Model Mathematics

\frac{d}{d time} (V) = \frac{- (i_CaL + i_CaT + i_Kr + i_Ks + i_to + i_sus + i_h + i_st + i_b_Na + i_K_ACh + i_NaK + i_NaCa)}{1}

E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai}) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki})

i_CaL = g_CaL (V - E_CaL) d f fCa

d_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 14.1)}{6}}} tau_d = \frac{1}{alpha_d + beta_d} alpha_d = \frac{(- 0.02839) (V + 35)}{ⅇ^{\frac{- (V + 35)}{2.5}} - 1} - (\frac{0.0849 V}{ⅇ^{\frac{- V}{4.808}} - 1}) beta_d = \frac{0.01143 (V - 5)}{ⅇ^{\frac{V - 5}{2.5}} - 1} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

f_infinity = \frac{1}{1 + ⅇ^{\frac{V + 30}{5}}} tau_f = 44.3 + 257.1 ⅇ^{- ({(\frac{V + 32.5}{13.9})}^{2})} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

alpha_fCa = Km_fCa beta_fCa fCa_infinity = \frac{Km_fCa}{Km_fCa + Ca_sub} tau_fCa = \frac{fCa_infinity}{alpha_fCa} \frac{d}{d time} (fCa) = \frac{fCa_infinity - fCa}{tau_fCa}

i_CaT = g_CaT (V - E_CaT) d f

d_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 26.3)}{6}}} tau_d = \frac{1}{1.068 ⅇ^{\frac{V + 26.3}{30}} + 1.068 ⅇ^{\frac{- (V + 26.3)}{30}}} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

f_infinity = \frac{1}{1 + ⅇ^{\frac{V + 61.7}{5.6}}} tau_f = \frac{1}{0.0153 ⅇ^{\frac{- (V + 61.7)}{83.3}} + 0.015 ⅇ^{\frac{V + 61.7}{15.38}}} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

g_Kr = 0.025 {(\frac{Ko}{1})}^{0.59} i_Kr = g_Kr (V - E_K) (0.6 paF + 0.4 paS) piy

pa_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 23.2)}{10.6}}} tau_paS = \frac{0.84655354}{0.0042 ⅇ^{\frac{V}{17}} + 0.00015 ⅇ^{\frac{- V}{21.6}}} tau_paF = \frac{0.84655354}{0.0372 ⅇ^{\frac{V}{15.9}} + 0.00096 ⅇ^{\frac{- V}{22.5}}} \frac{d}{d time} (paS) = \frac{pa_infinity - paS}{tau_paS} \frac{d}{d time} (paF) = \frac{pa_infinity - paF}{tau_paF}

pi_infinity = \frac{1}{1 + ⅇ^{\frac{V + 28.6}{17.1}}} tau_pi = \frac{1}{0.1 ⅇ^{\frac{- V}{54.645}} + 0.656 ⅇ^{\frac{V}{106.157}}} \frac{d}{d time} (piy) = \frac{pi_infinity - piy}{tau_pi}

E_Ks = \frac{R T}{F} \ln (\frac{Ko + 0.12 Nao}{Ki + 0.12 Nai}) i_Ks = g_Ks (V - E_Ks) n^{2}

n_infinity = \frac{alpha_n}{alpha_n + beta_n} tau_n = \frac{1}{alpha_n + beta_n} alpha_n = \frac{0.014}{1 + ⅇ^{\frac{- (V - 40)}{9}}} beta_n = 0.001 ⅇ^{\frac{- V}{45}} \frac{d}{d time} (n) = \frac{n_infinity - n}{tau_n}

i_to = g_to (V - E_K) q r i_sus = g_sus (V - E_K) r

q_infinity = \frac{1}{1 + ⅇ^{\frac{V + 49}{13}}} tau_q = 0.6 (\frac{65.17}{0.57 ⅇ^{(- 0.08) (V + 44)} + 0.065 ⅇ^{0.1 (V + 45.93)}} + 10.1) \frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q}

r_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 19.3)}{15}}} tau_r = 0.66 1.4 (\frac{15.59}{1.037 ⅇ^{0.09 (V + 30.61)} + 0.369 ⅇ^{(- 0.12) (V + 23.84)}} + 2.98) \frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r}

i_h_Na = g_h_Na (V - E_Na) y^{2} i_h_K = g_h_K (V - E_K) y^{2} i_h = i_h_Na + i_h_K

y_infinity = \frac{1}{1 + ⅇ^{\frac{V + 64}{13.5}}} tau_y = \frac{0.7166529}{ⅇ^{\frac{- (V + 386.9)}{45.302}} + ⅇ^{\frac{V - 73.08}{19.231}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

i_st = g_st (V - E_st) qa qi

qa_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 57)}{5}}} tau_qa = \frac{1}{alpha_qa + beta_qa} alpha_qa = \frac{1}{0.15 ⅇ^{\frac{- V}{11}} + 0.2 ⅇ^{\frac{- V}{700}}} beta_qa = \frac{1}{16 ⅇ^{\frac{V}{8}} + 15 ⅇ^{\frac{V}{50}}} \frac{d}{d time} (qa) = \frac{qa_infinity - qa}{tau_qa}

