DCM for Time Frequency

Will PennyWill Penny

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

SPM MEG/EEG Course, SPM MEG/EEG Course, May 11thMay 11th, 20, 201010

DCM for Induced ResponsesRegion 1

Region 3

Region 2

??

How does slow activityin one region affectfast activity in another ?

Relate change of powerin one region and frequency to power in others.

Single region 1 11 1 1z a z cu

u2

u1

z1

z2

z1

u1

a11c

DCM for fMRI

Multiple regions

1 11 1 1

2 21 22 2 2

00

z a z ucz a a z u

u2

u1

z1

z2

z1

z2

u1

a11

a22

c

a21

Modulatory inputs

1 11 1 1 12

2 21 22 2 21 2 2

0 0 00 0

z a z z ucu

z a a z b z u

u2

u1

z1

z2

u2

z1

z2

u1

a11

a22

c

a21

b21

Reciprocal connections

1 11 12 1 1 12

2 21 22 2 21 2 2

0 00 0

z a a z z ucu

z a a z b z u

u2

u1

z1

z2

u2

z1

z2

u1

a11

a22

c

a12

a21

b21

Single Region

dg(t)/dt=Ag(t)+Cu(t)

DCM for induced responses

Where g(t) is a K x 1 vector of spectral responsesA is a K x K matrix of frequency coupling parametersDiagonal elements of A (linear – within freq)Off diagonal (nonlinear – between freq)Also allow A to be changed by experimental condition

DCM for induced responses

Specify the DCM:• 2 areas (A1 and A2) • 2 frequencies (F and S)• 2 inputs (Ext. and Con.)• Extrinsic and intrinsic coupling

Integratethe state equations

A1 A2

2

1

2

1

),(),(),(),(

A

A

A

A

tSxtSxtFxtFx

Differential equation model for spectral energy

KKij

Kij

Kijij

ij

AA

AAA

1

111

Nonlinear (between-frequency) coupling

Linear (within-frequency) coupling

Extrinsic (between-source) coupling

)()()(1

1

1111

tuC

Ctg

AA

AA

g

gtg

JJJJ

J

J

Intrinsic (within-source) coupling

How frequency K in region j affects frequency 1 in region i

Modulatory connections

Intrinsic (within-source) coupling

Extrinsic (between-source) coupling

state equation

JJJJ

J

JJJ

J

JC

C

tutwx

BB

BB

tu

AA

AA

x

x

W, t

1

1

111

1

1111

)(),()()(g

Neural Mass Models

WeakInput

StrongInput

Single Region

G=USV’

Use of Frequency Modes

Where G is a K’ x T spectrogramU is K’xK matrix with K frequency modesV is K’ x T and contains spectral mode responsesHence A is only K x K, not K’ x K’

MEG Data

158139

zyx

158142

zyx

274542

zyx

245139

zyx

OFA OFA

FFAFFA

input

The “core” system

nonlinear (and linear)

linear

Forward

Back

ward

linear nonlinear

linea

rno

nlin

ear

FLBL FNBL

FLNB FNBN

OFA OFA

Input

FFAFFA

FLBL

Input

FNBL

OFA OFA

FFAFFA

FLBN

OFA OFA

Input

FFAFFA

FNBN

OFA OFA

Input

FFAFFA

Face selective effectsmodulate within hemisphereforward and backward cxs

FLBL FNBL FLBN *FNBN

-59890

-16308 -16306 -11895

-70000

-60000

-50000

-40000

-30000

-20000

-10000

0

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000 backward linear backward nonlinear

forward linearforward nonlinear

Model Inference

• Both forward and backward connections are nonlinear

Parameter Inference: gamma affects alpha

Right backward - inhibitory – suppressiveeffect of gamma-alpha coupling in backward connections

Left forward – excitatory - activating effect of gamma-alpha coupling in the forward connections

During face processing

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

Freq

uenc

y (H

z)

Right hemisphereLeft hemisphere-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Forward Backward

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Forward Backward

Parameter Inference: gamma affects alpha

“Gamma activity in input areas induces slower dynamics in higher areas as prediction error is accumulated. Nonlinear coupling in high-level area

induces gamma activity in that higher area which then accelerates the decay of activity in the lower level. This decay is manifest as damped alpha

oscillations.”

• C.C. Chen , S. Kiebel, KJ Friston , Dynamic causal modelling of induced responses. NeuroImage, 2008; (41):1293-1312.

 • C.C. Chen, R.N. Henson, K.E. Stephan, J.M. Kilner, and K.J. Friston.

Forward and backward connections in the brain: A DCM study of functional asymmetries in face processing. NeuroImage, 2009 Apr 1;45(2):453-62

 

For studying synchronization among brain regions Relate change of phase in one region to phase in others

Region 1

Region 3

Region 2

??

DCM for Phase Coupling

( )i i jj

g

One Oscillator

f1

Two Oscillators

f1

f2

Two Coupled Oscillators

f1

)sin(3.0 122 f

0.3

Different initial phases

f1

)sin(3.0 122 f

0.3

Stronger coupling

f1

)sin(3.0 122 f

0.6

Bidirectional coupling

)sin(3.0 122 f

0.30.3

)sin(3.0 211 f

3

2

1

DCM for Phase Coupling

)sin( jij

ijii af

3

2

1

DCM for Phase Coupling

)sin( jij

ijii af

])[sin( jij

ijkk

ii kaf

3

2

1

DCM for Phase Coupling

)sin( jij

ijii af

])[sin( jij

ijkk

ii kaf

])[cos(])[sin( jij

ijkk

jij

ijkk

ii kbkaf

Phase interaction function is an arbitrary order Fourier series

3

2

1

DCM for Phase Coupling

)sin( jij

ijii af

Allow connections to depend onexperimental condition

MEG Example

Fuentemilla et al, Current Biology, 2010

Delay activity (4-8Hz)

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] mm2. Right VIS [10,-100,0] mm3. Right IFG [39,28,-12] mm

• Fit models to control data (10 trials) and memory 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 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

MTL

VISIFG

MTL

VISIFG

MTL

VISIFG

MTL

VISIFG

MTL

VISIFG

MTL

VISIFG1

MTL

VISIFG2

3

4

5

6

7

Master-Slave

PartialMutualEntrainment

TotalMutualEntrainment

MTL Master VIS Master IFG Master

LogEv

Model

1 2 3 4 5 6 70

50

100

150

200

250

300

350

400

450

MTL

VISIFG

2.89

2.46

0.89

0.77

MTL-VIS

IFG

-V

IS

Control

MTL-VIS

IFG

-V

IS

Memory

Jones and Wilson, PLoS B, 2005

Recordings from rats doing spatial memory task:

Connection to Neurobiology:Septo-Hippocampal 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

dx x k x z w x Pdtdx x k x z w xdt

dx 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

Hippocampus

Septum

A

A

B

B

Hopf Bifurcation

cossin)( baz

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 orderVolterra kernelsFrom Neural Mass model.

Strong(saturating)input leads tocross-frequencycoupling