Model Mathematics

FoRT = \frac{F}{R T} RToF = \frac{R T}{F}

i_stim = \{\begin{matrix} stim_amplitude if time \geq stim_start \land time ≦ stim_end \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} bulk_volume = bulk_fraction cell_volume peripheral_volume = periphery_fraction cell_volume sr_volume = sr_fraction cell_volume diffusable_volume = (bulk_fraction + periphery_fraction) cell_volume \frac{d}{d time} (Vm) = \frac{(- 1) 1}{Cm} (i_k1 + i_to_fast + i_to_sustained + i_kr + i_ks + i_kb + i_nak + i_cal + i_cat + i_na_fast + i_na_late + i_pmca + i_cab + i_f_na + i_f_k + i_naca + i_nab + i_stim) \frac{d}{d time} (Ca_i_peripheral) = \frac{((-1) i_cal + (-1) i_cat + (-1) i_cab + 1 i_rel_per + (-1) i_serca_per + (-1) i_diff + 2 i_naca + (-1) i_pmca) 1000}{2 F peripheral_volume} \frac{d}{d time} (Ca_i_bulk) = \frac{((-1) i_serca_bulk + i_diff + i_leak) 1000}{2 F bulk_volume} \frac{d}{d time} (Ca_sr) = \frac{((-1) i_rel_per + (-1) i_leak + i_serca_bulk + i_serca_per) 1000}{2 F sr_volume} \frac{d}{d time} (K_i) = \frac{((-1) i_to_fast + (-1) i_to_sustained + (-1) i_kr + (-1) i_ks + (-1) i_k1 + (-1) i_kb + (-1) i_f_k + (-1) i_stim + 2 i_nak) 1000}{F diffusable_volume} \frac{d}{d time} (Na_i) = \frac{((-1) i_na_fast + (-1) i_na_late + (-3) i_nak + (-3) i_naca + (-1) i_f_na + (-1) i_nab) 1000}{F diffusable_volume}

x_inf_Ttype = \frac{1}{1 + ⅇ^{\frac{Vm + 47.8}{-5.5}}} \frac{d}{d time} (x_Ttype) = \frac{x_inf_Ttype - x_Ttype}{tau_x_Ttype}

y_inf_Ttype = \frac{1}{1 + ⅇ^{\frac{Vm + 67.9}{3.87}}} tau_y_Ttype = 1.42271 ⅇ^{(-0.05119) Vm} \frac{d}{d time} (y_Ttype) = \frac{y_inf_Ttype - y_Ttype}{tau_y_Ttype}

E_Ca = 0.5 RToF \ln (\frac{Ca_o}{Ca_i_peripheral}) i_cat = G_max_Ttype x_Ttype y_Ttype (Vm - E_Ca)

x_inf_Ltype = \frac{1}{1 + ⅇ^{\frac{Vm + 14.6}{-5.5}}} \frac{d}{d time} (x_Ltype) = \frac{x_inf_Ltype - x_Ltype}{tau_x_Ltype}

y_inf_Ltype = \frac{1}{1 + ⅇ^{\frac{Vm + 31.0}{5.54}}} tau_y_Ltype = \frac{25.1}{0.04 + 0.7 ⅇ^{(-1) {(0.028 (Vm + 14.5))}^{2.0}}} \frac{d}{d time} (y_Ltype) = \frac{y_inf_Ltype - y_Ltype}{tau_y_Ltype}

y_ca_inf_Ltype = 0.4 + \frac{0.6}{1 + {(\frac{Ca_i_peripheral}{0.0001})}^{2.0}} tau_y_ca_Ltype = 2.0 + \frac{80.0}{1 + {(\frac{Ca_i_peripheral}{0.0001})}^{2.0}} \frac{d}{d time} (y_ca_Ltype) = \frac{y_ca_inf_Ltype - y_ca_Ltype}{tau_y_ca_Ltype}

