F V_i} \frac{d}{d time} (K_i) = \frac{2 i_NaK - (i_K + i_f_K + i_B_K)}{F V_i} \frac{d}{d time} (Ca_i) = \frac{2 i_NaCa + i_rel - (i_Ca_L + i_Ca_T + i_Ca_P + i_B_Ca + i_up)}{2 V_i F} - phi_B

V_c = 0.136 Vol \frac{d}{d time} (Na_c) = \frac{Na_b - Na_c}{tau_p} + \frac{i_Na + 3 i_NaCa + 3 i_NaK + i_B_Na + i_f_Na}{F V_c} \frac{d}{d time} (K_c) = \frac{K_b - K_c}{tau_p} + \frac{(- 2) i_NaK + i_K + i_B_K + i_f_K}{F V_c} \frac{d}{d time} (Ca_c) = \frac{Ca_b - Ca_c}{tau_p} + \frac{(- 2) i_NaCa + i_Ca_L + i_Ca_T + i_Ca_P + i_B_Ca}{2 F V_c}

V_up = 0.01166 V_i V_rel = 0.001296 V_i K1 = \frac{k_cyca k_xcs}{k_SRCa} K2 = Ca_i + Ca_up K1 + k_cyca k_xcs + k_cyca r_act = 240 ⅇ^{(V - 40) 0.08} + 240 {(\frac{Ca_i}{Ca_i + k_rel})}^{4} r_inact = 40 + 240 {(\frac{Ca_i}{Ca_i + k_rel})}^{4} phi_Calse = 770 Ca_rel (1 - Ca_Calse) - (641 Ca_Calse) \frac{d}{d time} (Ca_Calse) = phi_Calse \frac{d}{d time} (F1) = 0.96 F3 - (r_act F1) \frac{d}{d time} (F2) = r_act F1 - (r_inact F2) \frac{d}{d time} (F3) = r_inact F2 - (0.96 F3) i_rel = alpha_rel {(\frac{F2}{F2 + 0.25})}^{2} Ca_rel i_tr = \frac{(Ca_up - Ca_rel) 2 F V_up}{0.06418} i_up = \frac{alpha_up Ca_i - (beta_up Ca_up K1)}{K2} \frac{d}{d time} (Ca_up) = \frac{i_up - i_tr}{2 V_up F} \frac{d}{d time} (Ca_rel) = \frac{i_tr - i_rel}{2 V_rel F} - (11.48 phi_Calse)

E_Na = \frac{R T}{F} \ln (\frac{Na_c}{Na_i}) E_K = \frac{R T}{F} \ln (\frac{K_c}{K_i}) E_Ca = \frac{0.5 R T}{F} \ln (\frac{Ca_c}{Ca_i})

\frac{d}{d time} (cAMP) = 1 (K_ADC (1 + \frac{Iso}{Iso + Km_Iso} - (\frac{P_M2_ADC ACh}{P_M2_ADC ACh + Km_ACh})) - (\frac{V_PDE cGMP cAMP}{cAMP + K_PDE cGMP}))