Model Mathematics

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Component: environment

Component: membrane

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) = - (i_Na + i_Ca + i_CaK + i_Kr + i_Ks + i_to + i_K1 + i_Kp + i_NaCa + i_NaK + i_p_Ca + i_Na_b + i_Ca_b + i_Stim)

Component: fast_sodium_current

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

Component: fast_sodium_current_m_gate

E0_m = V + 47.13 alpha_m = \frac{0.32 E0_m}{1 - (ⅇ^{(- 0.1) E0_m})} beta_m = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: fast_sodium_current_h_gate

alpha_h = 0.135 ⅇ^{\frac{V + 80 - shift_h}{- 6.8}} beta_h = \frac{7.5}{1 + ⅇ^{(- 0.1) (V + 11 - shift_h)}} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: fast_sodium_current_j_gate

alpha_j = \frac{0.175 ⅇ^{\frac{V + 100 - shift_j}{- 23}}}{1 + ⅇ^{0.15 (V + 79 - shift_j)}} beta_j = \frac{0.3}{1 + ⅇ^{(- 0.1) (V + 32 - shift_j)}} \frac{d}{d time} (j) = alpha_j (1 - j) - (beta_j j)

Component: time_independent_potassium_current

i_K1 = \frac{g_K1 K1_infinity K_o}{K_o + K_mK1} (V - E_K)

Component: time_independent_potassium_current_K1_gate

K1_infinity = \frac{1}{2 + ⅇ^{\frac{1.62 F}{R T} (V - E_K)}}

Component: rapid_activating_delayed_rectifiyer_K_current

E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) R_V = \frac{1}{1 + 2.5 ⅇ^{0.1 (V + 28)}} i_Kr = g_Kr R_V X_kr \sqrt{\frac{K_o}{4}} (V - E_K)

Component: rapid_activating_delayed_rectifiyer_K_current_X_kr_gate

X_kr_inf = \frac{1}{1 + ⅇ^{- 2.182 - (0.1819 V)}} tau_X_kr = 43 + \frac{1}{ⅇ^{(- 5.495) + 0.1691 V} + ⅇ^{- 7.677 - (0.0128 V)}} \frac{d}{d time} (X_kr) = \frac{X_kr_inf - X_kr}{tau_X_kr}

Component: slow_activating_delayed_rectifiyer_K_current

E_Ks = \frac{R T}{F} \ln (\frac{K_o + 0.01833 Na_o}{K_i + 0.01833 Na_i}) i_Ks = g_Ks {X_ks}^{2} (V - E_Ks)

Component: slow_activating_delayed_rectifiyer_K_current_X_ks_gate

X_ks_infinity = \frac{1}{1 + ⅇ^{\frac{V - 16}{- 13.6}}} tau_X_ks = \frac{1}{\frac{0.0000719 (V - 10)}{1 - (ⅇ^{(- 0.148) (V - 10)})} + \frac{0.000131 (V - 10)}{ⅇ^{0.0687 (V - 10)} - 1}} \frac{d}{d time} (X_ks) = \frac{X_ks_infinity - X_ks}{tau_X_ks}

Component: transient_outward_potassium_current

i_to = g_to X_to Y_to (V - E_K)

Component: transient_outward_potassium_current_X_to_gate

alpha_X_to = 0.04516 ⅇ^{0.03577 V} beta_X_to = 0.0989 ⅇ^{(- 0.06237) V} \frac{d}{d time} (X_to) = alpha_X_to (1 - X_to) - (beta_X_to X_to)

Component: transient_outward_potassium_current_Y_to_gate

alpha_Y_to = \frac{0.005415 ⅇ^{\frac{V + 33.5}{- 5}}}{1 + 0.051335 ⅇ^{\frac{V + 33.5}{- 5}}} beta_Y_to = \frac{0.005415 ⅇ^{\frac{V + 33.5}{5}}}{1 + 0.051335 ⅇ^{\frac{V + 33.5}{5}}} \frac{d}{d time} (Y_to) = alpha_Y_to (1 - Y_to) - (beta_Y_to Y_to)

Component: plateau_potassium_current

i_Kp = g_Kp Kp_V (V - E_K)

Component: plateau_potassium_current_Kp_gate

Kp_V = \frac{1}{1 + ⅇ^{\frac{7.488 - V}{5.98}}}

Component: sodium_potassium_pump

f_NaK = \frac{1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- V) F}{R T}}} sigma = \frac{1}{7} (ⅇ^{\frac{Na_o}{67.3}} - 1) i_NaK = \frac{\frac{i_NaK_max f_NaK}{1 + {(\frac{K_mNai}{Na_i})}^{1.5}} K_o}{K_o + K_mKo}

