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Dynamic Causal Modelling (DCM) for fMRI. Andre Marreiros. Wellcome Trust Centre for Neuroimaging University College London. Overview. Dynamic Causal Modelling of fMRI. Definitions & motivation. The neuronal model (bilinear dynamics) The Haemodynamic model. Estimation: Bayesian framework. - PowerPoint PPT Presentation
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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
a12
neuronalstates
Use differential equations to represent a neuronal system
)(
)()(
1
tz
tztz
n
system represented by state variables
• State vector – Changes with time
),,...(
),,...(
1
1111
nnn
n
n uzzf
uzzf
z
z
• Rate of change of state vector– Interactions between elements– External inputs, u
( , , )z f z u • System parameters
DCM parameters = rate constants
11
dz szdt
Decay function:
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
0.2
0.4
0.6
0.8
1
10.5 (0)z
1 1( ) (0)exp( )z t z at
1 1
1
( ) 0.5 (0)(0)exp( )
z zz s
1 1 1( ) (0)exp( ), (0) 1z t z st z
Linear dynamics: 2 nodes
1 1
2 21 1 2
1
2
1
2 21
21
(0) 1(0) 0
( ) exp( )( ) exp( )
0
z szz s a z z
zz
z t stz 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
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
czz
as
zz
z1
z2
Neurodynamics: positive modulation
000
001
01 2211
2
1221
22
1
212
1
bu
czz
bu
zz
as
zz
u2
u1
z1
z2
modulatory input u2 activity through the coupling a21
u1
u2
index, not squared
z1
z2
Neurodynamics: reciprocal connections
0,,00
001
1 22121121
2
1221
22
1
21
12
2
1
baau
czz
bu
zz
aa
szz
u2
u1
z1
z2
reciprocal connectiondisclosed by u2
u1
u2 z1
z2
0 20 40 60
024
0 20 40 60
024
seconds
Haemodynamics: reciprocal connections
blue: neuronal activityred: bold response
h1
h2
u1
u2 z1
z2
h(u,θ) represents the BOLD response (balloon model) to input
BOLD
(without noise)
BOLD
(without noise)
0 20 40 60
024
0 20 40 60
024
seconds
Haemodynamics: reciprocal connections
BOLD
with
Noise added
BOLD
with
Noise added
y1
y2
blue: neuronal activityred: bold response
u1
u2 z1
z2
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
CuzBuAzm
j
jj
)(1
mnmn
m
n
m
j jnn
jn
jn
j
j
nnn
n
n u
u
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cc
z
z
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bbu
aa
aa
z
z
1
1
1111
11
111
1
1111
modulation ofconnectivity
n regions m drv inputsm mod inputs
The haemodynamic “Balloon” model
sf
tionflow induc
q/vvf,Efqτ /α
dHbchanges in1)( /αvfvτ
volumechanges in1
)1( fγszsry signalvasodilato
( )neuronal input
z t
,)( signal BOLD
qvty
5 haemodynamic parameters: , , , ,h
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
uz
uFC
zz
uuzFB
zz
zFA
jj
j
2
fMRI data
Posterior densities of parameters
Neuronal dynamics Haemodynamics
Model comparison
DCM roadmap
Model inversion using
Expectation-maximization
State space Model
Priors
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
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.
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
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:
ηθ|y
Model comparison and selection
Given competing hypotheses, which model is the best?
Pitt & Miyung (2002), TICS
)()()|(logmcomplexity
maccuracymyp
)|()|(jmypimypBij
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
?
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
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.
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)
)(tu
ijij uBA
input
Single-state DCM
1x
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
NNNN
N
x
xtx
AA
AAA
CuxuBAtx
1
1
111
)(
)(
Two-state DCM
Ex1
)exp( ijij uBA
Ix1
11 11exp( )IE IEA uBIEx ,1
CuzBuAzz jj
Extension II: Two-state model
IN
EN
I
E
AA
AAA
AA
AAA
u
xx
xx
tx
eeeee
eeeee
A
CuxABtx
IINN
IENN
EINN
EENNN
IIIE
NEIEE
1
1
)(
000
000
)(
1
1111
11111
SPCSPC
SPCSPCVSPC
VV
SPCVVVVV
VV
VVVV
IIIEEIEEEE
IIIEEEEIEEEE
IIIEEEEIEE
A
0000000
0000000000000
5
55
55515
11
5111
Attention
SPCSPC
SPCSPCVSPC
VV
SPCVVVVV
VV
VVVV
IIIEEIEEEE
IIIEEEEIEEEE
IIIEEEEIEE
A
0000000
0000000000000
5
55
55515
11
5111
Attention
SPCSPC
SPCSPCVSPC
VV
SPCVVVVV
VV
VVVV
IIIEEIEEEE
IIIEEEEIEEEE
IIIEEEEIEE
A
0000000
0000000000000
5
55
55515
11
5111
Attention
DCM for Büchel & Friston
- FWD
- Intr
- BCW
b
Exam
ple:
Tw
o-st
ate
Mod
el
Com
pari
son
bilinear DCM
CuxDxBuAdtdx m
i
n
j
jj
ii
1 1
)()(CuxBuAdtdx 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
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)
)(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%)
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