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Mohamed Seghier
Wellcome Trust Centre for Neuroimaging,University College London, UK
DCM: Dynamic CausalModelling for fMRI
Wellcome Trust Centre for Neuroimaging
SPM-Course Edinburgh, April 2010
DCM [default] implementation:
Deterministic Stochastic [Daunizeau et al. 2009]
Bilinear Nonlinear [Stephan et al. 2008]
The one-state neuronal The two-state [Marreiros et al. 2008]
DCM is a generative model= a quantitative / mechanistic description of how observed data are generated.
Key features:1- Dynamic2- Causal3- Neuro-physiologically motivated4- Operate at hidden neuronal interactions5- Bayesian in all aspects.
The hemodynamics Deterministic dynamical systems
[Friston et al. 2000 Neuroimage] [Friston 2002 Neuroimage]
[Friston et al. 2003 Neuroimage]
“DCM is used to test the specific hypothesis thatmotivated the experimental design. It is not an exploratorytechnique […]; the results are specific to the tasks andstimuli employed during the experiment.”
“The central idea behind dynamic causal modelling(DCM) is to treat the brain as a deterministicnonlinear dynamic system that is subject to inputsand produces outputs.”
“DCM assumes the responses are driven by designedchanges in inputs.”
[Friston et al. 2003 Neuroimage]
Input u(t)
connectivity parameters
System state z(t)State changes of a systemare dependent on:
– the current state
– external inputs
– its connectivity
– time constants & delays
System =a set of elements whichinteract in a spatially andtemporally specific fashion
),,( uzFdt
dz
What is a system?
(evolution equation)
Basic idea of DCM for fMRI
λ
z
y
♣ Effective connectivity is parameterised in terms of coupling amongunobserved brain states (e.g., neuronal activity in different regions).The objective is to estimate these parameters by perturbing thesystem and measuring the response.
♣ A cognitive system is modelled as a bilinear model of neuralpopulation dynamics (z).
♣ The modelled neuronal dynamics (z) is transformed into area-
specific BOLD signals (y) by a hemodynamic forward model (λ).
Aim: to estimate the parameters of a reasonablyrealistic neural model such that the predictedregional blood oxygen level dependent (BOLD)signals, correspond as closely as possible to theobserved BOLD signals.
Neurodynamics: 2 nodes with input
u2
u1
z1
z2
00
0211
2
1
2221
11
2
1
au
c
z
z
aa
a
z
z
activity in is coupled to viacoefficient 21a
2z 1z
1212222
11111
zazaz
cuzaz
11a
22a
21a
R1
R2
Neurodynamics: positive modulation
u2
u1
z1
z2
000
0002211
2
1
221
2
2
1
2221
11
2
1
bu
c
z
z
bu
z
z
aa
a
z
z
modulatory input u2 activitythrough the coupling 21a
11a
22a
21a
R1
R2
122211212222
11111
zubzazaz
cuzaz
Neurodynamics: reciprocal connections
00000
0022112211
2
1
221
2
2
1
2221
1211
2
1
baau
c
z
z
bu
z
z
aa
aa
z
z
u2
u1
z1
z2
reciprocalconnectiondisclosed by u2
11a
22a
21a12a
bilineardynamicsystem R1
leftR2
right
R4right
R3left
z1 z2
z4z3
u2 u1CONTEXT
u3
3
2
1
12
21
4
3
2
1
334
312
3
444342
343331
242221
131211
4
3
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1
0
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0000
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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 for fMRI
statechanges
connectivityexternalinputs
statevector
directinputs
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
1
1
111
1
1111
modulation ofconnectivity
n regions m inputs (driv.)m inputs (mod.)
The neural state equation
“C”, the direct or driving effects:- extrinsic influences of inputs on neuronal activity.
“A”, the intrinsic coupling or the latent connectivity:- fixed or endogenous effective connectivity;- first order connectivity among the regions in the absence of input.
“B”, the bilinear term, modulatory effects, or the induced connectivity:- context-dependent change in connectivity;- eq. a second-order interaction between the input and activity in a sourceregion when causing a response in a target region.
