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![Page 1: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/1.jpg)
Dynamic Causal Modelling for fMRI
Dynamic Causal Modelling for fMRI
Friday 22nd Oct. 2010SPM fMRI course
Wellcome Trust Centre for Neuroimaging
London
André Marreiros
![Page 2: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/2.jpg)
Overview
Brain connectivity: types & definitionsAnatomical connectivity
Functional connectivity Effective connectivity
Brain connectivity: types & definitionsAnatomical connectivity
Functional connectivity Effective connectivity
Dynamic causal models (DCMs)Neuronal model
Hemodynamic model
Estimation: Bayesian framework
Dynamic causal models (DCMs)Neuronal model
Hemodynamic model
Estimation: Bayesian framework
Applications & extensions of DCM to fMRI data
Applications & extensions of DCM to fMRI data
![Page 3: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/3.jpg)
Functional specializationFunctional specialization Functional integrationFunctional integration
Principles of Organisation
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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
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Anatomical connectivity
Definition:
presence of axonal connections
• neuronal communication via synaptic contacts
• Measured with
– tracing techniques
– diffusion tensor imaging (DTI)
![Page 6: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/6.jpg)
Knowing anatomical connectivity is not enough...
• Context-dependent recruiting of connections :– Local functions depend on network
activity
• Connections show synaptic plasticity– change in the structure and
transmission properties of a synapse– even at short timescales
Look at functional and effective connectivity
![Page 7: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/7.jpg)
Definition: statistical dependencies between regional time series
• Seed voxel correlation analysis
• Coherence analysis
• Eigen-decomposition (PCA, SVD)
• Independent component analysis (ICA)
• any technique describing statistical dependencies amongst regional time series
Functional connectivity
![Page 8: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/8.jpg)
Seed-voxel correlation analyses
• hypothesis-driven choice of a seed voxel
• extract reference time series
• voxel-wise correlation with time series from all other voxels in the brain
seed voxel
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Pros & Cons of functional connectivity analysis
• Pros:– useful when we have no experimental control
over the system of interest and no model of what caused the data (e.g. sleep, hallucinations, etc.)
• Cons:– interpretation of resulting patterns is difficult /
arbitrary – no mechanistic insight– usually suboptimal for situations where we
have a priori knowledge / experimental control
Effective connectivity
![Page 10: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/10.jpg)
Effective connectivity
Definition: causal (directed) influences between neurons or neuronal populations
• In vivo and in vitro stimulation and recording
•
• Models of causal interactions among neuronal populations
– explain regional effects in terms of interregional connectivity
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Some models for computing effective connectivity from fMRI data
• Structural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000
• regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997
• Volterra kernels Friston & Büchel 2000
• Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003
• Dynamic Causal Modelling (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008
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Psycho-physiological interaction (PPI)
• bilinear model of how the psychological context A changes the influence of area B on area C :
B x A C
attention
no attention
V1 activity
V5
act
ivit
y
Friston et al. 1997, NeuroImageBüchel & Friston 1997, Cereb. Cortex
V1 x Att. V5
• A PPI corresponds to differences in regression slopes for different contexts.
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Pros & Cons of PPIs• Pros:
– given a single source region, we can test for its context-dependent connectivity across the entire brain
– easy to implement
• Cons:– only allows to model contributions from a single area – operates at the level of BOLD time series (SPM 99/2).
SPM 5/8 deconvolves the BOLD signal to form the proper interaction term, and then reconvolves it.
– ignores time-series properties of the data
Dynamic Causal Models
needed for more robust statements of effective connectivity.
![Page 14: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/14.jpg)
Dynamic causal models (DCMs)Basic idea
Neuronal modelHemodynamic model Parameter estimation, priors & inference
Dynamic causal models (DCMs)Basic idea
Neuronal modelHemodynamic model Parameter estimation, priors & inference
Brain connectivity: types & definitionsAnatomical connectivity
Functional connectivity Effective connectivity
Brain connectivity: types & definitionsAnatomical connectivity
Functional connectivity Effective connectivity
Overview
Applications & extensions of DCM to fMRI data
Applications & extensions of DCM to fMRI data
![Page 15: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/15.jpg)
Basics of Dynamic Causal Modelling
DCM allows us to look at how areas within a network interact:
Investigate functional integration & modulation of specific cortical pathways
– Temporal dependency of activity within and between areas (causality)
![Page 16: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/16.jpg)
Temporal dependence and causal relations
Seed voxel approach, PPI etc. Dynamic Causal Models
timeseries (neuronal activity)
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Basics of Dynamic Causal Modelling
DCM allows us to look at how areas within a network interact:
Investigate functional integration & modulation of specific cortical pathways
– Temporal dependency of activity within and between areas (causality)
– Separate neuronal activity from observed BOLD responses
![Page 18: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/18.jpg)
• Cognitive system is modelled at its underlying neuronal level (not directly accessible for fMRI).