E_Ca = 0.5 RToF \ln (\frac{Ca_o}{Ca_i_peripheral}) i_cal = G_max_Ltype x_Ltype y_Ltype y_ca_Ltype (Vm - E_Ca)

x_inf_to_fast = \frac{1}{1 + ⅇ^{\frac{Vm + (-7.0)}{-9.0}}} \frac{d}{d time} (x_to_fast) = \frac{x_inf_to_fast - x_to_fast}{tau_x_to_fast}

y_inf_to_fast = \frac{1}{1 + ⅇ^{\frac{Vm + 27.5}{8.0}}} \frac{d}{d time} (y_to_fast) = \frac{y_inf_to_fast - y_to_fast}{tau_y_to_fast}

E_k = RToF \ln (\frac{K_o}{K_i}) i_to_fast = G_max_to_fast x_to_fast y_to_fast (Vm - E_k)

x_to_sustained = \frac{1}{1 + ⅇ^{\frac{5.0 - Vm}{17.0}}}

E_k = RToF \ln (\frac{K_o}{K_i}) i_to_sustained = G_max_to_sustained x_to_sustained (Vm - E_k)

x_inf_na_fast = \frac{1}{1 + ⅇ^{\frac{Vm + 25.0}{-5.0}}} \frac{d}{d time} (x_na_fast) = \frac{x_inf_na_fast - x_na_fast}{tau_x_na_fast}

y_inf_na_fast = \frac{1}{1 + ⅇ^{\frac{Vm + 69.0}{3.96}}} \frac{d}{d time} (y_na_fast) = \frac{y_inf_na_fast - y_na_fast}{tau_y_na_fast}

E_na = RToF \ln (\frac{Na_o}{Na_i}) i_na_fast = G_max_na_fast x_na_fast y_na_fast (Vm - E_na)

x_inf_na_late = \frac{1}{1 + ⅇ^{\frac{Vm + 30.0}{-5.0}}} \frac{d}{d time} (x_na_late) = \frac{x_inf_na_late - x_na_late}{tau_x_na_late}

y_inf_na_late = 0.1 + \frac{0.9}{1 + ⅇ^{\frac{Vm + 75.6}{6.3}}} tau_y_na_late = 120.0 + 1.0 ⅇ^{\frac{Vm + 100.0}{25.0}} \frac{d}{d time} (y_na_late) = \frac{y_inf_na_late - y_na_late}{tau_y_na_late}

E_na = RToF \ln (\frac{Na_o}{Na_i}) i_na_late = G_max_na_late x_na_late y_na_late (Vm - E_na)

x_k1 = \frac{1}{1 + ⅇ^{\frac{92.0 + Vm}{10.0}}}

E_k = RToF \ln (\frac{K_o}{K_i}) i_k1 = G_max_k1 {(\frac{K_o}{5.4})}^{0.8} x_k1 (Vm - E_k)

x_kr = \frac{1}{1 + ⅇ^{\frac{33.0 + Vm}{22.4}}}

y_inf_kr = \frac{1}{1 + ⅇ^{\frac{Vm + 50.0}{-7.5}}} ykrv1 = \{\begin{matrix} \frac{0.00138 1 (Vm + 7)}{1 - (ⅇ^{(-0.123) (Vm + 7)})} if | Vm + 7 | > 0.001 \\ \frac{0.00138}{0.123} otherwise \end{matrix} ykrv2 = \{\begin{matrix} \frac{0.000061 1 (Vm + 10)}{ⅇ^{0.145 (Vm + 10)} - 1} if | Vm + 10 | > 0.001 \\ \frac{0.00061}{0.145} otherwise \end{matrix} tau_y_kr = \frac{1}{ykrv1 + ykrv2} \frac{d}{d time} (y_kr) = \frac{y_inf_kr - y_kr}{tau_y_kr}

E_k = RToF \ln (\frac{K_o}{K_i}) i_kr = G_max_kr {(\frac{K_o}{5.4})}^{1.0} x_kr y_kr (Vm - E_k)

x_inf_ks = \frac{1}{1 + ⅇ^{\frac{Vm - 1.5}{-16.7}}} tau_x_ks = \{\begin{matrix} 417.9462 if | Vm + 30 | < 0.0145 \\ \frac{1}{\frac{0.0000719 (Vm + 30)}{1 - (ⅇ^{(-0.148) (Vm + 30)})} + \frac{0.000131 (Vm + 30)}{ⅇ^{0.0687 (Vm + 30)} - 1}} otherwise \end{matrix} \frac{d}{d time} (x_ks) = \frac{x_inf_ks - x_ks}{tau_x_ks}

y_inf_ks = x_inf_ks tau_y_ks = 4 tau_x_ks \frac{d}{d time} (y_ks) = \frac{y_inf_ks - y_ks}{tau_y_ks}

