# Model Mathematics

### Component: wal_environment

$I_T = 1000 1 ⁢ vS - vT R_a$$dd time vS =- I_ionic_s + I_T C_m$$dd time vT =- I_ionic_t - I_T gam C_m$$I_ionic_s = I_Cl + I_IR + I_DR + I_Na + I_NaK +- I_HH$$I_ionic_t = I_Cl_t + I_IR_t + I_DR_t + I_Na_t + I_NaK_t$$dd time K_i =- f_T ⁢ I_IR_t + I_DR_t + I_K_rest +- 2 ⁢ I_NaK_t 1000 1 ⁢ FF ⁢ tsi - I_IR + I_DR + I_K_rest + -2 ⁢ I_NaK 1000 1 ⁢ FF ⁢ tsi2$$dd time K_t = I_IR_t + I_DR_t + I_K_rest +- 2 ⁢ I_NaK_t 1000 1 ⁢ FF ⁢ tsi - K_t - K_e tau_K$$dd time K_e = I_IR + I_DR + I_K_rest +- 2 ⁢ I_NaK 1000 1 ⁢ FF ⁢ tsi3 + K_t - K_e tau_K2$$dd time Na_i =- f_T ⁢ I_Na_t + I_Na_rest + 3 ⁢ I_NaK_t 1000 1 ⁢ FF ⁢ tsi - I_Na + I_Na_rest + 3 ⁢ I_NaK 1000 1 ⁢ FF ⁢ tsi2$$dd time Na_t = I_Na_t + I_Na_rest + 3 ⁢ I_NaK_t 1000 1 ⁢ FF ⁢ tsi - Na_t - Na_e tau_Na$$dd time Na_e = I_Na + I_Na_rest + 3 ⁢ I_NaK 1000 1 ⁢ FF ⁢ tsi3 + Na_t - Na_e tau_Na2$$E_K = RR ⁢ TT FF ⁢ln⁡ K_e K_i$$E_K_t = RR ⁢ TT FF ⁢ln⁡ K_t K_i$$Cl_i = 156.5 5 +ⅇ- FF ⁢ E_K RR ⁢ TT$$Cl_o = 156.5 - 5 ⁢ Cl_i$$Cl_i_t = 156.5 5 +ⅇ- FF ⁢ E_K_t RR ⁢ TT$$Cl_o_t = 156.5 - 5 ⁢ Cl_i_t$$J_K = vS ⁢ K_i - K_e ⁢ⅇ -1 ⁢ FF ⁢ vS RR ⁢ TT 1 -ⅇ -1 ⁢ FF ⁢ vS RR ⁢ TT$$J_K_t = vT ⁢ K_i - K_t ⁢ⅇ -1 ⁢ FF ⁢ vT RR ⁢ TT 1 -ⅇ -1 ⁢ FF ⁢ vT RR ⁢ TT$$I_HH = 150 if time ≥ 0 ∧ time < 0.5 150 if time ≥ 50 ∧ time < 50.5 150 if time ≥ 100 ∧ time < 100.5 150 if time ≥ 150 ∧ time < 150.5 150 if time ≥ 200 ∧ time < 200.5 150 if time ≥ 250 ∧ time < 250.5 150 if time ≥ 300 ∧ time < 300.5 150 if time ≥ 350 ∧ time < 350.5 150 if time ≥ 400 ∧ time < 400.5 0 otherwise$

### Component: sarco_Cl_channel

$a = 1 1 +ⅇ vS - V_a A_a$$J_Cl = vS ⁢ Cl_i - Cl_o ⁢ⅇ FF ⁢ vS RR ⁢ TT 1 -ⅇ FF ⁢ vS RR ⁢ TT$$g_Cl = g_Cl_bar ⁢ a 4$$I_Cl = g_Cl ⁢ J_Cl 45$

### Component: sarco_IR_channel

$K_R = K_e ⁢ⅇ- del ⁢ E_K ⁢ FF RR ⁢ TT$$g_IR_bar = G_K ⁢ K_R 2 K_K + K_R 2$$y = 1 - 1 + K_S ⁢ 1 + K_R 2 K_K S_i 2 ⁢ⅇ 2 ⁢ 1 - del ⁢ vS ⁢ FF RR ⁢ TT -1$$g_IR = g_IR_bar ⁢ y$$I_IR = g_IR ⁢ 1 if J_K > 0 0 otherwise⁢ J_K 50$