qi_infinity = \frac{alpha_qi}{alpha_qi + beta_qi} tau_qi = \frac{6.65}{alpha_qi + beta_qi} alpha_qi = \frac{1}{3100 ⅇ^{\frac{V}{13}} + 700 ⅇ^{\frac{V}{70}}} beta_qi = \frac{1}{95 ⅇ^{\frac{- V}{10}} + 50 ⅇ^{\frac{- V}{700}}} + \frac{0.000229}{1 + ⅇ^{\frac{- V}{5}}} \frac{d}{d time} (qi) = \frac{qi_infinity - qi}{tau_qi}

i_b_Na = g_b_Na (V - E_Na)

g_K_ACh = 0.0011 {(\frac{Ko}{1})}^{0.41} i_K_ACh = \frac{g_K_ACh (Ki - (Ko ⅇ^{\frac{(- V) F}{R T}}))}{1}

i_NaK = i_NaK_max {(1 + {(\frac{Km_Kp}{Ko})}^{1.2})}^{- 1} {(1 + {(\frac{Km_Nap}{Nai})}^{1.3})}^{- 1} {(1 + ⅇ^{\frac{- ((V - E_Na) + 120)}{30}})}^{- 1}

i_NaCa = \frac{kNaCa (x2 k21 - (x1 k12))}{x1 + x2 + x3 + x4} x1 = k41 k34 (k23 + k21) + k21 k32 (k43 + k41) x2 = k32 k43 (k14 + k12) + k41 k12 (k34 + k32) x3 = k14 k43 (k23 + k21) + k12 k23 (k43 + k41) x4 = k23 k34 (k14 + k12) + k14 k21 (k34 + k32) k43 = \frac{Nai}{K3ni + Nai} k12 = \frac{\frac{Ca_sub}{Kci} ⅇ^{\frac{(- Qci) V F}{R T}}}{di} k14 = \frac{\frac{\frac{Nai}{K1ni} Nai}{K2ni} (1 + \frac{Nai}{K3ni}) ⅇ^{\frac{Qn V F}{2 R T}}}{di} k41 = ⅇ^{\frac{(- Qn) V F}{2 R T}} di = 1 + \frac{Ca_sub}{Kci} (1 + ⅇ^{\frac{(- Qci) V F}{R T}} + \frac{Nai}{Kcni}) + \frac{Nai}{K1ni} (1 + \frac{Nai}{K2ni} (1 + \frac{Nai}{K3ni})) k34 = \frac{Nao}{K3no + Nao} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{Qco V F}{R T}}}{do} k23 = \frac{\frac{\frac{Nao}{K1no} Nao}{K2no} (1 + \frac{Nao}{K3no}) ⅇ^{\frac{(- Qn) V F}{2 R T}}}{do} k32 = ⅇ^{\frac{Qn V F}{2 R T}} do = 1 + \frac{Cao}{Kco} (1 + ⅇ^{\frac{Qco V F}{R T}}) + \frac{Nao}{K1no} (1 + \frac{Nao}{K2no}) (1 + \frac{Nao}{K3no})

j_Ca_dif = \frac{Ca_sub - Cai}{tau_dif_Ca} j_rel = \frac{P_rel (Ca_rel - Ca_sub)}{1 + {(\frac{K_rel}{Ca_sub})}^{2}} j_up = \frac{P_up}{1 + \frac{K_up}{Cai}} j_tr = \frac{Ca_up - Ca_rel}{tau_tr}

\frac{d}{d time} (Nai) = \frac{(- (i_h_Na + i_st + i_b_Na + 3 i_NaK + 3 i_NaCa)) Cm}{1 F (V_i + V_sub)} \frac{d}{d time} (Ki) = \frac{(- (i_Kr + i_Ks + i_to + i_sus + i_h_K + i_K_ACh + (- 2) i_NaK)) Cm}{1 F (V_i + V_sub)} \frac{d}{d time} (Cai) = \frac{j_Ca_dif V_sub - (j_up V_up)}{V_i} - (CM_tot delta_fCMi + TC_tot delta_fTC + TMC_tot delta_fTMC) \frac{d}{d time} (Ca_sub) = \frac{\frac{(- (i_CaL + i_CaT - (2 i_NaCa))) Cm}{1 2 F} + j_rel V_rel}{V_sub} - (j_Ca_dif + CM_tot delta_fCMs) \frac{d}{d time} (Ca_up) = j_up - (\frac{j_tr V_rel}{V_up}) \frac{d}{d time} (Ca_rel) = j_tr - (j_rel + CQ_tot delta_fCQ)

\frac{d}{d time} (fTC) = delta_fTC delta_fTC = kf_TC Cai (1 - fTC) - (kb_TC fTC) \frac{d}{d time} (fTMC) = delta_fTMC delta_fTMC = kf_TMC Cai (1 - (fTMC + fTMM)) - (kb_TMC fTMC) \frac{d}{d time} (fTMM) = delta_fTMM delta_fTMM = kf_TMM Mgi (1 - (fTMC + fTMM)) - (kb_TMM fTMM) \frac{d}{d time} (fCMi) = delta_fCMi delta_fCMi = kf_CM Cai (1 - fCMi) - (kb_CM fCMi) \frac{d}{d time} (fCMs) = delta_fCMs delta_fCMs = kf_CM Ca_sub (1 - fCMs) - (kb_CM fCMs) \frac{d}{d time} (fCQ) = delta_fCQ delta_fCQ = kf_CQ Ca_rel (1 - fCQ) - (kb_CQ fCQ)