[Units]: rates, [Hz];Strong connection = an effect that is influenced quicklyor with a small time constant.
CuzBuAzm
j
jj
)(1
DCM parameters = rate constants
dxax
dt 0( ) exp( )x t x at
The coupling parameter athus describes the speed of
the exponential change in x(t)0
0
( ) 0.5
exp( )
x x
x a
Integration of a first-order linear differential equation gives anexponential function:
/2lna
00.5x
a/2ln
Coupling parameter is inversely
proportional to the half life of x(t):
If AB is 0.10 s-1 this means that, per unit time, the increase in activity in Bcorresponds to 10% of the activity in A
a
hemodynamicmodelλ
z
y
integration
BOLDyyy
activityx1(t)
activityx2(t) activity
x3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
[Stephan & Friston (2007),Handbook of Brain Connectivity]
endogenousconnectivity
direct inputs
modulation ofconnectivity
Neural state equation CuzBuAz jj )( )(
u
zC
z
z
uB
z
zA
j
j
)(
hemodynamicmodel ??
λ
z
y
integration
BOLDyyy
activityx1(t)
activityx2(t) activity
x3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
endogenousconnectivity
direct inputs
modulation ofconnectivity
Neural state equation CuzBuAz jj )( )(
u
zC
z
z
uB
z
zA
j
j
)(
[Stephan & Friston (2007),Handbook of Brain Connectivity]
00000
00 22112211
2
1
221
2
2
1
2221
1211
2
1
baau
c
z
z
bu
z
z
aa
aa
z
z
Hemodynamics: the indirect link
21a
a11
a22
a12
Simulated responseneuronal activity bold response
00000
00 22112211
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2
2
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1
baau
c
z
z
bu
z
z
aa
aa
z
z
21a
a11
a22
a12
neuronal activity bold response
s
signal
inf
flow
q
dHb
signalBOLD
qvty ,)( )( tz
activity s
0
,
vfout
0inf
00
0,
E
EfEf inin
v
volume
0
,
v
qvfout
f
inf
1
s
s
The hemodynamic model
State Equations
[Friston 2000 Neuroimage]
Output function: a mixture of intra- and extra-vascular signal
Flow component:s : activity-dependent signal;
f : flow inducing signal
Balloon component:v : the rate of change of volume;
q : the change in deoxyhemoglobin
sf
tionflow induc
(rCBF)
s
v
inputs
v
q q/vvEf,EEfqτ /α
dHbchanges in
100)( /αvfvτ
volumechanges in
1
f
q
)1( fγsxs
signalryvasodilato
u
s
CuxBuAdt
dx m
j
jj
1
)(
t
neural state equation
1
3.4
111),(
3
002
001
3210
0
k
TEErk
TEEk
vkv
qkqkV
S
Svq
hemodynamicstate equationsf
Balloon model
BOLD signalchange equation
},,,,,{ h
important for model fitting,but of no interest forstatistical inference
• 6 hemodynamic parameters:
• Empirically determineda priori distributions.
• Area-specific estimates(like neural parameters) region-specific HRFs!
The hemodynamic model
[Friston et al. 2000, NeuroImage][Stephan et al. 2007, NeuroImage]
R1left
R2right
u2 u1
R4right
R3left
Example: modelled BOLD signal
black: observed BOLD signal
red: modelled BOLD signal
CuzBuAzm
j
jj
)(1
Multiple-input multiple-output system
Priors & parameter estimation
Bayesian statistics (inversion)
)()|()|( pypyp
posterior likelihood ∙ prior
)|( yp )(p
Bayes theorem allows us to express our prior knowledgeor “belief” about parameters of the model.
