# Model Mathematics

### Component: membrane

$dd time V =- 1.0 Cm ⁢ i_Na + i_Ca + i_to + i_Kr + i_Ks + i_K1 + i_NaCa + i_NaK + i_b_Na + i_b_Ca + I_st$

### 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

$alpha_h = 0.135 ⁢ⅇ 80.0 + V -6.8 if V < -40.0 0.0 otherwise beta_h = 3.56 ⁢ⅇ 0.079 ⁢ V + 31.0E5 ⁢ⅇ 0.35 ⁢ V if V < -40.0 1.0 0.13 ⁢ 1.0 +ⅇ- V + 10.66 11.1 otherwisedd time h = alpha_h ⁢ 1.0 - h - beta_h ⁢ h$

### 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: slow_calcium_current

$i_Ca = g_Ca_max ⁢ d ⁢ f ⁢ f_Ca ⁢ V - E_Ca E_Ca = R ⁢ T 2.0 ⁢ F ⁢ln⁡ Cao Cai$

### Component: slow_calcium_current_d_gate

$alpha_d = 14.98 16.68 ⁢ 2.0 ⁢π() 0.5 ⁢ⅇ- V - 22.36 16.68 2.0 2.0 beta_d = 0.1471 - 5.3 14.93 ⁢ 2.0 ⁢π() 0.5 ⁢ⅇ- V - 6.27 14.93 2.0 2.0 dd time d = alpha_d ⁢ 1.0 - d - beta_d ⁢ d$

### Component: slow_calcium_current_f_gate

$alpha_f = 6.87E-3 1.0 +ⅇ 6.1546 - V -6.12 beta_f = 0.069 ⁢ⅇ -0.11 ⁢ V + 9.825 + 0.011 1.0 +ⅇ -0.278 ⁢ V + 9.825 - 5.75E-4 dd time f = alpha_f ⁢ 1.0 - f - beta_f ⁢ f$

### Component: slow_calcium_current_f_Ca_gate

$f_Ca = 1.0 1.0 + Cai Km_Ca$

### Component: transient_outward_current

$i_to = g_to_max ⁢ r ⁢ t ⁢ V - E_to E_to = R ⁢ T F ⁢ln⁡ 0.043 ⁢ Nao + Ko 0.043 ⁢ Nai + Ki$

### Component: transient_outward_current_r_gate

$dd time r = alpha_r ⁢ 1.0 - r - beta_r ⁢ r F1_beta_r_e = 0.00006 ⁢ V alpha_r = F1_alpha_r F2_alpha_r F1_alpha_r = F1_alpha_r_a ⁢ⅇ F1_alpha_r_b ⁢ V + F1_alpha_r_c + F1_alpha_r_d + F1_alpha_r_e F2_alpha_r = F2_alpha_r_a ⁢ⅇ F2_alpha_r_b ⁢ V + F2_alpha_r_c + F2_alpha_r_d + F2_alpha_r_e beta_r = F1_beta_r F2_beta_r F1_beta_r = F1_beta_r_a ⁢ⅇ F1_beta_r_b ⁢ V + F1_beta_r_c + F1_beta_r_d + F1_beta_r_e F2_beta_r = F2_beta_r_a ⁢ⅇ F2_beta_r_b ⁢ V + F2_beta_r_c + F2_beta_r_d + F2_beta_r_e$

### Component: transient_outward_current_t_gate

$dd time t = alpha_t ⁢ 1.0 - t - beta_t ⁢ t F1_alpha_t_e = 0.00007 ⁢ V F1_beta_t_e = 0.0001215 ⁢ V alpha_t = F1_alpha_t F2_alpha_t F1_alpha_t = F1_alpha_t_a ⁢ⅇ F1_alpha_t_b ⁢ V + F1_alpha_t_c + F1_alpha_t_d + F1_alpha_t_e F2_alpha_t = F2_alpha_t_a ⁢ⅇ F2_alpha_t_b ⁢ V + F2_alpha_t_c + F2_alpha_t_d + F2_alpha_t_e beta_t = F1_beta_t F2_beta_t F1_beta_t = F1_beta_t_a ⁢ⅇ F1_beta_t_b ⁢ V + F1_beta_t_c + F1_beta_t_d + F1_beta_t_e F2_beta_t = F2_beta_t_a ⁢ⅇ F2_beta_t_b ⁢ V + F2_beta_t_c + F2_beta_t_d + F2_beta_t_e$

### Component: slowly_activating_delayed_rectifier_potassium_current

$i_Ks = g_Ks_max ⁢ Xs 2.0 ⁢ V - E_Ks E_Ks = R ⁢ T F ⁢ln⁡ 0.01833 ⁢ Nao + Ko ln⁡ 0.01833 ⁢ Nai + Ki$

