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• Neurodynamic model

• Hemodynamic model

• Model estimation and comparison

• Attention to visual motion

• 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

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

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

Hemodynamicvariables

For each region:

[ , , , ]s f v q=x

Hemodynamicparameters

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 neurodynamicsand hemodynamics ?

44332211 ucucucucazz ++++=

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

In a GLM for a single region, y=Xβ+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 PriorsHemodynamics

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 Estimation2( ) ( ; , )p pp Nθ θ µ σ=

Relative Precision Weighting

2( | ) ( ; , )ep y N y θ σθ =

2( | ) ( ; , )p y Nθ θ µ σ=

2 2 2

22 2

1 1 1

1 1

e p

pe p

y

σ σ σ

µ σ µσ σ

= +

= +

y eθ= +

pµ

yµ

Normal densities

Multiple parameters

(1)θ

(2)θ

( ) ( ; , )p pp N=θ θ µ C

( | ) ( ; , )ep N=y θ y Xθ C

( | ) ( ; , )p N=θ 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 µp are known

GeneralLinear Model

Nonlinear models( )b= +y θ e

( )( ) ( ) ( )

( )

( )

ii i

i

i

i

bb b

b

r y br

∂= + −

∂∂

=∂

∆ = −= −= ∆ +

µθ µ θ µθ

µJθ

θ θ µµ

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 bp N

=

∆ = ∆

y θ y θ Cr 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

θ∆

r

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

Model Comparison I

{ , , , }=θ A B C h

( | , ) ( | )( | , )( | )

p m p mp mp m

=y θ θθ y

y

V1

V5

SPC

( | ) ( | , ) ( | )p m p m p m d= ∫y 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 mp m

=y θ θθ y

y

V1

V5

SPC

Model, mParameters:

PriorPosterior Likelihood

( | ) ( )( | )( )

p m p mp mp

=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:

( | )( | )ijp m iBp m j

==

=yy

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

STIMULISTIMULI250 250 radiallyradially moving dots at 4.7 degrees/smoving dots at 4.7 degrees/s

PREPRE--SCANNINGSCANNING5 x 30s trials with 5 speed changes (reducing to 1%)5 x 30s trials with 5 speed changes (reducing to 1%)Task Task -- detect change in radial velocitydetect 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 1−10

−5

0

5

10

−4

−2

0

2

4

0 200 400 600 800 1000 1200−4

−2

0

2

4

V1

V5

Estimation

SPC

Time (seconds)

Posterior Inference

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

B321

P(B

3 21|y

)

γ

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• Neurodynamic model

• Hemodynamic model

• Bayesian estimation

• Attention to visual motion

• Single word processing

Single word processing

SPM{F}

“dog”“radio”“gate”….

10,15,30,60 and 90words perminute

A2

WA

A1

0.92

0.380.47

-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

Withor

without

A1

-0.62

Identifiability

−−

−=

11

1

3231

2321

1312

aaaaaa

A τ

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

Indeterminacy in neurodynamic and hemodynamic time constants is soaked up in τ

u2

z1

z2

u1

-τc

a12

a21

b21

-τ

Summary

• Neurodynamic model• Hemodynamic model• Bayesian estimation• Attention to visual motion• Single word processing

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