• The modelled neuronal dynamics (Z) are transformed into area-specific BOLD signals
(y) by a hemodynamic model (λ). λ
Z
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
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Neuronal systems are represented by differential equations
Input u(t)
connectivity parameters
systemz(t) state
State changes of the system states are dependent on:
– the current state z– external inputs u– its connectivity θ– time constants &
delays
A System is a set of elements zn(t) which interact in a spatially and temporally specific fashion
),,( uzFdt
dz
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DCM parameters = rate constants
11
dzsz
dt
Half-life
Generic solution to the ODEs in DCM:
ln 2 /s
1 1
1
( ) 0.5 (0)
(0)exp( )
z z
z s
1 1 1( ) (0)exp( ), (0) 1z t z st z z1
s
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
s/2ln
)0(5.0 1z
Decay function
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DCM parameters = rate constants
11
dzsz
dt
Generic solution to the ODEs in DCM:
1 1 1( ) (0)exp( ), (0) 1z t z st z z1
s
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
s/2ln
)0(5.0 1z
Decay function
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
A
B
0.10
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Linear dynamics: 2 nodes
1 1
2 21 1 2
1
2
1
2 21
21
(0) 1
(0) 0
( ) exp( )
( ) exp( )
0
z sz
z s a z z
z
z
z t st
z t sa t st
a
1;4 21 as
2;4 21 as
1;8 21 as
z2
21a
z1
s
s
z1 sa21t z2
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Neurodynamics: 2 nodes with input
u2
u1
z1
z2
activity in z2 is coupled to z1 via coefficient a21
u1
21a
001
01211
2
1
212
1
au
c
z
z
as
z
z
z1
z2
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Neurodynamics: positive modulation
000
00
1
01 2211
2
1221
22
1
212
1
bu
c
z
z
bu
z
z
as
z
z
u2
u1
z1
z2
modulatory input u2 activity through the coupling a21
u1
u2
index, not squared
z1
z2
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Neurodynamics: reciprocal connections
0,,00
00
1
1 22121121
2
1221
22
1
21
12
2
1
baau
c
z
z
bu
z
z
a
as
z
z
u2
u1
z1
z2
reciprocal connectiondisclosed by u2
u1
u2z1
z2
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0 20 40 60
0
2
4
0 20 40 60
0
2
4
seconds
Haemodynamics: reciprocal connections
blue: neuronal activity
red: bold response
h1
h2
u1
u2z1
z2
h(u,θ) represents the BOLD response (balloon model) to input
BOLD
(without noise)
BOLD
(without noise)
![Page 27: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/27.jpg)
0 20 40 60
0
2
4
0 20 40 60
0
2
4
seconds
Haemodynamics: reciprocal connections
BOLD
with
Noise added
BOLD
with
Noise added
y1
y2
blue: neuronal activity
red: bold response
u1
u2z1
z2
euhy ),( y represents simulated observation of BOLD response, i.e. includes noise
![Page 28: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/28.jpg)
Bilinear state equation in DCM for fMRI
state changes connectivity
externalinputs
state vector
direct inputs
CuzBuAzm
j
jj
)(1
mnmn
m
n
m
j jnn
jn
jn
j
j
nnn
n
n u
u
cc
cc
z
z
bb
bb
u
aa
aa
z
z
1
1
1111
11
111
1
1111
modulation ofconnectivity
n regions m drv inputsm mod inputs
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BOLDy
y
y
haemodynamicmodel
Inputu(t)
activityz2(t)
activityz1(t)
activityz3(t)
effective connectivity
direct inputs
modulation ofconnectivity
The bilinear model CuzBuAz jj )(
c1 b23
a12
neuronalstates
λ
z
y
integration
Neuronal state equation ),,( nuzFz Conceptual overview
Friston et al. 2003,NeuroImage
u
z
u
FC
z
z
uuz
FB
z
z
z
FA
jj
j
2
![Page 30: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/30.jpg)
The hemodynamic “Balloon” model
, , , ,h
sf
tionflow induc
q/vvf,Efqτ /α
dHbchanges in
1)( /αvfvτ
volumechanges in
1
)1( fγszs
ry signalvasodilato
( )
neuronal input
z t
,)(
signal BOLD
qvty
6 haemodynamic parameters
Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage
Region-specific HRFs!
