Model Mathematics

\frac{d}{d time} (V) = \frac{(- (i_Na + i_Ca + i_to + i_K + i_K1 + i_b_Na + i_b_Ca + i_NaCa + i_NaK)) + {(rho_x S)}^{-1.0} V_x2 + {(rho_y S)}^{-1.0} V_y2}{Cm} \frac{\partial^{2.0}}{\partial x^{2.0}} (V) = V_x2 \frac{\partial^{2.0}}{\partial y^{2.0}} (V) = V_y2

i_Na = g_Na m^{3.0} v^{2.0} (V - E_Na) E_Na = \frac{R T}{F} \ln (\frac{Na_e}{Na_i})

\frac{d}{d time} (m) = alpha_m (1.0 - m) - (beta_m m) alpha_m = \frac{0.32 (V + 47.13)}{1.0 - (ⅇ^{(-0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11.0}}

\frac{d}{d time} (v) = \frac{v_infinity - v}{tau_v} v_infinity = 0.5 (1.0 - (\tanh (7.74 + V 0.12))) tau_v = 0.25 + 2.24 \frac{1.0 - (\tanh (7.74 + V 0.12))}{1.0 - (\tanh (0.07 (V + 92.4)))}

i_Ca = g_Ca d_infinity f f_Ca (V - E_Ca) E_Ca = \frac{R T}{2.0 F} \ln (\frac{Ca_e}{Ca_i})

d_infinity = \frac{alpha_d}{alpha_d + beta_d} alpha_d = \frac{14.98 ⅇ^{(- 0.5) {(\frac{V - 22.36}{16.68})}^{2.0}}}{16.68 \sqrt{2.0 π}} beta_d = 0.1471 - (\frac{5.3 ⅇ^{(- 0.5) {(\frac{V - 6.27}{14.93})}^{2.0}}}{14.93 \sqrt{2.0 π}})

\frac{d}{d time} (f) = alpha_f (1.0 - f) - (beta_f f) alpha_f = \frac{6.87 e -3}{1.0 + ⅇ^{\frac{- (6.1546 - V)}{6.12}}} beta_f = \frac{0.069 ⅇ^{(-11.0) (V + 9.825)} + 0.011}{1.0 + ⅇ^{(-0.278) (V + 9.825)}} + 5.75 e -4

f_Ca = \frac{1.0}{1.0 + \frac{Ca_i}{0.0006}}

i_to = g_to r_infinity to (V - E_to) E_to = \frac{R T}{F} \ln (\frac{0.043 Na_e + K_e}{0.043 Na_i + K_i})

r_infinity = \frac{alpha_r}{alpha_r + beta_r} alpha_r = \frac{0.5266 ⅇ^{(- 0.0166) (V - 42.2912)}}{1.0 + ⅇ^{(- 0.0943) (V - 42.2912)}} beta_r = \frac{5.186 e -5 V + 0.5149 ⅇ^{(- 0.1344) (V - 5.0027)}}{1.0 + ⅇ^{(- 0.1348) (V - 5.186 e -5)}}

\frac{d}{d time} (to) = alpha_to (1.0 - to) - (beta_to to) alpha_to = \frac{5.612 e -5 V + 0.0721 ⅇ^{(-0.173) (V + 34.2531)}}{1.0 + ⅇ^{(-0.1732) (V + 34.2531)}} beta_to = \frac{1.215 e -4 V + 0.0767 ⅇ^{1.66 e -9 (V + 34.0235)}}{1.0 + ⅇ^{(- 0.1604) (V + 34.0235)}} tau_to = \frac{1.0}{p alpha_to V + p beta_to V} to_infinity = \frac{alpha_to (V - V_shift)}{alpha_to (V - V_shift) + beta_to (V - V_shift)}

i_K = g_K X^{2.0} (V - E_K) E_K = \frac{R T}{F} \ln (\frac{K_e}{K_i})

\frac{d}{d time} (X) = \frac{X_infinity - X}{tau_X} X_infinity = \frac{0.988}{1.0 + ⅇ^{-0.861 - (V 0.0620)}} tau_X = 240.0 ⅇ^{\frac{- ({(25.5 + V)}^{2.0})}{156.0}} + 182.0 (1.0 + \tanh (0.154 + 0.0116 V)) + tau_X_a tau_X_a = 40.0 (1.0 - (\tanh (160.0 + 2.0 V)))

i_K1 = g_K1 K1_infinity (V - E_K)

K1_infinity = \frac{alpha_K1}{alpha_K1 + beta_K1} alpha_K1 = \frac{0.1}{1.0 + ⅇ^{0.06 (V - (E_K + 200.0))}} beta_K1 = \frac{3.0 ⅇ^{2.0 e -4 (V + 100.0 + (- E_K))} + ⅇ^{0.1 (V - (10.0 + E_K))}}{1.0 + ⅇ^{(- 0.5) (V - E_K)}}

i_b_Ca = g_b_Ca (V - E_Ca)

i_b_Na = g_b_Na (V - E_Na)

i_NaK = g_NaK f_NaK f_NaK_a f_NaK = \frac{1.0}{1.0 + 0.1245 ⅇ^{(-0.0037) V} + 0.0365 sigma ⅇ^{(-0.037) V}} f_NaK_a = \frac{1.0}{1.0 + {(\frac{10.0}{Na_i})}^{1.5}} \frac{K_e}{K_e + 1.5} sigma = 0.1428 (ⅇ^{\frac{Na_e}{67.3}} - 1.0)

i_NaCa = g_NaCa f_NaCa f_NaCa = {({87.5}^{3.0} + {Na_e}^{3.0})}^{-1.0} {(1.38 + Ca_e)}^{-1.0} {(1.0 + 0.1 ⅇ^{(-0.024) V})}^{-1.0} ({Na_i}^{3.0} Ca_e ⅇ^{0.013 V} - ({Na_e}^{3.0} Ca_i ⅇ^{(-0.024) V}))