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The time evolution of the node variables is given by a set of coupled differential equations: dtdxi=fi(xi)+j̸=i∑Aijh(xi,xj)+ηi
External influences on the system are modelled by a Gaussian correlated noise ηi that is generated by the Ornstein-Uhlenbeck process: τndtdηi=−ηi+ξi
where ξi(t) is a zero-mean Gaussian white noise: <ξi(t)>=0,<ξi(t)ξj(t’)>=2Dijδ(t−t’)
Thus we have <ηi(t)>=0,<ηi(t)ηj(t’)>=τnDije−∣t−t’∣/τn
线性化可得: dtdδx=Qδx+ηi
对上式积分可得: δx(t)=eQtδx(0)+∫0te(t−t’)Qηi(t’)dt’
Hence we obtain Kτ=<δx(t+τ)δx(t)T>
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QK0+K0QT+D(I−τnQT)−1+(I−τnQ)−1DT=0
Kτ=eτQK0+J(τ)
J(τ)=(eτQ−e−τ/τnI)U+ττQV
U=τnRD(I−τnQT)−1
V=[I−(I+τnQ)R]RD(I−τnQT)−1
R=(I+τnQ)−1
Replacing τ by 2τ and then by 3τ and after some algebra, we obtain K2τ=SKτ−TK0
K3τ=SK2τ−TKτ
S=eQt+e−τ/τnI
T=e−τ/τneQt
Solving Eqs, we then obtain T=(K3τ−K2τKτ−1K2τ)(K0Kτ−1K2τ−Kτ)−1
S=(K2τ+TK))Kτ−1
S=eτ/τnT+e−τ/τnI
to estimate the value of eτ/τn and thus τn by a least-square fit of the diagonal elements of S, denoted by μ,