Model Mathematics

\frac{d}{d time} (V) = \frac{(1e-3) (I_app + (- I_leak) - I_Na - I_KD - I_KM - I_CaL - I_h)}{C_m}

tau = time - (period ⌊ \frac{time}{period} ⌋) I_app = \{\begin{matrix} i_stimAmplitude if tau \geq i_stimStart \land tau ≦ i_stimEnd \\ 0 otherwise \end{matrix}

I_leak = 1000 g_leak (V - E_leak)

I_Na = 1000 g_Na m^{3} h (V - E_Na)

alpha = \{\begin{matrix} 0.32 4 (1 - (\frac{13 + V_T - V}{2 4})) if | \frac{13 + V_T - V}{4} | < 1e-6 \\ \frac{0.32 (13 + V_T - V)}{ⅇ^{\frac{13 + V_T - V}{4}} - 1} otherwise \end{matrix} beta = \{\begin{matrix} (- 0.28) 5 (1 - (\frac{- (V - V_T - 40)}{2 5})) if | \frac{- (V - V_T - 40)}{5} | < 1e-6 \\ \frac{(- 0.28) (V - V_T - 40)}{ⅇ^{\frac{- (V - V_T - 40)}{5}} - 1} otherwise \end{matrix} tau_m = \frac{1}{alpha + beta} m_inf = \frac{alpha}{alpha + beta} \frac{d}{d time} (m) = \frac{m_inf - m}{tau_m}

alpha_h = 0.128 ⅇ^{\frac{17 + V_T + V_S - V}{18}} beta_h = \frac{4}{1 + ⅇ^{\frac{40 + V_S + V_T - V}{5}}} tau_h = \frac{1}{alpha_h + beta_h} h_inf = \frac{alpha_h}{alpha_h + beta_h} \frac{d}{d time} (h) = \frac{h_inf - h}{tau_h}

I_KD = 1000 g_KD n^{4} (V - E_K)

alpha_n = \{\begin{matrix} (- 0.032) 5 (1 - (\frac{V - V_T - 15}{2 5})) if | \frac{V - V_T - 15}{5} | < 1e-6 \\ \frac{(- 0.032) (V - V_T - 15)}{ⅇ^{\frac{V - V_T - 15}{5}} - 1} otherwise \end{matrix} beta_n = \{\begin{matrix} 0.5 40 (1 + \frac{V - V_T - 10}{2 40}) if | \frac{- (V - V_T - 10)}{40} | < 1e-6 \\ \frac{0.5 (- (V - V_T - 10))}{ⅇ^{\frac{- (V - V_T - 10)}{40}} - 1} otherwise \end{matrix} tau_n = \frac{1}{alpha_n + beta_n} n_inf = \frac{alpha_n}{alpha_n + beta_n} \frac{d}{d time} (n) = \frac{n_inf - n}{tau_n}

I_KM = 1000 g_KM p (V - E_K)

p_inf = \{\begin{matrix} \frac{1}{1 + ⅇ^{\frac{- (V + 35)}{10}}} if \frac{- (V + 35)}{10} < 25 \land \frac{- (V + 35)}{10} > - 25 \\ 1 otherwise \end{matrix} tau_p = \{\begin{matrix} \frac{tau_max}{3.3 ⅇ^{\frac{V + 35}{20}} + ⅇ^{\frac{- (V + 35)}{20}}} if \frac{V + 35}{20} < 25 \land \frac{V + 35}{20} > - 25 \\ 1 otherwise \end{matrix} \frac{d}{d time} (p) = \frac{p_inf - p}{tau_p}

I_CaL = 1000 P_Ca q^{2} G

alpha_q = \frac{6.32}{1 + ⅇ^{\frac{- (V - 5)}{13.89}}} beta_q = \{\begin{matrix} 0.02 (5.36 + \frac{1.31 - V}{2}) if | \frac{1.31 - V}{5.36} | < 1e-6 \\ \frac{0.02 (1.31 - V)}{1 - (ⅇ^{\frac{V - 1.31}{5.36}})} otherwise \end{matrix} tau_q = \frac{1}{alpha_q + beta_q} q_inf = \frac{1}{1 + ⅇ^{\frac{V + 10}{- 10}}} \frac{d}{d time} (q) = \frac{q_inf - q}{tau_q}

G = \frac{\frac{Z^{2} F^{2} (1e-3) V}{R T} (1e-6) (Ca_i - (Ca_o ⅇ^{\frac{Z F (1e-3) V}{R T}}))}{1 - (ⅇ^{\frac{(1e-3) Z F V}{R T}})}

drive_channel = \frac{k I_CaL}{2 F d} \frac{d}{d time} (Ca_i) = \{\begin{matrix} \frac{Ca_inf - Ca_i}{tau_r} if drive_channel ≦ 0 \\ drive_channel + \frac{Ca_inf - Ca_i}{tau_r} otherwise \end{matrix}

m = o_1 + g_inc + o_2 I_h = 1000 g_hbar m (V - E_h)

p_0 = \frac{p_1 k_2}{k_1Ca} p_1 = 1 - p_0 c_1 = \frac{beta}{alpha} o_1 o_1 = \frac{k_4}{k_3p} o_2 o_2 = 1 - c_1 - o_1

h_inf = \frac{1}{1 + ⅇ^{\frac{V + 75 - V_S}{5.5}}} tau_s = tau_m + \frac{1000}{ⅇ^{\frac{V + 71.55 - V_S}{14.2}} + ⅇ^{\frac{- (V + 89 - V_S)}{11.6}}} alpha = \frac{h_inf}{tau_s} beta = \frac{1 - h_inf}{tau_s} k_1Ca = k_2 {(\frac{Ca_i}{Ca_c})}^{n_Ca} k_3p = k_4 {(\frac{p_1}{p_C})}^{n_exp}