1. DCM for Phase Coupling Will Penny
Wellcome Trust Centre for Neuroimaging,
University College London, UK
Sir Peter Mansfield MR Centre, Nottingham University,
Wed Jan 28, 2009

2. Overall Aim To study long-range synchronization processes
Develop connectivity model for bandlimited data
Regions phase couple via changes in instantaneous frequency
Region 1
Region 2 ? ?
Region 3

3. Overview Phase Reduction Choice of Phase Interaction Function (PIF)
DCM for Phase Coupling
Ex 1: Finger movement
Ex 2: MEG Theta visual working memory
Conclusions

4. Overview Phase Reduction Choice of Phase Interaction Function (PIF)
DCM for Phase Coupling
Ex 1: Finger movement
Ex 2: MEG Theta visual working memory
Conclusions

5. Phase Reduction
Stable Limit Cycle
Perturbation

6. Isochrons of a Morris-Lecar Neuron
Same
Asymptotic
Phase
From Erm

7. Phase Reduction Stable Limit Cycle Perturbation ISOCHRON
Assume 1st order
Taylor expansion

8. Phase Reduction From a high-dimensional differential eq.
To a one dimensional
diff eq.
Phase Response Curve
Perturbation function

9. Example: Theta rhythm Denham et al. 2000: Hippocampus Septum
Wilson-Cowan style model

10. Four-dimensional state space

11. Now assume that
changes sufficiently slowly that 2nd term
can be replaced by a time average
over a single cycle
This is the ‘Phase Interaction Function’

12. Now assume that
changes sufficiently slowly that 2nd term
can be replaced by a time average
over a single cycle
Now 2nd term is only a function of phase difference
This is the ‘Phase Interaction Function’

13. Multiple Oscillators

14. Overview Phase Reduction Choice of Phase Interaction Function (PIF)
DCM for Phase Coupling
Ex 1: Finger movement
Ex 2: MEG Theta visual working memory
Conclusions

15. Choice of g We use a Fourier series approximation for the PIF
This choice is justified on the following grounds …

16. Phase Response Curves,
Experimentally – using perturbation method

17. Leaky Integrate and Fire Neuron
Type II
(pos and
neg)
Z is strictly positive: Type I response

18. Hopf Bifurcation
Stable Equilibrium Point
Stable Limit Cycle

19. For a Hopf bifurcation (Erm & Kopell…)

20. Septo-Hippocampal theta rhythm

21. Septo-Hippocampal Theta rhythm
Hippocampus
Septum
Theta from
Hopf bifurcation A B A B

22. Neural mass model of cortex
Jansen & Ritt

23. Oscillations from neural mass model

24. Bifurcation analysis of neural mass model
Output
Alpha Rhythm
From Hopf
Bifurcation
Input
Grimbert &
Faugeras

25. PIFs
Even if you have a type I PRC, if the perturbation is non-instantaneous,
then you’ll end up with a type II first order Fourier PIF (Van Vreeswijk, alpha function synapses)
… so that’s our justification.
… and then there are delays ….

26. Overview Phase Reduction Choice of Phase Interaction Function (PIF)
DCM for Phase Coupling
Ex 1: Finger movement
Ex 2: MEG Theta visual working memory
Conclusions

27. DCM for Phase Coupling Model
Where k denotes the kth trial. uq denotes qth modulatory input, a between trial effect
is the frequency in the ith region (prior mean f0, dev = 3fb)
has prior mean zero, dev=3fb

28. Sinusoidal coupling
-0.3
-0.6
-0.3
-0.3
is a stable fixed point

29. Overview Phase Reduction Choice of Phase Interaction Function (PIF)
DCM for Phase Coupling
Ex 1: Finger movement
Ex 2: MEG Theta visual working memory
Conclusions

30. MEG data from Visual Working Memory
1) No retention (control condition): Discrimination task +
2) Retention I (Easy condition): Non-configural task +
3) Retention II (Hard condition): Configural task +
1 sec
3 sec
5 sec
5 sec
ENCODING
MAINTENANCE
PROBE

31. Questions for DCM
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 Hipp [27,-18,-27] mm
2. Right Occ [10,-100,0] mm
3. 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 ?

32. 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
Use first order Fourier PIFs

33. 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

34. Model Comparison
LogEv
Model

35. Optimal model: Strength of connections
MTL
VIS
IFG
2.89
2.46
0.89
0.77
Numbers are norm of Fourier coeffs
Intrinsic connectivity (arrow end) established for control task (no memory)
Modulatory connections (dotted end) required for memory task

36. CONTROL Relative Phase Background Blue arrows=Flow field=
Gray-scale is
So black areas are
Fixed
Points
Blue arrows=Flow field=
fIFG-fVIS
Red dots
show
fitted
trajectories
of individual
trials
Global
Zero-Lag
Sync
fMTL-fVIS

37. MEMORY
fIFG-fVIS
fMTL-fVIS

38. Model Fit
MTL
VIS
IFG
Seconds
One memory trial

39. Overview Phase Reduction Choice of Phase Interaction Function (PIF)
DCM for Phase Coupling
Ex 1: Finger movement
Ex 2: MEG Theta visual working memory
Conclusions

40. Finger movement
Haken et al. 95
Low Freq
High Freq

41. (a)
PIF
Low Freq
Anti-Phase Stable
Ns=2, Nc=0
Anti-Phase Unstable
(b)
High Freq
Ns=1, Nc=0

42. Estimating coupling coefficient
Left
Finger
Right
a=0.5
EMA error
DCM error
Additive noise level

43. Inferring the order of the PIF
Multiple trials required
to adequately
sample state space
Left
Finger
Right
Distribution of
Initial Phase
Difference:
Narrow
Wide
# times true
model selected
High noise
s=0.2
Number of trials

44. Conclusions
Model is multivariate extension of bivariate work by Rosenblum & Pikovsky
(EMA approach)
On bivariate data DCM-P is more accurate than EMA
Additionally, DCM-P allows for inferences about master-slave versus
mutual entrainment mechanisms in multivariate (N>2) oscillator networks
Delay estimates from DTI
Use of phase response curves derived from specific neuronal models
using XPP or MATCONT
Stochastic dynamics (natural decoupling) … see Kuramoto 84, Brown 04
For within-trial inputs causing phase-sync and desync (Tass model)