# Model Mathematics

### Component: membrane

$dd time V =- 1.0 Cm ⁢ i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_K1 + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca + i_s + I_st$

### 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

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

### Component: fast_sodium_current_j_gate

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

### 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

$f_Ca = 1.0 1.0 + Cai Km_Ca$

### Component: T_type_Ca_channel

$i_Ca_T = g_Ca_T ⁢ b 2.0 ⁢ g ⁢ V - E_Ca E_Ca = R ⁢ T 2.0 ⁢ F ⁢ln⁡ Cao Cai$

### Component: T_type_Ca_channel_b_gate

$dd time b = b_infinity - b tau_b b_infinity = 1.0 1.0 +ⅇ- V + 14.0 10.8 tau_b = 3.7 + 6.1 1.0 +ⅇ 25.0 + V 4.5$

### Component: T_type_Ca_channel_g_gate

$dd time g = g_infinity - g tau_g g_infinity = 1.0 1.0 +ⅇ V + 60.0 5.6 tau_g = 12.0 if V > 0.0 -0.875 ⁢ V + 12.0 otherwise$

### Component: rapid_time_dependent_potassium_current

$g_Kr = 0.02614 ⁢ Ko 5.4 E_Kr = R ⁢ T F ⁢ln⁡ Ko Ki i_Kr = g_Kr ⁢ Xr ⁢ Rr ⁢ V - E_Kr$

### Component: rapid_time_dependent_potassium_current_Xr_gate

$dd time Xr = Xr_infinity - Xr tau_Xr Xr_infinity = 1.0 1.0 +ⅇ- V + 21.5 7.5 tau_Xr = 1.0 0.00138 ⁢ V + 14.2 1.0 -ⅇ -0.123 ⁢ V + 14.2 + 0.00061 ⁢ V + 38.9 ⅇ 0.145 ⁢ V + 38.9 - 1.0$

### Component: rapid_time_dependent_potassium_current_Rr_gate

$Rr = 1.0 1.0 +ⅇ V + 9.0 22.4$

### Component: slow_time_dependent_potassium_current

$g_Ks = 0.057 + 0.19 1.0 +ⅇ -7.2 + P_Ca 0.6 E_Ks = R ⁢ T F ⁢ln⁡ Ko + P_NaK ⁢ Nao Ki + P_NaK ⁢ Nai P_Ca =-log10⁡ Cai + 3.0 i_Ks = g_Ks ⁢ Xs 2.0 ⁢ V - E_Ks$

### Component: slow_time_dependent_potassium_current_Xs_gate

$dd time Xs = Xs_infinity - Xs tau_Xs Xs_infinity = 1.0 1.0 +ⅇ- V + -1.5 16.7 tau_Xs = 1.0 0.0000719 ⁢ V + 30.0 1.0 -ⅇ -0.148 ⁢ V + 30.0 + 0.000131 ⁢ V + 30.0 ⅇ 0.0687 ⁢ V + 30.0 - 1.0$

### Component: time_independent_potassium_current

$g_K1 = 0.75 ⁢ Ko 5.4 E_K1 = R ⁢ T F ⁢ln⁡ Ko Ki i_K1 = g_K1 ⁢ K1_infinity ⁢ V - E_K1$

### Component: time_independent_potassium_current_K1_gate

$alpha_K1 = 1.02 1.0 +ⅇ 0.2385 ⁢ V - E_K1 - 59.215 beta_K1 = 0.49124 ⁢ⅇ V + 5.476 - E_K1 12.45 +ⅇ V - E_K1 + 594.31 16.2 1.0 +ⅇ -0.5143 ⁢ V - E_K1 + 4.753 K1_infinity = alpha_K1 alpha_K1 + beta_K1$

### 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

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

### 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: stretch_activated_current

$i_s = g_s ⁢ V - E_s$

### Component: calcium_buffers_in_the_myoplasm

$TRPN_buff = TRPN_max ⁢ Cai Cai + K_mTRPN CMDN_buff = CMDN_max ⁢ Cai Cai + K_mCMDN Ca_JSR_new = b1 2.0 + 4.0 ⁢ c1 - b1 2.0 delta_Ca_JSR = Ca_JSR_old - Ca_JSR_new delta_Cai = Cai_old - Cai b1 = K_mCSQN - CSQN_max + CSQN_buff + delta_Ca_JSR + Ca_JSR_old c1 = K_mCSQN ⁢ CSQN_buff + delta_Ca_JSR + Ca_JSR_old Cai = 2.0 3.0 ⁢ b 2.0 - 3.0 ⁢ c ⁢cos⁡arccos⁡ 9.0 ⁢ b ⁢ c - 2.0 ⁢ b 3.0 + 27.0 ⁢ d 2.0 ⁢ b 2.0 - 3.0 ⁢ c 3.0 2.0 3.0 - b 3.0 b = CMDN_max + TRPN_max + K_mTRPN + K_mCMDN - Ca_total c = K_mCMDN ⁢ K_mTRPN + TRPN_max ⁢ K_mCMDN + CMDN_max ⁢ K_mTRPN - Ca_total ⁢ K_mTRPN + K_mCMDN d =- K_mTRPN ⁢ K_mCMDN ⁢ Ca_total Ca_total = TRPN_buff + CMDN_buff + delta_Cai + Cai_old$

### Component: calcium_fluxes_in_the_SR

$i_rel = G_rel ⁢ Ca_JSR_new - Cai G_rel = G_rel_max ⁢ delta_Ca_i2 - delta_Ca_ith K_mrel + delta_Ca_i2 - delta_Ca_ith ⁢ 1.0 -ⅇ- t tau_on ⁢ⅇ- t tau_off if calcium_overload = 0.0 G_rel_max ⁢ 1.0 -ⅇ- t tau_on ⁢ⅇ- t tau_off otherwise G_rel_max = 0.0 if delta_Ca_i2 < delta_Ca_ith 60.0 otherwiseif calcium_overload = 0.0 0.0 if CSQN_buff < CSQN_th 4.0 otherwiseotherwise CSQN_buff = CSQN_max ⁢ Ca_JSR_new Ca_JSR_new + K_mCSQN i_up = I_up ⁢ Cai Cai + K_mup i_leak = K_leak ⁢ Ca_NSR K_leak = I_up Ca_NSR_max i_tr = Ca_NSR - Ca_JSR_new tau_tr$

### Component: ionic_concentrations

$dd time Nai =- i_Na + i_CaNa + i_Na_b + i_ns_Na + i_NaCa ⁢ 3.0 + i_NaK ⁢ 3.0 ⁢ A_cap V_myo ⁢ F dd time Ki =- i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_ns_K +- i_NaK ⁢ 2.0 ⁢ A_cap V_myo ⁢ F dd time Ko = i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_ns_K +- i_NaK ⁢ 2.0 ⁢ A_cap V_cleft ⁢ F dd time Ca_NSR =- i_leak + i_tr - i_up dd time Ca_foot =- i_CaCa ⁢ A_cap 2.0 ⁢ V_myo ⁢ F ⁢ R_A_V$
Source
Derived from workspace Riemer, Sobie, Tung, 1998 at changeset 4ba9af6ef3b5.
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