Component: Na_Ca_exchanger

i_NaCa = \frac{K_NaCa}{({K_mNa}^{3} + {Na_o}^{3}) (K_mCa + Ca_o) (1 + K_sat ⅇ^{\frac{(eta - 1) V F}{R T}})} (ⅇ^{\frac{eta V F}{R T}} {Na_i}^{3} Ca_o - (ⅇ^{\frac{(eta - 1) V F}{R T}} {Na_o}^{3} Ca_i))

Component: sarcolemmal_calcium_pump

i_p_Ca = \frac{i_pCa_max Ca_i}{K_mpCa + Ca_i}

Component: calcium_background_current

E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_o}{Ca_i}) i_Ca_b = g_Cab (V - E_Ca)

Component: sodium_background_current

i_Na_b = g_Nab (V - E_Na)

Component: L_type_Ca_current

i_Ca = i_Ca_max f d f_Ca i_Ca_max = \frac{\frac{\frac{P_Ca}{C_sc} 4 V F^{2}}{R T} (Ca_i ⅇ^{\frac{2 V F}{R T}} - (0.341 Ca_o))}{ⅇ^{\frac{2 V F}{R T}} - 1} i_CaK = \frac{\frac{\frac{\frac{P_CaK}{C_sc} f d f_Ca}{1 + \frac{i_Ca_max}{i_Ca_half}} 1000 V F^{2}}{R T} (K_i ⅇ^{\frac{V F}{R T}} - K_o)}{ⅇ^{\frac{V F}{R T}} - 1}

Component: L_type_Ca_current_f_gate

f_infinity = \frac{1}{1 + ⅇ^{\frac{V + 12.5}{5}}} tau_f = 30 + \frac{200}{1 + ⅇ^{\frac{V + 20}{9.5}}} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

Component: L_type_Ca_current_d_gate

d_infinity = \frac{1}{1 + ⅇ^{\frac{V + 10}{- 6.24}}} E0_m = V + 40 tau_d = \frac{1}{\frac{0.25 ⅇ^{(- 0.01) V}}{1 + ⅇ^{(- 0.07) V}} + \frac{0.07 ⅇ^{(- 0.05) E0_m}}{1 + ⅇ^{0.05 E0_m}}} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

Component: L_type_Ca_current_f_Ca_gate

f_Ca_infinity = \frac{1}{1 + {(\frac{Ca_i}{K_mfCa})}^{3}} tau_f_Ca = 30 \frac{d}{d time} (f_Ca) = \frac{f_Ca_infinity - f_Ca}{tau_f_Ca}

Component: calcium_dynamics

J_up = \frac{V_up}{1 + {(\frac{K_mup}{Ca_i})}^{2}} gamma = \frac{1}{1 + {(\frac{2000}{Ca_SR})}^{3}} J_rel = \frac{P_rel f d f_Ca (gamma Ca_SR - Ca_i)}{1 + 1.65 ⅇ^{\frac{V}{20}}} J_leak = P_leak (Ca_SR - Ca_i) beta_SR = \frac{1}{1 + \frac{CSQN_tot K_mCSQN}{{(K_mCSQN + Ca_SR)}^{2}}} \frac{d}{d time} (Ca_SR) = \frac{beta_SR (J_up - J_leak - J_rel) V_myo}{V_SR} beta_i = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Ca_i)}^{2}}} \frac{d}{d time} (Ca_i) = beta_i (J_rel + J_leak - J_up - (\frac{A_Cap C_sc}{2 F V_myo} (i_Ca + i_Ca_b + i_p_Ca - (2 i_NaCa))))

Model Mathematics

Component: environment

Component: membrane

Component: fast_sodium_current

Component: fast_sodium_current_m_gate

Component: fast_sodium_current_h_gate

Component: fast_sodium_current_j_gate

Component: time_independent_potassium_current

Component: time_independent_potassium_current_K1_gate

Component: rapid_activating_delayed_rectifiyer_K_current

Component: rapid_activating_delayed_rectifiyer_K_current_X_kr_gate

Component: slow_activating_delayed_rectifiyer_K_current

Component: slow_activating_delayed_rectifiyer_K_current_X_ks_gate

Component: transient_outward_potassium_current

Component: transient_outward_potassium_current_X_to_gate

Component: transient_outward_potassium_current_Y_to_gate

Component: plateau_potassium_current

Component: plateau_potassium_current_Kp_gate

Component: sodium_potassium_pump

Component: Na_Ca_exchanger

Component: sarcolemmal_calcium_pump

Component: calcium_background_current

Component: sodium_background_current

Component: L_type_Ca_current

Component: L_type_Ca_current_f_gate

Component: L_type_Ca_current_d_gate

Component: L_type_Ca_current_f_Ca_gate

Component: calcium_dynamics

Component: standard_ionic_concentrations