# Model Mathematics

### Component: membrane

$dd time V = -1.0 Cm ⁢ i_Na + i_Ca + i_Ca_K + i_K + i_K1 + i_Kp + i_NaCa + i_NaK + i_ns_Ca + i_p_Ca + i_Ca_b + i_Na_b$

### 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: time_dependent_potassium_current

$g_K = 0.1128 ⁢ Ko 5.4 E_K = R ⁢ T F ⁢ln⁡ Ko + PNa_K ⁢ Nao Ki + PNa_K ⁢ Nai i_K = g_K ⁢ Xi ⁢ X 2.0 ⁢ V - E_K$

### 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

$Xi = 1.0 1.0 +ⅇ V - 40.0 40.0$

### 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

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

### 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: 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: 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

$i_ns_Na = I_ns_Na ⁢ 1.0 1.0 + K_m_ns_Ca Cai 3.0 i_ns_K = I_ns_K ⁢ 1.0 1.0 + K_m_ns_Ca Cai 3.0 i_ns_Ca = i_ns_Na + i_ns_K I_ns_Na = Pns_Na ⁢ V ⁢ F 2.0 R ⁢ T ⁢ 0.75 ⁢ Nai ⁢ⅇ V ⁢ F R ⁢ T - 0.75 ⁢ Nao ⅇ V ⁢ F R ⁢ T - 1.0 I_ns_K = Pns_K ⁢ V ⁢ F 2.0 R ⁢ T ⁢ 0.75 ⁢ Ki ⁢ⅇ V ⁢ F R ⁢ T - 0.75 ⁢ Ko ⅇ V ⁢ F R ⁢ T - 1.0$

### Component: sarcolemmal_calcium_pump

$i_p_Ca = I_pCa ⁢ Cai K_mpCa + Cai$

### Component: calcium_background_current

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

### Component: sodium_background_current

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

### Component: L_type_Ca_channel

$i_Ca = i_Ca_max ⁢ y ⁢ O + O_Ca i_Ca_max = P_Ca 4.0 ⁢ V ⁢ F 2.0 R ⁢ T ⁢ 0.001 ⁢ⅇ 2.0 ⁢ V ⁢ F R ⁢ T - 0.341 ⁢ Cao ⅇ 2.0 ⁢ V ⁢ F R ⁢ T - 1.0 i_Ca_K = p_k ⁢ y ⁢ O + O_Ca ⁢ V ⁢ F 2.0 R ⁢ T ⁢ Ki ⁢ⅇ V ⁢ F R ⁢ T - Ko ⅇ V ⁢ F R ⁢ T - 1.0 p_k = P_K 1.0 + i_Ca_max i_Ca_half alpha = 0.4 ⁢ⅇ V + 12.0 10.0 beta = 0.05 ⁢ⅇ- V + 12.0 13.0 alpha_a = alpha ⁢ a beta_b = beta b gamma = 0.5625 ⁢ Ca_SS dd time C0 = beta ⁢ C1 + omega ⁢ C_Ca0 - 4.0 ⁢ alpha + gamma ⁢ C0 dd time C1 = 4.0 ⁢ alpha ⁢ C0 + 2.0 ⁢ beta ⁢ C2 + omega b ⁢ C_Ca1 - beta + 3.0 ⁢ alpha + gamma ⁢ a ⁢ C1 dd time C2 = 3.0 ⁢ alpha ⁢ C1 + 3.0 ⁢ beta ⁢ C3 + omega b 2.0 ⁢ C_Ca2 - beta ⁢ 2.0 + 2.0 ⁢ alpha + gamma ⁢ a 2.0 ⁢ C2 dd time C3 = 2.0 ⁢ alpha ⁢ C2 + 4.0 ⁢ beta ⁢ C4 + omega b 3.0 ⁢ C_Ca3 - beta ⁢ 3.0 + alpha + gamma ⁢ a 3.0 ⁢ C3 dd time C4 = alpha ⁢ C3 + g ⁢ O + omega b 4.0 ⁢ C_Ca4 - beta ⁢ 4.0 + f + gamma ⁢ a 4.0 ⁢ C4 dd time O = f ⁢ C4 - g ⁢ O dd time C_Ca0 = beta_b ⁢ C_Ca1 + gamma ⁢ C_Ca0 - 4.0 ⁢ alpha_a + omega ⁢ C_Ca0 dd time C_Ca1 = 4.0 ⁢ alpha_a ⁢ C_Ca0 + 2.0 ⁢ beta_b ⁢ C_Ca2 + gamma ⁢ a ⁢ C1 - beta_b + 3.0 ⁢ alpha_a + omega b ⁢ C_Ca1 dd time C_Ca2 = 3.0 ⁢ alpha_a ⁢ C_Ca1 + 3.0 ⁢ beta_b ⁢ C_Ca3 + gamma ⁢ a 2.0 ⁢ C2 - beta_b ⁢ 2.0 + 2.0 ⁢ alpha_a + omega b 2.0 ⁢ C_Ca2 dd time C_Ca3 = 2.0 ⁢ alpha_a ⁢ C_Ca2 + 4.0 ⁢ beta_b ⁢ C_Ca4 + gamma ⁢ a 3.0 ⁢ C3 - beta_b ⁢ 3.0 + alpha_a + omega b 3.0 ⁢ C_Ca3 dd time C_Ca4 = alpha_a ⁢ C_Ca3 + g_ ⁢ O_Ca + gamma ⁢ a 4.0 ⁢ C4 - beta_b ⁢ 4.0 + f_ + omega b 4.0 ⁢ C_Ca4 dd time O_Ca = f_ ⁢ C_Ca4 - g_ ⁢ O_Ca$