### Component: slowly_activating_delayed_rectifier_potassium_current_Xs_gate

$dd time Xs = alpha_Xs ⁢ 1.0 - Xs - beta_Xs ⁢ Xs alpha_Xs = F1_alpha_Xs F2_alpha_Xs F1_alpha_Xs = F1_alpha_Xs_a ⁢ⅇ F1_alpha_Xs_b ⁢ V + F1_alpha_Xs_c + F1_alpha_Xs_d + F1_alpha_Xs_e F2_alpha_Xs = F2_alpha_Xs_a ⁢ⅇ F2_alpha_Xs_b ⁢ V + F2_alpha_Xs_c + F2_alpha_Xs_d + F2_alpha_Xs_e beta_Xs = F1_beta_Xs F2_beta_Xs F1_beta_Xs = F1_beta_Xs_a ⁢ⅇ F1_beta_Xs_b ⁢ V + F1_beta_Xs_c + F1_beta_Xs_d + F1_beta_Xs_e F2_beta_Xs = F2_beta_Xs_a ⁢ⅇ F2_beta_Xs_b ⁢ V + F2_beta_Xs_c + F2_beta_Xs_d + F2_beta_Xs_e$

### Component: rapidly_activating_delayed_rectifier_potassium_current

$i_Kr = g_Kr_max ⁢ 1.0 FCIKr ⁢ Xr ⁢ V - E_K E_K = R ⁢ T F ⁢ln⁡ Ko Ki FCIKr = FCIKr_a ⁢ⅇ FCIKr_b ⁢ V + FCIKr_c + FCIKr_d + FCIKr_e$

### Component: rapidly_activating_delayed_rectifier_potassium_current_Xr_gate

$dd time Xr = alpha_Xr ⁢ 1.0 - Xr - beta_Xr ⁢ Xr alpha_Xr = F1_alpha_Xr F2_alpha_Xr F1_alpha_Xr = F1_alpha_Xr_a ⁢ⅇ F1_alpha_Xr_b ⁢ V + F1_alpha_Xr_c + F1_alpha_Xr_d + F1_alpha_Xr_e F2_alpha_Xr = F2_alpha_Xr_a ⁢ⅇ F2_alpha_Xr_b ⁢ V + F2_alpha_Xr_c + F2_alpha_Xr_d + F2_alpha_Xr_e beta_Xr = F1_beta_Xr F2_beta_Xr F1_beta_Xr = F1_beta_Xr_a ⁢ⅇ F1_beta_Xr_b ⁢ V + F1_beta_Xr_c + F1_beta_Xr_d + F1_beta_Xr_e F2_beta_Xr = F2_beta_Xr_a ⁢ⅇ F2_beta_Xr_b ⁢ V + F2_beta_Xr_c + F2_beta_Xr_d + F2_beta_Xr_e$

### Component: inward_rectifier_potassium_current

$i_K1 = g_K1_max ⁢ alpha_K1 alpha_K1 + beta_K1 ⁢ V - E_K + Vs$

### Component: inward_rectifier_potassium_current_K1_gate

$F2_alpha_K1_c = 206.0 - E_K F2_beta_K1_c = -16.0 - E_K F1_beta_K1_c = 94.0 - E_K F3_beta_K1_c = 4.0 - E_K alpha_K1 = F1_alpha_K1 F2_alpha_K1 F1_alpha_K1 = F1_alpha_K1_a ⁢ⅇ F1_alpha_K1_b ⁢ V + F1_alpha_K1_c + F1_alpha_K1_d + F1_alpha_K1_e F2_alpha_K1 = F2_alpha_K1_a ⁢ⅇ F2_alpha_K1_b ⁢ V + F2_alpha_K1_c + F2_alpha_K1_d + F2_alpha_K1_e beta_K1 = F1_beta_K1 + F2_beta_K1 F3_beta_K1 F1_beta_K1 = F1_beta_K1_a ⁢ⅇ F1_beta_K1_b ⁢ V + F1_beta_K1_c + F1_beta_K1_d + F1_beta_K1_e F2_beta_K1 = F2_beta_K1_a ⁢ⅇ F2_beta_K1_b ⁢ V + F2_beta_K1_c + F2_beta_K1_d + F2_beta_K1_e F3_beta_K1 = F3_beta_K1_a ⁢ⅇ F3_beta_K1_b ⁢ V + F3_beta_K1_c + F3_beta_K1_d + F3_beta_K1_e$

### Component: calcium_background_current

$i_b_Ca = g_b_Ca_max ⁢ V - E_Ca$

### Component: sodium_background_current

$i_b_Na = g_b_Na_max ⁢ V - E_Na$

### 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: 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: CICR_of_JSR

$i_rel = G_rel ⁢ Ca_JSR - Cai G_rel = G_rel_max ⁢ delta_Ca_i2 - delta_Ca_ith K_mrel + delta_Ca_i2 - delta_Ca_ith ⁢ 1.0 -ⅇ- time tau_on ⁢ⅇ- time tau_off if calcium_overload = 0.0 G_rel_max ⁢ Ca_JSR - Cai ⁢ 1.0 -ⅇ- time tau_on ⁢ⅇ- time tau_off otherwise$

