DCM for fMRI Klaas Enno Stephan Laboratory for Social amp Neural Systems Research SNS University of Zurich Wellcome Trust Centre for Neuroimaging University College London SPM Course FIL ID: 927077
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Slide1
Dynamic Causal Modelling (DCM) for fMRI
Klaas Enno Stephan Laboratory for Social & Neural Systems Research (SNS) University of ZurichWellcome Trust Centre for NeuroimagingUniversity College London
SPM Course, FIL13 May 2011
Slide2Structural, functional & effective connectivityanatomical/structural connectivity= presence of axonal connectionsfunctional connectivity = statistical dependencies between regional time series
effective connectivity = directed influences between neurons or neuronal populationsSporns 2007, Scholarpedia
Slide3Some models of effective connectivity for fMRI dataStructural 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. 1997Volterra kernels Friston & Büchel 2000Time series models (e.g. MAR/VAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003Dynamic Causal Modelling (DCM)bilinear: Friston et al. 2003; nonlinear:
Stephan et al. 2008
Slide4Dynamic causal modelling (DCM)DCM framework was introduced in 2003 for fMRI by Karl Friston, Lee Harrison and Will Penny (NeuroImage 19:1273-1302)part of the SPM software package
currently more than 160 published papers on DCM
Slide5Neural state equation:
Electromagneticforward model:neural activityEEGMEGLFPDynamic Causal Modeling (DCM)simple neuronal model
complicated forward modelcomplicated neuronal model
simple forward model
fMRI
EEG/MEG
inputs
Hemodynamic
forward model:
neural activity
BOLD
Slide6LG
leftLGrightRVF
LVF
FG
right
FG
left
LG = lingual gyrus
FG = fusiform gyrus
Visual input in the
- left (LVF)
- right (RVF)
visual field.
x
1
x
2
x
4
x
3
u
2
u
1
Example:
a linear
model of interacting visual regions
Slide7Example: a linear model of interacting visual regionsLG = lingual gyrusFG = fusiform gyrusVisual input in the - left (LVF) - right (RVF)visual field.
state changeseffectiveconnectivity
externalinputs
system
state
input
parameters
LG
left
LG
right
RVF
LVF
FG
right
FG
left
x
1
x
2
x
4
x
3
u
2
u
1
Slide8Extension: bilinear model
LGleft
LG
right
RVF
LVF
FG
right
FG
left
x
1
x
2
x
4
x
3
u
2
u
1
CONTEXT
u
3
Slide9endogenous connectivity
direct inputsmodulation ofconnectivityNeural state equation
hemodynamic
model
λ
x
y
integration
BOLD
y
y
y
activity
x
1
(
t
)
activity
x
2
(
t
)
activity
x
3
(
t
)
neuronalstates
t
driving
input
u
1
(
t
)
modulatory
input
u
2
(
t
)
t
Slide10Bilinear DCM
Bilinear state equation:
driving
input
modulation
Two-dimensional Taylor series (around x
0
=0, u
0
=0):
Slide11DCM parameters = rate constants
The coupling parameter a thus describes the speed ofthe exponential change in x(t)Integration of a first-order linear differential equation gives anexponential function:
Coupling parameter a is inversely
proportional to the half life
of z(t):
Slide12-x2
stimuliu1contextu2
x1
+
+
-
-
-
+
u
1
Z
1
u
2
Z
2
Example:
context-dependent decay
u
1
u
2
x
2
x
1
Penny et al. 2004,
NeuroImage
Slide13The problem of hemodynamic convolutionGoebel et al. 2003, Magn. Res. Med.
Slide14He
modynamic forward models are important for connectivity analyses of fMRI dataDavid et al. 2008, PLoS Biol.Granger causalityDCM
Slide15stimulus functions
u
t
neural state
equation
hemodynamic
state
equations
Balloon model
BOLD signal change equation
The he
modynamic model in DCM
Stephan et al. 2007,
NeuroImage
Slide16A
B
C
h
ε
How interdependent are neural and hemodynamic parameter estimates?
Stephan et al. 2007,
NeuroImage
Slide17DCM is a Bayesian approachposterior
likelihood ∙ priorBayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities.In DCM: empirical, principled & shrinkage priors.The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.
new data
prior knowledge
Slide18stimulus function
umodelled BOLD response
observation model
hidden states
state equation
parameters
Combining the neural and h
e
modynamic
states gives the
complete
forward model
.
An
observation model
includes measurement
error
e
a
nd confounds
X
(e.g. drift).
Bayesian
inversion:
parameter
estimation
by means of
variational
EM under Laplace approximation
Result:
Gaussian a
posteriori
parameter distributions
, characterised by
mean
η
θ
|y
and
covariance
C
θ
|y
.
Overview:
parameter estimation
η
θ
|y
neural state
equation
Slide19VB in a nutshell (mean-field approximation)
Iterative updating of sufficient statistics of approx. posteriors by gradient ascent. Mean field approx. Neg. free-energy approx. to model evidence.
Maximise neg. free energy wrt. q = minimise divergence, by maximising variational
energies
Slide20Gaussian assumptions about the posterior distributions of the parametersposterior probability that a certain parameter (or contrast of parameters cT
ηθ|y) is above a chosen threshold γ:By default, γ
is chosen as zero ("does the effect exist?").
Inference about DCM parameters:
Bayesian single-subject analysis
Slide21Bayesian single subject inference
LGleft
LG
right
RVF
stim.
LVFstim.
