Flanagan, Bronac

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  ċ                                                         γ  γ  γ     γ                                                          (                αα                                                     CmdVmdt= −INa − IK − Ileak − Iapp        (3-1ci) ci =  cfast  +  cslow  ⇒dcidt=dcfastdt+dcslowdtdcfastdt= −JCa +  JPMleak  − JPMCa  dcslowdt= −Jchan −  JERpump −  JERleak JCa, JPMleak  JPMCaJchan, JERpump  IERleak   dGlucleftdt=  Jrelease −  Juptake −  Jdiffusion  (3-5)d[Glu]dt= [Glu]rel − g[Glu]                                                                                   Vmax Km)VEAAT = [Glu−]Vmax[Glu−] + KM         μd[Glu−]dt = {−[Glu−] + ([Ca2+] −  [Ca2+]th) − κλ    if [Ca2+] >  [Ca2+]th−[Glu−] − κλ                                                                  otherwise      μκλ,               φ)φ = −RT2Fln ([Glu−]𝑖 [Glu−]o)    [Glu−]o(eq)   =  [Glu−]i ([Na+]i[Na+]o)2([K+]o[K+]i) ([OH−]o[OH−]i)  exp (VFRT) V)              μS,i) μS,i = 𝜇S0 + RT ln([S]i) + zFViμS,e = μS0 + RT ln([S]e) + zFVe   μμ∆μS = μS,i − μS,e = RT ln[S]i[S]e+ zF(Vi − Ve) = RT ln[S]i[S]e+ zFV Vrev =RTzFln[S]e[S]i  Vrev =RT(z1+z2+z3+⋯ )Fln ([S1]e[S1]i){n1}([S2]e[S2]i){n2}([S3]e[S3]i){n3}…     d[X]astdt= −IX,astzFSAmemVolastd[X]syndt=IX,astzFSAmemVolsyn           γA,τAdγA(t)dt= −γA(t) + OM(1 − ζ)Glusyn(t)(1 − γa(t))τAτζ ββ δ δddtIP3(t) =  Jβ(γA(t)) + Jδ (Caast,soma(t), IP3(t)) −J3K (Caast,soma(t), IP3(t)) − J5P(IP3(t))  Jβ(γA(t)) = OβγA(t)Jδ (Caast,soma(t), IP3(t)) = Oδ κδκδ + IP3(t)ℋ(Caast,soma(t)2, Kδ)  J3K (Caast,soma(t), IP3(t)) =  O3Kℋ(Caast,soma(t)4, KD)ℋ(IP3(t), K3) J5P(IP3(t)) = Ω5PIP3(t)   ℋ(xn, K)  xnxn+Knμ𝛕𝐀      𝟎. 𝟓𝟓  𝐎𝐌     𝟎. 𝟑 × 𝟏𝟎𝟑 μ  𝛇  𝟎. 𝟕𝟓  𝐎𝛃    β  𝟏 μ  𝐎𝛅 δ  𝟎. 𝟎𝟓 μ  𝛋𝛅    δ  𝟏 μ  𝐊𝛅 δ  𝟎. 𝟓 μ  𝐎𝟑𝐊     𝟒. 𝟓 μ  𝐊𝐃      𝟎. 𝟓 μ  𝐊𝟑      𝟏 μ  𝛀𝟓𝐏     𝟎. 𝟏     ddtCaast,soma =  Jchan + Jleak − JSERCAhτhddth(t) =  h∞ − h(t)h∞ = d2IP3(t) + d1d2(IP3(t) + d1) + (IP3(t) + d3)Caast,soma(t) τh =IP3(t)+d3Ω2(IP3(t)+d1)+O2(IP3(t)+d3)Caast,soma(t) Jchan, Jchan = ΩCm(t)∞3 h(t)3(CT − (1 + ρA)Caast,soma(t))  m∞3 = ℋ(IP3(t), d1)ℋ(Caast,soma(t), d5)                      Jleak = ΩL (CT − (1 − ρA)Caast,soma(t))                        JSERCA = Opℋ(Caast,soma(t)2, Kp) ddtCaast,process =  Jchan + Jleak −1032F(23INCX)   𝛀𝐋   𝟎. 