# Model Mathematics

### Component: intracellular_space

$dd time V_i = Lie_p ⁢ pi_i - pi_e pi_i = R ⁢ T ⁢ Na_i + K_i + Cl_i + X_i V_i pi_e = R ⁢ T ⁢ Na_e + K_e + Cl_e + X_e V_e delta_init = delta_i - delta_e dd time delta = V_i ⁢ F ⁢ R ⁢ T ⁢ Na_i + K_i - Cl_i + z_i ⁢ X_i V_i C_m ⁢ A_s X_i = V_i ⁢ Na_i + K_i - Cl_i z_i$

### Component: intracellular_sodium

$dd time Na_i =- g_Na F ⁢ delta - delta_Na + 3.0 ⁢ P g_Na = g_b_Na + g_Na_max ⁢ m_infinity 3.0 ⁢ h_infinity delta_Na = R ⁢ T F ⁢ln⁡ Na_e Na_i$

### Component: intracellular_sodium_m_gate

$A_m = 0.1 ⁢ 25.0 - delta + delta_eq ⅇ 0.1 ⁢ 25.0 - delta + delta_eq - 1.0 B_m = 4.0 ⁢ⅇ- delta + delta_eq 18.0 m_infinity = A_m A_m + B_m$

### Component: intracellular_sodium_h_gate

$A_h = 0.07 ⁢ⅇ 0.05 ⁢- delta + delta_eq B_h = 1.0 ⅇ 0.1 ⁢ 30.0 - delta + delta_eq + 1.0 h_infinity = A_h A_h + B_h$

### Component: intracellular_potassium

$dd time K_i =- g_K F ⁢ delta - delta_K - 2.0 ⁢ P g_K = g_b_K + g_K_max ⁢ n_infinity 4.0 delta_K = R ⁢ T F ⁢ln⁡ K_e K_i$

### Component: intracellular_potassium_n_gate

$A_n = 0.01 ⁢ 10.0 - delta + delta_eq ⅇ 0.1 ⁢ 10.0 - delta + delta_eq - 1.0 B_n = 0.125 ⁢ⅇ- delta + delta_eq 80.0 n_infinity = A_n A_n + B_n$

### Component: intracellular_chloride

$dd time Cl_i = g_Cl F ⁢ delta + delta_Cl delta_Cl = R ⁢ T -1.0 ⁢ F ⁢ln⁡ Cl_e Cl_i$

### Component: extracellular_sodium

$dd time Na_e = g_Na F ⁢ delta - delta_Na + 3.0 ⁢ P - Jv_Na + Jh_Na$

### Component: extracellular_potassium

$dd time K_e = g_K F ⁢ delta - delta_K - 2.0 ⁢ P - Jv_K + Jh_K$

### Component: extracellular_chloride

$dd time Cl_e =- g_Cl F ⁢ delta + delta_Cl - Jv_Cl + Jh_Cl$

### Component: extracellular_space

$dd time V_e = Lie_p ⁢ pi_e - pi_i - Jv_H2O + Jh_H2O X_e = V_e ⁢ Na_e + K_e - Cl_e z_e$

### Component: water_flux

$Jh_H2O =- 1.0 tau_1 ⁢ V_infinity_e - V_e + 1.0 tau_2 ⁢ V_infinity_T - V_i + V_e$

### Component: sodium_flux

$Jh_Na = P_h ⁢ alpha_Na ⁢ Na_infinity_e - Na_e ⁢ⅇ alpha_Na 1.0 -ⅇ alpha_Na alpha_Na = Jh_H2O P_h - 1.0 ⁢ F R ⁢ T ⁢ delta_eh$

### Component: potassium_flux

$Jh_K = P_h ⁢ alpha_K ⁢ K_infinity_e - K_e ⁢ⅇ alpha_K 1.0 -ⅇ alpha_K alpha_K = Jh_H2O P_h - 1.0 ⁢ F R ⁢ T ⁢ delta_eh$

### Component: chloride_flux

$Jh_Cl = P_h ⁢ alpha_Cl ⁢ Cl_infinity_e - Cl_e ⁢ⅇ alpha_Cl 1.0 -ⅇ alpha_Cl alpha_Cl = Jh_H2O P_h - -1.0 ⁢ F R ⁢ T ⁢ delta_eh$

### Component: capillary_compartment_sodium

$Na_v = Qa ⁢ Na_a + Jv_Na Qv delta_ev_Na = R ⁢ T F ⁢ln⁡ Na_v Na_e Jv_Na = Qv ⁢ Na_v - Qa ⁢ Na_a$

### Component: capillary_compartment_potassium

$K_v = Qa ⁢ K_a + Jv_K Qv delta_ev_K = R ⁢ T F ⁢ln⁡ K_v K_e Jv_K = Qv ⁢ K_v - Qa ⁢ K_a$

### Component: capillary_compartment_chloride

$Cl_v = Qa ⁢ Cl_a + Jv_Cl Qv delta_ev_Cl = R ⁢ T F ⁢ln⁡ Cl_v Cl_e Jv_Cl = Qv ⁢ Cl_v - Qa ⁢ Cl_a$

### Component: capillary_space

$Qv = Qa + Jv_H2O Jv_H2O = Qv - Qa pi_v = Na_v + K_v + Cl_v + Pr_v Pr_v = Na_v + K_v +- Cl_v z_v$

### Component: arterial_space

$Qa = Qv ⁢ Pr_v Pr_a$

### Component: pump_rate

$P = a_t ⁢ P_max ⁢ f_NaK ⁢ 1.0 1.0 + Km_Na_i Na_i 1.5 ⁢ K_e Km_K_e + K_e f_NaK = 1.0 1.0 + 0.1245 ⁢ⅇ -0.1 ⁢ delta ⁢ F R ⁢ T + 0.0365 ⁢ 7.0 ⁢ⅇ -0.1 ⁢ delta ⁢ F R ⁢ T ⁢ⅇ Na_e 67.3 - 1.0 a_t = 0.35 time 15.0$
Source
Derived from workspace Yi, Fogelson, Keener, Peskin, 2003 at changeset 022023598e97.
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