Dynamic Causal Modelling THEORY

Dynamic Causal Modelling THEORY

SPM Course FIL, London22-24 October 2009

Hanneke den Ouden

Donders Centre for Cognitive NeuroimagingRadboud University Nijmegen

Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

Functional specializationFunctional specialization Functional integrationFunctional integration

Principles of Organisation

Overview

• Brain connectivity

• Dynamic causal models (DCMs)

– Basics

– Neural model

– Hemodynamic model

– Parameters & parameter estimation

– Inference & Model comparison

• Recent extentions to DCM

• Planning a DCM compatible study

Structural, functional & effective connectivity

• anatomical/structural connectivity= presence of axonal connections

• functional connectivity = statistical dependencies between regional time series

• effective connectivity = causal (directed) influences between neurons or neuronal populations

Sporns 2007, Scholarpedia

For understanding brain function mechanistically, we can use DCM to

create

models of causal interactions among neuronal populations

to explain regional effects in terms of interregional connectivity

Overview



– Basics

– Neural model






• Cognitive system is modelled at its underlying neuronal level (not directly accessible for fMRI).

• The modelled neuronal dynamics (x) are transformed into area-specific BOLD

signals (y) by a hemodynamic model (λ). λ

x

y

The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are optimally similar.

Basics of DCM: Neuronal and BOLD level

DCM: Linear Model

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DCM: Bilinear Model

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• Cognitive system is modelled at its underlying neuronal level (not directly accessible for fMRI).

• The modelled neuronal dynamics (x) are transformed into area-specific BOLD

signals (y) by a hemodynamic model (λ). λ

x

y

Basics of DCM: Neuronal and BOLD level

},,,,,{ h},,,,,{ h

important for model fitting, but of no interest for statistical inference

,)(

signal BOLD

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The hemodynamic model

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activity

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• 6 hemodynamic parameters:

• Computed separately for each area (like the neural parameters) region-specific HRFs!

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Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage

stimulus functionsut

neural state equation

hemodynamic state equations

Estimated BOLD response

Measured vs Modelled BOLD signalRecapThe aim of DCM is to estimate- neural parameters {A, B, C}- hemodynamic parameters such that the modelled (x) and measured (y) BOLD signals are maximally similar.

hemodynamicmodel

λx y

X1 X2 X3u1

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Overview



– Basics

– Neural model






DCM parameters = rate constants

dxax

dt 0( ) exp( )x t x at

The coupling parameter a determines the half life of x(t), and thus describes the speed of the exponential change

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

00.5x

a/2ln

If AB is 0.10 s-1 this means that, per unit time, the increase in activity in B corresponds to 10% of the activity in A

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Example: context-dependent decay

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Penny, Stephan, Mechelli, Friston NeuroImage (2004)

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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Conceptual overview

Neuronal states

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activityx2(t)

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Driving input(e.g. sensory stim)

Modulatory input(e.g. context/learning/drugs)

b12

BOLD Response

Parameters are optimised

so that the predicted

matches the measured

BOLD response

But how confident are

we in what these

parameters tell us?

Overview



– Basics

– Neural model






Model comparison and selection

Given competing hypotheses, which model is the best?

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Pitt & Miyung (2002) TICS

Inference about DCM parameters:

Bayesian single subject analysis

• The model parameters are distributions that have a mean ηθ|y and covariance Cθ|y.

– Use of the cumulative normal distribution to test the probability that a certain parameter is above a chosen threshold γ: ηθ|

y

Classical frequentist test across Ss

• Test summary statistic: mean ηθ|y

– One-sample t-test:Parameter > 0?

– Paired t-test:parameter 1 > parameter 2?

