18th February 2009

Stephanie Burnett

Christian Lambert

Methods for Dummies 2009

Dynamic Causal Modelling Part I: Theory

Last time, in MfD…

Psychophysiological interactions (PPI) and structural equation modelling (SEM)

Functional vs. effective connectivity Functional connectivity:

temporal correlation between spatially remote neurophysiological events

Effective connectivity: the influence that the elements of a neuronal system exert over each other

Standard fMRI analysis

PPIs, SEM, DCM

Introduction: DCM and its place in the methods family tree

Standard fMRI analysis: The BOLD signal (related to

brain activity in some implicit way) in some set of brain is correlated, and is also correlated with your task

Task BOLD signal

“This is a fronto-parietalnetwork collection of brain regions involved in activated while processing coffee”

V1V1V1 V5V5

attentionattention

PPIs Represent how the

(experimental) context modulates connectivity between a brain region of interest, and anywhere else

E.g. (Whatever gives rise to the) signal in one brain region (V1) will lead to a signal in V5, and the strength of this signal in V5 depends on attention

Introduction: DCM and its place in the methods family tree

V1

V5V5

attentionattention

V1V1

DCM models bidirectional and modulatory interactions, between multiple brain regions

DCM models how neuronal activitycauses the BOLD signal (forward model)

That is, your conclusions are about neural events

DCM Your experimental task

causes neuronal activity in an input brain region, and this generates a BOLD signal.

The neuronal activity in this input region, due to your task, then causes or modulates neuronal activity in other brain regions (with resultant patterns of BOLD signals across the brain)

“This sounds more like something I’d enjoy writing up!”

Introduction: DCM and its place in the methods family tree

DCM basics

DCM models interactions between neuronal populations

fMRI, MEG, EEG The aim is to estimate,

and make inferences about:

1. The coupling among brain areas

2. How that coupling is influenced by changes inexperimental context

DCM basics

DCM starts with a realistic model of how brain regions interact and where the inputs can come in

Adds a forward model of how neuronal activity causes the signals you observe (e.g. BOLD)

…and estimates the parameters in your model (effective connectivity), given your observed data

Neural and hemodynamic models

(more on this in a few minutes)

DCM basics

Inputs State variables Outputs

DCM basics

Inputs In functional connectivity

models (e.g. standard fMRI analysis), conceptually your input could have entered anywhere

In effective connectivity models (e.g. DCM), input only enters at certain places

DCM basics

Inputs can exert their influence in two ways: 1. Direct influence

e.g. visual input to V1 2. Vicarious (indirect)

influence e.g. attentional

modulation of the coupling between V1 and V5

DCM basics

State variables Neuronal activities,

and other neuro- or bio-physical variables needed to form the outputs Neuronal priors Haemodynamic priors

What you’re modelling is how the inputs modulate the coupling among these state variables

DCM basics

Output The BOLD signal (for

example) that you’ve measured in the brain regions specified in your model

Dynamic Modelling (i)

Generate equations to model the dynamics of physical systems.

These will be LINEAR or NON-LINEAR

Linear models provide good approximation

However neuronal dynamics are non-linear in nature

Linear Dynamic Model

X1= A11X1 + A21X2 + C11U1 X2= A22X2 + A12X1 + C22U2

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The Linear Approximation

fL(x,u)=Ax + Cu

Intrinsic Connectivity Extrinsic (input) Connectivity

INPUT U1 INPUT U2

C11 C22

X1 X2A11

A12

A21

A22

Dynamic Modelling (ii) In DCM we are modelling the brain as a:

“Deterministic non-linear dynamic system”

Effective connectivity is parameterised in terms of coupling between unobserved brain states

Bilinear approximation is useful: Reduces the parameters of the model to three sets

1) Direct/extrinsic 2) Intrinsic/Latent 3) Changes in intrinsic coupling induced by inputs

The idea behind DCM is not limited to bilinear forms

AIM:Estimate the parameters by perturbing the system

and observing the response.

Important in experimental design:

1) One factor controls sensory perturbation

2) One factor manipulates the context of sensory evoked responses

INPUT U1 INPUT U2

C11 C22

X1 X2A11

A12

A21

A22

INPUT U1 INPUT U2

C11 C22

X1 X2A11

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The Bilinear Approximation

fB(x,u)=(A+jUjBj)x + Cu

Intrinsic

Connectivity

Extrinsic (input)

ConnectivityINDUCED CONNECTIVITY

Bi-Linear Dynamic Model (DCM)

INPUT U1 INPUT U2

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Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI).

The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ).

λ

z

y

The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals

are maximally similar to the experimentally measured BOLD signals.

DCM for fMRI: the basic ideaDCM for fMRI: the basic idea

Priors on biophysical parametersPriors on biophysical parameters

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• Computed separately for each area (like the neural parameters).

