Model Mathematics

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Component: environment

Component: membrane

FCell = \frac{1.07 (3 dCell - 0.1)}{3 (1 + 0.7745 ⅇ^{\frac{- (3 dCell - 2.05)}{0.295}})} Cm = CmCentre + FCell (CmPeriphery - CmCentre) \frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Na + i_Ca_L + i_Ca_T + i_to + i_sus + i_K_r + i_K_s + i_f_Na + i_f_K + i_b_Na + i_b_Ca + i_b_K + i_NaCa + i_p + i_Ca_P)

Component: sodium_current

g_Na = g_Na_Centre + FCell (g_Na_Periphery - g_Na_Centre) i_Na = \frac{\frac{g_Na m^{3} h Na_o F^{2}}{R T} (ⅇ^{\frac{(V - E_Na) F}{R T}} - 1)}{ⅇ^{\frac{V F}{R T}} - 1} V

Component: sodium_current_m_gate

\frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m} m_infinity = {(\frac{1}{1 + ⅇ^{\frac{- V}{5.46}}})}^{\frac{1}{3}} tau_m = \frac{0.0006247}{0.832 ⅇ^{(- 0.335) (V + 56.7)} + 0.627 ⅇ^{0.082 (V + 65.01)}} + 4 e -5

Component: sodium_current_h_gate

F_Na = \frac{0.0952 ⅇ^{(- 0.063) (V + 34.4)}}{1 + 1.66 ⅇ^{(- 0.225) (V + 63.7)}} + 0.0869 h = (1 - 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}{1 + ⅇ^{\frac{V + 66.1}{6.4}}} h2_infinity = h1_infinity tau_h1 = \frac{3.717 e -6 ⅇ^{(- 0.2815) (V + 17.11)}}{1 + 0.003732 ⅇ^{(- 0.3426) (V + 37.76)}} + 0.0005977 tau_h2 = \frac{3.186 e -8 ⅇ^{(- 0.6219) (V + 18.8)}}{1 + 7.189 e -5 ⅇ^{(- 0.6683) (V + 34.07)}} + 0.003556

Component: L_type_Ca_channel

g_Ca_L = g_Ca_L_Centre + FCell (g_Ca_L_Periphery - g_Ca_L_Centre) i_Ca_L = g_Ca_L (f_L d_L + \frac{0.006}{1 + ⅇ^{\frac{- (V + 14.1)}{6}}}) (V - E_Ca_L)

Component: L_type_Ca_channel_d_gate

\frac{d}{d time} (d_L) = \frac{d_L_infinity - d_L}{tau_d_L} alpha_d_L = \frac{(- 28.38) (V + 35)}{ⅇ^{\frac{- (V + 35)}{2.5}} - 1} - (\frac{84.9 V}{ⅇ^{(- 0.208) V} - 1}) beta_d_L = \frac{11.42 (V - 5)}{ⅇ^{0.4 (V - 5)} - 1} tau_d_L = \frac{2}{alpha_d_L + beta_d_L} d_L_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 23.1)}{6}}}

Component: L_type_Ca_channel_f_gate

\frac{d}{d time} (f_L) = \frac{f_L_infinity - f_L}{tau_f_L} alpha_f_L = \frac{3.12 (V + 28)}{ⅇ^{\frac{V + 28}{4}} - 1} beta_f_L = \frac{25}{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 + 45}{5}}}

Component: T_type_Ca_channel

g_Ca_T = g_Ca_T_Centre + FCell (g_Ca_T_Periphery - g_Ca_T_Centre) i_Ca_T = g_Ca_T d_T f_T (V - E_Ca_T)

Component: T_type_Ca_channel_d_gate

\frac{d}{d time} (d_T) = \frac{d_T_infinity - d_T}{tau_d_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 + 37)}{6.8}}}

Component: T_type_Ca_channel_f_gate

\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 ⅇ^{\frac{V + 71.7}{15.38}} tau_f_T = \frac{1}{alpha_f_T + beta_f_T} f_T_infinity = \frac{1}{1 + ⅇ^{\frac{V + 71}{9}}}

Component: four_AP_sensitive_currents

g_to = g_to_Centre + FCell (g_to_Periphery - g_to_Centre) g_sus = g_sus_Centre + FCell (g_sus_Periphery - g_sus_Centre) i_to = g_to q r (V - E_K) i_sus = g_sus r (V - E_K)

Component: four_AP_sensitive_currents_q_gate

\frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q} q_infinity = \frac{1}{1 + ⅇ^{\frac{V + 59.37}{13.1}}} tau_q = 0.0101 + \frac{0.06517}{0.57 ⅇ^{(- 0.08) (V + 49)}} + 2.4 e -5 ⅇ^{0.1 (V + 50.93)}

