Dynamic Causal Modelling Introduction SPM Course (fMRI), October 2015 Peter Zeidman Wellcome Trust Centre for Neuroimaging University College London.

Dynamic Causal ModellingIntroduction

SPM Course (fMRI), October 2015

Peter ZeidmanWellcome Trust Centre for NeuroimagingUniversity College London

is a framework

for creating, estimating and comparing generative models

of neuroimaging timeseries

Dynamic Causal Modelling

We use these models to investigate effective connectivity of neuronal populations

Contents

• Overview of DCM– Effective connectivity, DCM framework, generative

models

• Model specification– Neural model, haemodynamic model

• Model estimation– Model inversion, parameter inference

• Example

Contents


models



• Example

The system of interest

Stimulus from Buchel and Friston, 1997Brain by Dierk Schaefer, Flickr, CC 2.0

Experimental Stimulus (Hidden) Neural Activity Observations (BOLD)

time

Vector y

BO

LD

?off

on

time

Vector u

https://creativecommons.org/licenses/by/2.0/

Connectivity

• Structural ConnectivityPhysical connections of the brain

• Functional ConnectivityDependencies between BOLD observations

• Effectivity ConnectivityCausal relationships between brain regions

"Connectome" by jgmarcelino. CC 2.0 via Wikimedia CommonsFigure 1, Hong et al. 2013 PLOS ONE.KE Stefan, SPM Course 2011

DCM Framework

Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0

Experimental Stimulus (u)

Observations (y)

z = f(z,u,θn).

How brain activity z

changes over time

y = g(z, θh)

What we would see in the scanner, y, given the

neural model?

Neural Model Observation Model


DCM Framework



Observations (y)Neural Model Observation Model

Generative model p(u,y)


DCM Framework



Model Inversion(Variational EM)

1. Gives parameter estimates:Given our observations y, and stimuli u, what parameters θ make the

model best fit the data?

2. Gives an approximation to the log model evidence:

Free energy = accuracy - complexity

DCM Framework





Model 1:

Model 2:

Model comparison: Which model best explains my observed data?

DCM Framework1. We embody each of our hypotheses in a generative

model.The generative model separates neural activity from haemodynamics

2. We perform model estimation (inversion)This identifies parameters θ = {θn,θh} which make the model best fit the data and the free energy (log model evidence)

3. We inspect the estimated parameters and / or we compare models to see which best explains the data.

Contents


models



• Example

The Neural Model“How does brain activity, z, change over time?”

Driving input u1

V1z1a

c

u1

z1

z2

�̇�1=𝑎𝑧+𝑐 𝑢1 Inhibitory self-connection (Hz).Rate constant: controls rate of decay in region 1. More negative = faster decay.


V5

V1

Driving input u1

z1

z2

a11

a22

a21

c11

�̇�1=𝑎11 𝑧1+𝑐11𝑢1

Change of activity in V1:

�̇� 2=𝑎22𝑧 2+𝑎21𝑧1


Self decay V1 input


V5

V1z1

z2

a11

a22

c11�̇�=𝐴𝑧+𝐶𝑢1

[ �̇�1�̇�2]=[𝑎11 0𝑎21 𝑎22 ] [𝑧1𝑧 2]+[𝑐110 ]𝑢1

Columns are outgoing connectionsRows are incoming connections

Driving input u1

a21


V5

V1z1

z2

a11

a22

c11

u1

z1

z2

�̇�=𝐴𝑧+𝐶𝑢1

Driving input u1

a21

“How does brain activity, z, change over time?”

The Neural Model

V5

V1z1

z2

a11

a22

c11

Driving input u1

a21u2

u1

z1

z2

b21

Attention u2

Could model be used to model a main effect and interaction


V5

V1z1

z2

a11

a22

c11

Driving input u1

a21

�̇�1=𝑎11 𝑧1+𝑐11𝑢1


b21

Attention u2

𝑧 2=𝑎22𝑧 2+𝑎21𝑧1+¿ ¿


Self decay V1 input Modulatory input


V5

V1z1

z2

a11

a22

c11

Driving input u1

a21

b21

Attention u2

�̇�=(𝐴+∑𝑗=1

𝑚

𝑢 𝑗𝐵𝑗) 𝑧+𝐶𝑢

For m inputs:

[ �̇�1�̇�2]=([𝑎11 0𝑎21 𝑎22 ]+𝑢2[ 0 0

𝑏21 0 ])[𝑧 1𝑧 2]+[𝑐11 00 0 ][𝑢1𝑢2]

A: Structure B: ModulatoryInput

C: DrivingInput

Change in activity per

region

External input 2(attention)

Currentactivity

per region

All external input

DCM Framework



Observations (y)

z = f(z,u,θn).


changes over time

y = g(z, θh)


neural model?



The Haemodynamic Model

Contents


models



• Example

DCM Framework



Observations (y)

z = f(z,u,θn).


changes over time

y = g(z, θh)


neural model?



