Model Mathematics

\frac{d}{d time} (V) = \frac{-1.0}{C} (i_Na + i_Ca_L + i_Ca_T + i_to + i_sus + i_K_r + i_K_s + i_f + i_b_Na + i_b_Ca + i_b_K + i_NaCa + i_p + i_st + i_Ca_P)

i_Na = g_Na m^{3.0} h Na_o \frac{F^{2.0}}{R T} \frac{ⅇ^{\frac{(V - E_Na) F}{R T}} - 1.0}{ⅇ^{\frac{V F}{R T}} - 1.0} V

\frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m} m_infinity = {(\frac{1.0}{1.0 + ⅇ^{\frac{- V}{5.46}}})}^{\frac{1.0}{3.0}} tau_m = \frac{0.6247 e -3}{0.832 ⅇ^{(-0.335) (V + 56.7)} + 0.627 ⅇ^{0.082 (V + 65.01)}} + 0.00004

h = (1.0 - F_Na) h1 + F_Na h2 \frac{d}{d time} (h1) = \frac{h1_infinity - h1}{tau_h1} \frac{d}{d time} (h2) = \frac{h2_infinity - h2}{tau_h2} h1_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 66.1}{6.4}}} h2_infinity = h1_infinity tau_h1 = \frac{0.000003171 ⅇ^{(-0.2815) (V + 17.11)}}{1.0 + 0.003732 ⅇ^{(-0.3426) (V + 37.76)}} + 0.0005977 tau_h2 = \frac{0.00000003186 ⅇ^{(-0.6219) (V + 18.8)}}{1.0 + 0.00007189 ⅇ^{(-0.6683) (V + 34.07)}} + 0.003556

i_Ca_L = g_Ca_L (f_L d_L + \frac{0.006}{1.0 + ⅇ^{\frac{- (V + 14.1)}{6.0}}}) (V - E_Ca_L)

\frac{d}{d time} (d_L) = \frac{d_L_infinity - d_L}{tau_d_L} alpha_d_L = (-14.19) \frac{V + 35.0}{ⅇ^{\frac{- (V + 35.0)}{2.5}} - 1.0} - (\frac{42.45 V}{ⅇ^{(-0.208) V} - 1.0}) beta_d_L = \frac{5.71 (V - 5.0)}{ⅇ^{0.4 (V - 5.0)} - 1.0} tau_d_L = \frac{1.0}{alpha_d_L + beta_d_L} d_L_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 23.1)}{6.0}}}

\frac{d}{d time} (f_L) = \frac{f_L_infinity - f_L}{tau_f_L} alpha_f_L = \frac{3.12 (V + 28.0)}{ⅇ^{\frac{V + 28.0}{4.0}} - 1.0} beta_f_L = \frac{25.0}{1.0 + ⅇ^{\frac{- (V + 28.0)}{4.0}}} tau_f_L = \frac{1.0}{alpha_f_L + beta_f_L} f_L_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 45.0}{5.0}}}

i_Ca_T = g_Ca_T d_T f_T (V - E_Ca_T)

\frac{d}{d time} (d_T) = \frac{d_T_infinity - d_T}{tau_d_T} alpha_d_T = 1068.0 ⅇ^{\frac{V + 26.3}{30.0}} beta_d_T = 1068.0 ⅇ^{\frac{- (V + 26.3)}{30.0}} tau_d_T = \frac{1.0}{alpha_d_T + beta_d_T} d_T_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 37.0)}{6.8}}}

\frac{d}{d time} (f_T) = \frac{f_T_infinity - f_T}{tau_f_T} alpha_f_T = 15.3 ⅇ^{\frac{- (V + 71.7)}{83.3}} beta_f_T = 15.0 ⅇ^{\frac{V + 71.7}{15.38}} tau_f_T = \frac{1.0}{alpha_f_T + beta_f_T} f_T_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 71.0}{9.0}}}

i_to = g_to q r (V - E_K) i_sus = g_sus r (V - E_K)

\frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q} q_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 59.37}{13.1}}} tau_q = 10.1 e -3 + \frac{0.006517}{0.57 ⅇ^{(-0.08) (V + 49.0)}} + 0.000024 ⅇ^{0.1 (V + 50.93)}

\frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r} r_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V - 10.93)}{19.7}}} tau_r = 0.00298 + \frac{0.001559}{1.037 ⅇ^{0.09 (V + 30.61)} + 0.369 ⅇ^{(-0.12) (V + 23.84)}}

i_K_r = g_K_r P_a P_i (V - E_K) P_a = (1.0 - F_K_r) P_af + F_K_r P_as

\frac{d}{d time} (P_af) = \frac{P_af_infinity - P_af}{tau_P_af} P_af_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 14.2)}{10.6}}} tau_P_af = \frac{1.0}{37.2 ⅇ^{\frac{V - 9.0}{15.9}} + 0.96 ⅇ^{\frac{- (V - 9.0)}{22.5}}}

\frac{d}{d time} (P_as) = \frac{P_as_infinity - P_as}{tau_P_as} P_as_infinity = P_af_infinity tau_P_as = \frac{1.0}{4.2 ⅇ^{\frac{V - 9.0}{17.0}} + 0.15 ⅇ^{\frac{- (V - 9.0)}{21.6}}}

\frac{d}{d time} (P_i) = \frac{P_i_infinity - P_i}{tau_P_i} P_i_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 18.6}{10.1}}}

i_K_s = g_K_s {xs}^{2.0} (V - E_K_s)