E_k = RToF \ln (\frac{K_o}{K_i}) i_ks = G_max_ks x_ks y_ks (Vm - E_k)

i_naca = \frac{512.009113204 g_NaCa (ⅇ^{\frac{gamma (n_NaCa - 2) Vm}{RToF}} {Nai}^{n_NaCa} Cao - (Cai ⅇ^{\frac{(gamma - 1) (n_NaCa - 2) Vm}{RToF}} {Nao}^{n_NaCa}))}{(1 + d_NaCa (Cai {Nao}^{n_NaCa} + Cao {Nai}^{n_NaCa})) (1 + \frac{Cai}{0.0069})}

x_nak = \frac{1}{1 + ⅇ^{\frac{Vm + 80.0}{-45.0}}}

y_nak = \frac{1}{1 + ⅇ^{\frac{Vm + 0.0}{125.0}}}

i_nak = g_nak x_nak y_nak \frac{1.0}{1.0 + {(\frac{1.9}{K_o})}^{1.45}} \frac{1.0}{1.0 + {(\frac{31.98}{Na_i})}^{1.0}}

E_k = RToF \ln (\frac{K_o}{K_i}) i_kb = G_max_kb (Vm - E_k)

E_na = RToF \ln (\frac{Na_o}{Na_i}) i_nab = G_max_nab (Vm - E_na)

E_ca = 0.5 RToF \ln (\frac{Ca_o}{Ca_i_peripheral}) i_cab = G_max_cab (Vm - E_ca)

i_pmca = PMCA_max \frac{1.0}{1.0 + {(\frac{Kpmca}{Ca_i_peripheral})}^{Hpmca}}

E_k = RToF \ln (\frac{K_o}{K_i}) i_f_k = G_f_k y_gate_f_k (Vm - E_k)

E_na = RToF \ln (\frac{Na_o}{Na_i}) i_f_na = G_f_na y_gate_f_na (Vm - E_na)

y_inf_f_gate = \frac{1.0}{1.0 + ⅇ^{\frac{Vm + 109.0}{10.0}}} tau_y_f_gate = \frac{6000.0}{ⅇ^{(-1.0) (2.9 + 0.04 Vm)} + ⅇ^{1.0 (3.6 + 0.11 Vm)}} \frac{d}{d time} (y_gate_f_k) = \frac{y_inf_f_gate - y_gate_f_k}{tau_y_f_gate}

y_inf_f_gate = \frac{1.0}{1.0 + ⅇ^{\frac{Vm + 109.0}{10.0}}} tau_y_f_gate = \frac{6000.0}{ⅇ^{(-1.0) (2.9 + 0.04 Vm)} + ⅇ^{1.0 (3.6 + 0.11 Vm)}} \frac{d}{d time} (y_gate_f_na) = \frac{y_inf_f_gate - y_gate_f_na}{tau_y_f_gate}

i_rel_per = REL_max \frac{Ca_sr}{1.0 + {(\frac{Krel}{Ca_i_peripheral})}^{2}}

i_serca_per = SERCA_max \frac{{(\frac{Ca_i_peripheral}{Kmf})}^{H} - ({(\frac{Ca_sr}{Kmr})}^{H})}{1.0 + {(\frac{Ca_i_peripheral}{Kmf})}^{H} + {(\frac{Ca_sr}{Kmr})}^{H}}

i_serca_bulk = SERCA_max \frac{{(\frac{Ca_i_bulk}{Kmf})}^{H} - ({(\frac{Ca_sr}{Kmr})}^{H})}{1.0 + {(\frac{Ca_i_bulk}{Kmf})}^{H} + {(\frac{Ca_sr}{Kmr})}^{H}}

i_leak = LEAK_max (Ca_sr - Ca_i_bulk)

i_diff = DIFF_max (Ca_i_peripheral - Ca_i_bulk)