𝟏𝐂𝐓     𝟐 μ  𝛒𝐀      𝟎. 𝟏𝟖  𝐝𝟏     𝟎. 𝟏𝟑 μ  𝐝𝟐     𝟏. 𝟎𝟓 μ  𝐝𝟑     𝟎. 𝟗𝟒𝟑𝟒 μ  𝐝𝟓    𝟎. 𝟎𝟖 μ  𝐎𝟐  𝟎. 𝟐 μ  𝛀𝟐  𝟎. 𝟐𝟏 μ  𝛀𝐂  𝟔𝐊𝐏     𝟎. 𝟎𝟓 μ  𝐎𝐏   𝟎. 𝟗 μ   α βdrNMDAdt= αNMDAGlusyn(1 − rNMDA) − βNMDArNMDAMGV  =  (1 + e−0.062Vm[MG]3.57 )−1   INMDA  =  gNMDA rNMDA (Vm − ENMDA) MGV α β  drAMPAdt= αAMPAGlusyn(1 − rAMPA) − βAMPArAMPA IAMPA  =  gAMPA rAMPA (Vm − EAMPA) α βdrGABAAdt= αGABAAGABAsyn(1 − rGABAA) − βGABAArGABAA                    IGABAA  =  gGABAA rGABAA (Vm − EGABAA)   𝐠𝐍𝐌𝐃𝐀 𝟎. 𝟎𝟏𝟖𝐠𝐀𝐌𝐏𝐀 𝟎. 𝟎𝟐𝟔𝐠𝐆𝐀𝐁𝐀𝐀 𝟎. 𝟎𝟓𝐄𝐆𝐀𝐁𝐀𝐀 −𝟖𝟓𝐄𝐀𝐌𝐏𝐀 𝟎𝐄𝐍𝐌𝐃𝐀 𝟎𝛂𝐆𝐀𝐁𝐀𝐀 𝟓 × 𝟏𝟎𝟐𝛂𝐀𝐌𝐏𝐀 𝟏. 𝟏 × 𝟏𝟎𝟑𝛂𝐍𝐌𝐃𝐀 𝟕𝟐𝛃𝐆𝐀𝐁𝐀𝐀 𝟎. 𝟎𝟒𝟓  𝛃𝐀𝐌𝐏𝐀 𝟎. 𝟏𝟗𝟎𝛃𝐍𝐌𝐃𝐀 𝟎. 𝟎𝟎𝟔𝟔    dxdt=zτrec− Usex δ(t − tsp)  dydt= −yτin+ Usex δ(t − tsp)  dzdt=yτin−ztaurec  𝛕𝐟𝐚𝐜𝐢𝐥 𝟎. 𝟓𝟑 𝐬𝐞𝐜𝛕𝐢𝐧 𝟎. 𝟎𝟎𝟑 𝐬𝐞𝐜𝛕𝐫𝐞𝐜 𝟎. 𝟖𝟎𝟎 𝐬𝐞𝐜𝐔𝐬𝐞 𝟎. 𝟓 ~   θ μ tjδrA(t) = UAxA(t) τGddtxA(t) = 1 − xA(t) − rA(t)δ(t − tj)τG τGGrel(t) = ρeGTrA(t)                                                          ρe  GTGT = ρGG                                                                         ρG =Gluast,eqGluast,norm          τEddtGluA(t) = −GluA(t) + Grel(t)δ(t − tj)τE τ   INKA(Naast, Ksyn) =PATPAse,maxF(Naast1.5Naast1.5 +KNai1.5 ) (KsynKsyn+ KKE)       IKir = gKir√Ksyn(Va − EKir)     INCX = INCX,max ((NaastNasyn)3exp (γFVa103RT) − (Caast,process⌈Ca2+⌉syn) exp ((1 − γ)FVa103RT)) γ 𝐍𝐊𝐀  𝟏. 𝟏𝟐 × 𝟏𝟎−𝟔𝟏𝟎𝟏. 𝟓𝟎. 𝟎𝟏γ 𝟎. 𝟓  𝟏𝟒𝟒𝟎. 𝟎𝟎𝟐𝟓   𝟐. 𝟏 × 𝟏𝟎𝟐    VrevGAT  =RTF log (NasynNaast)2 GABAsynGABAast (ClsynClast)−1        IGAT  =  ggat(Va −  VrevGAT)                              CdVm(t)dt= −INa(t) − IK(t) − Ileak(t) − Isyn(t) − Iexsyn(t) μμ INa (t) =    gNam(t)∞3  b(t)(Vm(t)  − ENa)                   IK(t)  =   gkn(t)4(Vm(t)  −  EK)  Ileak(t) = gleak(Vm(t) − Eleak)        m(t)∞  = (1 +  exp (−Vm(t)+309.5))−1    τb(t) =  0.1 + 0.75 (1 + exp (−Vm(t)+40.5−6))−1      db(t)dt =b(t)∞ − b(t)τb(t)       b∞ = (1 +  exp (Vm(t)+457))−1   K+ τn(t)  =  0.1 + 0.5 (1 + exp (Vm(t)+2715))−1     