Dynamic Causal Models

Will Penny

Olivier David, Karl Friston, Lee Harrison, Stefan Kiebel, Andrea Mechelli,

Klaas Stephan

MultiModal Brain Imaging, Copenhagen, October 25-26, 2005

V1

V5

SPC

V1

V5

SPC

Wellcome Department of Imaging Neuroscience, ION, UCL, UK.

Contents

• Neurodynamic model

• Hemodynamic model

• Model estimation and comparison

• Attention to visual motion

Friston et al.(2003) Neuro-Image, 19 (4), pp. 1273-1302.

Contents

• Neurodynamic model

• Hemodynamic model

• Model estimation and comparison

• Attention to visual motion

Single region

1 11 1 1z a z cu

u2

u1

z1

z2

z1

u1

a11c

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

Modulatory inputs

1 11 1 1 12

2 21 22 2 21 2 2

0 0 0

0 0

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z a a z b z u

u2

u1

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u2

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z2

u1

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a22

c

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Reciprocal connections

1 11 12 1 1 12

2 21 22 2 21 2 2

0 0

0 0

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z a a z b z u

u2

u1

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2

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Neurodynamics

i ii

z Az u B z Cu

InputsChange inNeuronalActivity

NeuronalActivity

IntrinsicConnectivityMatrix

ModulatoryConnectivityMatrices

InputConnectivityMatrix

V1

V5

SPC

Contents

• Neurodynamic model

• Hemodynamic model

• Model estimation and comparison

• Attention to visual motion

• Single word processing

Hemodynamics

( , , )

( )

g z

y b

x x h

x

Hemodynamic variables

For each region:

[ , , , ]s f v qx

Hemodynamic parameters

Seconds

Dynamics

Why have explicit models for neurodynamics and hemodynamics ?

44332211 ucucucucazz

For 4 event types u1, u2, u3, u4 :

In a GLM for a single region, y=X+e, with 3 basis functions per event type (canonical,shifter, stretcher) there are 12 parameters to estimate. These relate hemodynamics directly to each stimulus.

In a (single region) DCM there are 4 neuronal efficacy parameters relating neuronal activity to each stimulus

And 5 hemodynamic parameters relating neuronal activity to the BOLDsignal.

A total of 9 parameters.

),( hzby

DCM Priors

Hemodynamics

Rate of signal decay: 0.65Elimination rate: 0.41Transit time: 0.98Grubbs exponent: 0.32Oxygenation fraction: 0.34

E[h]Cov[h]

Neurodynamics

Stability priors ensure principal Lyapunov exponent is less than zerowith high probability.

Contents

• Neurodynamic model

• Hemodynamic model

• Model estimation and comparison

• Attention to visual motion

• Single word processing

Bayesian Estimation

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Relative Precision Weighting

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2( | ) ( ; , )p y N

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22 2

1 1 1

1 1

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pe p

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y e

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Normal densities

Multiple parameters

(1)θ

(2)θ

( ) ( ; , )p pp Nθ θ μ C

( | ) ( ; , )ep Ny θ y Xθ C

( | ) ( ; , )p Nθ y θ μ C

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1 1

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GeneralLinear Model

Nonlinear models( )b y θ e

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Gauss-Newton ascent with priors

Linearization

CurrentEstimates

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r

Friston et al.(2002) Neuro-Image, 16 (2), pp. 513-530.

Model Comparison I

{ , , , }θ A B C h

( | , ) ( | )( | , )

( | )

p m p mp m

p m

y θ θθ y

y

V1

V5

SPC

( | ) ( | , ) ( | )p m p m p m dy y θ θ θ

Model, m Parameters:

PriorPosterior Likelihood

Evidence

Laplace, AIC, BIC approximations

Model fit + complexityPenny et al. (2004) NeuroImage, 22(3), pp. 1157-1172.

Model Comparison II

{ , , , }θ A B C h

( | , ) ( | )( | , )

( | )

p m p mp m

p m

y θ θθ y

y

V1

V5

SPC

Model, mParameters:

PriorPosterior Likelihood

( | ) ( )( | )

( )

p m p mp m

p

yy

y

PriorPosterior Evidence

Parameter Parameter

Model Model

Model Comparison III

V1

V5

SPC

( | ) ( | , ) ( | )p m i p m i p m i d y y θ θ θ

( | ) ( | , ) ( | )p m j p m j p m j d y y θ θ θ

Model, m=i

V1

V5

SPC

Model, m=j

Model Evidences:

Bayes factor:

( | )

( | )ij

p m iB

p m j

y

y

1 to 3: Weak3 to 20: Positive20 to 100: Strong>100: Very Strong

Contents

• Neurodynamic model

• Hemodynamic model

• Bayesian estimation

• Attention to visual motion

• Single word processing

Attention to Visual MotionAttention to Visual Motion

STIMULISTIMULI

250 radially moving dots at 4.7 degrees/s250 radially moving dots at 4.7 degrees/s

PRE-SCANNINGPRE-SCANNING

5 x 30s trials with 5 speed changes (reducing to 1%)5 x 30s trials with 5 speed changes (reducing to 1%)

Task - detect change in radial velocityTask - detect change in radial velocity

SCANNINGSCANNING (no speed changes)(no speed changes)

6 normal subjects, 4 100 scan sessions;6 normal subjects, 4 100 scan sessions;

each session comprising 10 scans of 4 different conditioneach session comprising 10 scans of 4 different condition

1. Photic2. Motion3. Attention

Experimental Factors

Buchel et al. 1997

Specify regions of interest

Identify regions of Interest eg. V1, V5, SPC

GLM analysis

V1

V5

SPC

Motion

Photic

Att

Model 1

V1

V5

SPC

Motion

Photic

Att

Model 1V1

V5

Estimation

SPC

Time (seconds)

Posterior Inference

B321

P(B

3 21|y

)

How muchattention (input 3) changes connection fromV1 (region 1) to V5 (region(2)

V1

V5

SPC

Motion

Photic

Att

Model 1

Motion

Photic

Att

V1

V5

SPC

Model 2

Bayes Factor B12 > 1019

VeryStrong

V1

V5

SPC

Motion

Photic

Att

Model 1

V1

V5

SPC

Motion

Photic

Att

Model 3

Bayes Factor B13=3.6

Positive

V1

V5

SPC

Motion

Photic

Att

Model 1

Motion

Photic

Att

Att

V1

V5

SPC

Model 4

Bayes Factor B14=2.8

Weak

Penny et al. (2004) NeuroImage, Special Issue.

Summary

• Neurodynamic model

• Hemodynamic model

• Bayesian estimation

• Attention to visual motion