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

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 3 Region 2 ? ?

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

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

Phase Reduction Stable Limit Cycle Perturbation ISOCHRON Assume 1 st 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 2 nd 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 2 nd term can be replaced by a time average over a single cycle This is the ‘Phase Interaction Function’ Now 2 nd term is only a function of phase difference

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 Z is strictly positive: Type I response Type II (pos and neg)

Hopf Bifurcation Stable Equilibrium Point Stable Limit Cycle

For a Hopf bifurcation (Erm & Kopell…)

Septo-Hippocampal theta rhythm

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

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. u q denotes qth modulatory input, a between trial effect is the frequency in the ith region (prior mean f 0, dev = 3f b ) has prior mean zero, dev=3f b

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 FreqHigh Freq

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

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

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

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 sec 3 sec 5 sec 1) No retention (control condition): Discrimination task 2) Retention I (Easy condition): Non-configural task 3) Retention II (Hard condition): Configural task ENCODING MAINTENANCEPROBE

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

Hipp Occ IFG Hipp Occ IFG Hipp Occ IFG Hipp Occ IFG Hipp Occ IFG Hipp Occ IFG 1 Hipp Occ IFG Master- Slave Partial Mutual Entrainment Total Mutual Entrainment Hippocampal source Occipital source Frontal source

LogEv Model Model Comparison

Hipp OccIFG 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 f=6.0Hz f=5.7Hz

Seconds Model Fit

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

Conclusions 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) 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

Neural Mass model

Input Output Grimbert & Faugeras Alpha Rhythm From Hopf Bifurcation

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