measures of neuronal signal synchrony
TRANSCRIPT
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 111
+4 Recomendar esto en Googl
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 211
τ = minus ( N minus 1 ) 0 N minus 1
983160
983150
983161
983150
N
( τ ) = 983163 C
X Y
1
N minus τ
sum
N minus τ
983150 = 1
983160
983150 + τ
983161
983150
( minus τ ) C
X Y
τ ge 0
τ lt 0
( 1 )
X Y 1
minus 1 0
( ω ) =
E [ (
ω ) (
ω ) ]
C
X Y
F
X
F
lowast
Y
( 2 )
E [ a ] a ( ω ) F
X
983160 ω
( ω ) = Γ
X Y
| ( ω ) C
X Y
|
2
| ( ω ) | | ( ω ) | C
X X
C
Y Y
( 3 )
( ω ) Γ
X Y
δ θ α β γ
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 311
X Y
I ( X Y ) = H ( X ) + H ( Y ) minus H ( X Y ) ( 4 )
H ( X ) = minus (
983145 ) l o g (
983145 ) sum
983145 = 1
M
983160
983152
983160
983152
983160
( 5 )
H ( X Y ) = minus (
983145 983146 ) l o g (
983145 983146 ) sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983152
983160 983161
( 6 )
( 983145 )
983145 = 1 [ (
983146 )
983146 = 1 ]
983152
983160
M
983160
983152
983161
M
983161
983145 983146
X Y ( 983145 983146 ) 983152
983160 983161
M
983160
M
983161
X Y X
Y 2
I ( X Y )
( 983145 ) ( 983146 ) 983152
983160
983152
983161
X Y ( 983145 983146 ) 983152
983160 983161
I ( X Y ) = minus ( 983145 983146 ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
( 983145 983146 ) 983152
983160 983161
( 983145 ) ( 983146 ) 983152
983160
983152
983161
( 7 )
983160 983161
X Y
( ) ( )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 411
T ( X Y ) = minus ( ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160
983145
983150 + 1
983145
( 983147 )
983150
( 8 )
T (
Y
X )
( | )
983152
983160
983145
983150 + 1
983145
( 983147 )
983150
983145
983150 + 1
X
983145
( 983147 )
983150
983147 ( | )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
983145
983150 + 1
X 983145
( 983147 )
983150
983147 X 983146
( 983148 )
983150
983148 Y
983145
983150 + 1
X 983145
( 983147 )
983150
X 983146
( 983148 )
983150
Y
( )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
= + 983160
983150
sum
983147 = 1
K
a
983160
983147
983160
983150 minus 983147
983157
983160
983150
( 9 )
= + 983161
983150
sum
983147 = 1
K
a
983161
983147
983161
983150 minus 983147
983157
983161
983150
= + + 983160
983150
sum
983147 = 1
K
a
983160 983161
983147
983160
983150 minus 983147
sum
983147 = 1
K
b
983160 983161
983147
983161
983150 minus 983147
983157
983160 983161
983150
( 1 0 )
= + + 983161
983150
sum
983147 = 1
K
a
983161 983160
983147
983161
983150 minus 983147
sum
983147 = 1
K
b
983161 983160
983147
983160
983150 minus 983147
983157
983161 983160
983150
K a
983160
983147
a
983161
983147
a
983160 983161
983147
a
983161 983160
983147
b
983160 983161
983147
b
983161 983160
983147
983157
983160
983150
983157
983161
983150
983157
983160 983161
983150
983157
983161 983160
983150
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 511
X Y
M
= ( ) 983160
rarr
983150
983160
983150
983160
983150 minus (
983149 minus 1 )
983140
= ( ) 983161
rarr
983150
983161
983150
983161
983150 minus (
983149 minus 1 )
983140
983149 983140
Y X
983147 983160
rarr
983150
983146 = 1 983147 983154
983150 983146
983160
rarr
983150
983147
( X ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983154
983150 983146
)
2
( 1 1 )
983161
rarr
983150
983155
983150 983146
983161
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 611
( X | Y ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983155
983150 983146
)
2
( 1 2 )
N minus (
983149 minus 1 )
983140
983160
rarr
983150
( X ) = ( minus
R
