# Model Mathematics

### Component: membrane

$dV_dt= I_st - i_Na + i_Ca_L + i_K + i_K1 + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca Cm$$dd time V =dV_dt$$IStimC=I_st$

### Component: fast_sodium_current

$i_Na = g_Na ⁢ m 3.0 ⁢ h ⁢ j ⁢ V - E_Na$$E_Na = R ⁢ T F ⁢ln⁡ Nao Nai$

### Component: fast_sodium_current_m_gate

$alpha_m = 0.32 ⁢ V + 47.13 1.0 -ⅇ -0.1 ⁢ V + 47.13$$beta_m = 0.08 ⁢ⅇ- V 11.0$$dd time m = alpha_m ⁢ 1.0 - m - beta_m ⁢ m$

### Component: fast_sodium_current_h_gate

$alpha_h = 0.135 ⁢ⅇ 80.0 + V -6.8 if V < -40.0 0.0 otherwise$$beta_h = 3.56 ⁢ⅇ 0.079 ⁢ V + 310000.0 ⁢ⅇ 0.35 ⁢ V if V < -40.0 1.0 0.13 ⁢ 1.0 +ⅇ V + 10.66 -11.1 otherwise$$dd time h = alpha_h ⁢ 1.0 - h - beta_h ⁢ h$

### Component: fast_sodium_current_j_gate

$alpha_j = -127140.0 ⁢ⅇ 0.2444 ⁢ V - 0.00003474 ⁢ⅇ -0.04391 ⁢ V ⁢ V + 37.78 1.0 +ⅇ 0.311 ⁢ V + 79.23 if V < -40.0 0.0 otherwise$$beta_j = 0.1212 ⁢ⅇ -0.01052 ⁢ V 1.0 +ⅇ -0.1378 ⁢ V + 40.14 if V < -40.0 0.3 ⁢ⅇ -0.0000002535 ⁢ V 1.0 +ⅇ -0.1 ⁢ V + 32.0 otherwise$$dd time j = alpha_j ⁢ 1.0 - j - beta_j ⁢ j$

### Component: L_type_Ca_channel

$i_CaCa = d ⁢ f ⁢ f_Ca ⁢ I_CaCa$$i_CaNa = d ⁢ f ⁢ f_Ca ⁢ I_CaNa$$i_CaK = d ⁢ f ⁢ f_Ca ⁢ I_CaK$$I_CaCa = P_Ca ⁢ 2.0 2.0 ⁢ V ⁢ F 2.0 R ⁢ T ⁢ gamma_Cai ⁢ Cai ⁢ⅇ 2.0 ⁢ V ⁢ F R ⁢ T - gamma_Cao ⁢ Cao ⅇ 2.0 ⁢ V ⁢ F R ⁢ T - 1.0$$I_CaNa = P_Na ⁢ 1.0 2.0 ⁢ V ⁢ F 2.0 R ⁢ T ⁢ gamma_Nai ⁢ Nai ⁢ⅇ 1.0 ⁢ V ⁢ F R ⁢ T - gamma_Nao ⁢ Nao ⅇ 1.0 ⁢ V ⁢ F R ⁢ T - 1.0$$I_CaK = P_K ⁢ 1.0 2.0 ⁢ V ⁢ F 2.0 R ⁢ T ⁢ gamma_Ki ⁢ Ki ⁢ⅇ 1.0 ⁢ V ⁢ F R ⁢ T - gamma_Ko ⁢ Ko ⅇ 1.0 ⁢ V ⁢ F R ⁢ T - 1.0$$i_Ca_L = i_CaCa + i_CaK + i_CaNa$

### Component: L_type_Ca_channel_d_gate

$alpha_d = d_infinity tau_d$$d_infinity = 1.0 1.0 +ⅇ- V + 10.0 6.24$$tau_d = d_infinity ⁢ 1.0 -ⅇ- V + 10.0 6.24 0.035 ⁢ V + 10.0$$beta_d = 1.0 - d_infinity tau_d$$dd time d = alpha_d ⁢ 1.0 - d - beta_d ⁢ d$

### Component: L_type_Ca_channel_f_gate

$alpha_f = f_infinity tau_f$$f_infinity = 1.0 1.0 +ⅇ V + 35.06 8.6 + 0.6 1.0 +ⅇ 50.0 - V 20.0$$tau_f = 1.0 0.0197 ⁢ⅇ- 0.0337 ⁢ V + 10.0 2.0 + 0.02$$beta_f = 1.0 - f_infinity tau_f$$dd time f = alpha_f ⁢ 1.0 - f - beta_f ⁢ f$

### Component: L_type_Ca_channel_f_Ca_gate

$f_Ca = 1.0 1.0 + Cai Km_Ca 2.0$

### Component: time_dependent_potassium_current

$g_K =g_K_max⁢ Ko 5.4$$E_K = R ⁢ T F ⁢ln⁡ Ko + PR_NaK ⁢ Nao Ki + PR_NaK ⁢ Nai$$i_K = g_K ⁢ X 2.0 ⁢ Xi ⁢ V - E_K$

### Component: time_dependent_potassium_current_X_gate

$alpha_X = 0.0000719 ⁢ V + 30.0 1.0 -ⅇ -0.148 ⁢ V + 30.0$$beta_X = 0.000131 ⁢ V + 30.0 -1.0 +ⅇ 0.0687 ⁢ V + 30.0$$dd time X = alpha_X ⁢ 1.0 - X - beta_X ⁢ X$

