Dynamic Causal Models

Dynamic Causal Models

Will Penny

Olivier David, Karl Friston, Lee Harrison, Andrea Mechelli, Klaas Stephan

Mathematics in Brain Imaging, IPAM, UCLA, USA, July 22 2004.

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

• Single word processingFriston et al.(2003) Neuro-Image, 19 (4), pp. 1273-1302.

Contents





• Single word processing

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

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 0

0 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

a1

2

a21

b21

Neurodynamics

i ii

z Az u B z Cu

InputsChange inNeuronalActivity

NeuronalActivity

IntrinsicConnectivityMatrix

ModulatoryConnectivityMatrices

InputConnectivityMatrix

V1

V5

SPC

Contents






Hemodynamics

( , , )

( )

g z

y b

x x h

x

Hemodynamic variables

For each region:

[ , , , ]s f v qx

Hemodynamic parameters

Seconds

Dynamics

Hemodynamic saturation

Sub-linear and super-linearresponses to pairs of stimuli

Equivalent input-outputfunctions

Neuronal impulse

Neuronal impulses

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






Bayesian Estimation

2( ) ( ; , )p pp N

Relative Precision Weighting

2( | ) ( ; , )ep y N y

2( | ) ( ; , )p y N

2 2 2

22 2

1 1 1

1 1

e p

pe p

y

y e

p

y

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

1 1 1

1 1

Te p

Te p p

C X C X C

μ C X C y C μ

y Xθ e

One-step if Ce, Cp and p are known

GeneralLinear Model

Nonlinear models( )b y θ e

( )( ) ( ) ( )

( )

( )

ii i

i

i

i

bb b

b

r y b

r

μθ μ θ μ

θμ

Jθ

θ θ μ

μ

J θ e

( ) ( ; , )

( ) ( ; , )

p p

p i p

p N

p N

θ θ μ C

θ θ μ μ C

( | ) ( ; ( ), )

( | ) ( ; , )e

e

p N b

p N

y θ y θ C

r θ r J θ C

,i iμ C

1 1 1

1

1 11 1

Ti e p

Ti i i e p p i

C J C J C

μ μ C J C r C μ μ

Gauss-Newton ascent with priors

Linearization

CurrentEstimates

{ , , , }θ A B C h

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



• Bayesian estimation



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.

Contents






Single word processing

SPM{F}

“dog”“radio”“gate”….

10,15,30,60 and 90words perminute

A2

WA

A1

0.92

0.38

0.47

-0.62

-0.51

0.37

Input 1:Word Presentation

Input 1

Estimated Model

Intrinsic connections: Full connectivityassumed – only significant connectionsshown.

Modulatory connections: modulation of all self-connections, only A1 and WA significant

Hemodynamic saturation in A1

Seconds

y

Summed individual responses

Responseto pair

IndividualStimuli

Pair ofStimuli

Neuronal Saturation in A1

Seconds

zt modulationof A1 selfconnection

With

or

without

A1

-0.62

Identifiability

1

1

1

3231

2321

1312

aa

aa

aa

A

Estimation of relative intrinsic connections and modulatory connections is robust to errors inestimation of hemodynamics due toeg. slice timing problems

Indeterminacy in neurodynamic and hemodynamic time constants is soaked up in

u2

z1

z2

u1

-c

a1

2

a21

b21

-

Summary






Neuronal Transients and BOLD

300ms 500ms

More enduring transients produce bigger BOLD signals

SecondsSeconds

BOLD is sensitive to frequencycontent of transients

Seconds

SecondsRelative timings of transients areamplified in BOLD

Neurodynamics Hemodynamics

Dynamic Causal Models

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