Recommended
More Related Content
What's hot
What's hot (20)
Similar to Phase Resetting Curves and Synchrony in Type I Membranes
Similar to Phase Resetting Curves and Synchrony in Type I Membranes (20)
Recently uploaded
Recently uploaded (20)
Phase Resetting Curves and Synchrony in Type I Membranes
- 6. dX dt = F(X) + p(X, t) dϕ dt = ω + Z(ϕ) ⋅ p(X, t)
- 9. X(t) = (V, m, h, n)T dX(t) dt = F(X(t))
- 10. dynamical system
- 11. XT(t + T) = XT(t) dϕ dt = 2π T ≡ ω
- 13. dϕ dt = ∇Xϕ ⋅ dX dt = ∇Xϕ ⋅ F(X) = ω ϕ(X) = ϕ0 = Const .
- 14. dϕ dt = ∇Xϕ ⋅ dX dt dϕ(X) = ∂ϕ ∂x1 dx1 + ∂ϕ ∂x2 dx2 + ⋯ = ∇Xϕ ⋅ dX
- 15. dW dt = (1 + iω)W − (1 + iα)|W|2 W W = Reiθ R = 1 θ − α log R = Const .
- 16. dX dt = F(X) + p(X, t) p = 0 XT(t) dϕ dt = ∇Xϕ ⋅ (F(X) + p(X, t) = ω + ∇Xϕ ⋅ p(X, t)
- 17. dϕ dt = ∇Xϕ ⋅ (F(X) + p(X, t)) = ω + ∇Xϕ ⋅ p(X, t) XT(ϕ) Z(ϕ) ≡ ∇X(ϕ)|X=XT(ϕ) dX dt = F(X) + p(X, t) dϕ dt = ω + Z(ϕ) ⋅ p(X, t)
- 18. dX(t) dt = F(X(t)) dX dt = F(X) + p(X, t)
- 19. dϕ dt = ω + ZV(ϕ) ⋅ I(VT(ϕ), t) I(t) = Aδ(t − ts) ϕ = ωT + AZV(ϕs) dZ(ϕ) dϕ = − {DF(XT(ϕ))}T Z(ϕ)
- 20. dV dt = I − gV if V > 1, then V = 0 VT(t) = I g (1 − e−gT ) T = 1 g log I I − g ϕ = 2π T t ZV(ϕ) = dϕ dV V=VT = 2π T dt dV V=VT = 2π T 1 I − gVT(t) = 2π Tg (1 − e−gT )e Tg 2π ϕ
- 21. dX1 dt = F(X1) + δF1(X1) + V12(X1, X2) dX2 dt = F(X2) + δF2(X2) + V21(X2, X1)
- 22. dϕi dt = ω + Z(ϕi) ⋅ δFi(XT(ϕi)) + Z(ϕi) ⋅ Vij(XT(ϕi), XT(ϕj)) ϕi − ϕj ϕi = ωt + φi dϕi dt = ω + δωi + Γij(ϕi − ϕj) δωi = 1 2π ∫ 2π 0 Z(θ) ⋅ δFi(XT(θ))dθ Γij(ϕ) = 1 2π ∫ 2π 0 Z(θ) ⋅ Vij(XT(θ), XT(θ − ϕ))dθ
- 23. Γij(ϕ) = Γ(ϕ) dΔϕ dt = Γ(Δϕ) − Γ(−Δϕ) = Γodd(Δϕ) Δϕeq = 0 Δϕ = ϕ1 − ϕ2 Γ′odd(Δϕeq) < 0
- 25. Γij(ϕ) = 1 2π ∫ 2π 0 Z(θ) ⋅ Vij(XT(θ), XT(θ − ϕ))dθ Isyn = gsynm(t)(V − Esyn) Esyn
- 26. Isyn = gsynm(t)(V − Esyn) m(t) = ∑ ts<t α(t − ts) α(t) = t τ exp ( − t − τ τ ) Γij(ϕ) Γij(ϕ) = 1 2π ∫ 2π 0 ZV(θ)gsynmT(θ − ϕ)(VT(θ) − Esyn)dθ ωt = ϕ
- 28. dV dt = I − gV if V > 1, then V = 0 ZV(ϕ) = 2π Tg (1 − e−gT )e Tg 2π ϕ
- 29. dV dt = 1 C (−INa − IK(t) − IL(t) + I)
- 30. dV dt = I + V2 ZV(ϕ) = 2 ω2 (1 − cos ϕ) V = tan(θ/2) dθ dt = 1 − cos θ + (1 + cos θ)I
- 31. Z(ϕ) = A sin(ϕ + β) dW dt = (1 + iω)W − (1 + iα)|W|2 W
- 33. I < 0 I = 0 I > 0 dθ dt = 1 − cos θ + (1 + cos θ)I
- 34. dW dt = (μ + i)W − |W|2 W W = Reiθ dR dt = μR − R3 = f(R), dθ dt = 1 R = 0 f′(R = 0) μ > 0 R = μ
- 35. dϕi dt = ωi + N ∑ j=1 Γij(ϕi − ϕj) dϕi dt = ωi + 1 N N ∑ j=1 Γ(ϕi − ϕj)
- 36. ωi
- 37. ωi Γij(ϕ) = − K sin(ϕ) dϕi dt = ωi − K N N ∑ j=1 sin(ϕi − ϕj) N → ∞ KC K > KC r = 1 N N ∑ j=1 eiϕj K < KC K > KC r ∼ O( K − KC) r = 0
- 42. Type I Membranes, Phase Resetting Curves, and Synchrony http://www.f.waseda.jp/atsuko_ta/BioSystems/15BioSys.pdf