### Component: L_type_Ca_channel_y_gate

$dd time y = y_infinity - y tau_y y_infinity = 1.0 1.0 +ⅇ V + 55.0 7.5 + 0.1 1.0 +ⅇ- V + 21.0 6.0 tau_y = 20.0 + 600.0 1.0 +ⅇ V + 30.0 9.5$

### Component: RyR_channel_states

$dd time P_C1 =- k_a_plus ⁢ Ca_SS n ⁢ P_C1 + k_a_minus ⁢ P_O1 dd time P_O1 = k_a_plus ⁢ Ca_SS n ⁢ P_C1 - k_a_minus ⁢ P_O1 + k_b_plus ⁢ Ca_SS m ⁢ P_O1 + k_c_plus ⁢ P_O1 + k_b_minus ⁢ P_O2 + k_c_minus ⁢ P_C2 dd time P_O2 = k_b_plus ⁢ Ca_SS m ⁢ P_O1 - k_b_minus ⁢ P_O2 dd time P_C2 = k_c_plus ⁢ P_O1 - k_c_minus ⁢ P_C2$

### Component: SERCA_pump

$J_up = Vmax_f ⁢ fb - Vmax_r ⁢ rb 1.0 + fb + rb fb = Cai k_fb n_fb rb = Ca_NSR k_rb n_rb$

### Component: intracellular_Ca_fluxes

$J_rel = v1 ⁢ P_O1 + P_O2 ⁢ Ca_JSR - Ca_SS J_tr = Ca_NSR - Ca_JSR tau_tr J_xfer = Ca_SS - Cai tau_xfer J_trpn = k_htrpn_plus ⁢ Cai ⁢ HTRPN_tot - HTRPNCa - k_htrpn_minus ⁢ HTRPNCa + k_ltrpn_plus ⁢ Cai ⁢ LTRPN_tot - LTRPNCa - k_ltrpn_minus ⁢ LTRPNCa$

### Component: intracellular_ionic_concentrations

$betai = 1.0 1.0 + CMDN_tot ⁢ K_mCMDN K_mCMDN + Cai 2.0 beta_SS = 1.0 1.0 + CMDN_tot ⁢ K_mCMDN K_mCMDN + Ca_SS 2.0 beta_JSR = 1.0 1.0 + CSQN_tot ⁢ K_mCSQN K_mCSQN + Ca_JSR 2.0 dd time Cai = betai ⁢- J_xfer + J_up + J_trpn + i_Ca_b +- 2.0 ⁢ i_NaCa + i_p_Ca ⁢ A_cap 2.0 ⁢ V_myo ⁢ F dd time Ca_SS = beta_SS ⁢ J_rel ⁢ V_JSR V_SS - J_xfer ⁢ V_myo V_SS - i_Ca ⁢ A_cap 2.0 ⁢ V_SS ⁢ F dd time Ca_JSR = beta_JSR ⁢ J_tr - J_rel dd time Ca_NSR = J_up ⁢ V_myo V_NSR - J_tr ⁢ V_JSR V_NSR dd time Nai =- i_Na + i_Na_b + i_ns_Na + i_NaCa ⁢ 3.0 + i_NaK ⁢ 3.0 ⁢ A_cap V_myo ⁢ F dd time Ki =- i_Ca_K + i_K + i_K1 + i_Kp + i_ns_K +- i_NaK ⁢ 2.0 ⁢ A_cap V_myo ⁢ F$

