This document discusses synchronization of oscillatory systems. It introduces a model where the phase of an oscillator φi evolves according to its natural frequency ωi plus a coupling term dependent on the phase differences between oscillators φi - φj. The coupling is described by a phase response curve Z(φ) which can depend on the system. Synchronization occurs when the oscillators lock to a common frequency and their phases cluster.

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