DCM for Time Frequency
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![Page 1: DCM for Time Frequency](https://reader036.fdocuments.in/reader036/viewer/2022062400/56812aba550346895d8e7fa3/html5/thumbnails/1.jpg)
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, SPM MEG/EEG Course, May 11thMay 11th, 20, 201010
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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 power in others.
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Single region 1 11 1 1z a z cu
u2
u1
z1
z2
z1
u1
a11c
DCM for fMRI
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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
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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
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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
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
tSxtSxtFxtFx
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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
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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, t
1
1
111
1
1111
)(),()()(g
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Neural Mass Models
WeakInput
StrongInput
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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’
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MEG Data
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158139
zyx
158142
zyx
274542
zyx
245139
zyx
OFA OFA
FFAFFA
input
The “core” system
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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
Face selective effectsmodulate within hemisphereforward and backward cxs
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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
0
1000 backward linear backward nonlinear
forward linearforward nonlinear
Model Inference
• Both forward and backward connections are nonlinear
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Parameter Inference: gamma affects alpha
Right backward - inhibitory – suppressiveeffect of gamma-alpha coupling in backward connections
Left forward – excitatory - activating effect of gamma-alpha coupling in the forward connections
During face processing
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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 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
Parameter Inference: 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 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. Friston.
Forward and backward connections in the brain: A DCM study 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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One Oscillator
f1
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Two Oscillators
f1
f2
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Two Coupled Oscillators
f1
)sin(3.0 122 f
0.3
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Different initial phases
f1
)sin(3.0 122 f
0.3
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Stronger coupling
f1
)sin(3.0 122 f
0.6
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Bidirectional coupling
)sin(3.0 122 f
0.30.3
)sin(3.0 211 f
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3
2
1
DCM for Phase Coupling
)sin( jij
ijii af
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3
2
1
DCM for Phase Coupling
)sin( jij
ijii af
])[sin( jij
ijkk
ii kaf
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3
2
1
DCM for Phase Coupling
)sin( jij
ijii af
])[sin( jij
ijkk
ii kaf
])[cos(])[sin( jij
ijkk
jij
ijkk
ii kbkaf
Phase interaction function is an arbitrary order Fourier series
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3
2
1
DCM for Phase Coupling
)sin( jij
ijii af
Allow connections to depend onexperimental condition
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MEG Example
Fuentemilla et al, Current Biology, 2010
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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 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 memory 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 memory task ?
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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
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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
-V
IS
Control
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MTL-VIS
IFG
-V
IS
Memory
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Jones and Wilson, PLoS B, 2005
Recordings from rats doing spatial memory task:
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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
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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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Connection to Neural Mass Models
First and Second orderVolterra kernelsFrom Neural Mass model.
Strong(saturating)input leads tocross-frequencycoupling