### Component: sarco_DR_channel

$alpha_n = alpha_n_bar ⁢ vS - V_n 1 -ⅇ- vS - V_n K_alpha_n$$beta_n = beta_n_bar ⁢ⅇ- vS - V_n K_beta_n$$h_K_inf = 1 1 +ⅇ vS - V_h_K_inf A_h_K_inf$$tau_h_K = 1000 ⁢ⅇ- vS + 40 25.75$$dd time n = alpha_n ⁢ 1 - n - beta_n ⁢ n$$dd time h_K = h_K_inf - h_K tau_h_K$$g_DR = g_K_bar ⁢ n 4 ⁢ h_K$$I_DR = g_DR ⁢ J_K 50$

### Component: sarco_Na_channel

$alpha_h = alpha_h_bar ⁢ⅇ- vS - V_h K_alpha_h$$beta_h = beta_h_bar 1 +ⅇ- vS - V_h K_beta_h$$alpha_m = alpha_m_bar ⁢ vS - V_m 1 -ⅇ- vS - V_m K_alpha_m$$beta_m = beta_m_bar ⁢ⅇ- vS - V_m K_beta_m$$S_inf = 1 1 +ⅇ vS - V_S_inf A_S_inf$$tau_S = 8571 0.2 + 5.65 ⁢ vS + V_tau 100 2$$J_Na = vS ⁢ Na_i - Na_e ⁢ⅇ -1 ⁢ FF ⁢ vS RR ⁢ TT 1 -ⅇ -1 ⁢ FF ⁢ vS RR ⁢ TT$$dd time m = alpha_m ⁢ 1 - m - beta_m ⁢ m$$dd time h = alpha_h ⁢ 1 - h - beta_h ⁢ h$$dd time S = S_inf - S tau_S$$g_Na = g_Na_bar ⁢ m 3 ⁢ h ⁢ S$$I_Na = g_Na ⁢ J_Na 75$

### Component: sarco_NaK_channel

$sig = 1 7 ⁢ⅇ Na_e 67.3 - 1$$f1 = 1 + 0.12 ⁢ⅇ -0.1 ⁢ vS ⁢ FF RR ⁢ TT + 0.04 ⁢ sig ⁢ⅇ- vS ⁢ FF RR ⁢ TT -1$$I_NaK_bar = FF ⁢ J_NaK_bar 1 + K_m_K K_e 2 ⁢ 1 + K_m_Na Na_i 3$$I_NaK = I_NaK_bar ⁢ f1$

### Component: t_Cl_channel

$a_t = 1 1 +ⅇ vT - V_a A_a$$J_Cl_t = vT ⁢ Cl_i_t - Cl_o_t ⁢ⅇ FF ⁢ vT RR ⁢ TT 1 -ⅇ FF ⁢ vT RR ⁢ TT$$g_Cl_t = g_Cl_bar ⁢ a_t 4$$I_Cl_t = eta_Cl ⁢ g_Cl_t ⁢ J_Cl_t 45$

### Component: t_IR_channel

$K_R_t = K_t ⁢ⅇ- del ⁢ E_K_t ⁢ FF RR ⁢ TT$$g_IR_bar_t = G_K ⁢ K_R_t 2 K_K + K_R_t 2$$y_t = 1 - 1 + K_S ⁢ 1 + K_R_t 2 K_K S_i 2 ⁢ⅇ 2 ⁢ 1 - del ⁢ vT ⁢ FF RR ⁢ TT -1$$g_IR_t = g_IR_bar_t ⁢ y_t$$I_IR_t = eta_IR ⁢ g_IR_t ⁢ J_K_t 50$

