DCM for Time-Frequency
1. DCM for Induced Responses
2. DCM for Phase Coupling
Bernadette van Wijk
Dynamic Causal Models
Physiological
Phenomenological
Neurophysiological model
Models a particular data feature
inhibitory
interneurons
Pyramidal
Cells
Frequency
spiny
stellate
cells
Phase
Time
Electromagnetic forward model
included
Source locations not optimized
• DCM for ERP
• DCM for SSR
• DCM for Induced Responses
• DCM for Phase Coupling
1. DCM for Induced Responses
?
?
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?
cf. Neural state equations in DCM for fMRI
z1  a11 z1  cu1
Single region
u1
c
a11
z1
u1
u2
z1
z2
cf. DCM for fMRI
Multiple regions
 z1   a11
  
 z 2   a 21
0   z1   c   u 1 
   
a 22   z 2   0   u 2 
u1
c
a11
z1
u2
a21
z2
a22
u1
z1
z2
cf. DCM for fMRI
Modulatory inputs
 z1   a11
  
 z 2   a 21
u1
0   z1 
    u2
a 22   z 2 
u2
c
u1
a11
z1
b21
a21
z2
a22
u2
z1
z2
 0

 b 21
0   z1   c   u 1 
   
0   z2  0  u2 
cf. DCM for fMRI
Reciprocal connections
 z1   a11
  
 z 2   a 21
u1
a12   z1 
    u2
a 22   z 2 
u2
c
u1
a11
z1
b21
a12
a21
z2
a22
u2
z1
z2
 0

 b 21
0   z1   c   u 1 
   
0   z2  0  u2 
DCM for induced responses
Frequency
Single Region
dg(t)/dt=A∙g(t)+C∙u(t)
Time
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
Use of Frequency Modes
Frequency
Single Region
G=USV’
Time
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
Differential equation model
Intrinsic (within-source) coupling
 g 1

 g ( t )  



 g J

 A11









 AJ 1



A1 J 
 C1 




g (t ) 
 u (t )




A JJ 

C J 

Extrinsic (between-source) coupling
Linear (within-frequency) coupling
Aij
11
 Aij

  
 A K1
 ij
Nonlinear (between-frequency) coupling
1K

Aij



A ij
KK





How frequency K in
region j affects
frequency 1 in region i
Modulatory connections
Intrinsic (within-source) coupling
 g 1    A11
  
 g (t )  

  
 g J    A J 1
A1 J 
 B11


v


 B J 1
A JJ 
Extrinsic (between-source) coupling
B1 J  
 C1 

 
g
(
t
)

u (t )

 
 C J 
B JJ  
Example: MEG Data
Motor imagery through mental hand rotation
De Lange et al. 2008
• Do trials with fast and slow reaction times differ in timefrequency modulations?
• Are slow reaction times associated with altered forward and/or
backward information processing?
• How do (cross-)frequency couplings lead to the observed timefrequency modulations?
van Wijk et al, Neuroimage, 2013
Sources in Motor and Occipital areas
M
O
MNI coordinates
[34 -28 37]
[-37 -25 39]
[14 -69 -2]
[-18 -71 -5]
• Do trials with fast and slow reaction times differ in timefrequency modulations?
Slow reaction times: - Stronger increase in gamma power in O
- Stronger decrease in beta power in O
• Are slow reaction times
associated with altered forward
and/or backward information
processing?
Results for Model Bforward/backward
Good correspondence between observed and
predicted time-frequency spectra
Decomposing contributions to the time-frequency spectra
Feedback loop with M acts to attenuate gamma
and beta modulations in O
Attenuation is weaker for slow reaction times
• How do (cross-)frequency couplings lead to
the observed time-frequency modulations?
O
M
Interactions are mainly
within frequency bands
Slow reaction times
accompanied by a
negative beta to
gamma coupling
from M to O
2. DCM for Phase Coupling
 i      ( i   j )
j
Region 2
Region 1
?
?
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
One oscillator
1  f
Two oscillators
1  f
2  f
Two coupled oscillators
1  f
0.3
2  f  0 . 3 sin(  2  1 )
Different initial phases
1  f
0.3
2  f  0 . 3 sin(  2  1 )
Stronger coupling
1  f
0.6
 2  f  0.6 sin( 2   1 )
Bidirectional coupling
1  f  0 . 3 sin( 1   2 )
0.3
0.3
2  f  0 . 3 sin(  2  1 )
DCM for Phase Coupling
Allow connections to
depend on experimental
condition
j
i  f i   a ij sin(  i   j )
j
a ij
 j
i
a ij 
 i  fi  
K
a
j
ijK
sin( K [ i   j ])  
K
b
ijK
cos( K [ i   j ])
j
Phase interaction function is an arbitrary order Fourier series
Example: MEG data
Fuentemilla et al, Current Biology, 2010
Delay activity (4-8Hz)
Visual Cortex (VIS)
Medial Temporal Lobe (MTL)
Inferior Frontal Gyrus (IFG)
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] mm
2. Right VIS [10,-100,0] mm
3. 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 ?
MTL Master
MasterSlave
1
IFG
VIS
VIS Master
3 IFG
IFG
5
IFG
MTL
MTL
2
Partial
Mutual
Entrainment
VIS
IFG Master
VIS
4
IFG
MTL
MTL
VIS
IFG
VIS
MTL
6
IFG
VIS
MTL
MTL
7
Total
Mutual
Entrainment
VIS
Analysis
• Source reconstruct activity in areas of interest
• Bandpass data into frequency range of interest
• Hilbert transform data to obtain instantaneous phase
• Use multiple trials per experimental condition
• Model inversion
3 IFG
VIS
MTL
450
400
350
300
LogEv
250
200
150
100
50
0
1
2
3
4
Model
5
6
7
0.77
2.46
IFG
VIS
0.89
2.89
MTL
 i  f i   a ij sin([ i   j ])   bij cos([ i   j ])
j
j