983150
1
N minus ( 983149 minus 1 ) 983140
sum
983146 = 1
983146 ne 983150
N minus (
983149 minus 1 )
983140
983160
rarr
983150
983160
rarr
983146
)
2
( 1 3 )
M ( X | Y ) =
1
N
sum
983150 = 1
N
( X ) minus ( X | Y ) R
983150
R
( 983147 )
983150
( X ) minus (
X ) R
983150
R
( 983147 )
983150
( 1 4 )
Y X ( X
) asymp ( X | Y
) ≪ ( X
) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 1
M ( X | Y ) = 1
( X ) ≪ (
X | Y ) asymp (
X ) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 0
X Y M ( Y | X )
983160 983161
( 983156 ) ϕ
983160
( 983156 ) ϕ
983161
983160 (
983156 )
( 983156 ) = 983160
( 983156 ) + 983145
( 983156 ) = Z
983160
983160
~
A
H
983160
983141
983145 ( 983156 ) ϕ
H
983160
( 1 5 )
( 983156 ) 983160
~
983160 ( 983156 )
( 983156 ) = p v 983140 983160
~
1
π
int
infin
minus infin
983160 ( ) 983156
prime
983156 minus 983156
prime
983156
prime
( 1 6 )
Z
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 711
( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 811
[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 911
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1011
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1111
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 211
τ = minus ( N minus 1 ) 0 N minus 1
983160
983150
983161
983150
N
( τ ) = 983163 C
X Y
1
N minus τ
sum
N minus τ
983150 = 1
983160
983150 + τ
983161
983150
( minus τ ) C
X Y
τ ge 0
τ lt 0
( 1 )
X Y 1
minus 1 0
( ω ) =
E [ (
ω ) (
ω ) ]
C
X Y
F
X
F
lowast
Y
( 2 )
E [ a ] a ( ω ) F
X
983160 ω
( ω ) = Γ
X Y
| ( ω ) C
X Y
|
2
| ( ω ) | | ( ω ) | C
X X
C
Y Y
( 3 )
( ω ) Γ
X Y
δ θ α β γ
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 311
X Y
I ( X Y ) = H ( X ) + H ( Y ) minus H ( X Y ) ( 4 )
H ( X ) = minus (
983145 ) l o g (
983145 ) sum
983145 = 1
M
983160
983152
983160
983152
983160
( 5 )
H ( X Y ) = minus (
983145 983146 ) l o g (
983145 983146 ) sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983152
983160 983161
( 6 )
( 983145 )
983145 = 1 [ (
983146 )
983146 = 1 ]
983152
983160
M
983160
983152
983161
M
983161
983145 983146
X Y ( 983145 983146 ) 983152
983160 983161
M
983160
M
983161
X Y X
Y 2
I ( X Y )
( 983145 ) ( 983146 ) 983152
983160
983152
983161
X Y ( 983145 983146 ) 983152
983160 983161
I ( X Y ) = minus ( 983145 983146 ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
( 983145 983146 ) 983152
983160 983161
( 983145 ) ( 983146 ) 983152
983160
983152
983161
( 7 )
983160 983161
X Y
( ) ( )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 411
T ( X Y ) = minus ( ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160
983145
983150 + 1
983145
( 983147 )
983150
( 8 )
T (
Y
X )
( | )
983152
983160
983145
983150 + 1
983145
( 983147 )
983150
983145
983150 + 1
X
983145
( 983147 )
983150
983147 ( | )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
983145
983150 + 1
X 983145
( 983147 )
983150
983147 X 983146
( 983148 )
983150
983148 Y
983145
983150 + 1
X 983145
( 983147 )
983150
X 983146
( 983148 )
983150
Y
( )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
= + 983160
983150
sum
983147 = 1
K
a
983160
983147
983160
983150 minus 983147
983157
983160
983150
( 9 )
= + 983161
983150
sum
983147 = 1
K
a
983161
983147
983161
983150 minus 983147
983157
983161
983150
= + + 983160
983150
sum
983147 = 1
K
a
983160 983161
983147
983160
983150 minus 983147
sum
983147 = 1
K
b
983160 983161
983147
983161
983150 minus 983147
983157
983160 983161
983150
( 1 0 )
= + + 983161
983150
sum
983147 = 1
K
a
983161 983160
983147
983161
983150 minus 983147