### Component: t_DR_channel

$alpha_n_t = alpha_n_bar ⁢ vT - V_n 1 -ⅇ- vT - V_n K_alpha_n$$beta_n_t = beta_n_bar ⁢ⅇ- vT - V_n K_beta_n$$h_K_inf_t = 1 1 +ⅇ vT - V_h_K_inf A_h_K_inf$$tau_h_K_t = 1 ⁢ⅇ- vT + 40 25.75$$dd time n_t = alpha_n_t ⁢ 1 - n_t - beta_n_t ⁢ n_t$$dd time h_K_t = h_K_inf_t - h_K_t tau_h_K_t$$g_DR_t = g_K_bar ⁢ n_t 4 ⁢ h_K_t$$I_DR_t = eta_DR ⁢ g_DR_t ⁢ J_K_t 50$

### Component: t_Na_channel

$alpha_h_t = alpha_h_bar ⁢ⅇ- vT - V_h K_alpha_h$$beta_h_t = beta_h_bar 1 +ⅇ- vT - V_h K_beta_h$$alpha_m_t = alpha_m_bar ⁢ vT - V_m 1 -ⅇ- vT - V_m K_alpha_m$$beta_m_t = beta_m_bar ⁢ⅇ- vT - V_m K_beta_m$$S_inf_t = 1 1 +ⅇ vT - V_S_inf A_S_inf$$tau_S_t = 8571 0.2 + 5.65 ⁢ vT + V_tau 100 2$$J_Na_t = vT ⁢ Na_i - Na_t ⁢ⅇ -1 ⁢ FF ⁢ vT RR ⁢ TT 1 -ⅇ -1 ⁢ FF ⁢ vT RR ⁢ TT$$dd time m_t = alpha_m_t ⁢ 1 - m_t - beta_m_t ⁢ m_t$$dd time h_t = alpha_h_t ⁢ 1 - h_t - beta_h_t ⁢ h_t$$dd time S_t = S_inf_t - S_t tau_S_t$$g_Na_t = g_Na_bar ⁢ m_t 3 ⁢ h_t ⁢ S_t$$I_Na_t = eta_Na ⁢ g_Na_t ⁢ J_Na_t 75$

### Component: t_NaK_channel

$sig_t = 1 7 ⁢ⅇ Na_t 67.3 - 1$$f1_t = 1 + 0.12 ⁢ⅇ -0.1 ⁢ vT ⁢ FF RR ⁢ TT + 0.04 ⁢ sig_t ⁢ⅇ- vT ⁢ FF RR ⁢ TT -1$$I_NaK_bar_t = FF ⁢ J_NaK_bar 1 + K_m_K K_t 2 ⁢ 1 + K_m_Na Na_i 3$$I_NaK_t = eta_NaK ⁢ I_NaK_bar_t ⁢ f1_t$

### Component: sternrios

$k_C = 0.5 ⁢ alpha1 ⁢ⅇ vT - Vbar 8 ⁢ K$$k_Cm = 0.5 ⁢ alpha1 ⁢ⅇ Vbar - vT 8 ⁢ K$$dd time C_0 =- k_L ⁢ C_0 + k_Lm ⁢ O_0 + -4 ⁢ k_C ⁢ C_0 + k_Cm ⁢ C_1$$dd time O_0 = k_L ⁢ C_0 +- k_Lm ⁢ O_0 + -4 ⁢ k_C ⁢ O_0 f + f ⁢ k_Cm ⁢ O_1$$dd time C_1 = 4 ⁢ k_C ⁢ C_0 +- k_Cm ⁢ C_1 +- k_L ⁢ C_1 f + f ⁢ k_Lm ⁢ O_1 + -3 ⁢ k_C ⁢ C_1 + 2 ⁢ k_Cm ⁢ C_2$$dd time O_1 = k_L ⁢ C_1 f +- k_Lm ⁢ f ⁢ O_1 + 4 ⁢ k_C ⁢ O_0 f +- f ⁢ k_Cm ⁢ O_1 + -3 ⁢ k_C ⁢ O_1 f + 2 ⁢ f ⁢ k_Cm ⁢ O_2$$dd time C_2 = 3 ⁢ k_C ⁢ C_1 + -2 ⁢ k_Cm ⁢ C_2 +- k_L ⁢ C_2 f 2 + f 2 ⁢ k_Lm ⁢ O_2 + -2 ⁢ k_C ⁢ C_2 + 3 ⁢ k_Cm ⁢ C_3$$dd time O_2 = 3 ⁢ k_C ⁢ O_1 f + -2 ⁢ f ⁢ k_Cm ⁢ O_2 + k_L ⁢ C_2 f 2 +- k_Lm ⁢ f 2 ⁢ O_2 + -2 ⁢ k_C ⁢ O_2 f + 3 ⁢ f ⁢ k_Cm ⁢ O_3$$dd time C_3 = 2 ⁢ k_C ⁢ C_2 + -3 ⁢ k_Cm ⁢ C_3 +- k_L ⁢ C_3 f 3 + k_Lm ⁢ f 3 ⁢ O_3 +- k_C ⁢ C_3 + 4 ⁢ k_Cm ⁢ C_4$$dd time O_3 = k_L ⁢ C_3 f 3 +- k_Lm ⁢ f 3 ⁢ O_3 + 2 ⁢ k_C ⁢ O_2 f + -3 ⁢ k_Cm ⁢ f ⁢ O_3 +- k_C ⁢ O_3 f + 4 ⁢ f ⁢ k_Cm ⁢ O_4$$dd time C_4 = k_C ⁢ C_3 + -4 ⁢ k_Cm ⁢ C_4 +- k_L ⁢ C_4 f 4 + k_Lm ⁢ f 4 ⁢ O_4$$dd time O_4 = k_C ⁢ O_3 f + -4 ⁢ f ⁢ k_Cm ⁢ O_4 + k_L ⁢ C_4 f 4 +- k_Lm ⁢ f 4 ⁢ O_4$

