DCM for Time Frequency
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Transcript of DCM for Time Frequency
DCM for Time Frequency
Will PennyWill Penny
Wellcome Trust Centre for Neuroimaging,Wellcome Trust Centre for Neuroimaging,University College London, UKUniversity College London, UK
SPM MEG/EEG Course, Oct 27, 2009SPM MEG/EEG Course, Oct 27, 2009
DCM for Induced ResponsesRegion 1
Region 3
Region 2
??
How does slow activityin one region affectfast activity in another ?
Relate change of powerin one region and frequency to change in power in others.
Single region 1 11 1 1z a z cu
u2
u1
z1
z2
z1
u1
a11c
DCM for fMRI
Multiple regions
1 11 1 1
2 21 22 2 2
00
z a z ucz a a z u
u2
u1
z1
z2
z1
z2
u1
a11
a22
c
a21
Modulatory inputs
1 11 1 1 12
2 21 22 2 21 2 2
0 0 00 0
z a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u1
a11
a22
c
a21
b21
Reciprocal connections
1 11 12 1 1 12
2 21 22 2 21 2 2
0 00 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
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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
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
tSxtSxtFxtFx
Differential equation model for spectral energy
KKij
Kij
Kijij
ij
AA
AAA
1
111
Nonlinear (between-frequency) coupling
Linear (within-frequency) coupling
Extrinsic (between-source) coupling
)()()(1
1
1111
tuC
Ctg
AA
AA
g
gtg
JJJJ
J
J
Intrinsic (within-source) coupling
How frequency K in region j affects frequency 1 in region i
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
)(),()()(.
Connection to Neurobiology
First and Second orderVolterra kernelsFrom Neural Mass model.
Strong(saturating)input leads tocross-frequencycoupling
Single Region
G=USV’
Use of Frequency Modes
Where G is a K’ x T spectrogramU is K’xK matrix with K frequency modesV is K’ x T and contains spectral mode responsesHence A is only K x K, not K’ x K’
MEG Data
158139
zyx
158142
zyx
274542
zyx
245139
zyx
OFA OFA
FFAFFA
input
The “core” system
nonlinear (and linear)
linear
Forward
Back
ward
linear nonlinear
linea
rno
nlin
ear
FLBL FNBL
FLNB FNBN
OFA OFA
Input
FFAFFA
FLBL
Input
FNBL
OFA OFA
FFAFFA
FLBN
OFA OFA
Input
FFAFFA
FNBN
OFA OFA
Input
FFAFFA
FLBL FNBL FLBN *FNBN
-59890
-16308 -16306 -11895
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1000 backward linear backward nonlinear
forward linearforward nonlinear
Inference on model I
• Both forward and backward connections are nonlinear
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 t df 72; FWHM 7.8 x 6.5 Hz
Freq
uenc
y (H
z)
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 of gamma- alpha coupling in backward connections
Gamma affects Alpha
“Gamma activity in input areas induces slowerdynamics in higher areas as prediction error is accumulated. Nonlinear coupling
in high-level area induces gamma activity in that higher area 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 of induced 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 DCM study of functional asymmetries in face processing. NeuroImage, 2009 Apr 1;45(2):453-62
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
Connection to Neurobiology:Septo-Hippocampal theta rhythm
Denham et al. 2000: Hippocampus
Septum
11 1 1 13 3 3
22 2 2 21 1
13 3 3 34 4 3
44 4 4 42 2
( ) ( )
( ) ( )
( ) ( )
( ) ( )
e e CA
i i
i e CA
i i S
dx x k x z w x Pdtdx x k x z w xdt
dx x k x z w x Pdtdx x k x z w x Pdt
1x
2x 3x
4xWilson-Cowan style model
Four-dimensional state space
Hippocampus
Septum
A
A
B
B
Hopf Bifurcation
cossin)( baz
For a generic Hopf bifurcation (Erm & Kopell…)
See Brown et al. 04, for PRCs corresponding to other bifurcations
01 sin( ) cos( )2
iij i j ij i j
j
d f a bdt
3
2
1
12a
13a
DCM for Phase Coupling
12b
13b
MEG Example Fuentemilla et al 09
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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
Delay activity (4-8Hz)
Questions
• Duzel et al. find different patterns of theta-coupling in the delay period dependent on task.
• Pick 3 regions based on [previous source reconstruction]
1. Right MTL [27,-18,-27] mm2. 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 trial comprises 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 ?
Data Preprocessing
• Source reconstruct activity in areas of interest (with fewer sources than sensors 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
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG
MTL
VISIFG1
MTL
VISIFG2
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Master-Slave
PartialMutualEntrainment
TotalMutualEntrainment
MTL Master VIS Master IFG Master
LogEv
Model
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MTL
VISIFG
2.89
2.46
0.89
0.77
MTL-VIS
IFG
-V
IS
Control
MTL-VIS
IFG
-V
IS
Memory