dcm for resting state fmri - fil.ion.ucl.ac.ukarazi/webapge/spectraldcm_razi_spm... · dcm for...

DCM for resting state fMRI

SPM Course May 2017

Adeel Razi

Wellcome Trust Centre for Neuroimaging

Institute of Neurology

University College London

[email protected]

www.adeelrazi.org

@adeelrazi

Introduction and background(Spectral) Dynamic Causal

Modelling

What’s new?

OU

TL

INE

Applications

Technical /

Mathematical

Content

DMN connectivity

Large DCMs

Cognitive /

Psychiatric

Worked example using SPM

Brain connectivitystructural, functional and effective

Structural connectivity

presence of axonal connections

Functional connectivity

statistical dependencies between regional time series

Effective connectivity

causal (directed) influences between neuronal populations

• Relationship between functional and effective connectivity

Hidden causes Measured consequences

Mapping

Brain connectivitystructural, functional and effective

( )u tThe forward (dynamic

causal) model

Externalstimulus

Observed timeseries

DCM for task fMRIclassic DCM

)()),(()(

)()(

),),(()(

1

tetxhty

CutxBuA

utxftx

jm

j

j

Deterministic system

Ordinary differential equation (ODE)

),,()( uxftx

exhy ),(

The forward (dynamic

causal) model

Externalstimulus

Observed timeseries

DCM for task fMRIclassic DCM

exhy ),(

)()),(()(

)()()(

),,),(()(

1

tetxhty

tvCutxBuA

vutxftx

jm

j

j

Stochastic system

Stochastic differential equation (SDE)

),,()( uxftx

+

Endogenous fluctuations

( )u t

DCM for resting state fMRIclassic DCM


causal) model

External stimulus

Observed timeseries

0)( tu

),,()( uxftx


+

)()),(()(

)()()(

),,),(()(

1

tetxhty

tvCutxBuA

vutxftx

jm

j

j

exhy ),(


causal) model


Observed timeseries

)(tv


),,()( vxftx )()),(()(

)()()(

tetxhty

tvtAxtx

Stochastic system


More parameters to estimate

Slow estimation!

exhy ),(

Quick detour..Fourier transform, cross covariance, cross spectra

)()(

)())()(

Yty

dtetytyY ti

F(

)()()()( 2121 bYaYtbytay

Some properties…

)()()()( 2121 YYtyty

Linearity:

Convolution:

)(ty )(Y

function Sinc

)()( 21 tyty

function Sinc2

)()( 21 YY

t

Quick detour..Fourier transform, cross covariance, cross spectra

Now we are interested in situation where we have multiple

time series and explore relationship between them

)]()([E)( 2121 tyyyy

)0()0(

)()(

2211

21

21

yyyy

yy

yy

Cross covariance

Cross correlation

)]()([)(*

2121 YYg yy E

Cross spectral density

)()(

)()(

2211

21

21

2

yyyy

yy

yygg

gC

Coherence

Measures of functional connectivity

)(1 ty

)(2 ty

)(3 ty

)(4 ty


causal) model


Observed timeseries

)(tv


),,()( uxftx )()),(()(

)()()(

tetxhty

tvtAxtx

Stochastic system


More parameters to estimate

Slow estimation!

exhy ),(



),( v

),,()( vxftx


causal) model

)()),(()(

)()()(

tetxhty

tvtAxtx

)()()()(),(

)()()()(

)exp()(

evy

xx

tetvtty

fh

),( y

DCM for resting state fMRIspectral DCM


Complex cross-spectra

)(yg

),( vg


causal) model

)()),(()(

)()()(

tetxhty

tvtAxtx

),,()( vxftx

)()()()()(

))()(

*

evy gKgKg

tK

F(

},,,{ CA

e

v

ee

vv

g

g

),(

),(

Power law

With amplitude and exponent

Fast estimation

http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=Gyv-UBuMK2jUnM&tbnid=UOEzDqSXCTnPQM:&ved=0CAUQjRw&url=http://article.sapub.org/10.5923.j.ijis.20120204.04.html&ei=EA4DVIGeEoqUapGdgPgI&bvm=bv.74115972,d.d2k&psig=AFQjCNH9FU66c7LcGmunE3oRTBjfKO7dzQ&ust=1409572670514390



causal) modelEndogenous fluctuations



Observed timeseries

),( yg


),( vg

),,()( vxftx

exhy ),(



( )y t

Bayesian model

inversionEndogenous fluctuations



Observed timeseries

Posterior density

Log model evidence

(Free energy)