sum
983147 = 1
K
b
983161 983160
983147
983160
983150 minus 983147
983157
983161 983160
983150
K a
983160
983147
a
983161
983147
a
983160 983161
983147
a
983161 983160
983147
b
983160 983161
983147
b
983161 983160
983147
983157
983160
983150
983157
983161
983150
983157
983160 983161
983150
983157
983161 983160
983150
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 511
X Y
M
= ( ) 983160
rarr
983150
983160
983150
983160
983150 minus (
983149 minus 1 )
983140
= ( ) 983161
rarr
983150
983161
983150
983161
983150 minus (
983149 minus 1 )
983140
983149 983140
Y X
983147 983160
rarr
983150
983146 = 1 983147 983154
983150 983146
983160
rarr
983150
983147
( X ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983154
983150 983146
)
2
( 1 1 )
983161
rarr
983150
983155
983150 983146
983161
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 611
( X | Y ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983155
983150 983146
)
2
( 1 2 )
N minus (
983149 minus 1 )
983140
983160
rarr
983150
( X ) = ( minus
R
983150
1
N minus ( 983149 minus 1 ) 983140
sum
983146 = 1
983146 ne 983150
N minus (
983149 minus 1 )
983140
983160
rarr
983150
983160
rarr
983146
)
2
( 1 3 )
M ( X | Y ) =
1
N
sum
983150 = 1
N
( X ) minus ( X | Y ) R
983150
R
( 983147 )
983150
( X ) minus (
X ) R
983150
R
( 983147 )
983150
( 1 4 )
Y X ( X
) asymp ( X | Y
) ≪ ( X
) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 1
M ( X | Y ) = 1
( X ) ≪ (
X | Y ) asymp (
X ) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 0
X Y M ( Y | X )
983160 983161
( 983156 ) ϕ
983160
( 983156 ) ϕ
983161
983160 (
983156 )
( 983156 ) = 983160
( 983156 ) + 983145
( 983156 ) = Z
983160
983160
~
A
H
983160
983141
983145 ( 983156 ) ϕ
H
983160
( 1 5 )
( 983156 ) 983160
~
983160 ( 983156 )
( 983156 ) = p v 983140 983160
~
1
π
int
infin
minus infin
983160 ( ) 983156
prime
983156 minus 983156
prime
983156
prime
( 1 6 )
Z
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 711
( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 811
[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 911
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1011
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1111
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 311
X Y
I ( X Y ) = H ( X ) + H ( Y ) minus H ( X Y ) ( 4 )
H ( X ) = minus (
983145 ) l o g (
983145 ) sum
983145 = 1
M
983160
983152
983160
983152
983160
( 5 )
H ( X Y ) = minus (
983145 983146 ) l o g (
983145 983146 ) sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983152
983160 983161
( 6 )
( 983145 )
983145 = 1 [ (
983146 )
983146 = 1 ]
983152
983160
M
983160
983152
983161
M
983161
983145 983146
X Y ( 983145 983146 ) 983152
983160 983161
M
983160
M
983161
X Y X
Y 2
I ( X Y )
( 983145 ) ( 983146 ) 983152
983160
983152
983161
X Y ( 983145 983146 ) 983152
983160 983161
I ( X Y ) = minus ( 983145 983146 ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
( 983145 983146 ) 983152
983160 983161
( 983145 ) ( 983146 ) 983152
983160
983152
983161
( 7 )
983160 983161
X Y
( ) ( )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 411
T ( X Y ) = minus ( ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160
983145
983150 + 1
983145
( 983147 )
983150
( 8 )
T (
Y
X )
( | )
983152
983160
983145
983150 + 1
983145
( 983147 )
983150
983145
983150 + 1
X
983145
( 983147 )
983150
983147 ( | )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
983145
983150 + 1
X 983145
( 983147 )
983150
983147 X 983146
( 983148 )
983150
983148 Y
983145
983150 + 1
