Will PennyWill Penny

DCM for Time-Frequency

DCM Course, Paris, 2012DCM Course, Paris, 2012

1. DCM for Induced Responses

2. DCM for Phase Coupling

Dynamic Causal Models

Neurophysiological Phenomenological

• DCM for ERP

• DCM for SSR

• DCM for Induced Responses• DCM for Phase Coupling

spiny stellate

cells

inhibitory interneuron

s

PyramidalCells Time

Freq

uenc

y

Phase

Source locations not optimizedElectromagnetic forward model included

Region 1 Region 2

?

?

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 ?

1. DCM for Induced Responses

Time

Freq

uenc

y

Freq

uenc

y

Time

Connection to Neural Mass Models

First and Second orderVolterra kernelsFrom Neural Mass model.

Strong(saturating)input leads tocross-frequencycoupling

Single region 1 11 1 1z a z cu

u2

u1

z1

z2

z1

u1

a11c

cf. Neural state equations in DCM for fMRI

Multiple regions

1 11 1 1

2 21 22 2 2

0

0

z a z uc

z a a z u

u2

u1

z1

z2

z1

z2

u1

a11

a22

c

a21

cf. DCM for fMRI

Modulatory inputs

1 11 1 1 12

2 21 22 2 21 2 2

0 0 0

0 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

cf. DCM for fMRI

u1 u2

z1

z2

a11

a22

c

a12

a21

b21

Reciprocal connections

1 11 12 1 1 12

2 21 22 2 21 2 2

0 0

0 0

z a a z z ucu

z a a z b z u

u2

u1

z1

z2

cf. DCM for fMRI

dg(t)/dt=A g(t)∙+C u(t)∙

DCM for induced responses

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

Time

Freq

uenc

y

G=USV’

Use of Frequency Modes

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

Time

Freq

uenc

y

Connection to Neurobiology

From Neural Mass model.

Strong(saturating)input leads tocross-frequencycoupling

Connection to Neurobiology

Strong(saturating)input leads tocross-frequencycoupling

Connection to Neurobiology

Weak input does not

Differential equation model for spectral energy

KKij

Kij

Kijij

ij

AA

AA

A

1

111

Nonlinear (between-frequency) coupling

Linear (within-frequency) coupling

Extrinsic (between-source) coupling

)()()(1

1

1111

tu

C

C

tg

AA

AA

g

g

tg

JJJJ

J

J

Intrinsic (within-source) coupling

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

Modulatory connections

Extrinsic (between-source) coupling

1 11 1 11 1 1

1 1

( ) ( ) ( )J J

J J JJ J JJ J

g A A B B C

g t v g t u t

g A A B B C

Intrinsic (within-source) coupling

Example: MEG Data

15

81

39

z

y

x

15

81

42

z

y

x

27

45

42

z

y

x

24

51

39

z

y

x

OFA OFA

FFAFFA

input

The “core” system

nonlinear (and linear)

linear

Forward

Bac

kwar

d

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

1000backward linear backward nonlinear

forward linearforward nonlinear

Model Inference

Winning model: FNBN

Both forward and backward connections are nonlinear

Parameter Inference: gamma affects alpha

Right backward - inhibitory - suppressive effect of gamma-alpha coupling in backward connections

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

Freq

uenc

y (H

z)

From 30Hz

To 10Hz

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

Region 1

Region 3

Region 2

??

2. DCM for Phase Coupling2. DCM for Phase Coupling

( )i i jj

g PhaseInteractionFunction

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

Region 1 Region 2

( )i i jj

?

?

Synchronization

Gamma sync synaptic plasticity, forming ensembles

Theta sync system-wide distributed control (phase coding)

Pathological (epilepsy, Parkinsons)

Phase Locking Indices, Phase Lag etc are useful characterising systems in their steady state

Sync – Steve Strogatz

One Oscillator

f1

Two Oscillators

f1

f2

Two Coupled Oscillators

f1

)sin(3.0 122 f

0.3

Here we assume the Phase Interaction Function (PIF) is a sinewave

Different initial phases

f1

)sin(3.0 122 f

0.3

Stronger coupling

f1

2 2 10.6sin( )f

0.6

Bidirectional coupling

)sin(3.0 122 f

0.30.3

)sin(3.0 211 f

j

j

i

DCM for Phase Coupling

)sin( jij

ijii af

sin( [ ]) cos( [ ])i i ijK i j ijK i jK j K j

f a K b K

Phase interaction function is an arbitrary order Fourier series

Allow connections to depend on experimental condition

ija

ija

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] mm2. Right VIS [10,-100,0] mm3. 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

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

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. Data that we model are unwrapped phase time series in multiple regions.

• Use multiple trials per experimental condition

• Model inversion

LogEv

Model

1 2 3 4 5 6 70

50

100

150

200

250

300

350

400

450MTL

VISIFG3

MTL

VISIFG

2.89

2.46

0.89

0.77

sin([ ]) cos([ ])i i ij i j ij i jj j

f a b

Connection Strengths

Jones and Wilson, PLoS B, 2005

Recordings from rats doing spatial memory task:

MTL-VIS

IF

G-

VIS

Control

MTL-VIS

IF

G-

VIS

Memory

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

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

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