)),((minarg ygF

)(yg

),( vg


Bayesian model

inversion

Posterior density

Log model evidence

)),(()|)(

)(),(

yy

y

gFmg

qmg

p( ln

|)|p( ),,()( vxftx



0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

model

prob

abili

ty

0 10 20 30 40 50 60-600

-500

-400

-300

-200

-100

0

100

model

log-

prob

abili

ty

Bayesian model

comparison

)|)( mg y p( ln ))(| ygmp(

( )y t


)),(()|)(

)(),(

yy

y

gFmg

qmg

p( ln

|)|p(

Bayesian model

inversion

Posterior density

Log model evidence

Bayesian model averaging

m

yyy gmmgg ))(|)),(|)( p(p()|p()(yg

),( vg


),,()( vxftx



0 50 100 150 200 250 300-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Frequency and time (bins)

real

Prediction and response

0 50 100 150 200 250 300-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Frequency and time (bins)

imag

inar

y

Network or graph generating data

-0.3

-0.20.4

0.2

Friston, Kahan, Biswal, Razi, NeuroImage, 2014DCM for resting state fMRIface validity

128 256 384 512 640 768 896 10240

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Session length (scans)

RM

S

Root mean square error (Spectral)

128 256 384 512 640 768 896 10240

0.05

0.1

0.15

0.2

0.25

Session length (scans)R

MS

Root mean square error (Stochastic)

Network or graph generating data

4

32

-0.1

1

-0.3

0.3

0.4

0.2

0.2

0.1

-0.1

Razi, Kahan, Rees, Friston, NeuroImage, 2015

DCM for resting state fMRIconstruct validity

4

32

-0.3

1-0.3

0.3

0.4

0.5

0.2

0.1

-0.1

Network for second set of 24 subjectsNetwork for first set of 24 subjects

4

32

-0.1

1-0.3

0.3

0.4

0.2

0.20.1

-0.1

Razi, Kahan, Rees, Friston, NeuroImage, 2015

DCM for resting state fMRIconstruct validity

State-space model

𝐱 𝑡 = 𝑓 𝐱(𝑡), 𝛉 + 𝐰 𝑡𝐲 𝑡 = ℎ 𝐱(𝑡), 𝛉 + 𝐞 𝑡

𝚺𝐰 t = 𝐰 𝑡 𝐰(𝑡 − 𝜏)𝑇 and 𝚺𝐞 𝑡 = 𝐞 𝑡 𝐞(𝑡 − 𝜏)𝑇

E.g., Bilinear DCM: 𝐱 𝑡 = 𝐀 + 𝑗 𝐮𝑗𝐁𝑗 𝐱 + 𝐂𝑢 + 𝐰(𝑡)

Vector autoregressive model

(VAR)(assuming 𝐱 𝑡 = 𝐲 𝑡 )

𝐲 𝑡 =

𝑖=1

𝑁

𝑎𝑖𝐲(𝑡 − 𝑖) + 𝐳 𝑡

Convolution kernel

𝐲 𝑡 = 𝛋 𝜏 ∗ 𝐰 𝑡 + 𝐞 𝑡𝛋 𝜏 = 𝜕𝐱ℎ . exp (𝜏 𝜕𝐱𝑓)

Cross covariance

𝚺 𝜏 = 𝐲 𝑡 . 𝐲(𝑡 − 𝜏)𝑇

= 𝛋 𝑡 ∗ 𝚺𝐰 𝑡 ∗ 𝛋 −𝑡 +𝚺𝐞 𝑡

Cross correlation

𝑐𝑖𝑗(𝜏) =Σ𝑗𝑘(𝜏)

𝛴𝑗𝑗(0)𝛴𝑘𝑘(0)

Convolution theorem

𝒀 𝜔 = 𝑲 𝜔 𝑾 𝜔 +𝑬 𝜔

𝑲 𝜔 = FT 𝛋 𝜏

Cross spectral density

𝒈𝐲 𝜔 = 𝒀 𝜔 𝒀 𝜔 †

= 𝑲 𝜔 .𝒈𝐰 (𝜔). 𝑲 𝜔 † + 𝒈e(𝜔)

Coherence

𝐶𝑖𝑗(𝜔) =𝑔𝑗𝑘(𝜔)

2

𝑔𝑗𝑗(𝜔)𝑔𝑘𝑘(𝜔)

Directed transfer functions

𝒀 𝜔 = 𝑺 𝜔 .𝒁 𝜔𝑺 𝜔 = (𝐈 − 𝑨(𝜔))−1

Granger causality

𝐺𝑗𝑘(𝜔) = −ln 1 −𝑆𝑗𝑘(𝜔)

2

𝑔𝑗𝑗(𝜔)

Auto-regression coefficients

𝐀 = 𝐘𝐓 𝐘−1 𝐘T 𝐘

= 𝛒−𝟏 𝛒1, … 𝛒𝐩𝑇

Auto- correlation

𝑐𝑖𝑖 = (𝐈 − 𝐀)−𝟏(𝐈 − 𝐀𝑻)−𝟏

Spectral representations

Structural equation models (SEM)

(assuming 𝐱 𝑡 = y 𝑡 and 𝐲 𝒕 = 0 𝐮 𝑡 = 0 )