FG
right
FG
left
LD|RVF
LD|LVF
LD
LD
0.34
0.14
-0.08
0.16
0.13
0.19
0.01
0.17
0.44
0.14
0.29
0.14
Contrast:
Modulation LG right
LG links by LD|LVF
vs.
modulation LG left
LG right by LD|RVF
p(c
T
>0|y)
= 98.7%
Stephan et al. 2005,
Ann. N.Y. Acad. Sci.
Slide22Likelihood distributions from different subjects are independent one can use the posterior from one subject as the prior for the next
Under Gaussian assumptions this is easy to compute:groupposterior covariance
individualposterior
covariances
group
posterior
mean
individual posterior
covariances and means
“Today’s posterior is tomorrow’s prior”
Inference about DCM parameters:
Bayesian parameter averaging (FFX group analysis)
Slide23Inference about DCM parameters:RFX group analysis (frequentist)In analogy to “random effects” analyses in SPM,
2nd level analyses can be applied to DCM parameters:Separate fitting of identical models for each subjectSelection of (bilinear) parameters of interest
one-sample t-test:
parameter > 0 ?
paired t-test:
parameter 1 > parameter 2 ?
rmANOVA:
e.g. in case of multiple sessions per subject
Slide24inference on
model structure
or inference on
model parameters?
inference on
individual models
or model space partition
?
comparison of model families using
FFX or RFX BMS
optimal model structure assumed to be identical across subjects?
FFX BMS
RFX BMS
yes
no
inference on
parameters of an optimal model
or
parameters of all models
?
BMA
definition of model space
FFX analysis of parameter estimates
(e.g. BPA)
RFX analysis of parameter estimates
(e.g. t-test, ANOVA)
optimal model structure assumed to be identical across subjects?
FFX BMS
yes
no
RFX BMS
Stephan et al. 2010,
NeuroImage
Slide25Any design that is good for a GLM of fMRI data.What type of design is good for DCM?
Slide26GLM vs. DCMDCM tries to model the same phenomena (i.e. local BOLD responses) as a GLM, just in a different way (via connectivity and its modulation).No activation detected by a GLM → no motivation to include this region in a deterministic DCM.However, a stochastic DCM could be applied despite the absence of a local activation.
Stephan 2004, J. Anat.
Slide27Multifactorial design: explaining interactions with DCM
Task factorTask ATask BStim 1
Stim 2
Stimulus factor
T
A
/S1
T
B/S1
T
A
/S2
T
B/S2
X
1
X
2
Stim2/
Task A
Stim1/
Task A
Stim 1/
Task B
Stim 2/
Task B
GLM
X
1
X
2
Stim2
Stim1
Task A
Task B
DCM
Let’s assume that an SPM analysis shows a main effect of stimulus in X
1
and a stimulus
task interaction in X
2
.
How do we model this using DCM?
Slide28Stim 1
Task A
Stim 2
Task A
Stim 1
Task B
Stim 2
Task B
Simulated data
X
1
X
2
+++
X
1
X
2
Stimulus 2
Stimulus 1
Task A
Task B
+
+++
+
+++
+
–
–
Stephan et al. 2007,
J. Biosci.
Slide29Stim 1
Task A
Stim 2
Task A
Stim 1
Task B
Stim 2
Task B
plus added noise (SNR=1)
X
1
X
2
Slide30DCM10 in SPM8DCM10 was released as part of SPM8 in July 2010 (version 4010).Introduced many new features, incl. two-state DCMs and stochastic DCMsThis led to various changes in model defaults, e.g.inputs mean-centredchanges in coupling priorsself-connections: separately estimated for each areaFor details, see: www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf
Further changes in version 4290 (released April 2011) to accommodate new developments and give users more choice (e.g. whether or not to mean-centre inputs).
Slide31The evolution of DCM in SPMDCM is not one specific model, but a framework for Bayesian inversion of dynamic system modelsThe default implementation in SPM is evolving over timebetter numerical routines for inversionchange in priors to cover new variants (e.g., stochastic DCMs, endogenous DCMs etc.)To enable replication of your results, you should ideally state which SPM version you are using when publishing papers.
Slide32Factorial structure of model specification in DCM10Three dimensions of model specification:bilinear vs. nonlinearsingle-state vs. two-state (per region)deterministic vs. stochasticSpecification via GUI.
Slide33bilinear DCM
Bilinear state equation:
driving
input
modulation
driving
input
modulation
non-linear DCM
Two-dimensional Taylor series (around x
0
=0, u
0
=0):
Nonlinear state equation:
Slide34Neural population activity
fMRI signal change (%)
x
1
x
2
x
3
Nonlinear dynamic causal model (DCM
)
Stephan et al. 2008,
NeuroImage
u
1
u
2
Slide35V1
V5stim
PPC
attention
motion
1.25
0.13
0.46
0.39
0.26
0.50
0.26
0.10
MAP = 1.25
Stephan et al. 2008,
NeuroImage
Slide36V1
V5PPCobserved
fitted
motion &attention
motion &
no attention
static
dots
Slide37input
Single-state DCM
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
Two-state DCM
Two-state DCM
Marreiros et
al. 2008,
NeuroImage
Slide38Stochastic DCM
Friston et al. (2008, 2011) NeuroImageDaunizeau et al. (2009) Physica Daccounts for stochastic neural fluctuationscan be fitted to resting state data
has unknown precision and smoothness additional
hyperparameters
Li et
al. (2011) NeuroImage
Slide39Thank you