Component: four_AP_sensitive_currents_r_gate

\frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r} r_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 10.93)}{19.7}}} tau_r = 0.001 (2.98 + \frac{15.59}{1.037 ⅇ^{0.09 (V + 30.61)} + 0.369 ⅇ^{(- 0.12) (V + 23.84)}})

Component: rapid_delayed_rectifying_potassium_current

g_K_r = g_K_r_Centre + FCell (g_K_r_Periphery - g_K_r_Centre) i_K_r = g_K_r P_a P_i (V - E_K) P_a = 0.6 P_af + 0.4 P_as

Component: rapid_delayed_rectifying_potassium_current_P_af_gate

\frac{d}{d time} (P_af) = \frac{P_af_infinity - P_af}{tau_P_af} P_af_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 14.2)}{10.6}}} tau_P_af = \frac{1}{37.2 ⅇ^{\frac{V - 9}{15.9}} + 0.96 ⅇ^{\frac{- (V - 9)}{22.5}}}

Component: rapid_delayed_rectifying_potassium_current_P_as_gate

\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}{4.2 ⅇ^{\frac{V - 9}{17}} + 0.15 ⅇ^{\frac{- (V - 9)}{21.6}}}

Component: rapid_delayed_rectifying_potassium_current_P_i_gate

\frac{d}{d time} (P_i) = \frac{P_i_infinity - P_i}{tau_P_i} P_i_infinity = \frac{1}{1 + ⅇ^{\frac{V + 18.6}{10.1}}}

Component: slow_delayed_rectifying_potassium_current

g_K_s = g_K_s_Centre + FCell (g_K_s_Periphery - g_K_s_Centre) i_K_s = g_K_s {xs}^{2} (V - E_K_s)

Component: slow_delayed_rectifying_potassium_current_xs_gate

\frac{d}{d time} (xs) = alpha_xs (1 - xs) - (beta_xs xs) alpha_xs = \frac{14}{1 + ⅇ^{\frac{- (V - 40)}{9}}} beta_xs = 1 ⅇ^{\frac{- V}{45}}

Component: hyperpolarisation_activated_current

g_f_Na = g_f_Na_Centre + FCell (g_f_Na_Periphery - g_f_Na_Centre) i_f_Na = g_f_Na y (V - E_Na) g_f_K = g_f_K_Centre + FCell (g_f_K_Periphery - g_f_K_Centre) i_f_K = g_f_K y (V - E_K)

Component: hyperpolarisation_activated_current_y_gate

\frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y) alpha_y = 1 ⅇ^{\frac{- (V + 78.91)}{26.62}} beta_y = 1 ⅇ^{\frac{V + 75.13}{21.25}}

Component: sodium_background_current

g_b_Na = g_b_Na_Centre + FCell (g_b_Na_Periphery - g_b_Na_Centre) i_b_Na = g_b_Na (V - E_Na)

Component: potassium_background_current

g_b_K = g_b_K_Centre + FCell (g_b_K_Periphery - g_b_K_Centre) i_b_K = g_b_K (V - E_K)

Component: calcium_background_current

g_b_Ca = g_b_Ca_Centre + FCell (g_b_Ca_Periphery - g_b_Ca_Centre) i_b_Ca = g_b_Ca (V - E_Ca)

Component: sodium_calcium_exchanger

k_NaCa = k_NaCa_Centre + FCell (k_NaCa_Periphery - k_NaCa_Centre) i_NaCa = \frac{k_NaCa ({Na_i}^{3} Ca_o ⅇ^{0.03743 V gamma_NaCa} - ({Na_o}^{3} Ca_i ⅇ^{0.0374 V (gamma_NaCa - 1)}))}{1 + d_NaCa (Ca_i {Na_o}^{3} + Ca_o {Na_i}^{3})}

Component: sodium_potassium_pump

i_p_max = i_p_max_Centre + FCell (i_p_max_Periphery - i_p_max_Centre) i_p = \frac{i_p_max {(\frac{Na_i}{K_m_Na + Na_i})}^{3} {(\frac{K_o}{K_m_K + K_o})}^{2} 1.6}{1.5 + ⅇ^{\frac{- (V + 60)}{40}}}

Component: ionic_concentrations

Component: reversal_and_equilibrium_potentials

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_Ca = \frac{R T}{2 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})

Component: sustained_inward_current

i_st = g_st d_s f_s (V - 18.0)

Component: sustained_inward_current_d_gate

\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}

Component: sustained_inward_current_f_gate

\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}

Component: intracellular_calcium_handling

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)