𝐲 𝑡 = 𝐀𝐲(𝑡) + 𝐰 𝑡 = 𝚯 − 𝐈 𝐲(𝑡) + 𝐞 𝑡𝐲 𝑡 = 𝚯𝐲 𝑡 + 𝐞 𝑡

Convolution theorem

𝒀 𝜔 = 𝑨 𝜔 .𝒀 𝜔 + 𝒁 𝜔A 𝜔 = FT 𝑎1, 𝑎2. . 𝑎𝑁

(Inverse) Fourier Transform

Transformation under assumptions

𝒀 =

0𝑦(𝑡 − 1) 0𝑦(𝑡 − 2) 𝑦(𝑡 − 1) 0

⋮ ⋱ ⋱

𝐀 =

0𝑎1 0𝑎2 𝑎1 0⋮ ⋱ ⋱

Brain connectivitymeasures of connectivity

Razi and Friston. IEEE Sig. Proc. Mag, 2016

DCM for resting state fMRIinterim summary

• State space models – all connectivity measures can

be derived from them as approximations

• Effective connectivity as what causes observations

(functional connectivity)

• Spectral DCM is accurate, (computationally) efficient

and sensitive relative to stochastic DCM


Modelling

What’s new?

OU

TL

INE

Applications

Technical /

Mathematical

Content

DMN connectivity

Large DCMs

Cognitive /

Psychiatric


Worked example

1. GLM estimation – to get SPM.mat

2. CSF/WM signal extraction

3. GLM estimation – to remove confounds

4. Extraction of time series from ROIs

Worked example

PCC [0 -52 26]

mPFC [3 54 -2]

L-IPC [-50 -63 32]

R-IPC [48 -69 35]

Di & Biswal, NeuroImage 2014 and Razi et al NeuroImage, 2015

1. GLM estimation – to get SPM.mat

2. CSF/WM signal extraction

3. GLM estimation – to remove confounds

4. Extraction of time series from ROIs

5. Specify DCM

6. Estimate DCM

7. Review DCM

Worked example

Worked exampleDefault mode network

0 50 100 150 200 250 300 350-2

-1

0

1

2

PCC: responses

time {seconds}

0 50 100 150 200 250 300 350-2

-1

0

1

2

MPFC: responses

time {seconds}

0 50 100 150 200 250 300 350-2

-1

0

1

2

LIPC: responses

time {seconds}

0 50 100 150 200 250 300 350-3

-2

-1

0

1

2

RIPC: responses

time {seconds}


Input time series Data fits for CSD

Connectivity parameters: DCM.Ep.A

Neural fluctuation parameters: DCM.Ep.a



Modelling

What’s new?

OU

TL

INE

Applications

Technical /

Mathematical

Content

DMN connectivity

Large DCMs

Cognitive /

Psychiatric


Raichle Annual Reviews, 2015

36 nodes network

Large-scale DCMs for resting state fMRI

36 nodes network

Stochastic DCM

Spectral DCM

Number of modes (m)


Razi, Seghier, Yuan, McColgan, Zeidman, Park, Sporns, Rees, Friston, Net. Neurosci. In press

34

A B

C

Default mode network

Dorsal attention network

Control executive network

Salience network

Sensorimotor network

Auditory network

Visual network

Inhibitory connections

Excitatory connections



35


Averaged functional connectivity Averaged effective connectivity

Averaged binarized adjacency matrix

after BMR

Averaged weighted adjacency matrix

after BMR


Averaged functional connectivity Averaged effective connectivity

Averaged binarised after BMR Averaged weighted after BMR

36




Modelling

What’s new?

OU

TL

INE

Applications

Technical /

Mathematical

Content

DMN connectivity

Large DCMs

Cognitive /

Psychiatric


Admon & Pizzagalli,(2016) Nat. Comm.

Major Depressive DisorderNatural time course of positive mood

These findings suggest:

1) that corticostriatal pathways

contribute to the natural time course

of positive mood fluctuations

2) and that disturbances of those neural

interactions may characterize

individuals with a past history of mood

disorders

Spectral DCM analysis

Admon and Pizzagalli,(2016) Nat. Comm.

Major Depressive DisorderNatural time course of positive mood

Hierarchical organization of intrinsic brain modesanticorrelated brain modes

– Functional network organization

Kelly et al. 2008

Dorsal Attention Network

Default Mode NetworkPower et al. 2011

– Network interactivity important for

cognitive function

Salience Network

(SN)

Default Mode Network

(DMN)

Dorsal Attention Network

(DAN)

Tsvetanov et al. 2016

VOIs identified using spatial independent

component analysis (ICA) (N=404)Functional connectivity

Yuan, Friston, Zeidman, Chen, Li, Razi, Submitted


1. Regions belonging to the same

network grouped together

2. The task-positive mode (SN, DAN)

inhibits the cDN

3. The task negative mode (cDN) excites

the task positive mode (SN, DAN)


Yuan, Friston, Zeidman, Chen, Li, Razi, Submitted

Thank you

And thanks to

FIL Methods Group

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