Formulario BEM

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FLUSSO jd = −Dk∇Ck Jd = jdZkF = −DkZkF ∇Ck ji = −uk Zk |Zk|Ck∇Φ Ji = jiZkF = −uk|Zk|Ck F ∇Φ = −σk∇Φ Dk = uk RT |Zk|F J = Jd + Ji = −DkZkF(∇Ck + ZkCkF RT ∇Φ) POTENZIALE equazione di equilibrio di Nernst Veq = Φin − Φout = RT FZk ln [Ck]out [Ck]in equilibrio di Donnan Vr = RT F lnd→ d = [K]out [K]in = [Na]out [Na]in = [Cl]in [Cl]out 𝐸𝐴 = 𝐸𝐵 → ( [A]out [A]in ) 1 𝑍𝐴 = ( [B]out [B]in ) 1 𝑍𝐵 equilibrio dinamico Vr = gKEK+gNaENa+gClECl gK+gNa+gCl equazione di Goldman-Hodgkin-Katz Vm = RT F ln PK[K]out+PNa[Na]out+PCl[Cl]in PK[K]in+PNa[Na]in+PCl[Cl]out equazione di riposo di Goldman-Hodgkin-Katz Vr = RT F ln [K]out+PNa/PK [Na]out [K]in+PNa/PK [Na]in CARICA Q𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑎 = C𝑚 ∙ A𝑙𝑎𝑡 ∙ V𝑚 Q𝑖𝑛𝑡𝑟𝑎𝑐𝑒𝑙𝑙 = [X]𝑖𝑛 ∙ Vol∙ F sfera→ Alat = 4πr2 − Vol= 4πr3 3 cilindro→ Alat = 2πr l − Vol= πr2 l CORRENTE Im = gX(Vm − EX) Im = gX(Vclamp− EX) VOLTAGE-CLAMP EX = Vm + Vclamp DINAMICA DI CANALI n= E[nopen] N = n(V1)− [n(V1)− n(Vr)]e−t/τ neq = αn αn+βn τ = 1 αn+βn MODELLO HODGKIN-HUXLEY gK(t,vm) = gK̅̅̅n4 dn dt = αn(1 − n)− βnn αn = 0.01(10−vm) exp(10−vm 10 )−1 ↔ βn = 0.125exp(− vm 80) gNa(t,vm) = gNa̅̅̅̅̅m3h dm dt = αm(1 − m)− βmm αm = 0.1(25−vm) exp(25−vm 10 ) − 1 ↔ βm = 4exp(− vm 18) dh dt = αh(1 − h)− βhh αh = 0.07exp(− vm 20) ↔ βh = 1 exp(30−vm 10 )+1 MODELLI RIDOTTI Cm dV dt = −gK( W a)4(V− VK)− gNam∞ 3 (b− W)(V− VNa)− gL(V− VL)+ Iapp W= an= b− h dW dt = αW(a− W)− βWW modello di Fitzhugh-Nagumo dW dt = Φ(V+ a− bW) = 0.08(V+ 0.7 − 0.8W) dV dt = V− V3 3 − W+ I Weq = V+a b = V− V3 3 + I NEURONE ui(t) = ∑ ∑ εij(t− tk)k + urestj u(t) = ∑ PSPj(t,I0)+ r(t)j η(t− t0) = { 1/∆t t0 < t < t0 + ∆t −ηexp(− t−t0 τ ) t> t0 + ∆t ν = 1 E(s) = ∫H(s)ds ∫sH(s)ds → νmax = 1 Trefrattarietà modello di spike-response ui(t) = η(t− ti)+ ∑ wij∑ εij(t− tk)k + ∫K(s)Iext(t− s)dsj modello integrate-and-fire Ii(t) = ui R + C…