DCM for Time-Frequency

1. DCM for Time-Frequency 1. DCM for Induced Responses 2. DCM for Phase Coupling Will Penny DCM Course, Paris, 2012

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

3. 1. DCM for Induced Responses Region 2 Region 1 ? 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. Connection to Neural Mass Models First and Second order Volterra kernels From Neural Mass model. Strong (saturating) input leads to cross-frequency coupling

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

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

7. cf. DCM for fMRI Modulatory inputs u1 u2 c u1 a11 z1 u2 b21 z1 a21 z2 z2 a22

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

9. 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

10. Use of Frequency Modes G=USV’ Frequency 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

11. Connection to Neurobiology From Neural Mass model. Strong (saturating) input leads to cross-frequency coupling

12. Connection to Neurobiology Strong (saturating) input leads to cross-frequency coupling

13. Connection to Neurobiology Weak input does not

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

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

16. Example: MEG Data

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

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

20. Model Inference Winning model: FNBN Both forward and backward connections are nonlinear FLBL FNBL FLBN *FNBN 1000 backward linear backward nonlinear 0 0 -1000 -10000 -2000 -16306 -16308 -11895 -20000 -3000 -30000 -4000 -5000 -40000 forward linear -6000 forward nonlinear -50000 -7000 -60000 -8000 -59890 -70000

21. Parameter Inference: gamma affects alpha SPM tdf 72; FWHM 7.8 x 6.5 Hz From 30Hz Left forward - excitatory- activating effect of gamma-alpha coupling in the forward connections 4 12 20 28 36 44 To 10Hz Frequency (Hz) 44 36 28 20 12 4 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

23. 2. DCM for Phase Coupling For studying synchronization among brain regions Relate change of phase in one region to phase in others Region 2 Region 1 ? ? Phase Interaction Function Region 3

24. 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

25. Synchronization Gamma sync  synaptic plasticity, forming ensembles Theta sync  system-wide distributed control (phase coding) Pathological (epilepsy, Parkinsons) Phase Locking Indices, Phase Lag etc are useful characterising systems in their steady state Sync – Steve Strogatz

26. One Oscillator

27. Two Oscillators

28. Two Coupled Oscillators 0.3 Here we assume the Phase Interaction Function (PIF) is a sinewave

29. Different initial phases 0.3

30. Stronger coupling 0.6

31. Bidirectional coupling 0.3 0.3

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

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

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

36. 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 ?

37. 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

38. 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. Data that we model are • unwrapped phase time series in multiple regions. • Use multiple trials per experimental condition • Model inversion

39. 3 IFG VIS MTL LogEv Model

40. 0.77 2.46 IFG VIS 0.89 2.89 MTL Connection Strengths

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

44. Control fIFG-fVIS fMTL-fVIS

45. Memory fIFG-fVIS fMTL-fVIS

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

48. Four-dimensional state space

49. Hippocampus Septum Hopf Bifurcation A B A B

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