dynamic causal modelling (dcm): theory demis hassabis & hanneke den ouden thanks to klaas enno...
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Dynamic Causal Modelling (DCM): Theory
Demis Hassabis & Hanneke den Ouden
Thanks to
Klaas Enno Stephan
Functional Imaging LabWellcome Dept. of Imaging NeuroscienceInstitute of NeurologyUniversity College London
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
• Interpretation of parameters– Statistical inference– Bayesian model selection
System analyses in functional neuroimaging
System analyses in functional neuroimaging
Functional integrationAnalyses of Analyses of inter-regional effectsinter-regional effects: : what are the interactions what are the interactions between the elements of a given between the elements of a given neuronal system?neuronal system?
Functional integrationAnalyses of Analyses of inter-regional effectsinter-regional effects: : what are the interactions what are the interactions between the elements of a given between the elements of a given neuronal system?neuronal system?
Functional connectivity= the temporal correlation = the temporal correlation betweenbetween spatially remote spatially remote neurophysiologicalneurophysiological events events
Functional connectivity= the temporal correlation = the temporal correlation betweenbetween spatially remote spatially remote neurophysiologicalneurophysiological events events
Effective connectivity= the influence that the elements = the influence that the elements
of a neuronal system exert over of a neuronal system exert over anotheranother
Effective connectivity= the influence that the elements = the influence that the elements
of a neuronal system exert over of a neuronal system exert over anotheranother
Functional specialisationAnalyses of Analyses of regionally specific regionally specific effectseffects: which areas constitute a : which areas constitute a neuronal system?neuronal system?
Functional specialisationAnalyses of Analyses of regionally specific regionally specific effectseffects: which areas constitute a : which areas constitute a neuronal system?neuronal system?
MECHANISM-FREE MECHANISTIC
Models of effective connectivity
• Structural Equation Modelling (SEM)
• Psycho-physiological interactions (PPI)
• Multivariate autoregressive models (MAR)& Granger causality techniques
• Kalman filtering
• Volterra series
• Dynamic Causal Modelling (DCM)Friston et al., NeuroImage 2003
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
• Interpretation of parameters– Statistical inference– Bayesian model selection
Models of effective connectivity = system models.But what precisely is a system?
• System = set of elements which interact in a spatially and temporally specific fashion.
• System dynamics = change of state vector in time
• Causal effects in the system:– interactions between elements– external inputs u
• System parameters :specify the nature of the interactions
• general state equation for non-autonomous systems
)(
)(
)(1
tz
tz
tz
n
),,...(
),,...(
1
1111
nnn
n
n uzzf
uzzf
z
z
),,( uzFz
overall system staterepresented by state variables
nz
z
zdt
dz
1
change ofstate vectorin time
Example: linear dynamic system
LGleft
LGright
RVF LVF
FGright
FGleft
LG = lingual gyrusFG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF)visual field.z1 z2
z4z3
u2 u1
state changes
effectiveconnectivity
externalinputs
systemstate
inputparameters
2
1
12
21
4
3
2
1
444342
343331
242221
131211
4
3
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u
u
c
c
z
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z
z
aaa
aaa
aaa
aaa
z
z
z
z
CuAzz
},{ CA
Extension:
bilinear dynamic system
LGleft
LGright
RVF LVF
FGright
FGleft
z1 z2
z4z3
u2 u1
CONTEXTu3
CuzBuAzm
j
jj
)(1
3
2
112
21
4
3
2
1
334
312
3
444342
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0000
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0
0
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u
u
uc
c
z
z
z
z
b
b
u
aaa
aaa
aaa
aaa
z
z
z
z
Bilinear state equation in DCM
state changes
intrinsicconnectivity
m externalinputs
systemstate
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
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
• Interpretation of parameters– Statistical inference– Bayesian model selection
DCM for fMRI: the basic idea
• 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.
BOLDy
y
y
hemodynamicmodel
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
Neural 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
-
Z2
stimuliu1
contextu2
Z1
+
+
-
-
-+
u1
Z1
u2
Z2
2
112
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2
112
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12
u
uczuz
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a
z
z
CuzBuAzz
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2
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12
2
1
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u
uczuz
a
a
z
z
CuzBuAzz
bb
Example: generated neural data
u1
u2
z2
z1
sf
tionflow induc
s
v
f
v
q q/vvf,Efqτ /α
dHbchanges in
1)( /αvfvτ
volumechanges in
1
f
q
)1( fγszs
ry signalvasodilato
},,,,{ h},,,,{ h
important for model fitting, but of no interest for statistical inference
signal BOLD
qvty ,)(
The hemodynamic “Balloon” model
)(tz
activity • 5 hemodynamic parameters:
• Empirically determineda priori distributions.
• Computed separately for each area (like the neural parameters).
LGleft
LGright
RVF LVF
FGright
FGleft
Example: modelled BOLD signal
Underlying model(modulatory inputs not shown)
LG = lingual gyrus Visual input in the FG = fusiform gyrus - left (LVF)
- right (RVF)visual field.
blue: observed BOLD signalred: modelled BOLD signal (DCM)
left LG
right LG
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
• Interpretation of parameters– Statistical inference– Bayesian model selection
Bayesian rule in DCM
)()|()|( pypyp )()|()|( pypyp posterior likelihood ∙ prior
Bayes Theorem• Likelihood derived from error
and confounds (eg. drift)
• Priors – empirical (haemodynamic parameters) and non-empirical (eg. shrinkage priors, temporal scaling)
• Posterior probability for each effect calculated and probability that it exceeds a set threshold expressed as a percentage
sf (rCBF)induction -flow
s
v
f
stimulus function u
modelled BOLD response
vq q/vvf,Efqτ /α1)(
dHbin changes
/αvfvτ 1
in volume changes
f
q
)1(
signalry vasodilatodependent -activity
fγszs
s
)(xy )(xy eXuhy ),(
observation model
hidden states},,,,{ qvfszx
state equation),,( uxFx
parameters
},{
},...,{
},,,,{1
nh
mn
h
CBBA
• Combining the neural and hemodynamic states gives the complete forward model.
