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Bernadette van WijkBernadette van Wijk
DCM for Time-Frequency
VU University Amsterdam, The NetherlandsVU University Amsterdam, The Netherlands
1. DCM for Induced Responses
2. DCM for Phase Coupling
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Dynamic Causal Models
Neurophysiological Phenomenological
• DCM for ERP• DCM for SSR
• DCM for Induced Responses• DCM for Phase Coupling
spiny stellate
cells
inhibitory interneuron
s
PyramidalCells Time
Freq
uenc
y
Phase
Source locations not optimizedElectromagnetic forward model included
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Region 1 Region 2
?
?
Changes in power caused by external input and/or coupling with other regions
Model comparisons: Which regions are connected? E.g. Forward/backward connections
(Cross-)frequency coupling: Does slow activity in one region affect fast activity in another ?
1. DCM for Induced Responses
Time
Freq
uenc
y
Freq
uenc
y
Time
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Single region 1 11 1 1z a z cu
u2
u1
z1
z2
z1
u1
a11c
cf. Neural state equations in DCM for fMRI
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Multiple regions
1 11 1 1
2 21 22 2 2
0
0
z a z uc
z a a z u
u2
u1
z1
z2
z1
z2
u1
a11
a22
c
a21
cf. DCM for fMRI
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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
u1
a11
a22
c
a21
b21
cf. DCM for fMRI
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u1 u2
z1
z2
a11
a22
c
a12
a21
b21
Reciprocal connections
1 11 12 1 1 12
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
cf. DCM for fMRI
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dg(t)/dt=A g(t)∙+C u(t)∙
DCM for induced responses
Where g(t) is a K x 1 vector of spectral responses
A is a K x K matrix of frequency coupling parameters
Also allow A to be changed by experimental condition
Time
Freq
uenc
y
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G=USV’
Use of Frequency Modes
Where G is a K x T spectrogram
U is K x K’ matrix with K frequency modes
V is K x T and contains spectral mode responses over time
Hence A is only K’ x K’, not K x K
Time
Freq
uenc
y
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Differential equation model for spectral energy
KKij
Kij
Kijij
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 in region j affects frequency 1 in region i
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Modulatory connections
Extrinsic (between-source) coupling
1 11 1 11 1 1
1 1
( ) ( ) ( )J J
J J JJ J JJ J
g A A B B C
g t v g t u t
g A A B B C
Intrinsic (within-source) coupling
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Example: 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
Bac
kwar
d
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
1000backward linear backward nonlinear
forward linearforward nonlinear
Model Inference
Winning model: FNBN
Both forward and backward connections are nonlinear
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Parameter Inference: gamma affects alpha
Right backward - inhibitory - suppressive effect of gamma-alpha coupling in backward connections
Left forward - excitatory - activating effect of gamma-alpha coupling in the forward connections
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)
From 30Hz
To 10Hz
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Synchronization achieved by phase coupling between regions
Model comparisons: Which regions are connected? E.g. ‘master-slave’/mutual connections
Parameter inference: (frequency-dependent) coupling values
Region 1 Region 2
( )i i jj
2. DCM for Phase Coupling
?
?
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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
2 2 10.6sin( )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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j
j
i
DCM for Phase Coupling
)sin( jij
ijii af
sin( [ ]) cos( [ ])i i ijK i j ijK i jK j K j
f a K b K
Phase interaction function is an arbitrary order Fourier series
Allow connections to depend on experimental condition
ija
ija
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Example: MEG data
Fuentemilla et al, Current Biology, 2010
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Delay activity (4-8Hz)
Visual Cortex (VIS)Medial Temporal Lobe (MTL)Inferior Frontal Gyrus (IFG)
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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
• 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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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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Analysis
• 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
• Model inversion
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LogEv
Model
1 2 3 4 5 6 70
50
100
150
200
250
300
350
400
450MTL
VISIFG3
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MTL
VISIFG
2.89
2.46
0.89
0.77
sin([ ]) cos([ ])i i ij i j ij i jj j
f a b
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MTL-VIS
IFG-VIS
Control
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MTL-VIS
IFG-VIS
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
dxx k x z w x P
dtdx
x k x z w xdtdx
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