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

DCM for Phase CouplingDCM for Phase Coupling

Will PennyWill Penny

Wellcome Trust Centre for Neuroimaging,Wellcome Trust Centre for Neuroimaging,University College London, UKUniversity College London, UK

Brain Modes, Dec 12, 2008Brain Modes, Dec 12, 2008

Overall Aim

To study long-range synchronization processesDevelop connectivity model for bandlimited dataRegions 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 Reduction0 0

0 0

0

( )

( ) ( )

( )

X F X

X t T X t

X

( ) ( )

( ) _

X F X P X

X Asymptotic Phase

Stable Limit Cycle

Perturbation

n

Isochrons of a Morris-Lecar Neuron

From Erm

Isochron=SameAsymptotic Phase

Phase Reduction0 0

0 0

0

( )

( ) ( )

( )

X F X

X t T X t

X

0 00 0

0

( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )

X F X P X

d X d X d XX X F X P X

dX dX dXd X d X

X F X P XdX dX

z p

Stable Limit Cycle

Perturbation

ISOCHRON

Assume 1st orderTaylor expansion

Phase Reduction

( ) ( )z p

( ) ( )X F X P X From a high-dimensionaldifferential eq.

To a one dimensionaldiff eq.

Phase Response Curve

( )z

Perturbation function

( )p

Example: Theta rhythm

Denham et al. 2000: Hippocampus

Septum

11 1 1 13 3 3

22 2 2 21 1

13 3 3 34 4 3

44 4 4 42 2

( ) ( )

( ) ( )

( ) ( )

( ) ( )

e e CA

i i

i e CA

i i S

dxx k x z w x P

dtdx

x k x z w xdtdx

x k x z w x Pdtdx

x k x z w x Pdt

1x

2x 3x

4xWilson-Cowan style model

Four-dimensional state space

( ) ( )z p

1 1 1 2( ) ( , )z p

2 2 2 1( ) ( , )z p

Now assume thatchanges sufficiently slowly that 2nd term can be replaced by a time averageover a single cycle

1( ) ,g z t p t t dt

T

2 1

This is the ‘Phase Interaction Function’

1 1 1 2( ) ( , )z p

2 2 2 1( ) ( , )z p

Now assume thatchanges sufficiently slowly that 2nd term can be replaced by a time averageover a single cycle

1( ) ,g z t p t t dt

T

2 1

This is the ‘Phase Interaction Function’

1 2 1( )g

2 1 2( )g

Now 2nd term is only a function of phase difference

( )i i jj

g

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

1

1( ) ,

( ) sin cosfN

n nn

g z t p t t dtT

g a n b n

We use a Fourier series approximation for the PIF

This choice is justified on the following grounds …

Phase Response Curves,

• Experimentally – using perturbation method

0 1

0

T T

T

)(z

Leaky Integrate and Fire Neuron

0 5 10 15 20 25 30

-60-55-50

t (ms)

V (m

V)0 5 10 15 20 25 30

01020

t(ms)dt

/dV

0 0.2 0.4 0.6 0.8 1024

z( )

VVVdV

dt

V

VVt

tVVV

Ra

a

R

aR

1log

))/exp(1(

Z is strictly positive: Type I response

Type II(pos and neg)

Hopf Bifurcation

Stable Equilibrium Point Stable Limit Cycle

cossin)( baz

For a Hopf bifurcation (Erm & Kopell…)

Septo-Hippocampal theta rhythm

Hippocampus

Septum

A

A

B

B

Septo-Hippocampal Theta rhythm

Theta fromHopf bifurcation

PIFs

1( ) ,g z t p t t dt

T

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

1

1 1

1

1

2

sin( ) cos( )

r

s c

q

Nki

i ij ki kjj

N Ns c

ij ijn ijnn n

Nq

ijn ijn c ijnq

df g

dt

g a n a n

a a u b

DCM for Phase Coupling Model

Where k denotes the kth trial. uq denotes qth modulatory input, a between trial effect

if is the frequency in the ith region (prior mean f0, dev = 3fb)

ijna

ijnb

has prior mean zero, dev=3fb

has prior mean zero, dev=3fb

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

0 1 2 3 4 5 6-0.5

0

0.5

V(

)

0 1 2 3 4 5 6-0.5

0

0.5

G(

)

tt t

dVg

dt

(b) High Freq

Ns=2, Nc=0

Ns=1, Nc=0

Anti-Phase Unstable

0 1 2 3 4 5 6-1

-0.5

0

0.5

V(

)

0 1 2 3 4 5 6-1

-0.5

0

0.5

1G

()

(a)

PIF

Low Freq

Anti-Phase Stable

Estimating coupling coefficient

2ˆ( )E a a

LeftFinger

RightFinger

a=0.5

DCM error

EMA error

sin( )

left

right right left

f

f a

Additive noise level

Inferring the order of the PIF

LeftFinger

RightFinger

Number of trials

p(est=2|true=2)

Multiple trials requiredto 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

+

+

+

1 sec 3 sec 5 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 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] mm2. Right Occ [10,-100,0] mm3. 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

OccIFG

Hipp

OccIFG

Hipp

OccIFG

Hipp

OccIFG

Hipp

OccIFG

Hipp

OccIFG1

Hipp

OccIFG2

3

4

5

6

7

Master-Slave

PartialMutualEntrainment

TotalMutualEntrainment

Hippocampalsource

Occipitalsource

Frontalsource

1 2 3 4 5 6 70

50

100

150

200

250

LogEv

Model

Model Comparison

Hipp

OccIFG

0.17

0.03

0.99

0.65

0.030.03

0.00

0.13

• 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=5.7Hzf=5.7Hz

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6-1

0

1Region Right-Hipp

Data

Fitted

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6-1

0

1Region Right-Occ

Data

Fitted

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6-1

0

1

Trial 20

Region Right-IFG

Data

Fitted

Seconds

Model Fit

Estimated Phase Interaction Functions, g

From

To

Hipp Occ IFG

Hipp

Occ

IFG

Hard

Control

0 5-0.1

0

0.1

0 5-1

0

1

0 5-0.2

0

0.2

0 5-1

0

1

0 5-1

0

1

0 5-2

0

2

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

Neural Mass model

Input

Output

Grimbert &Faugeras

Alpha RhythmFrom HopfBifurcation

Eg. Leaky Integrate and Fire Neuron

0 5 10 15 20 25 30

-60-55-50

t (ms)V

(mV)

0 5 10 15 20 25 300

1020

t(ms)

dt/d

V

0 0.2 0.4 0.6 0.8 1024

z( )

Z is strictly positive: Type I response

Type II(pos and neg)