### Component: time_dependent_potassium_current_Xi_gate

$Xi = 1.0 1.0 +ⅇ V - 56.26 32.1$

### Component: time_independent_potassium_current

$g_K1 =g_K1_max⁢ Ko 5.4$$E_K1 = R ⁢ T F ⁢ln⁡ Ko Ki$$i_K1 = g_K1 ⁢ K1_infinity ⁢ V - E_K1$

### Component: time_independent_potassium_current_K1_gate

$alpha_K1 = 1.02 1.0 +ⅇ 0.2385 ⁢ V - E_K1 - 59.215$$beta_K1 = 0.49124 ⁢ⅇ 0.08032 ⁢ V + 5.476 - E_K1 +ⅇ 0.06175 ⁢ V - E_K1 + 594.31 1.0 +ⅇ -0.5143 ⁢ V - E_K1 + 4.753$$K1_infinity = alpha_K1 alpha_K1 + beta_K1$

### Component: plateau_potassium_current

$E_Kp = E_K1$$Kp = 1.0 1.0 +ⅇ 7.488 - V 5.98$$i_Kp = g_Kp ⁢ Kp ⁢ V - E_Kp$

### Component: sarcolemmal_calcium_pump

$i_p_Ca = I_pCa ⁢ Cai K_mpCa + Cai$

### Component: sodium_background_current

$E_NaN = E_Na$$i_Na_b = g_Nab ⁢ V - E_NaN$

### Component: calcium_background_current

$E_CaN = R ⁢ T 2.0 ⁢ F ⁢ln⁡ Cao Cai$$i_Ca_b = g_Cab ⁢ V - E_CaN$

### Component: sodium_potassium_pump

$f_NaK = 1.0 1.0 + 0.1245 ⁢ⅇ -0.1 ⁢ V ⁢ F R ⁢ T + 0.0365 ⁢ sigma ⁢ⅇ- V ⁢ F R ⁢ T$$sigma = 1.0 7.0 ⁢ⅇ Nao 67.3 - 1.0$$i_NaK = I_NaK ⁢ f_NaK ⁢ 1.0 1.0 + K_mNai Nai 1.5 ⁢ Ko Ko + K_mKo$

### Component: non_specific_calcium_activated_current

$EnsCa = R ⁢ T F ⁢ln⁡Ko+NaoKi+Nai$$Vns=V-EnsCa$$i_ns_Na = I_ns_Na ⁢ 1.0 1.0 + K_m_ns_Ca Cai 3.0$$i_ns_K = I_ns_K ⁢ 1.0 1.0 + K_m_ns_Ca Cai 3.0$$i_ns_Ca = i_ns_Na + i_ns_K$$I_ns_Na = P_ns_Ca ⁢ 1.0 2.0 ⁢ Vns ⁢ F 2.0 R ⁢ T ⁢ gamma_Nai ⁢ Nai ⁢ⅇ 1.0 ⁢ Vns ⁢ F R ⁢ T - gamma_Nao ⁢ Nao ⅇ 1.0 ⁢ Vns ⁢ F R ⁢ T - 1.0$$I_ns_K = P_ns_Ca ⁢ 1.0 2.0 ⁢ Vns ⁢ F 2.0 R ⁢ T ⁢ gamma_Ki ⁢ Ki ⁢ⅇ 1.0 ⁢ Vns ⁢ F R ⁢ T - gamma_Ko ⁢ Ko ⅇ 1.0 ⁢ Vns ⁢ F R ⁢ T - 1.0$

### Component: Na_Ca_exchanger

$i_NaCa = K_NaCa ⁢ 1.0 K_mNa 3.0 + Nao 3.0 ⁢ 1.0 K_mCa + Cao ⁢ 1.0 1.0 + K_sat ⁢ⅇ eta - 1.0 ⁢ V ⁢ F R ⁢ T ⁢ⅇ eta ⁢ V ⁢ F R ⁢ T ⁢ Nai 3.0 ⁢ Cao -ⅇ eta - 1.0 ⁢ V ⁢ F R ⁢ T ⁢ Nao 3.0 ⁢ Cai$

### Component: calcium_fluxes_in_the_SR

$i_rel = G_rel ⁢ Ca_JSR - Cai$$G_rel = G_rel_peak ⁢ delta_Ca_i2 - delta_Ca_ith K_mrel + delta_Ca_i2 - delta_Ca_ith ⁢ 1.0 -ⅇ- t_CICR tau_on ⁢ⅇ- t_CICR tau_off$$G_rel_peak = 0.0 if delta_Ca_i2 < delta_Ca_ith G_rel_max otherwise$$i_up = I_up ⁢ Cai Cai + K_mup$$i_leak = K_leak ⁢ Ca_NSR$$K_leak = I_up Ca_NSR_max$$i_tr = Ca_NSR - Ca_JSR tau_tr$

### Component: ionic_concentrations

$dd time Nai =- i_Na + i_CaNa + i_Na_b + i_ns_Na + i_NaCa ⁢ 3.0 + i_NaK ⁢ 3.0 ⁢ Am V_myo ⁢ F$$dd time Cai =- i_CaCa + i_p_Ca + i_Ca_b - i_NaCa ⁢ Am 2.0 ⁢ V_myo ⁢ F + i_rel ⁢ V_JSR V_myo + i_leak - i_up ⁢ V_NSR V_myo$$dd time Ki =- i_CaK + i_K + i_K1 + i_Kp + i_ns_K +- i_NaK ⁢ 2.0 ⁢ Am V_myo ⁢ F$$dd time Ca_JSR =- i_rel - i_tr ⁢ V_NSR V_JSR$$dd time Ca_NSR =- i_leak + i_tr - i_up$
Source
Derived from workspace Luo, Rudy, 1994 at changeset e58e5df49727.
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