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}{C} (I_stim + i_Na + i_si + i_K + i_K1 + i_Kp + i_b)

E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai}) i_Na = g_Na m^{3} h j (V - E_Na)

alpha_m = \frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{80 + V}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

alpha_j = \{\begin{matrix} \frac{((- 127140) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.3 ⅇ^{(- 0.0000002535) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} \frac{d}{d time} (j) = alpha_j (1 - j) - (beta_j j)

E_si = 7.7 - (13.0287 \ln (\frac{Cai}{1})) i_si = 0.09 d f (V - E_si)

alpha_d = \frac{0.095 ⅇ^{(- 0.01) (V - 5)}}{1 + ⅇ^{(- 0.072) (V - 5)}} beta_d = \frac{0.07 ⅇ^{(- 0.017) (V + 44)}}{1 + ⅇ^{0.05 (V + 44)}} \frac{d}{d time} (d) = alpha_d (1 - d) - (beta_d d)

alpha_f = \frac{0.012 ⅇ^{(- 0.008) (V + 28)}}{1 + ⅇ^{0.15 (V + 28)}} beta_f = \frac{0.0065 ⅇ^{(- 0.02) (V + 30)}}{1 + ⅇ^{(- 0.2) (V + 30)}} \frac{d}{d time} (f) = alpha_f (1 - f) - (beta_f f)

g_K = 0.282 \sqrt{\frac{Ko}{5.4}} E_K = \frac{R T}{F} \ln (\frac{Ko + PR_NaK Nao}{Ki + PR_NaK Nai}) i_K = g_K X Xi (V - E_K)

alpha_X = \frac{0.0005 ⅇ^{0.083 (V + 50)}}{1 + ⅇ^{0.057 (V + 50)}} beta_X = \frac{0.0013 ⅇ^{(- 0.06) (V + 20)}}{1 + ⅇ^{(- 0.04) (V + 20)}} \frac{d}{d time} (X) = alpha_X (1 - X) - (beta_X X)

Xi = \{\begin{matrix} \frac{2.837 (ⅇ^{0.04 (V + 77)} - 1)}{(V + 77) ⅇ^{0.04 (V + 35)}} if V > - 100 \\ 1 otherwise \end{matrix}

g_K1 = 0.6047 \sqrt{\frac{Ko}{5.4}} E_K1 = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_K1 = g_K1 K1_infinity (V - E_K1)

alpha_K1 = \frac{1.02}{1 + ⅇ^{0.2385 (V - E_K1 - 59.215)}} beta_K1 = \frac{0.49124 ⅇ^{0.08032 (V + 5.476 - E_K1)} + 1 ⅇ^{0.06175 (V - (E_K1 + 594.31))}}{1 + ⅇ^{(- 0.5143) ((V - E_K1) + 4.753)}} K1_infinity = \frac{alpha_K1}{alpha_K1 + beta_K1}

E_Kp = E_K1 Kp = \frac{1}{1 + ⅇ^{\frac{7.488 - V}{5.98}}} i_Kp = g_Kp Kp (V - E_Kp)

i_b = g_b (V - E_b)

\frac{d}{d time} (Cai) = \frac{- 0.0001}{1} i_si + 0.07 (0.0001 - Cai)