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DCM for Phase Coupling

This study aims to develop a connectivity model for bandlimited data by studying long-range synchronization processes through phase coupling and changes in instantaneous frequency between regions.

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DCM for Phase Coupling

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  1. DCM for Phase Coupling Will Penny Wellcome Trust Centre for Neuroimaging, University College London, UK Brain Modes, Dec 12, 2008

  2. Overall Aim To study long-range synchronization processes Develop connectivity model for bandlimited data Regions phase couple via changes in instantaneous frequency Region 2 Region 1 ? ? 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. n Isochrons of a Morris-Lecar Neuron Isochron= 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. Hippocampus Septum Example: Theta rhythm Denham et al. 2000: 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 Limit Cycle Stable Equilibrium Point

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

  20. Septo-Hippocampal theta rhythm

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

  22. 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 ….

  23. 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

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

  25. 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

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

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

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

  29. Left Finger Right Finger Inferring the order of the PIF Multiple trials required to adequately sample state space p(est=2|true=2) High noise s=0.2 Number of trials

  30. 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

  31. 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 + 5 sec 3 sec 5 sec 1 sec MAINTENANCE PROBE ENCODING

  32. 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 ?

  33. 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

  34. 1 3 5 Occ Occ Occ Occ Occ Occ Occ IFG IFG IFG IFG IFG IFG IFG Hipp Hipp Hipp Hipp Hipp Hipp Hipp 6 2 4 7 Hippocampal source Occipital source Frontal source Master- Slave Partial Mutual Entrainment Total Mutual Entrainment

  35. Model Comparison LogEv Model

  36. 0.99 0.65 0.13 IFG Occ 0.00 0.03 0.03 0.17 Hipp 0.03 f=5.7Hz f=5.7Hz f=6.0Hz • Intrinsic connectivity established for control task (no memory requirement) • Modulatory connections required for ‘hard’ memory task • Fronto-occipital connections increased most strongly esp. Occ->IFG

  37. Model Fit Seconds

  38. Estimated Phase Interaction Functions, g From IFG Hipp Occ Hipp Hard Control Occ To IFG

  39. 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)

  40. Neural Mass model

  41. Neural Mass model Output Alpha Rhythm From Hopf Bifurcation Input Grimbert & Faugeras

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

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