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

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

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 ?

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

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

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

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

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

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

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

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

Example: MEG Data

The “core” system FFA FFA OFA OFA input

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

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

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 4 12 20 28 36 44 44 36 28 20 12 4 SPM t df 72; FWHM 7.8 x 6.5 Hz Frequency (Hz)

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

One Oscillator

Two Oscillators

Two Coupled Oscillators 0.3

Different initial phases 0.3

Stronger coupling 0.6

Bidirectional coupling 0.3 0.3

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

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

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

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 ?

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

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

3 IFG VIS MTL LogEv Model

MTL VIS IFG 2.89 2.46 0.89 0.77

Control fIFG-fVIS fMTL-fVIS

Memory fIFG-fVIS fMTL-fVIS

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

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

Four-dimensional state space

Hopf Bifurcation Hippocampus Septum A B A B

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

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