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8/3/2019 08 DCM Induced Phase
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DCM for Time Frequency
Will Penny
Wellcome Trust Centre for Neuroimaging,
University College London, UK
SPM MEG/EEG Course, Oct 27, 2009
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DCM for Induced Responses
Region 1
Region 3
Region 2
??
How does slow activityin one region affectfast activity in another ?
Relate change of powerin one region and frequencyto change in power in others.
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Single region 1 11 1 1z a z cu
u2
u1
z1
z2
z
1
u1
a11c
DCM for fMRI
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Multiple regions
1 11 1 1
2 21 22 2 2
0
0z a z ucz a a z u
u2
u1
z1
z2
z
1
z2
u1
a11
a22
c
a21
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Modulatory inputs
1 11 1 1 12
2 21 22 2 21 2 2
0 0 0
0 0
z a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u
1
a11
a22
c
a21
b21
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Reciprocal connections
1 11 12 1 1 1
2
2 21 22 2 21 2 2
0 0
0 0
z a a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u1
a11
a22
c
a12
a21
b21
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Single Region
dg(t)/dt=Ag(t)+Cu(t)
DCM for induced responses
Where g(t) is a K x 1 vector of spectral responsesA is a K x K matrix of frequency coupling parametersDiagonal elements of A (linear within freq)Off diagonal (nonlinear between freq)Also allow A to be changed by experimental condition
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DCM for induced responses
Specify the DCM: 2 areas (A1 and A2) 2 frequencies (F and S) 2 inputs (Ext. and Con.) Extrinsic and intrinsic coupling
Integratethe state equations
A1 A2
2
1
2
1
),(
),(
),(
),(
A
A
A
A
tSx
tSx
tFx
tFx
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Differential equation model for spectral energy
KK
ij
K
ij
K
ijij
ij
AA
AA
A
1
111
Nonlinear (between-frequency) coupling
Linear (within-frequency) coupling
Extrinsic (between-source) coupling
)()()(
1
1
1111
tu
C
C
tg
AA
AA
g
g
tg
JJJJ
J
J
Intrinsic (within-source) coupling
How frequency K inregion j affectsfrequency 1 in region i
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Modulatory connections
Intrinsic (within-source) coupling
Extrinsic (between-source) coupling
state equation
JJJJ
J
JJJ
J
JC
C
tutwx
BB
BB
tu
AA
AA
x
x
W, tx
1
1
111
1
1111
)(),()()(.
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Connection to Neurobiology
First and
Second orderVolterra kernelsFrom NeuralMass model.
Strong
(saturating)input leads tocross-frequencycoupling
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Single Region
G=USV
Use of Frequency Modes
Where G is a K x T spectrogramU is KxK matrix with K frequency modesV is K x T and contains spectral mode responsesHence A is only K x K, not K x K
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MEG Data
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15
81
39
z
y
x
15
81
42
z
y
x
27
45
42
z
y
x
24
51
39
z
y
x
OFA OFA
FFAFFA
input
The core system
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nonlinear (and linear)
linear
Forward
Backwar
d
linear nonlinear
linear
nonlinear
FLBL FNBL
FLNB FNBN
OFA OFA
Input
FFAFFA
FLBL
Input
FNBL
OFA OFA
FFAFFA
FLBN
OFA OFA
Input
FFAFFA
FNBN
OFA OFA
Input
FFAFFA
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FLBL FNBL FLBN *FNBN
-59890
-16308 -16306 -11895
-70000
-60000
-50000
-40000
-30000
-20000
-10000
0
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
01000 backward linear backward nonlinear
forward linear
forward nonlinear
Inference on model I
Both forward and backward connections arenonlinear
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From 32 Hz (gamma) to 10 Hz (alpha)
t= 4.72; p = 0.002
4 12 20 28 36
44
44
36
28
20
12
4
SPM tdf 72; FWHM 7.8 x 6.5 Hz
Frequency(Hz)
Right hemisphereLeft hemisphere
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Forward Backward
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Forward Backward
Left forward excitatory -- activating effect of
gamma- alpha coupling in the forward connections
Right backward - inhibitory -- suppressive effect ofgamma- alpha coupling in backward connections
Gamma affects Alpha
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Gamma activity in input areas induces slower
dynamics in higher areas as prediction error is accumulated. Nonlinear
coupling in high-level area induces gamma activity in that higherarea which then accelerates the decay of activity in the lower level.
This decay is manifest as damped alpha oscillations.
C.C. Chen , S. Kiebel, KJ Friston , Dynamic causal modelling ofinduced responses. NeuroImage, 2008; (41):1293-1312.
C.C. Chen, R.N. Henson, K.E. Stephan, J.M. Kilner, and K.J.Friston1. Forward and backward connections in the brain: A DCMstudy of functional asymmetries in face processing. NeuroImage,2009 Apr 1;45(2):453-62
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For studying synchronization among brain regions
Relate change of phase in one region to phase in others
Region 1
Region 3
Region 2
?
?
DCM for Phase Coupling
( )i i jj
g
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Connection to Neurobiology:Septo-Hippocampal theta rhythm
Denham et al. 2000: Hippocampus
Septum
1
1 1 1 13 3 3
2
2 2 2 21 1
1
3 3 3 34 4 3
4
4 4 4 42 2
( ) ( )
( ) ( )
( ) ( )
( ) ( )
e e CA
i i
i e CA
i i S
dxx k x z w x P
dt
dxx k x z w x
dt
dxx k x z w x P
dt
dxx k x z w x P
dt
1x
2x 3x
4xWilson-Cowan style model
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Four-dimensional state space
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Hippocampus
Septum
A
A
B
B
Hopf Bifurcation
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cossin)( baz
For a generic Hopf bifurcation (Erm & Kopell)
See Brown et al. 04, for PRCs corresponding to other bifurcations
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0
1sin( ) cos( )
2
iij i j ij i j
j
df a b
dt
3
2
1
12a
13a
DCM for Phase Coupling
12b
13b
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MEG Example Fuentemilla et al 09
+
+
+
1 sec 3 sec 5 sec 5 sec
1) No retention (control condition): Discrimination task
2) Retention I (Easy condition): Non-configural task
3) Retention II (Hard condition): Configural task
ENCODING MAINTENANCE PROBE
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Delay activity (4-8Hz)
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Questions
Duzel et al. find different patterns of theta-coupling in the delay perioddependent on task.
Pick 3 regions based on [previous source reconstruction]
1. Right MTL [27,-18,-27] mm
2. Right VIS [10,-100,0] mm3. Right IFG [39,28,-12] mm
Fit models to control data (10 trials) and hard data (10 trials). Each trialcomprises first 1sec of delay period.
Find out if structure of network dynamics is Master-Slave (MS) or(Partial/Total) Mutual Entrainment (ME)
Which connections are modulated by (hard) memory task ?
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Data Preprocessing
Source reconstruct activity in areas of interest (with fewer sources thansensors and known location, then pinv will do; Baillet 01)
Bandpass data into frequency range of interest
Hilbert transform data to obtain instantaneous phase
Use multiple trials per experimental condition
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MTL
VISIFG
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG1
MTL
VISIFG2
3
4
5
6
7
Master-Slave
PartialMutualEntrainment
TotalMutualEntrainment
MTL Master VIS Master IFG Master
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LogEv
Model
1 2 3 4 5 6 70
50
100
150
200
250
300
350
400
450
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MTL
VISIFG
2.89
2.46
0.89
0.77
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MTL-VIS
IFG-VIS
Control
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MTL-VIS
IFG-VIS
Memory
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