X 983145
( 983147 )
983150
X 983146
( 983148 )
983150
Y
( )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
= + 983160
983150
sum
983147 = 1
K
a
983160
983147
983160
983150 minus 983147
983157
983160
983150
( 9 )
= + 983161
983150
sum
983147 = 1
K
a
983161
983147
983161
983150 minus 983147
983157
983161
983150
= + + 983160
983150
sum
983147 = 1
K
a
983160 983161
983147
983160
983150 minus 983147
sum
983147 = 1
K
b
983160 983161
983147
983161
983150 minus 983147
983157
983160 983161
983150
( 1 0 )
= + + 983161
983150
sum
983147 = 1
K
a
983161 983160
983147
983161
983150 minus 983147
sum
983147 = 1
K
b
983161 983160
983147
983160
983150 minus 983147
983157
983161 983160
983150
K a
983160
983147
a
983161
983147
a
983160 983161
983147
a
983161 983160
983147
b
983160 983161
983147
b
983161 983160
983147
983157
983160
983150
983157
983161
983150
983157
983160 983161
983150
983157
983161 983160
983150
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 511
X Y
M
= ( ) 983160
rarr
983150
983160
983150
983160
983150 minus (
983149 minus 1 )
983140
= ( ) 983161
rarr
983150
983161
983150
983161
983150 minus (
983149 minus 1 )
983140
983149 983140
Y X
983147 983160
rarr
983150
983146 = 1 983147 983154
983150 983146
983160
rarr
983150
983147
( X ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983154
983150 983146
)
2
( 1 1 )
983161
rarr
983150
983155
983150 983146
983161
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 611
( X | Y ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983155
983150 983146
)
2
( 1 2 )
N minus (
983149 minus 1 )
983140
983160
rarr
983150
( X ) = ( minus
R
983150
1
N minus ( 983149 minus 1 ) 983140
sum
983146 = 1
983146 ne 983150
N minus (
983149 minus 1 )
983140
983160
rarr
983150
983160
rarr
983146
)
2
( 1 3 )
M ( X | Y ) =
1
N
sum
983150 = 1
N
( X ) minus ( X | Y ) R
983150
R
( 983147 )
983150
( X ) minus (
X ) R
983150
R
( 983147 )
983150
( 1 4 )
Y X ( X
) asymp ( X | Y
) ≪ ( X
) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 1
M ( X | Y ) = 1
( X ) ≪ (
X | Y ) asymp (
X ) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 0
X Y M ( Y | X )
983160 983161
( 983156 ) ϕ
983160
( 983156 ) ϕ
983161
983160 (
983156 )
( 983156 ) = 983160
( 983156 ) + 983145
( 983156 ) = Z
983160
983160
~
A
H
983160
983141
983145 ( 983156 ) ϕ
H
983160
( 1 5 )
( 983156 ) 983160
~
983160 ( 983156 )
( 983156 ) = p v 983140 983160
~
1
π
int
infin
minus infin
983160 ( ) 983156
prime
983156 minus 983156
prime
983156
prime
( 1 6 )
Z
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 711
( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 811
[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 911
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1011
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1111
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 411
T ( X Y ) = minus ( ) l o g sum
983145 = 1
M
983160
sum
983146 = 1
M
983161
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
( | ) 983152
983160
983145
983150 + 1
983145
( 983147 )
983150
( 8 )
T (
Y
X )
( | )
983152
983160
983145
983150 + 1
983145
( 983147 )
983150
983145
983150 + 1
X
983145
( 983147 )
983150
983147 ( | )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
983145
983150 + 1
X 983145
( 983147 )
983150
983147 X 983146
( 983148 )
983150
983148 Y
983145
983150 + 1
X 983145
( 983147 )
983150
X 983146
( 983148 )
983150
Y
( )
983152
983160 983161
983145
983150 + 1
983145
( 983147 )
983150
983146
( 983148 )
983150
= + 983160
983150
sum
983147 = 1
K
a
