Model Mathematics

\frac{d}{d time} (Vm_n) = \frac{0.001 ((- (i_NaT + i_NaP + i_leakNa + i_KDR + i_KA + i_leakK + i_leakf + i_NaKATPase_n)) + i_app)}{Cm} i_app = \{\begin{matrix} 4500 if time \geq t0 \land time ≦ t1 \\ 0 otherwise \end{matrix}

i_NaT = gNaT m^{3} h (Vm_n - ENa_n) J_NaT = \frac{gNaT m^{3} h (Vm_n - ENa_n)}{F}

alpha_m = \frac{0.32 (- Vm_n - 51.9)}{ⅇ^{- (0.25 Vm_n + 12.975)} - 1} beta_m = \frac{0.28 (Vm_n + 24.89)}{ⅇ^{0.2 Vm_n + 4.978} - 1} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = 0.128 ⅇ^{- (0.056 Vm_n + 2.94)} beta_h = \frac{4}{ⅇ^{- (0.2 Vm_n + 6)} + 1} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

i_NaP = gNaP m^{2} h (Vm_n - ENa_n) J_NaP = \frac{gNaP m^{2} h (Vm_n - ENa_n)}{F}

alpha_m = \frac{\frac{1}{tau_activation} 1}{ⅇ^{- (0.143 Vm_n + 5.67)} + 1} beta_m = \frac{\frac{1}{tau_activation} ⅇ^{- (0.143 Vm_n + 5.67)}}{ⅇ^{- (0.143 Vm_n + 5.67)} + 1} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = 5.12 e -8 ⅇ^{- (0.056 Vm_n + 2.94)} beta_h = \frac{1.6 e -6}{ⅇ^{- (0.2 Vm_n + 8)} + 1} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

i_KDR = gKDR n^{2} (Vm_n - EK_n) J_KDR = \frac{gKDR n^{2} (Vm_n - EK_n)}{F}

alpha_n = \frac{0.016 (- Vm_n - 34.9)}{ⅇ^{- (0.2 Vm_n + 6.98)} - 1} beta_n = 0.25 ⅇ^{- (0.025 Vm_n + 1.25)} \frac{d}{d time} (n) = alpha_n (1 - n) - (beta_n n)

i_KA = gKA m^{2} h (Vm_n - EK_n) J_KA = \frac{gKA m^{2} h (Vm_n - EK_n)}{F}

alpha_m = \frac{0.02 (- Vm_n - 56.9)}{ⅇ^{- (0.1 Vm_n + 5.69)} - 1} beta_m = \frac{0.0175 (Vm_n + 29.9)}{ⅇ^{0.1 Vm_n + 2.99} - 1} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = 0.016 ⅇ^{- (0.056 Vm_n + 4.61)} beta_h = \frac{0.5}{ⅇ^{- (0.2 Vm_n + 11.98)} + 1} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

i_NaKATPase_n = J_NaKATPase_n F J_NaKATPase_n = \frac{\frac{I_NaKATPase_n_max {Nan}^{1.5}}{{Nan}^{1.5} + {KmNa}^{1.5}} Ko}{Ko + KmK}

i_leakNa = gleakNa (Vm_n - ENa_n) J_leakNa = \frac{gleakNa (Vm_n - ENa_n)}{F}

i_leakK = gleakK (Vm_n - EK_n) J_leakK = \frac{gleakK (Vm_n - EK_n)}{F}

i_leakf = gleakf (Vm_n - Ef_n)

Vm_g = \frac{gNa ENa_g + gK EK_g + gCl ECl_g + gNBC ENBC_g}{gNa + gK + gCl + gNBC} - (\frac{F J_NaKATPase_g}{gNa + gK + gCl + gNBC})

J_Na = \frac{gNa (Vm_g - ENa_g)}{F}

J_K = \frac{gK (Vm_g - EK_g)}{F}

J_NaKATPase_g = \frac{\frac{I_NaKATPase_g_max {Nag}^{1.5}}{{Nag}^{1.5} + {KmNa}^{1.5}} Ko}{Ko + KmK}

J_NBC = \frac{gNBC (Vm_g - ENBC_g)}{F}

J_NKCC1 = \{\begin{matrix} \frac{\frac{{(Ko - P_Ko)}^{10}}{{(Ko - P_Ko)}^{10} + {0.03}^{10}} Psi gNKCC1 \ln (\frac{\frac{Ko}{Kg} Nao}{Nag} {(\frac{Clo}{Clg})}^{2})}{F} if Ko > P_Ko \\ 0 otherwise \end{matrix}

\frac{d}{d time} (N_Nag) = dN_Nag_dt dN_Nag_dt = 0.01 ((- J_Na - (3 J_NaKATPase_g)) + J_NKCC1 + J_NBC)

\frac{d}{d time} (N_Kg) = dN_Kg_dt dN_Kg_dt = 0.01 ((- J_K) + 2 J_NaKATPase_g + J_NKCC1)

\frac{d}{d time} (wg) = 10 Lp (Nag + Kg + Clg + HCO3g + \frac{Xg}{wg} - (Nao + Ko + Clo + HCO3o))

wo = P_wg + P_wo - wg

\frac{d}{d time} (N_Nao) = 0.01 (3 J_NaKATPase_n + J_NaT + J_NaP + J_leakNa - (100 dN_Nag_dt))

\frac{d}{d time} (N_Ko) = (-0.01) (J_NaT + J_NaP + J_leakNa + 3 J_NaKATPase_n) - dN_Kg_dt

\frac{d}{d time} (N_HCO3o) = (- 0.01) 2 J_NBC

ENa_n = Psi \ln (\frac{Nao}{Nan}) EK_n = Psi \ln (\frac{Ko}{Kn}) ENa_g = Psi \ln (\frac{Nao}{Nag}) EK_g = Psi \ln (\frac{Ko}{Kg}) ECl_g = (- 1) Psi \ln (\frac{Clo}{Clg}) ENBC_g = (- Psi) \ln (\frac{Nao}{Nag} {(\frac{HCO3o}{HCO3g})}^{2})

P_Clo = P_Nao + P_Ko - P_HCO3o P_HCO3g = P_HCO3o \sqrt{\frac{P_Nao}{P_Nag} ⅇ^{\frac{P_Vm_g}{Psi}}} P_Clg = P_Clo ⅇ^{\frac{P_Vm_g}{Psi}} Kg = \frac{N_Kg}{wg} Ko = \frac{N_Ko}{wo} Kn = \frac{P_wn P_Kn + P_wg P_Kg + P_wo P_Ko - (N_Ko + N_Kg)}{P_wn} Nag = \frac{N_Nag}{wg} Nao = \frac{N_Nao}{wo} Nan = \frac{P_wn P_Nan + P_wg P_Nag + P_wo P_Nao - (N_Nao + N_Nag)}{P_wn} Clg = Nag + Kg - (HCO3g + \frac{rho Xg}{wg}) Cln = (P_Cln + Nan - P_Nan) + (Kn - P_Kn) Clo = \frac{P_wg P_Clg + P_wo P_Clo + P_wn P_Cln - (wg Clg + P_wn Cln)}{wo} HCO3o = \frac{N_HCO3o}{wo} HCO3g = \frac{P_wg P_HCO3g + P_wo P_HCO3o - N_HCO3o}{wg}

Psi = \frac{1000 R T}{F}