• An observation model includes measurement error e and confounds X (e.g. drift).
• Bayesian parameter estimation: minimise difference between data and model
• Result:Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.
Parameter estimation in DCM
ηθ|y
neural stateequation
CuzBuAz jj )( CuzBuAz j
j )(
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
• Interpretation of parameters– Statistical inference– Bayesian model selection
DCM parameters:interpretation & inference
Hypothesis: modulation by context > 0
–How can we make inference about effects represented by these parameters
–At a single subject level?
–At a group level?
– How do we select between different models?
- DCM gives gaussian posterior densities of parameters (intrinsic connectivity, effective connectivity and inputs)
LGleft
LGright
RVF LVF
FGright
FGleft
z1 z2
z4z3
u2 u1
CONTEXTu3
• Assumption: posterior distribution of the parameters is gaussian
• Use of the cumulative normal distribution to test the probability by which a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ:
• γ can be chosen as zero ("does the effect exist?") or as a function of the expected half life τ of the neural process: γ = ln 2 / τ
Bayesian single-subject analysis
ηθ|y
ηθ|y
Probability
Group analysis
• In analogy to “random effects” analyses in SPM, 2nd level analyses can be applied to DCM parameters:
Separate fitting of identical models for each subject
Separate fitting of identical models for each subject
Selection of bilinear parameters of interestSelection of bilinear
parameters of interest
one-sample t-test:
parameter > 0 ?
one-sample t-test:
parameter > 0 ?
paired t-test: parameter 1 > parameter 2 ?
paired t-test: parameter 1 > parameter 2 ?
rmANOVA: e.g. in case of
multiple sessions per subject
rmANOVA: e.g. in case of
multiple sessions per subject
Model comparison and selection
Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?
For which model i does p(y|mi) become maximal?
Which model represents thebest balance between model fit and model complexity?
Pitt & Miyung (2002), TICS
Bayesian Model Selection
Bayes theorem:
Model evidence:
The log model evidence can be represented as:
Bayes factor:
)|(
)|(),|(),|(
myp
mpmypmyp
dmpmypmyp )|(),|()|(
)(
)()|(log
mcomplexity
maccuracymyp
)|(
)|(
jmyp
imypBij
Penny et al. 2004, NeuroImage
The DCM cycle
Design a study thatallows to investigatethat system
Extraction of time seriesfrom SPMs
Parameter estimationfor all DCMs considered
Bayesian modelselection of optimal DCM
Statistical test on
parameters of optimal
model
Hypothesis abouta neural system
Definition of DCMs as systemmodels
Data acquisition
Inference about DCM parameters: Bayesian fixed-effects group analysis
Because the likelihood distributions from different subjects are independent, one can combine their posterior densities by using the posterior of one subject as the prior for the next:
)|()...|()|(),...,|(
...
)|()|(
)()|()|(),|(
)()|( )|(
111
12
1221
11
ypypypyyp
ypyp
pypypyyp
pypyp
NNN
)|()...|()|(),...,|(
...
)|()|(
)()|()|(),|(
)()|( )|(
111
12
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11
ypypypyyp
ypyp
pypypyyp
pypyp
NNN
1,...,|
1|
1|,...,|
1
1|
1,...,|
11
1
NiiN
iN
yy
N
iyyyy
N
iyyy
CC
CC
1,...,|
1|
1|,...,|
1
1|
1,...,|
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iN
yy
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iyyyy
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iyyy
CC
CC
Under Gaussian assumptions this is easy to compute:
groupposterior covariance
individualposterior covariances
groupposterior mean
individual posterior covariances and means
See:spm_dcm_average.m Neumann & Lohmann, NeuroImage 2003
Approximations to model evidence
Laplace approximation:
Akaike information criterion (AIC):
Bayesian information criterion (BIC):
pmaccuracymyAIC )()|(
Penny et al. 2004, NeuroImage
yppypT
py
eT
e hh
mcomplexitymaccuracyF
||1
|
1
log2
1log
2
1)()(
2
1
))(())((2
1log
2
1
)()(
CCθθCθθ
θyCθyC
yppypT
py
eT
e hh
mcomplexitymaccuracyF
||1
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log2
1log
2
1)()(
2
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))(())((2
1log
2
1
)()(
CCθθCθθ
θyCθyC
SNp
maccuracymyBIC log2
)()|(
Unfortunately, the complexity term depends on the prior density, which is determined individually for each model to ensure stability. Therefore, we need other approximations to the model evidence.
DCM parameters = rate constants
azdt
dz )exp()( 0 atztz
The coupling parameter a thus describes the speed ofthe exponential growth/decay:
)exp(
5.0)(
0
0
az
zz
Integration of a first order linear differential equation gives an exponential function:
/2lna
05.0 z
a/2ln
The coupling parameter a is inversely proportional to the half life of z(t):