1. DCM for Time-Frequency
1. DCM for Induced Responses
2. DCM for Phase Coupling
Bernadette van Wijk
VU University Amsterdam, The Netherlands

2. inhibitory interneurons
Dynamic Causal Models
Neurophysiological
Phenomenological
spiny stellate cells
inhibitory interneurons
Pyramidal
Cells
Phase
Frequency
Time
Electromagnetic forward model included
Source locations not optimized
DCM for ERP
DCM for SSR
DCM for Induced Responses
DCM for Phase Coupling

3. 1. DCM for Induced Responses
Region 1
Region 2 ?
Frequency
Frequency ?
Time
Time
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 ?

4. cf. Neural state equations in DCM for fMRI
Single region u1 c
a11 u1 z1 u2 z1 z2

5. cf. DCM for fMRI
Multiple regions z1 z2 u1
a11
a22 c
a21 u1 u2 z1 z2

6. Modulatory inputs u1 u2 z1 z2 u1 u2 z1 z2 cf. DCM for fMRI c a11 b21

7. Reciprocal connections
cf. DCM for fMRI
Reciprocal connections u1 u2 c
a11 u1 z1 u2
a12
b21 z1
a21 z2 z2
a22

8. DCM for induced responses
dg(t)/dt=A∙g(t)+C∙u(t)
Frequency
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

9. Use of Frequency Modes G=USV’ Where G is a K x T spectrogram
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

10. Differential equation model for spectral energy
Extrinsic (between-source) coupling
Intrinsic (within-source) coupling
Nonlinear (between-frequency) coupling
Linear (within-frequency) coupling
How frequency K in
region j affects
frequency 1 in region i

11. Modulatory connections
Extrinsic (between-source) coupling
Intrinsic (within-source) coupling

13. Example: MEG Data

14. The “core” system
FFA
FFA
OFA
OFA
input

15. FLBL FNBL FLBN FNBN FLBL FNBL FLNB FNBN Forward Backward
Face selective effects
modulate within hemisphere
forward and backward cxs
Forward
linear
nonlinear
linear
FLBL
FNBL
nonlinear (and linear)
linear
Backward
nonlinear
FLNB
FNBN
OFA
Input
FFA
FLBL
Input
FNBL
OFA
FFA
FLBN
OFA
Input
FFA
FNBN
OFA
Input
FFA

17. Model Inference Winning model: FNBN
Both forward and backward connections are nonlinear
FLBL
FNBL
FLBN
*FNBN
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backward linear
backward nonlinear
forward linear
forward nonlinear

18. From 32 Hz (gamma) to 10 Hz (alpha)
Parameter Inference: gamma affects alpha
From 30Hz
Left forward - excitatory - activating effect of gamma-alpha coupling in the forward connections
To 10Hz
Right backward - inhibitory - suppressive effect of gamma-alpha coupling in backward connections
From 32 Hz (gamma) to 10 Hz (alpha)
t = 4.72; p = 0.002
SPM t df 72; FWHM 7.8 x 6.5 Hz
Frequency (Hz)

19. 2. DCM for Phase Coupling
Region 1
Region 2 ? ?
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

20. One Oscillator

21. Two Oscillators

22. Two Coupled Oscillators
0.3

23. Different initial phases
0.3

24. Stronger coupling
0.6

25. Bidirectional coupling
0.3
0.3

27. DCM for Phase Coupling
Allow connections to depend on experimental condition
Phase interaction function is an arbitrary order Fourier series

28. Example: MEG data
Fuentemilla et al, Current Biology, 2010

29. Delay activity (4-8Hz) Visual Cortex (VIS) Medial Temporal Lobe (MTL)
Inferior Frontal Gyrus (IFG)

31. 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 ?

32. MTL Master
VIS Master
IFG Master 1
IFG
VIS 3
IFG
VIS 5
IFG
VIS
Master-
Slave
MTL
MTL
MTL 2 6
IFG
VIS
IFG
VIS 4
IFG
VIS
Partial
Mutual
Entrainment
MTL
MTL
MTL 7
IFG
VIS
Total
Mutual
Entrainment
MTL

33. 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

34. 3
IFG
VIS
MTL
LogEv
Model

35. MTL
VIS
IFG
2.89
2.46
0.89
0.77

38. Control
fIFG-fVIS
fMTL-fVIS

39. Memory
fIFG-fVIS
fMTL-fVIS

40. Recordings from rats doing spatial memory task:
Jones and Wilson, PLoS B, 2005

41. Connection to Neurobiology: Septo-Hippocampal theta rhythm
Denham et al. 2000:
Hippocampus
Septum
Wilson-Cowan style model

43. Four-dimensional state space

44. Hopf Bifurcation
Hippocampus
Septum A B A B

45. For a generic Hopf bifurcation (Erm & Kopell…)
See Brown et al. 04, for PRCs corresponding to other bifurcations

46. Connection to Neural Mass Models
First and
Second order
Volterra kernels
From Neural
Mass model.
Strong
(saturating)
input leads to
cross-frequency
coupling