### Component: razumova

$V_o = 0.95 ⁢ L_x ⁢π⁢ R_R 2$$V_1 = 0.01 ⁢ V_o$$V_2 = 0.99 ⁢ V_o$$V_SR = 0.05 ⁢ L_x ⁢π⁢ R_R 2$$V_SR1 = 0.01 ⁢ V_SR$$V_SR2 = 0.99 ⁢ V_SR$$T_0 = T_tot +- Ca_T_2 +- Ca_CaT2 +- D_0 +- D_1 +- D_2 +- A_1 +- A_2$$dd time Ca_1 = i2 ⁢ O_0 + O_1 + O_2 + O_3 + O_4 ⁢ Ca_SR1 - Ca_1 V_1 - nu_SR ⁢ Ca_1 Ca_1 + K_SR V_1 + L_e ⁢ Ca_SR1 - Ca_1 V_1 +- tau_R ⁢ Ca_1 - Ca_2 V_1 +- k_P_on ⁢ Ca_1 ⁢ P_tot +- Ca_P1 +- Mg_P1 +- k_P_off ⁢ Ca_P1 +- k_CATP_on ⁢ Ca_1 ⁢ ATP1 +- k_CATP_off ⁢ Ca_ATP1$$dd time Ca_SR1 =- i2 ⁢ O_0 + O_1 + O_2 + O_3 + O_4 ⁢ Ca_SR1 - Ca_1 V_SR1 + nu_SR ⁢ Ca_1 Ca_1 + K_SR V_SR1 +- L_e ⁢ Ca_SR1 - Ca_1 V_SR1 +- tau_SR_R ⁢ Ca_SR1 - Ca_SR2 V_SR1 +- k_Cs_on ⁢ Ca_SR1 ⁢ Cs_tot - Ca_Cs1 +- k_Cs_off ⁢ Ca_Cs1$$dd time Ca_2 =- nu_SR ⁢ Ca_2 Ca_2 + K_SR V_2 + L_e ⁢ Ca_SR2 +- Ca_2 V_2 + tau_R ⁢ Ca_1 - Ca_2 V_2 +- k_T_on ⁢ Ca_2 ⁢ T_0 +- k_T_off ⁢ Ca_T_2 + k_T_on ⁢ Ca_2 ⁢ Ca_T_2 +- k_T_off ⁢ Ca_CaT2 + k_T_on ⁢ Ca_2 ⁢ D_0 +- k_T_off ⁢ D_1 + k_T_on ⁢ Ca_2 ⁢ D_1 +- k_T_off ⁢ D_2 +- k_P_on ⁢ Ca_2 ⁢ P_tot +- Ca_P2 +- Mg_P2 +- k_P_off ⁢ Ca_P2 +- k_CATP_on ⁢ Ca_2 ⁢ ATP2 +- k_CATP_off ⁢ Ca_ATP2$$dd time Ca_SR2 = nu_SR ⁢ Ca_2 Ca_2 + K_SR V_SR2 +- L_e ⁢ Ca_SR2 +- Ca_2 V_SR2 + tau_SR_R ⁢ Ca_SR1 +- Ca_SR2 V_SR2 +- k_Cs_on ⁢ Ca_SR2 ⁢ Cs_tot +- Ca_Cs2 +- k_Cs_off ⁢ Ca_Cs2 - 1000 1 ⁢ A_p ⁢ P_SR ⁢ 0.001 1 ⁢ Ca_SR2 - PP ⁢ 1 if P_SR ⁢ 0.001 1 ⁢ Ca_SR2 - PP > 0 0 otherwise⁢ 0.001 1 ⁢ P_SR ⁢ Ca_SR2 - B_p ⁢ P_C_SR ⁢ PP - P_SR ⁢ 0.001 1 ⁢ Ca_SR2 ⁢ 1 if PP - P_SR ⁢ 0.001 1 ⁢ Ca_SR2 > 0 0 otherwise$$dd time Ca_T_2 = k_T_on ⁢ Ca_2 ⁢ T_0 +- k_T_off ⁢ Ca_T_2 +- k_T_on ⁢ Ca_2 ⁢ Ca_T_2 + k_T_off ⁢ Ca_CaT2 +- k_0_on ⁢ Ca_T_2 + k_0_off ⁢ D_1$$dd time Ca_P1 = k_P_on ⁢ Ca_1 ⁢ P_tot +- Ca_P1 +- Mg_P1 +- k_P_off ⁢ Ca_P1$$dd time Ca_P2 = k_P_on ⁢ Ca_2 ⁢ P_tot +- Ca_P2 +- Mg_P2 +- k_P_off ⁢ Ca_P2$$dd time Mg_P1 = k_Mg_on ⁢ P_tot +- Ca_P1 +- Mg_P1 ⁢ Mg1 +- k_Mg_off ⁢ Mg_P1$$dd time Mg_P2 = k_Mg_on ⁢ P_tot +- Ca_P2 +- Mg_P2 ⁢ Mg2 +- k_Mg_off ⁢ Mg_P2$$dd time Ca_Cs1 = k_Cs_on ⁢ Ca_SR1 ⁢ Cs_tot +- Ca_Cs1 +- k_Cs_off ⁢ Ca_Cs1$$dd time Ca_Cs2 = k_Cs_on ⁢ Ca_SR2 ⁢ Cs_tot +- Ca_Cs2 +- k_Cs_off ⁢ Ca_Cs2$$dd time Ca_ATP1 = k_CATP_on ⁢ Ca_1 ⁢ ATP1 +- k_CATP_off ⁢ Ca_ATP1 +- tau_ATP ⁢ Ca_ATP1 +- Ca_ATP2 V_1$$dd time Ca_ATP2 = k_CATP_on ⁢ Ca_2 ⁢ ATP2 +- k_CATP_off ⁢ Ca_ATP2 + tau_ATP ⁢ Ca_ATP1 +- Ca_ATP2 V_2$$dd time Mg_ATP1 = k_MATP_on ⁢ Mg1 ⁢ ATP1 +- k_MATP_off ⁢ Mg_ATP1 +- tau_ATP ⁢ Mg_ATP1 +- Mg_ATP2 V_1$$dd time Mg_ATP2 = k_MATP_on ⁢ Mg2 ⁢ ATP2 +- k_MATP_off ⁢ Mg_ATP2 + tau_ATP ⁢ Mg_ATP1 +- Mg_ATP2 V_2$$dd time ATP1 =- k_CATP_on ⁢ Ca_1 ⁢ ATP1 +- k_CATP_off ⁢ Ca_ATP1 +- k_MATP_on ⁢ Mg1 ⁢ ATP1 +- k_MATP_off ⁢ Mg_ATP1 +- tau_ATP ⁢ ATP1 +- ATP2 V_1$$dd time ATP2 =- k_CATP_on ⁢ Ca_2 ⁢ ATP2 +- k_CATP_off ⁢ Ca_ATP2 +- k_MATP_on ⁢ Mg2 ⁢ ATP2 +- k_MATP_off ⁢ Mg_ATP2 + tau_ATP ⁢ ATP1 +- ATP2 V_2$$dd time Mg1 =- k_Mg_on ⁢ P_tot +- Ca_P1 +- Mg_P1 ⁢ Mg1 +- k_Mg_off ⁢ Mg_P1 +- k_MATP_on ⁢ Mg1 ⁢ ATP1 +- k_MATP_off ⁢ Mg_ATP1 +- tau_Mg ⁢ Mg1 +- Mg2 V_1$$dd time Mg2 =- k_Mg_on ⁢ P_tot +- Ca_P2 +- Mg_P2 ⁢ Mg2 +- k_Mg_off ⁢ Mg_P2 +- k_MATP_on ⁢ Mg2 ⁢ ATP2 +- k_MATP_off ⁢ Mg_ATP2 + tau_Mg ⁢ Mg1 +- Mg2 V_2$$dd time Ca_CaT2 = k_T_on ⁢ Ca_2 ⁢ Ca_T_2 +- k_T_off ⁢ Ca_CaT2 +- k_Ca_on ⁢ Ca_CaT2 + k_Ca_off ⁢ D_2$$dd time D_0 =- k_T_on ⁢ Ca_2 ⁢ D_0 + k_T_off ⁢ D_1 + k_0_on ⁢ T_0 +- k_0_off ⁢ D_0$$dd time D_1 = k_T_on ⁢ Ca_2 ⁢ D_0 +- k_T_off ⁢ D_1 + k_0_on ⁢ Ca_T_2 +- k_0_off ⁢ D_1 +- k_T_on ⁢ Ca_2 ⁢ D_1 + k_T_off ⁢ D_2$$dd time D_2 = k_T_on ⁢ Ca_2 ⁢ D_1 +- k_T_off ⁢ D_2 + k_Ca_on ⁢ Ca_CaT2 +- k_Ca_off ⁢ D_2 +- f_o ⁢ D_2 + f_p ⁢ A_1 + g_o ⁢ A_2$$dd time A_1 = f_o ⁢ D_2 +- f_p ⁢ A_1 + h_p ⁢ A_2 +- h_o ⁢ A_1$$dd time A_2 =- h_p ⁢ A_2 + h_o ⁢ A_1 +- g_o ⁢ A_2$$dd time P = 0.001 1 ⁢ h_o ⁢ A_1 - h_p ⁢ A_2 + -1 ⁢ b_p ⁢ P + -1 ⁢ k_p ⁢ P - P_SR V_2$$dd time P_SR = k_p ⁢ P - P_SR V_SR2 - 1 ⁢ A_p ⁢ P_SR ⁢ 0.001 1 ⁢ Ca_SR2 - PP ⁢ 1 if P_SR ⁢ 0.001 1 ⁢ Ca_SR2 - PP > 0 0 otherwise⁢ 0.001 1 ⁢ P_SR ⁢ Ca_SR2 - B_p ⁢ P_C_SR ⁢ PP - P_SR ⁢ 0.001 1 ⁢ Ca_SR2 ⁢ 1 if PP - P_SR ⁢ 0.001 1 ⁢ Ca_SR2 > 0 0 otherwise$$dd time P_C_SR = 1 ⁢ A_p ⁢ P_SR ⁢ 0.001 1 ⁢ Ca_SR2 - PP ⁢ 1 if P_SR ⁢ 0.001 1 ⁢ Ca_SR2 - PP > 0 0 otherwise⁢ 0.001 1 ⁢ P_SR ⁢ Ca_SR2 - B_p ⁢ P_C_SR ⁢ PP - P_SR ⁢ 0.001 1 ⁢ Ca_SR2 ⁢ 1 if PP - P_SR ⁢ 0.001 1 ⁢ Ca_SR2 > 0 0 otherwise$
Source
Derived from workspace Shorten, Ocallaghan, Davidson, Soboleva, 2007 at changeset 33944b1d8ee3.
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