– rmANOVA: e.g. in case of multiple sessions per subject

DCM roadmap

fMRI data

Posterior densities of parameters

Neuronal dynamics

Haemodynamics

Model comparison

Bayesian Model inversion

State space Model

Priors

Overview



– Basics

– Neural model






Two-state DCM

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Extensions to DCM

• Ext. 1: two state model– excitatory & inhibitory

• Ext. 2: Nonlinear DCM– Gating of connections by

other areas

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Planning a DCM-compatible study

• Suitable experimental design:– any design that is suitable for a GLM – preferably multi-factorial (e.g. 2 x 2)

• e.g. one factor that varies the driving (sensory) input• and one factor that varies the contextual input

• Hypothesis and model:– Define specific a priori hypothesis– Which parameters are relevant to test this hypothesis?– If you want to verify that intended model is suitable to

test this hypothesis, then use simulations– Define criteria for inference– What are the alternative models to test?

So, DCM….

• enables one to infer hidden neuronal processes from fMRI data

• tries to model the same phenomena as a GLM

– explaining experimentally controlled variance in local responses

– based on connectivity and its modulation

• allows one to test mechanistic hypotheses about observed effects

• is informed by anatomical and physiological principles.

• uses a Bayesian framework to estimate model parameters

• is a generic approach to modeling experimentally perturbed dynamic

systems.

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

– DCM is not model or modality specific (Models will change and the method

extended to other modalities e.g. ERPs)

Some useful references• The first DCM paper: Dynamic Causal Modelling (2003). Friston et

al. NeuroImage 19:1273-1302.

• Physiological validation of DCM for fMRI: Identifying neural drivers

with functional MRI: an electrophysiological validation (2008). David et

al. PLoS Biol. 6 2683–2697

• Hemodynamic model: Comparing hemodynamic models with DCM

(2007). Stephan et al. NeuroImage 38:387-401

• Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008).

Stephan et al. NeuroImage 42:649-662

• Two-state model: Dynamic causal modelling for fMRI: A two-state

model (2008). Marreiros et al. NeuroImage 39:269-278

• Group Bayesian model comparison: Bayesian model selection for

group studies (2009). Stephan et al. NeuroImage 46:1004-10174

• Watch out for: 10 Simple Rules for DCM, Stephan et al (in prep).

Time to do a DCM!

Dynamic Causal ModellingPRACTICAL

SPM Course FIL, London22-24 October 2009

Andre Marreiros

Hanneke den Ouden

Donders Centre for Cognitive NeuroimagingRadboud University Nijmegen

Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

DCM – Attention to MotionParadigm

Parameters - blocks of 10 scans

- 360 scans total

- TR = 3.22 seconds

Stimuli 250 radially moving dots at 4.7 degrees/s

Pre-Scanning

5 x 30s trials with 5 speed changes (reducing to 1%)

Task - detect change in radial velocity

Scanning (no speed changes)

F A F N F A F N S ….

F - fixation

S - observe static dots + photic

N - observe moving dots + motion

A - attend moving dots + attention

Attention to Motion in the visual system

Results

Büchel & Friston 1997, Cereb. CortexBüchel et al. 1998, Brain

V5+

SPCV3A

Attention – No attention

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


Paradigm

V1

V5

SPC

Motion

Photic

Attention

V1

V5

SPC

Motion

PhoticAttention

Model 1attentional modulationof V1→V5: forward

Model 2attentional modulationof SPC→V5: backward

Bayesian model selection: Which model is optimal?

DCM: comparison of 2 models

Ingredients for a DCM

Specific hypothesis/question

Model: based on hypothesis

Timeseries: from the SPM

Inputs: from design matrix


Paradigm

V1

V5

SPC

Motion

Photic

Attention

V1

V5

SPC

Motion

PhoticAttention

Model 1attentional modulationof V1→V5: forward

Model 2attentional modulationof SPC→V5: backward

DCM – GUI basic steps

1 – Extract the time series (from all regions of interest)

2 – Specify the model

3 – Estimate the model

4 – Review the estimated model

5 – Repeat steps 2 and 3 for all models in model space

6 – Compare models


Dynamic Causal Modelling THEORY

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