983160
983147
983160
983150 minus 983147
983157
983160
983150
( 9 )
= + 983161
983150
sum
983147 = 1
K
a
983161
983147
983161
983150 minus 983147
983157
983161
983150
= + + 983160
983150
sum
983147 = 1
K
a
983160 983161
983147
983160
983150 minus 983147
sum
983147 = 1
K
b
983160 983161
983147
983161
983150 minus 983147
983157
983160 983161
983150
( 1 0 )
= + + 983161
983150
sum
983147 = 1
K
a
983161 983160
983147
983161
983150 minus 983147
sum
983147 = 1
K
b
983161 983160
983147
983160
983150 minus 983147
983157
983161 983160
983150
K a
983160
983147
a
983161
983147
a
983160 983161
983147
a
983161 983160
983147
b
983160 983161
983147
b
983161 983160
983147
983157
983160
983150
983157
983161
983150
983157
983160 983161
983150
983157
983161 983160
983150
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 511
X Y
M
= ( ) 983160
rarr
983150
983160
983150
983160
983150 minus (
983149 minus 1 )
983140
= ( ) 983161
rarr
983150
983161
983150
983161
983150 minus (
983149 minus 1 )
983140
983149 983140
Y X
983147 983160
rarr
983150
983146 = 1 983147 983154
983150 983146
983160
rarr
983150
983147
( X ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983154
983150 983146
)
2
( 1 1 )
983161
rarr
983150
983155
983150 983146
983161
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 611
( X | Y ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983155
983150 983146
)
2
( 1 2 )
N minus (
983149 minus 1 )
983140
983160
rarr
983150
( X ) = ( minus
R
983150
1
N minus ( 983149 minus 1 ) 983140
sum
983146 = 1
983146 ne 983150
N minus (
983149 minus 1 )
983140
983160
rarr
983150
983160
rarr
983146
)
2
( 1 3 )
M ( X | Y ) =
1
N
sum
983150 = 1
N
( X ) minus ( X | Y ) R
983150
R
( 983147 )
983150
( X ) minus (
X ) R
983150
R
( 983147 )
983150
( 1 4 )
Y X ( X
) asymp ( X | Y
) ≪ ( X
) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 1
M ( X | Y ) = 1
( X ) ≪ (
X | Y ) asymp (
X ) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 0
X Y M ( Y | X )
983160 983161
( 983156 ) ϕ
983160
( 983156 ) ϕ
983161
983160 (
983156 )
( 983156 ) = 983160
( 983156 ) + 983145
( 983156 ) = Z
983160
983160
~
A
H
983160
983141
983145 ( 983156 ) ϕ
H
983160
( 1 5 )
( 983156 ) 983160
~
983160 ( 983156 )
( 983156 ) = p v 983140 983160
~
1
π
int
infin
minus infin
983160 ( ) 983156
prime
983156 minus 983156
prime
983156
prime
( 1 6 )
Z
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 711
( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 811
[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 911
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1011
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 1111
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 511
X Y
M
= ( ) 983160
rarr
983150
983160
983150
983160
983150 minus (
983149 minus 1 )
983140
= ( ) 983161
rarr
983150
983161
983150
983161
983150 minus (
983149 minus 1 )
983140
983149 983140
Y X
983147 983160
rarr
983150
983146 = 1 983147 983154
983150 983146
983160
rarr
983150
983147
( X ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983154
983150 983146
)
2
( 1 1 )
983161
rarr
983150
983155
983150 983146
983161
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 611
( X | Y ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983155
983150 983146
)
2
( 1 2 )
N minus (
983149 minus 1 )
983140
983160
rarr
983150
( X ) = ( minus
R
983150
1
N minus ( 983149 minus 1 ) 983140
sum
983146 = 1
983146 ne 983150
N minus (
983149 minus 1 )
983140
983160
rarr
983150
983160
rarr
983146
)
2
( 1 3 )
M ( X | Y ) =
1
N
sum
