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Weakly Coupled Oscillators

Weakly Coupled Oscillators. Will Penny, Vladimir Litvak, Lluis Fuentemilla, Emrah Duzel, Karl Friston. Wellcome Trust Centre for Neuroimaging, University College London, UK. Symposium on Multimodal Brain Imaging, University of Birmingham, May 15 th , 2009. Transient Synchronization.

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Weakly Coupled Oscillators

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  1. Weakly Coupled Oscillators Will Penny, Vladimir Litvak, Lluis Fuentemilla, Emrah Duzel, Karl Friston Wellcome Trust Centre for Neuroimaging, University College London, UK Symposium on Multimodal Brain Imaging, University of Birmingham, May 15th, 2009

  2. Transient Synchronization • Neuronal oscillations are the key to linking computational to imaging neuroscience and different imaging modalities to each other • Short range (zero-lag) sync (within single cortical area). Hebbian learning (STDP) results in gamma bursts signalling recognition events (Hopfield). Local sync necessary for long range signal transmission (Fries). • Long range (zero-lag) sync (between cortical areas) mediated by oscillations (Singer, Varela,..). Triplets of regions (Vicente, Lumer …). Long range sync at lower freqs ? Information relayed via pattern and/or phase coding (O’Keefe, Panzeri et al.) or sync necessary for chunking (Jensen/Lisman). • Sync is Transient

  3. 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

  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. 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

  6. Phase Reduction Stable Limit Cycle Perturbation

  7. n Isochrons of a Morris-Lecar Neuron Isochron= Same Asymptotic Phase From Erm

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

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

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

  11. Four-dimensional state space

  12. 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’

  13. 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’

  14. Multiple Oscillators

  15. 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

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

  17. Phase Response Curves, • Experimentally – using perturbation method

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

  19. Hopf Bifurcation Stable Limit Cycle Stable Equilibrium Point

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

  21. Septo-Hippocampal theta rhythm

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

  23. 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 ….

  24. 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

  25. 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

  26. -0.3 -0.6 -0.3 -0.3

  27. 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

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

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

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

  31. 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

  32. 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

  33. 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 MTL [27,-18,-27] mm • 2. Right VIS [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 ?

  34. 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

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

  36. LogEv Model

  37. 0.77 2.46 IFG VIS 0.89 2.89 MTL

  38. MTL VIS IFG Seconds

  39. Control fIFG-fVIS fMTL-fVIS

  40. Memory fIFG-fVIS fMTL-fVIS

  41. 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. Brunel and Wang, 03 ! • Stochastic dynamics (natural decoupling) … see Kuramoto 84, Brown 04 • For within-trial inputs causing phase-sync and desync (Tass model) • What would we see in fMRI ?

  42. Neural Mass model

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

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

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