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

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

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

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

Phase Reduction Stable Limit Cycle Perturbation

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

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

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

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

Four-dimensional state space

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’

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’

Multiple Oscillators

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

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

Phase Response Curves, Experimentally – using perturbation method

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

Hopf Bifurcation Stable Equilibrium Point Stable Limit Cycle

For a Hopf bifurcation (Erm & Kopell…)

Septo-Hippocampal theta rhythm

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

Neural mass model of cortex Jansen & Ritt

Oscillations from neural mass model

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

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 ….

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

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

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

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

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

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 ?

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

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

Model Comparison LogEv Model

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

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

MEMORY fIFG-fVIS fMTL-fVIS

Model Fit MTL VIS IFG Seconds One memory trial

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

Finger movement Haken et al. 95 Low Freq High Freq

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

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

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

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)