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Friston et al. 2003,NeuroImage

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Estimating model parameters

DCMs are biologically plausible (i.e. complicated) - they have lots of free parameters

A Bayesian framework is a good way to embody the constraints on these parameters

Bayes Theorem

posterior likelihood ∙ prior

)()|()|( pypyp

Use Bayes’ theorem to estimate model parameters

Priors – empirical (haemodynamic parameters) and non-empirical (eg. shrinkage priors, temporal scaling)

Likelihood derived from error and confounds (eg. drift)

Calculate the Posterior probability for each effect, and the probability that it exceeds a set threshold

Bayes Theorem

posterior likelihood ∙ prior

)()|()|( pypyp

Inferences about the strength (= speed) of connections between the brain regions in your model

- EM algorithm – works out the parameters in a model

- Bayesian model selection to test between alternative models

Single subject analysis Use the cumulative normal

distribution to test the probability with which a certain parameter is above a chosen threshold γ:

ηθ|y

Interpretation of parametersInterpretation of parameters

A good model of your data will balance model fit with complexity (overfitting models noise)

You find this by taking evidence ratios (the “Bayes factor”)

The “Bayes factor” is a summary of the evidence in favour of one model as opposed to another

Model comparison and selectionModel comparison and selection

Bayes’ theorem:

Model evidence:

The log model evidence can be represented as:

Bayes factor:

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Bayesian Model SelectionBayesian Model Selection

- Group analysis:• Like “random effects” analysis in SPM, 2nd level

analysis can be applied to DCM parameters:

Separate fitting of identical models for each subject

Selection of bilinear parameters of interest

One sample t-test:Parameter > 0?

Paired t-test:Parameter 1 > parameter 2?

rm ANOVA:For multiple sessionsper subject

Interpretation of parametersInterpretation of parameters

1. DCM now accounts for the slice timing problem

New stuff in DCMNew stuff in DCM

Potential timing problem in DCM:

Temporal shift between regional time

series because of multi-slice

acquisition

Solution:

Modelling of (known) slice timing of each area.

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Extension I: Slice timing model

Slice timing extension now allows for any slice timing differences.

Long TRs (> 2 sec) no longer a limitation.

(Kiebel et al., 2007)

1. DCM now accounts for the slice timing problem (SPM5)

2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.)

New stuff in DCMNew stuff in DCM

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Extension II: Two-state model

1. DCM now accounts for the slice timing problem (SPM5)

2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.)

3. Biological plausibility: more complex balloon model (SPM5)

4. Non-linear version of DCM as well as bilinear (SPM8)

New stuff in DCMNew stuff in DCM

Dynamic Causal Modelling of fMRIDynamic Causal Modelling of fMRI

Model inversion using

Expectation-maximization

State space Model

fMRI data (y)

Posterior distribution of parameters

Network dynamics

Haemodynamicresponse

Model comparison

Priors

Practical steps of a DCM study - I1. Definition of the hypothesis & the model (on

paper)• Structure: which areas, connections and inputs?• Which parameters in the model concern my hypothesis?• How can I demonstrate the specificity of my results?• What are the alternative models to test?

2. Defining criteria for inference:• single-subject analysis: stat. threshold? contrast?• group analysis: which 2nd-level model?

3. Conventional SPM analysis (subject-specific)• DCMs are fitted separately for each session (subject)

→ for multi-session experiments, consider concatenation of sessions or adequate 2nd level analysis

Practical steps of a DCM study - II4. Extraction of time series, e.g. via VOI tool in SPM

• caveat: anatomical & functional standardisation important for group analyses

5. Possibly definition of a new design matrix, if the “normal” design matrix does not represent the inputs appropriately.• NB: DCM only reads timing information of each input

from the design matrix, no parameter estimation necessary.

6. Definition of model• via DCM-GUI or directly

in MATLAB

7. DCM parameter estimation• caveat: models with many regions & scans can crash

MATLAB!

8. Model comparison and selection:• Which of all models considered is the optimal one?

Bayesian model selection

9. Testing the hypothesis Statistical test onthe relevant parametersof the optimal model

Practical steps of a DCM study - III

DCM button

‘specify’NB: in order!

Summary

DCM is NOT EXPLORATORY

Used to test the hypothesis that motivated the experimental design BUILD A MODEL TO EXPRESS HYPOTHESIS IN TERMS

OF NEURAL CONNECTIVITY

The GLM used in typical fMRI data analysis uses the same architecture as DCM but embodies more assumptions

Note: In DCM a “Strong Connection” means an influence that is expressed quickly or with a small time constant.

When constructing experiments, consider whether you want to use DCM early

When in doubt, ask the experts………

Karl J. Friston. Dynamic Causal Modelling. Human brain function. Chapter 22. Second Edition.

http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/

K.J Friston, L. Harrison and W. Penny. Dynamic Causal Modelling. Neuroimage 2003; 19:1273-1302.

SPM Manual

Last year’s presentation

REFERENCESREFERENCES

ANY QUESTIONS???