### Component: Ca_uptake_of_NSR

$i_up = I_up ⁢ Cai Cai + K_mup$

### Component: Ca_leakage_of_NSR

$i_leak = K_leak ⁢ Ca_NSR$

### Component: translocation_of_Ca_from_NSR_to_JSR

$i_tr = Ca_NSR - Ca_JSR tau_tr$

### Component: calcium_buffers_in_the_myoplasm

$Tn_buff = Tn_max ⁢ Cai Cai + K_mTn CMDN_buff = CMDN_max ⁢ Cai Cai + K_mCMDN$

### Component: calcium_buffers_in_the_JSR

$CSQN_buff = CSQN_max ⁢ Ca_JSR Ca_JSR + K_mCSQN$

### Component: ionic_concentrations

$dd time Nai =- i_Na + i_b_Na + i_NaCa ⁢ 3.0 + i_NaK ⁢ 3.0 ⁢ A_cap V_myo ⁢ F dd time Cai = i_Ca + i_b_Ca - i_NaCa ⁢ A_cap 2.0 ⁢ V_myo ⁢ F + i_rel ⁢ V_JSR V_myo + i_leak - i_up ⁢ V_NSR V_myo dd time Ki =- i_to + i_Kr + i_K1 + i_Ks +- i_NaK ⁢ 2.0 ⁢ A_cap V_myo ⁢ F dd time Ko = i_to + i_Kr + i_K1 + i_Ks +- i_NaK ⁢ 2.0 ⁢ A_cap V_cleft ⁢ F dd time Ca_JSR =- i_rel - i_tr ⁢ V_NSR V_JSR dd time Ca_NSR =- i_leak + i_tr - i_up dd time Ca_foot =- i_Ca ⁢ A_cap 2.0 ⁢ V_myo ⁢ F ⁢ R_A_V$

### Component: TCa

$∂∂timeTCa= kon ⁢ 1.0 + S_A_M + lamda TCa_lamda ⁢ Cai ⁢ T - koff ⁢ TCa$

### Component: TMon

$∂∂timeTMon= kmon ⁢ TCa ⁢ 1.0 + TMoncoop + lamda ⁢ TMon TMonpow ⁢ TMoff - kmoff ⁢ TMon$

### Component: M_ATP

$∂∂timeM_ATP= t2 + t14 - t3$

$∂∂timeM_ADP_Pi= t3 + t13 - t4$

$∂∂timeA_M__ADP_Pi= t4 - t5$

$∂∂timeA_M_ADP_Pi= t5 - t6 + t13$

$∂∂timeA_M__ADP= t6 - t7 + t11$

$∂∂timeA_M_ADP= t7 - t8 + t10$

### Component: A_M

$∂∂timeA_M= t8 - t9 + t1$

### Component: A__M_ATP

$∂∂timeA__M_ATP= t1 - t2$

### Component: M

$∂∂timeM= t9 + t12 - t14$

$∂∂timeM_ADP= t10 + t11 - t12$

### Component: S_A_M

$S_A_M= A_M_ADP_Pi + A_M__ADP + A_M_ADP + A_M$

### Component: t1

$t1= k1 ⁢ ATP ⁢ A_M - k_1 ⁢ A__M_ATP$

### Component: t2

$t2= k2 ⁢ 1.0 + v_factor ⁢ v_detach ⁢ A__M_ATP$

### Component: t3

$t3= k3 ⁢ M_ATP - k_3 ⁢ M_ADP_Pi$

### Component: t4

$t4= k4 ⁢ M_ADP_Pi - k_4 ⁢ 1.0 + v_factor ⁢ v_detach ⁢ A_M__ADP_Pi$

### Component: t5

$t5= k5 ⁢ TMon ⁢ k5_lamda ⁢ lamda + 0.4 ⁢ 1.0 + k5_xb ⁢ S_A_M 2.0 ⁢ A_M__ADP_Pi - k_5 ⁢ A_M_ADP_Pi$

### Component: t6

$t6= k6 ⁢ A_M_ADP_Pi - k_6 ⁢ A_M__ADP$

### Component: t7

$t7= k7 ⁢ A_M__ADP ⁢ k7_base - k7_lamda ⁢ lamda + v 1.0 + k7_force ⁢ F$

### Component: t8

$t8= k8 ⁢ A_M_ADP - k_8 ⁢ A_M$

### Component: t9

$t9= k9 ⁢ v_factor ⁢ A_M$

### Component: t10

$t10= k10 ⁢ v_factor ⁢ A_M_ADP$

### Component: t11

$t11= k11 ⁢ v_factor ⁢ A_M__ADP$

### Component: t12

$t12= k12 ⁢ M_ADP$

### Component: t13

$t13= k13 ⁢ v_factor ⁢ A_M_ADP_Pi$

### Component: t14

$t14= k14 ⁢ ATP ⁢ M$

### Component: v_factor

$v_factor= v Nv v Nv + v50 Nv$

### Component: F

$F= Fmax ⁢ Cai nh Cai nh + Ca50 nh$

### Component: parameters

Source
Derived from workspace Seemann, Sachse, Weib, Dossel, 2003 at changeset 77a005a9163c.
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