brain connectivity methods in sensorimotor network

7/24/2019 Brain connectivity methods in Sensorimotor network

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A B C

A C A B

C

B



X = (x1,x2,...,xN ) Y = (y1,y,...,yN ) N

xt

yt

A B

xt

yt

s

s

s

xs,τ xt

xs,τ = (xt, xt−τ , xt−2τ ,...,xt−(s−1)τ )T

s τ

t = 1, 2,...,N

τ = 1



xs = xs,1

yt

ys,τ

= (yt, yt−τ , yt−2τ ,...,yt−(s−1)τ )T

ys,τ xs,τ B A

|∆φ| = |nφ(f m) − mφ(f z)| = const

nφ(f m)

mφ(f n)

f m

f n

nf m = mf n

∆φ

f 1 → f 1

f 1 → 3f 1

{f 2, f 3} → 2f 2 − f 3

d y = xd

R f 1, f 2,...,f R dth

f Σ =R

Σ r=1

arf r

ar

R

Σ r=1

|ar| = d

|∆φ| = | RΣ r=1

arφ(f r) − φ(f Σ )| = const

X (f )

Y (f )

dth ΨXY

ΨXY (f 1, f 2,...,f R; a1, a2,...,aR)d = 1

K

K

Σ k=1

exp( j(R

Σ r=1

arφXk(f r) − φY k(f Σ )))

K

φXk(f r)

φY k(f r)

X (f r) Y (f r) kth

ψXY = |Ψ|

K



yt = AT Φ(yst−1) + BT Ψ(xst−1) + y|x

Φ Ψ

ys p

≤ p≤

apriori

σ2

G(−→yt , σ2) = 1

(√

2π)2exp(−|

−→yt |22σ2

)

Y

yt = bs1yst−1 + bq2xqt−1 + bs3yst−1yst−1 + bq4xqt−1xqt−1 + bq5xqt−1yst−1 + ey

s

q

b1

Y

b2

X

Y

b3

Y

b4

b5

X

Y

ey

Y −n

Y

X −n

X (XY )−n

X

Y

X

Y

NARXGC nX→Y = ln var(Y |Y −l , Y −n , Y −l , X −l )

var(Y |Y −l Y −n , X −l , X −l , (XY )−n )

var(·)



s xt+1

xst≡(xt,...,xt−(s−1))T

p(xt+1|xst

xst

hX =

p(xt+1,xst )logp(xt+1|xst )

H X(s+1)−H X(s)

hX

xt+1 xst

X

Y

p(xt+1

|xst ) = p(xt+1

|xst , ylt)

Y

X

p(xt+1, xst )

p(xt+1|xst , yst ),

Y

X

T Y →X =

p(xt+1, xst , ylt) × log p(xt+1|xst , ylt)

p(xt+1|x)

p(xt+1, xst , ylt)

xt

yt

xst,f = Atexp(i(f t + θx,st ))

ylt,f = Atexp(i(f t + θy,lt ))

θx,st

θy,lt

s

l

xst,f ylt,f

pT E Y →X =

p(θxt+v, θx,st , θy,lt ) × log p(θxt+u|θx,st , θy,lt )

p(θxt+u|θx,st )

v

θxt+v

Y X

pT E Y →X = H θxt+v,θx,st

+ H θx,st ,θ

y,lt− H θx,st

− H θxt+v,θx,st ,θ

y,lt



x(t)

y(t)

l

s

yt = bs1(k1)xs(t−k1)+ bs2(k1, k2)xs(t−k1)xs(t−k2)+ · · ·+bsl (k1, · · · , kl)xs(t−k1) · · · xs(t−kl)

b(k1, ·· ·, kl)

l

yt = θM ϕM t

θ N

ϕt

ϕt

ϕt = {xt, xt−1, · · · , x(t − d)}T

Y = P Θ

Y

P

1xN

Θ

1xM

P T (t) =(ϕ1(t),...,ϕ2(t))

P = A ×W

W

W =

w1(1) · · · wN (1)

w1(M ) · · · wN (M )

A

A =

1 a12 a13 · · · a131 a23 · · · a2N

1

1 aN −1N 1

Y = P Θ

Y = W G

G = AΘ = [g1,...,gN ]T

θN = gN

θl = gi−N k=i+1

aik θk, i = N − 1, ..., 1

PESRn =

1

(1 − λn/M )2 (1−n

i=1 [ERR]i)



n M

ERR

ERRi =

giE [w2i (t)]

E [y2(t)]

PESRn

ϕi(i = 1,...,N l) ϕj( j = N l+1,...,N N l+N n

N l, N n(N l + N n = N )

ERRC (l)X→Y =

N li=1

ERRi

ERRC (n)X→Y =

N l+N ni=N l+1

ERRi

ERRC (t)X→Y =

N l+N ni=1

ERRi

x

u

·x = f (x,u,θ)

·x

x

f

x θ

f

x

y

y = z(x, ϕ)

z

ϕ

y φ = (θ, ϕ) φ



X X X

X

X X X X X X X X ∗ X

X X X X X

∗

X N O Y ES

brain connectivity methods in sensorimotor network

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