983150 = 1
N
( X ) minus ( X | Y ) R
983150
R
( 983147 )
983150
( X ) minus (
X ) R
983150
R
( 983147 )
983150
( 1 4 )
Y X ( X
) asymp ( X | Y
) ≪ ( X
) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 1
M ( X | Y ) = 1
( X ) ≪ (
X | Y ) asymp (
X ) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 0
X Y M ( Y | X )
983160 983161
( 983156 ) ϕ
983160
( 983156 ) ϕ
983161
983160 (
983156 )
( 983156 ) = 983160
( 983156 ) + 983145
( 983156 ) = Z
983160
983160
~
A
H
983160
983141
983145 ( 983156 ) ϕ
H
983160
( 1 5 )
( 983156 ) 983160
~
983160 ( 983156 )
( 983156 ) = p v 983140 983160
~
1
π
int
infin
minus infin
983160 ( ) 983156
prime
983156 minus 983156
prime
983156
prime
( 1 6 )
Z
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 711
( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 811
[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
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( X | Y ) = ( minus R
( 983147 )
983150
1
983147
sum
983146 = 1
983147
983160
rarr
983150
983160
rarr
983155
983150 983146
)
2
( 1 2 )
N minus (
983149 minus 1 )
983140
983160
rarr
983150
( X ) = ( minus
R
983150
1
N minus ( 983149 minus 1 ) 983140
sum
983146 = 1
983146 ne 983150
N minus (
983149 minus 1 )
983140
983160
rarr
983150
983160
rarr
983146
)
2
( 1 3 )
M ( X | Y ) =
1
N
sum
983150 = 1
N
( X ) minus ( X | Y ) R
983150
R
( 983147 )
983150
( X ) minus (
X ) R
983150
R
( 983147 )
983150
( 1 4 )
Y X ( X
) asymp ( X | Y
) ≪ ( X
) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 1
M ( X | Y ) = 1
( X ) ≪ (
X | Y ) asymp (
X ) R
( 983147 )
983150
R
( 983147 )
983150
R
983150
M ( X | Y ) asymp 0
X Y M ( Y | X )
983160 983161
( 983156 ) ϕ
983160
( 983156 ) ϕ
983161
983160 (
983156 )
( 983156 ) = 983160
( 983156 ) + 983145
( 983156 ) = Z
983160
983160
~
A
H
983160
983141
983145 ( 983156 ) ϕ
H
983160
( 1 5 )
( 983156 ) 983160
~
983160 ( 983156 )
( 983156 ) = p v 983140 983160
~
1
π
int
infin
minus infin
983160 ( ) 983156
prime
983156 minus 983156
prime
983156
prime
( 1 6 )
Z
7242019 Measures of Neuronal Signal Synchrony
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( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
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[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
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( 983156 ) = a r c t a n
ϕ
H
983160
( 983156 ) 983160
~
983160 (
983156 )
( 1 7 )
( 983156 ) ϕ
H
983161
983161 ( 983156 )
Ψ ( 983156 ) = ( minus )
983141
983145 983156 ω
c
983141
2 ω
2
c
σ
2
983141
minus 2 983156
2
σ
2
( 1 8 )
ω
c
σ
983150
c
983160 ( 983156 ) Ψ ( 983156 ) ω
c
( 983156 ) = ( Ψ ∘ 983160 ) ( 983156 ) = int | Ψ ( ) 983160 ( 983156 minus ) | 983140 = ( 983156 ) W
983160
983156
prime
983156
prime
983156
prime
A
W
983160
983141
983145 ( 983156 ) ϕ
W
983160
( 1 9 )
( 983156 ) = a r c t a n ϕ
W
983160
I m ( 983156 )
W
983160
R e ( 983156 )
W
983160
( 2 0 )
( 983156 )
W
983161
( 983156 )
ϕ
W
983161
983161 ( 983156 )
[ 2 π )
R =
∣
∣
∣
∣
1
N
sum
983146 = 1
N
983141
983145 [ ( ) minus ( ) ]
ϕ
983160
983156
983146
ϕ
983161
983156
983146
∣
∣
∣
∣
( 2 1 )
7242019 Measures of Neuronal Signal Synchrony
httpslidepdfcomreaderfullmeasures-of-neuronal-signal-synchrony 811
[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
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[ 0 1 ]
A A ( X | Y ) A ( Y | X )
= A
S
A ( X | Y ) + A ( Y | X )
2
( 2 2 )
= A
A
A ( X | Y ) minus A ( Y | X )
A ( X | Y ) + A ( Y | X )
( 2 3 )
A
A
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