Generative Models for Brain Imaging

Will Penny

Wellcome Department of Imaging Neuroscience, UCLhttp://www.fil.ion.ucl.ac.uk/~wpenny

Natural Computing Applications Forum (NCAF), ‘Cognitive Systems –from Neural Imaging to Neurocomputing’, 18-19 May, 2005,

York University, UK.

2D fMRI time-series

Time

Serial snapshotsof blood –

oxygenation levels

Time

Even without applied spatial smoothing, activation maps (and maps of eg. AR coefficients) have spatial structure

Motivation

We can increase the sensitivity of our inferences by smoothing data with Gaussian kernels (SPM2). This is worthwhile, but crude.Can we do better with a spatial model (SPM5) ?

AR(1)

Aim: For SPM5 to remove the need for spatial smoothing just as SPM2 removed the need for temporal smoothing

Contrast

β

A

q1 q2

α

W

Y

( ) ( )

( ) ( )1

1 2; ,

K

kk

k k

p p

p Ga q q

α

α α=

=

=

∏α

( ) ( )

( ) ( )( )1

11; ,

KTk

k

T T Tk k k

p p

p N α

=

−−

=

=

∏W w

w w 0 S S

u1 u2

λ

( ) ( )

( ) ( )1

1 2; ,

N

nn

n n

p p

p Ga u u

λ

λ λ=

=

=

∏λ

( )1

1 1

( ) ( )

( ) ; , ( )

P

pp

Tp p p

p p

p N β=

− −

=

=

∏A a

a a 0 S S

Y=XW+E[TxN] [TxK] [KxN] [TxN]

r1 r2

1

1 2

( ) ( )

( ) ( ; , )

P

pp

p p

p p

p Ga r r

β

β β=

=

=

∏β

0pβ →

pβ →∞

Voxel-wise AR:

Spatial pooled AR:

The Model

t

Synthetic Data 1 : from Laplacian Prior

reshape(w1,32,32)

Prior, Likelihood and PosteriorIn the prior, W factorises over k and A factorises over p:

1 2 1 2( , , , , ) ( | ) ( | , ) ( | ) ( | , )k k k p p pk p

p p p q q p p r rα α β β

= ∏ ∏W A λ α β w a

1 2( | , )nn

p u uλ ∏

The likelihood factorises over n:

( | , , ) ( | , , )n n n nn

p p λ=∏Y W A λ y w a

The posterior over W therefore does’nt factor over k or n. It is a Gaussianwith an NK-by-NK full covariance matrix. This is unwieldy to even store, let alone invert ! So exact inference is intractable.

Variational Bayes

L F KL= +

( , )( )( | )ppp

=Y θYθ Y

log ( ) log ( , ) log ( | )p p p= −Y Y θ θ Y

log ( ) ( ) log ( , ) ( ) log ( | )p q p d q p d= −∫ ∫Y θ Y θ θ θ θ Y θ

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

p qp q d q dq p

= +∫ ∫Y θ θY θ θ θ θθ θ Y

L

F

KL

Variational Bayes

( ) ( )ii

q q=∏θ θ

If you assume posterior factorises

then F can be maximised by letting

exp[ ( )]( )exp[ ( )]

ii

i i

IqI d

=∫

θθθ θ

where

/ /( ) ( ) log ( , )i i iI q p d= ∫θ θ Y θ θ

Variational Bayes

( , , , | ) ( | ) ( | ) ( | ) ( | ) ( | )k p n n nk p n

q q q q q qα β λ

= ∏ ∏ ∏W A λ α Y Y Y w Y a Y Y

In chosen approximate posterior, W and A factorise over n:

In the prior, W factorises over k and A factorises over p:

1 2 1 2( , , , , ) ( | ) ( | , ) ( | ) ( | , )k k k p p pk p

p p p q q p p r rα α β β

= ∏ ∏W A λ α β w a

1 2( | , )nn

p u uλ ∏

So, in the posterior for W we only have to store and invert N K-by-K covariance matrices.

( )ˆˆ( ) ; ,n n n nq N=w w w Σ

[ ]( )T Tdiag= ⊗B H α S S H

1,

ˆN

n ni ii i n= ≠

= − ∑r B w

( )( ) 1

Tnn n n n

n n n nn

λ

λ−

= +

= +

w Σ b r

Σ A B

( ) ( )

( ) ( )

( )

1

1

2

; ,1 1 1

2

2

K

kk

k k k k

TT Tk k k

k

k

k k k

q q

q Ga g h

Trg q

Nh q

g h

α

α α

α

=

=

=

= + +

= +

=

∏α

Σ S S w S Sw

Regression coefficients, W

Spatial precisions for W

1

2

( ) ( ; , )

1 12

2

n n n n

n

n

n

q Ga b c

Gb u

Tc u

λ λ=

= +

= +

Observation noise

AR coefficients, A

( )( )( )

1

1,

( ) ( ; , )

( )

n n n n

nn n nn

nn n n n

T Ta a

N

n ni ii i n

q N

diag

λ

λ

−

= ≠

=

= +

= +

= ⊗

= − ∑

a a m V

V C J

m V d j

J H β S S H

j J m

( )

1

1 2

1 1

2 2

1 2

( ) ( )

( ) ( ; , )

1 1 1( )2

2

P

pp

p p p p

T T Tp p p

p

p

p pp

q q

q Ga r r

Trr r

Pr r

r r

β

β β

β

=

=

=

= + +

= +

=

∏β

V S S m S Sm

Spatial precisions for A

Updating approximate posterior

y

x t

Synthetic Data 1 : from Laplacian Prior

Iteration Number

F

y

x

y

x

Least Squares VB – Laplacian Prior

`Coefficient RESELS’ = 366Coefficients = 1024

Synthetic Data II : blobs

True Smoothing

Global prior Laplacian prior

1-Specificity

Sen

sitiv

ity

Event-related fMRI: Faces versus chequerboard

Smoothing

Global prior Laplacian Prior

Event-related fMRI: Familiar faces versus unfamiliar faces

Smoothing

Global prior Laplacian Prior