![Page 31: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/31.jpg)
fMRI data
Posterior densities of parameters
Neuronal dynamics
Hemodynamics
Model selection
DCM roadmap
Model inversion using
Expectation-maximization
State space Model
Priors
![Page 32: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/32.jpg)
Constraints on• Haemodynamic parameters
• Connections
Models of• Haemodynamics in a single region
• Neuronal interactions
Bayesian estimation
)(p
)()|()|( pypyp
)|( yp
posterior
priorlikelihood term
Estimation: Bayesian framework
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sf (rCBF)induction -flow
s
v
f
stimulus function u
modeled BOLD response
vq q/vvf,Efqτ /α1)(
dHbin changes
/αvfvτ 1
in volume changes
f
q
)1(
signalry vasodilatodependent -activity
fγszs
s
( , , )h x u ( , , )y h x u X e
observation model
hidden states},,,,{ qvfszx
state equation( , , )x F x u
parameters
},{
},...,{
},,,,{1
nh
mn
h
CBBA
Overview:parameter estimation
ηθ|y
neuronal stateequation CuzBuAz j
j )(
• 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.
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0 20 40 60-10123
0 20 40 60-10123
seconds
Forward coupling, a21
21a
Input coupling, c1
1c
Prior density Posterior density true values
Parameter estimation: an example
u1
21a
z1
z2
Simulated response
![Page 35: Dynamic Causal Modelling for fMRI Friday 22 nd Oct. 2010 SPM fMRI course Wellcome Trust Centre for Neuroimaging London André Marreiros.](https://reader038.fdocuments.us/reader038/viewer/2022110103/56649ee05503460f94bf0323/html5/thumbnails/35.jpg)
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 groups
• 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
Inference about DCM parameters
Bayesian parameter averaging
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Model comparison and selection
Given competing hypotheses, which model is the best?
Pitt & Miyung (2002), TICS
)(
)()|(log
mcomplexity
maccuracymyp
)|(
)|(
jmyp
imypBij
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Inference on model space
Model evidence: The optimal balance of fit and complexity
Comparing models
• Which is the best model?
1 2 3 4 5 6 7 8 9 10model
lme
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Inference on model space
Model evidence: The optimal balance of fit and complexity
Comparing models
• Which is the best model?
Comparing families of models
• What type of model is best?
• Feedforward vs feedback
• Parallel vs sequential processing
• With or without modulation
1 2 3 4 5 6 7 8 9 10model
lme
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Model evidence: The optimal balance of fit and complexity
Comparing models
• Which is the best model?
Comparing families of models
• What type of model is best?
• Feedforward vs feedback
• Parallel vs sequential processing
• With or without modulation
Only compare models with the same data
1 2 3 4 5 6 7 8 9 10model
lme
A
D
B
C
A
B
C
Inference on model space
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Applications & extensions of DCM to fMRI data
Applications & extensions of DCM to fMRI data
Brain connectivity: types & definitionsAnatomical connectivity
Functional connectivity Effective connectivity
Brain connectivity: types & definitionsAnatomical connectivity
Functional connectivity Effective connectivity
Overview
Dynamic causal models (DCMs)Neuronal model
Hemodynamic model
Estimation: Bayesian framework
Dynamic causal models (DCMs)Neuronal model
Hemodynamic model
Estimation: Bayesian framework
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V1
V5
SPCPhotic
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
SPC
Motion
Photic
Attention
0.85
0.57 -0.02
1.360.70
0.84
0.23
Model 1:attentional modulationof V1→V5
V1
V5
SPC
Motion
PhoticAttention
0.86
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
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• 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.
1
2
slic
e ac
quis
ition
visualinput
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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)(tu
ijij uBA
input
Single-state DCM
1x
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
NNNN
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x
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tx
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Two-state DCM
Ex1
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Extension II: Two-state model
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SPCSPC
SPCSPCVSPC
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SPCSPC
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SPCSPC
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DCM for Büchel & Friston
- FWD
- Intr
- BCW
b
Exam
ple
: Tw
o-s
tate
Mod
el C
om
pari
son
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bilinear DCM
CuxDxBuAdt
dx m
i
n
j
jj
ii
1 1
)()(CuxBuA
dt
dx m
i
ii
1
)(
Bilinear state equation
u1
u2
nonlinear DCM
Nonlinear state equation
u2
u1
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
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Extension III: Nonlinear DCM for fMRI
.
The posterior density of indicates that this gating existed with 97% confidence.
(The D matrix encodes which of the n neural units gate which connections in the system)
)(1,5
SPCVVD
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%)
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So, DCM….
• enables one to infer hidden neuronal processes from fMRI data
• allows one to test mechanistic hypotheses about observed effects
– uses a deterministic differential equation to model neuro-dynamics (represented
by matrices A,B and C).
• is informed by anatomical and physiological principles.
• uses a Bayesian framework to estimate model parameters
• is a generic approach to modelling 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, LFPs)
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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
• 10 Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.
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Thank you for your attention!!!