dn(t)dt=n(t)∞ − n(t)τn(t)         n(t)∞ = (1 +  exp (−Vm(t)+3510))−1       3Naout+ + Gluout− + Hout+ + Kin+ ↔ 3Nain+ + Gluin− + Hin+ + Kout+     Vrev =RT2Fln ((NasynNaast)3 [H+]out[H+]inGlusynGluastKastKsyn )           μ   IEAAT(V − Vrev) = 0                                               μ    IEAAT →IEAATA =IEAAT1764π×10−12pA m2⁄                                 IEAAT = αeβ(V−Vrev)                                             ln(IEAAT) = ln α + ln eβ(V−Vrev) = ln α + β(V − Vrev)               α βα β IEAAT = −α(e−β(V−Vrev))                                          α β  α  βαβ   μ μ           ddtGlusyn(t) =  Yrelδ(t − tk) −VEAAT(Vrev(t), Va)Volsyn   αα  δ ddtGluast(t) = −Gluast(t)τg+VEAAT(Vrev(t), Va)Volast+ c                 τ VolastddtNaast(t) = 3VEAAT(Vrev(t), Va) + 3VNKA(t) + VNCX(t))  VolsynddtKsyn(t) = VEAAT(t) + 2VNKA(t)  + 0.5δ(t − tk)               CadVa(t)dt= F(−VNKA(t) − VEAAT(t) − VNCX(t))    θ μ Isic(t) = INMDA(t) + IAMPA(t)                                     CmdVm(t)dt= −INa(t) − IK(t) − Isic(t) − Ileak(t) − Isyn(t)                     β                                            CMdVm,predt= −(INa,Pre + IK,Pre + Ileak,Pre + IPreGABAA + Iapp)       INa,ast     =       1.5IEAAT + 3INKA + 3INCX + 2IGAT + INa,leak   IK,ast       =  − 0.5IEAAT − 2INKA + IKir + IK,leak   IGlu,ast    =    −0.5IEAAT +   IGlu,leak                                    ICa,ast      =     −2 INCX    + ICa,leak                                 IGABA,ast =          IGAT + IGABA,leak              Ileak = gX(Va − EX)    dXastdt= −IX,astzFSAmemVolast                                   dXsyndt=IX,astzFSAmemVolsyn                                        CMdVmdt= −(INa,neuron + IK,neuron + Ileak,neuron + INMDA + IAMPA + IGABAA)       μ             μ                                                                                               

A Biophysical Model of the Role of Astrocytes in Hyper-Excitability

https://pure.ulster.ac.uk/ws/files/77729155/2019FlanaganBPhD.pdf

A Biophysical Model of the Role of Astrocytes in Hyper-Excitability

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