\frac{d}{d time} (xs) = \frac{xs_infinity - xs}{tau_xs} alpha_xs = \frac{14.0}{1.0 + ⅇ^{\frac{- (V - 40.0)}{9.0}}} beta_xs = ⅇ^{\frac{- V}{45.0}} xs_infinity = \frac{alpha_xs}{alpha_xs + beta_xs} tau_xs = \frac{1.0}{alpha_xs + beta_xs}

i_f = i_f_Na + i_f_K i_f_Na = g_f_Na y (V - E_Na) i_f_K = g_f_K y (V - E_K)

\frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y} alpha_y = ⅇ^{\frac{- (V + 78.91)}{26.62}} beta_y = ⅇ^{\frac{V + 75.13}{21.25}} y_infinity = \frac{alpha_y}{alpha_y + beta_y} tau_y = \frac{1.0}{alpha_y + beta_y}

i_b_Na = g_b_Na (V - E_Na)

i_b_K = g_b_K (V - E_K)

i_b_Ca = g_b_Ca (V - E_Ca)

i_NaCa = K_NaCa \frac{{Na_i}^{3.0} Ca_o ⅇ^{0.03743 V gamma_NaCa} - ({Na_o}^{3.0} Ca_i ⅇ^{0.0374 V (gamma_NaCa - 1.0)})}{1.0 + d_NaCa (Ca_i {Na_o}^{3.0} + Ca_o {Na_i}^{3.0})}

i_p = i_p_max {(\frac{Na_i}{K_m_Na + Na_i})}^{3.0} {(\frac{K_o}{K_m_K + K_o})}^{2.0} \frac{1.6}{1.5 + ⅇ^{\frac{- (V + 60.0)}{40.0}}}

i_st = g_st d_s f_s (V - 18.0)

\frac{d}{d time} (d_s) = \frac{d_s_infinity - d_s}{tau_d_s} alpha_d_s = \frac{1000.0}{0.15 ⅇ^{\frac{- V}{11.0}} + 0.2 ⅇ^{\frac{- V}{700.0}}} beta_d_s = \frac{1000.0}{16.0 ⅇ^{\frac{V}{8.0}} + 0.2 ⅇ^{\frac{V}{50.0}}} tau_d_s = \frac{1.0}{alpha_d_s + beta_d_s} d_s_infinity = \frac{alpha_d_s}{alpha_d_s + beta_d_s}

\frac{d}{d time} (f_s) = \frac{f_s_infinity - f_s}{tau_f_s} alpha_f_s = \frac{1000.0}{3100.0 ⅇ^{\frac{- V}{13.0}} + 700.0 ⅇ^{\frac{- V}{70.0}}} beta_f_s = \frac{1000.0}{16.0 ⅇ^{\frac{V}{8.0}} + 0.2 ⅇ^{\frac{V}{50.0}}} tau_f_s = \frac{1.0}{alpha_f_s + beta_f_s} f_s_infinity = \frac{alpha_f_s}{alpha_f_s + beta_f_s}

U_d = 1.0 - (\frac{B_d}{K_m_b + Ca_d + B_d}) J_Ca_ds = alpha_ds Vol_d (Ca_d - Ca_s) i_Ca_P = i_Ca_P_max \frac{Ca_s}{Ca_s + 0.0004} J_Ca_r = alpha_r f_R \frac{{Ca_d}^{2.0}}{{K_m_r}^{2.0} + {Ca_d}^{2.0}} Vol_r Ca_r \frac{d}{d time} (f_R) = (- alpha_fR) Ca_d f_R + beta_fR (1.0 - f_R) U_s = 1.0 - (\frac{B_s}{K_m_b + Ca_s + B_s}) J_Ca_P = J_Ca_P_max \frac{Ca_s}{Ca_s + 0.0004} J_Ca_u = J_Ca_u_max \frac{{Ca_s}^{2.0}}{{K_m_u}^{2.0} + {Ca_s}^{2.0}} J_Ca_ur = alpha_ur Vol_u (Ca_u - Ca_r) J_Ca_1 = alpha_1 Vol_u Ca_u i_Ca = i_Ca_L + i_Ca_T \frac{d}{d time} (Ca_d) = \frac{U_d}{Vol_d} (J_Ca_ds - (\frac{0.95 i_Ca}{2.0 F})) \frac{d}{d time} (Ca_s) = \frac{U_s}{Vol_s} ((J_Ca_ds - (\frac{(0.05 i_Ca - (2.0 i_NaCa)) + i_b_Ca}{2.0 F} + J_Ca_u)) + J_Ca_r + J_Ca_1) \frac{d}{d time} (Ca_u) = \frac{J_Ca_u - (J_Ca_1 + J_Ca_ur)}{Vol_u} \frac{d}{d time} (Ca_r) = \frac{J_Ca_ur - J_Ca_r}{Vol_r} Vol_u = f_u Vol_c Vol_r = f_r Vol_c Vol_d = f_d Vol_c Vol_s = Vol_c - (Vol_u + Vol_d)

E_Na = \frac{R T}{z F} \ln (\frac{Na_o}{Na_i}) E_K = \frac{R T}{z F} \ln (\frac{K_o}{K_i}) E_Ca = \frac{R T}{z F} \ln (\frac{Ca_o}{Ca_i}) E_K_s = \frac{R T}{F} \ln (\frac{K_o + 0.12 Na_o}{K_i + 0.12 Na_i})