Bayesian Models

posterior likelihood ∙ prior

new data prior knowledge

parameter estimates

pypyp ||

Priors• All parameters have prior distributions,

• Between-regions neuronal coupling parameters have shrinkage priors.

• Haemodynamic parameters have empirical priors.

-1 -0.5 0 0.5 1Connection strength (Hz)

Prior on between-region coupling

N(0,1/64)

Prior means stored in DCM.M.pE, covariance in DCM.M.pC

Model Estimation

• Inverting the model (via Variational EM) gives:

• Posterior probability distribution for each parameter,

• Posterior estimate of the noise precision

• Approximation of the model evidence,

Posterior mean stored in DCM.Ep

Posterior variance stored in DCM.Vp.

Noise precision stored in DCM.Ce

Free energy stored in DCM.F

Bayesian Model Reduction

Option 1:

Individually fit each model to the data (then inspect or compare)

Option 2:

Fit only the full model (model 1) then use ‘post-hoc model reduction’ (Bayesian Model Reduction) to estimate the others

Model 1 Model 2 Model 3

Contents


models



• Example

PREPARING DATA

0

2

4

6

8

10

12

Choosing Regions of Interest

We generally start with SPM results

+

ROI Options

1. A sphere with given radius

Positioned at the group peak

Allowed to vary for each subject, within a radius of the group peak

or

2. An anatomical mask

Pre-processing

1. Regress out nuisance effects (anything not specified in the ‘effects of interest f-contrast’)

2. Remove confounds such as low frequency drift

3. Summarise the ROI by performing PCA and retaining the first component

200 400 600 800 1000-4

-3

-2

-1

0

1

2

3

1st eigenvariate: test

time \{seconds\}

230 voxels in VOI from mask VOI_test_mask.niiVariance: 81.66%

New in SPM12: VOI_xx_eigen.nii(When using the batch only)

EXAMPLE

Reading > fixation (29 controls)Lesion (Patient AH)

1. Extracted regions of interest. Spheres placed at the peak SPM coordinates from two contrasts:

A. Reading in patient > controls B. Reading in controls

2. Asked which region should receive the driving input

Seghier et al., Neuropsychologia, 2012

Key:ControlsPatient

Bayesian Model Averaging

Seghier et al., Neuropsychologia, 2012

TROUBLESHOOTING

0 200 400 600 800 1000 1200-2

-1

0

1

2

3

4

responses and predictionsvariance explained 87%

time {seconds}

1 2 3 4 5 6 7 8 9-1

-0.5

0

0.5

1

intrinsic and extrinsic connectionslargest connection strength 0.58

parameter}

posterior correlationsestimable parameters 13

10 20 30 40 50

10

20

30

40

50

spm_dcm_fmri_check(DCM) spm_dcm_explore (DCM)

From Jean Daunizeau’s website

Further ReadingThe original DCM paper Friston et al. 2003, NeuroImage

Descriptive / tutorial papers

Role of General Systems Theory Stephan 2004, J Anatomy

DCM: Ten simple rules for the clinician Kahan et al. 2013, NeuroImage

Ten Simple Rules for DCM Stephan et al. 2010, NeuroImage

DCM Extensions

Two-state DCM Marreiros et al. 2008, NeuroImage

Non-linear DCM Stephan et al. 2008, NeuroImage

Stochastic DCM Li et al. 2011, NeuroImageFriston et al. 2011, NeuroImageDaunizeau et al. 2012, Front Comput Neurosci

Post-hoc DCM Friston and Penny, 2011, NeuroImageRosa and Friston, 2012, J Neuro Methods

A DCM for Resting State fMRI Friston et al., 2014, NeuroImage

EXTRAS

Approximates: The log model evidence:Posterior over parameters:

The log model evidence is decomposed:

The difference between the true and approximate posterior

Free energy (Laplace approximation)

Accuracy Complexity-

Variational Bayes

The Free Energy

Accuracy Complexity-

Complexity

posterior-prior parameter means

Prior precisions

Occam’s factor

Volume of posterior parameters

Volume of prior parameters

(Terms for hyperparameters not shown)

Distance between prior and posterior means

DCM parameters = rate constants

dx axdt

0( ) exp( )x t x at

The coupling parameter ‘a’ thus describes the speed ofthe exponential change in x(t)

0

0

( ) 0.5exp( )

x xx a

Integration of a first-order linear differential equation gives anexponential function:

/2lna

00.5x

a/2ln

Coupling parameter a is inverselyproportional to the half life of x(t):

Main effects → driving inputsInteractions → modulatory inputs

FFA

Amy

Face

ValenceFrom a factorial design:

A factorial design translates easily to DCMA (fictitious!) example of a 2x2 design:

Factor 1:Stimulus (face or inverted face)

Factor 2:Valence (neutral or angry)

Main effect of face: FFA

Interaction of Stimulus x Valence: Amygdala

Dynamic Causal Modelling Introduction SPM Course (fMRI), October 2015 Peter Zeidman Wellcome Trust Centre for Neuroimaging University College London.

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