### Component: troponin

$dd time HTRPNCa = k_htrpn_plus ⁢ Cai ⁢ HTRPN_tot - HTRPNCa - k_htrpn_minus ⁢ HTRPNCa dd time LTRPNCa = k_ltrpn_plus ⁢ Cai ⁢ LTRPN_tot - LTRPNCa - k_ltrpn_minus ⁢ LTRPNCa ⁢ 0.333 + 0.667 ⁢ 1.0 - Force_norm$

### Component: tropomyosin_cross_bridges

$f_01 = 3.0 ⁢ f_XB f_12 = 10.0 ⁢ f_XB f_23 = 7.0 ⁢ f_XB g_01_SL = 1.0 ⁢ g_XB_SL g_12_SL = 2.0 ⁢ g_XB_SL g_23_SL = 3.0 ⁢ g_XB_SL g_XB_SL = g_XB ⁢ 1.0 + 1.0 - SL_norm 1.6 SL_norm = SL - 1.7 0.7 k_trop_np = k_trop_pn ⁢ LTRPNCa LTRPN_tot K_trop_half N_trop N_trop = 3.5 + 2.5 ⁢ SL_norm K_trop_half = 1.0 + K_trop_Ca 1.7 - 0.9 ⁢ SL_norm -1.0 K_trop_Ca = k_ltrpn_minus k_ltrpn_plus dd time N0 = k_trop_pn ⁢ P0 - k_trop_np ⁢ N0 + g_01_SL ⁢ N1 dd time P0 =- k_trop_pn + f_01 ⁢ P0 + k_trop_np ⁢ N0 + g_01_SL ⁢ P1 dd time P1 =- k_trop_pn + f_12 + g_01_SL ⁢ P1 + k_trop_np ⁢ N1 + k_trop_np ⁢ N1 + f_01 ⁢ P0 + g_12_SL ⁢ P2 dd time P2 =- f_23 + g_12_SL ⁢ P2 + f_12 ⁢ P1 + g_23_SL ⁢ P3 dd time P3 =- g_23_SL ⁢ P3 + f_23 ⁢ P2$

### Component: force_computation

$Force = zeta ⁢ Force_norm Force_norm = phi_SL ⁢ P1 + N1 + 2.0 ⁢ P2 + 3.0 ⁢ P3 Force_max Force_max = P1_max + 2.0 ⁢ P2_max + 3.0 ⁢ P3_max phi_SL = SL - 0.6 1.4 if SL < 2.0 ∧ SL > 1.7 1.0 if SL < 2.2 ∧ SL > 2.0 3.6 - SL 1.4 if SL < 2.3 ∧ SL > 2.2 P1_max = f_01 ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB g_XB ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB + f_01 ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB + f_01 ⁢ f_12 ⁢ 3.0 ⁢ g_XB + f_01 ⁢ f_12 ⁢ f_23 P2_max = f_01 ⁢ f_12 ⁢ 3.0 ⁢ g_XB g_XB ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB + f_01 ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB + f_01 ⁢ f_12 ⁢ 3.0 ⁢ g_XB + f_01 ⁢ f_12 ⁢ f_23 P3_max = f_01 ⁢ f_12 ⁢ f_23 g_XB ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB + f_01 ⁢ 2.0 ⁢ g_XB ⁢ 3.0 ⁢ g_XB + f_01 ⁢ f_12 ⁢ 3.0 ⁢ g_XB + f_01 ⁢ f_12 ⁢ f_23$
Source
Derived from workspace Rice, Jafri, Winslow, 2000 at changeset 4533913da7a3.
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