Dynamic Causal Modelling (DCM) Dynamic Causal Modelling (DCM) for fMRIfor fMRI

Wellcome Trust Centre for NeuroimagingUniversity College London

Andre MarreirosAndre Marreiros

Overview

Dynamic Causal Modelling of fMRI

Definitions & motivation

The neuronal model (bilinear dynamics)

The Haemodynamic model

Estimation: Bayesian framework

DCM latest Extensions

Principles of organisation

Functional specialization Functional integration

Conceptual overview

BOLDy

y

y

Inputu(t)

activityz2(t)

activityz1(t)

activityz3(t)

c1 b23

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neuronalstates

Use differential equations to represent a neuronal system

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DCM parameters = rate constants

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Half-life:

Generic solution to the ODEs in DCM:

ln 2 /s -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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Linear dynamics: 2 nodes

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0 20 40 60

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Haemodynamics: reciprocal connections

blue: neuronal activityred: bold response

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h2

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h(u,θ) represents the BOLD response (balloon model) to input

BOLD

(without noise)

BOLD

(without noise)

0 20 40 60

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0 20 40 60

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seconds

Haemodynamics: reciprocal connections

BOLD

with

Noise added

BOLD

with

Noise added

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y2

blue: neuronal activityred: bold response

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euhy ),( y represents simulated observation of BOLD response, i.e. includes noise

Bilinear state equation in DCM for fMRI

state changes connectivity external

inputsstate vector

direct inputs

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effective connectivity

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Friston et al. 2003,NeuroImage

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Posterior densities of parameters

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Model comparison

DCM roadmap

Model inversion using

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State space Model

Priors

Constraints on•Haemodynamic parameters

•Connections

Models of•Haemodynamics in a single region

•Neuronal interactions

Bayesian estimation

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Estimation: Bayesian framework

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Overview:parameter estimation

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• Specify model (neuronal and haemodynamic level)

• Make it an observation model by adding measurement error e and confounds X (e.g. drift).

• Bayesian parameter estimation using expectation-maximization.

• Result:(Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y.

0 20 40 60-10123

0 20 40 60-10123

seconds

Forward coupling, a21

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Input coupling, c1

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Prior density Posterior density true values

Parameter estimation: an example

u1

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z1

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Simulated response

Inference about DCM parameters:single-subject analysis

• Bayesian parameter estimation in DCM: Gaussian assumptions about the posterior distributions of the parameters

• Quantify the probability that a parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ:

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Model comparison and selection

Given competing hypotheses, which model is the best?

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Motion

Time [s]

Attention

We used this model to assess the site of attention modulation during visual motion processing in an fMRI paradigm reported by Büchel & Friston.

Friston et al. 2003,NeuroImage

Attention to motion in the visual system

- fixation only- observe static dots+ photic V1- observe moving dots + motion V5- task on moving dots + attention V5 + parietal cortex

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V1

V5

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Photic

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Model 1:attentional modulationof V1→V5

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V5

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Motion

Photic Attention0.86

0.56 -0.02

1.42

0.550.75

0.89

Model 2:attentional modulationof SPC→V5

Comparison of two simple models

Bayesian model selection: Model 1 better than model 2

→ Decision for model 1: in this experiment, attention

primarily modulates V1→V5

1 2log ( | ) log ( | )p y m p y m

• potential timing problem in DCM:temporal shift between regional time series because of multi-slice acqisition

• 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)

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Extrinsic (between-region) coupling

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DCM for Büchel & Friston

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Bilinear state equation

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nonlinear DCM

Nonlinear state equation

u2

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Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas.

Extension III: Nonlinear DCM for fMRI

Extension III: Nonlinear DCM for fMRI

.

The posterior density of indicates that this gating existed with 97.4% confidence.

(The D matrix encodes which of the n neural units gate which connections in the system)

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Can V5 activity during attention to motion be explained by allowing activity in SPC to modulate the V1-to-V5 connection?

V1 V5

SPC

visualstimulation

attention

0.03(100%)

motion

0.04(100%)

1.65(100%)

0.19(100%)

0.01(97.4%)

Conclusions

Dynamic Causal Modelling (DCM) of fMRI is mechanistic model that is informed by anatomical and physiological principles.

DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. ERPs)

DCM uses a deterministic differential equation to model neuro-dynamics (represented by matrices A,B and C)

DCM uses a Bayesian framework to estimate model parameters

DCM provides an observation model for neuroimaging data, e.g. fMRI, M/EEG