The posterior probability of the parametersgiven the data is an optimal combination ofprior knowledge and new data, weighted bytheir relative precision.
new data prior knowledge
Priors in DCM
- hemodynamic parameters: empirical priors
- coupling parameters of self-connections: principled priors
- coupling parameters other connections: shrinkage priors
Constraints on parameter estimation:
Inference about DCM parameters:Bayesian inversion
• Gaussian assumptions about the posterior distributions of theparameters
• Use of the cumulative normal distribution to test the probability thata certain parameter (or contrast of parameters cT ηθ|y) is above achosen threshold γ:
• By default, γ is chosen as zero ("does the effect exist?").
cCc
cp
y
T
y
T
N
ηθ|y
Bayesian parameter estimation by means of expectation-maximisation (EM)
[Friston 2002 Neuroimage]
yy
BOLD
DCM: practical stepsSelect areas you want to model
• Extract timeseries of these areas(x(t))
• Specify at neuronal level
– what drives areas (c)
– how areas interact (a)
– what modulates interactions (b)
• State-space model with 2 levels:
– Hidden neural dynamics
– Predicted BOLD response
• Estimate model parameters:
Gaussian a posteriori parameterdistributions, characterised bymean ηθ|y and covariance Cθ|y.
neuronalstates activity
x1(t) a12 activityx2(t)
c2
c1
Driving input(e.g. sensory stim)
Modulatory input(e.g. context/learning/drugs)
b12
ηθ|y
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)
6 normal subjects, 4 x 100 scan sessions;
each session comprising 10 scans of 4 differentconditions
F A F N F A F N S .................
F - fixation point only
A - motion stimuli with attention (detect changes)
N - motion stimuli without attention
S - no motion
[Büchel & Friston 1997, Cereb. Cortex][Büchel et al. 1998, Brain]
Attention – No attention
Attention to motion in the visual system
V5
SPC
Attention – No attention
How we can interpret, mechanistically,the increase in activity of area V5 byattention when motion is physicallyunchanged.
Choice of areas and time series extraction. Three ROIs: V1, V5, and SPC.
Definition of driving inputs. All visual stimuli/conditions (photic: A N S)
Definition of modulatory inputs. The effects of motion and attention (A N)
Building the model:1- how to connect regions (intrinsic connections “A”);2- how the driving inputs enter the system (extrinsic effects “C”);3- define the context-dependent connections (modulatory effects “B”).
V1
V5
SPC
Motion
Photic
Attention• Visual inputs drive V1.
• Activity then spreads tohierarchically arranged visualareas.
• Motion modulates the strength ofthe V1→V5 forward connection.
• Attention modualtes the strengthof the SPC→V5 backwardconnection.
Re-analysis of data from[Friston et al., 2003 NeuroImage]
• Motion modulates thestrength of the V1→V5forward connection.
• The intrinsic connectionV1→V5 is insignificant in theabsence of motion (a21=-0.05 Hz).
• Attention increases thebackward-connectionSPC→V5.
V1
V5
SPC
Motion
Photic
Attention
0.88
0.48
0.37
0.42
0.66
0.56
-0.05
Re-analysis of data fromFriston et al., NeuroImage 2003
After DCM estimation:
Are there otherplausible/alternative models?
V1
V5
SPC
Motion
PhoticAttention
0.86
0.56 -0.02
1.42
0.55
0.75
0.89
Model 1:attentional modulationof V1→V5
V1
V5
SPC
Motion
Photic
Attention
0.85
0.57 -0.02
1.360.70
0.84
0.23
Model 2:attentional modulationof SPC→V5
V1
V5
SPC
Motion
PhoticAttention
0.85
0.57 -0.02
1.36
0.03
0.70
0.85
Attention
0.23
Model 3:attentional modulationof V1→V5 and SPC→V5
How we can compare between competing hypotheses? BMS (Bayesian Model Selection)
Alternative models (hypothesis-driven approach):
Model evidence and selection
Given competing hypotheseson functional mechanisms ofa system, which model is thebest?
For which model m does p(y|m)become maximal?
Which model represents thebest balance between modelfit and model complexity?
[Pitt and Miyung 2002 TICS]
dmpmypmyp )|(),|()|( Model evidence:
Bayesian model selection (BMS)
)|(
)|(),|(),|(
myp
mpmypmyp
Bayes’ rule:
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
integral usually not analytically solvable, approximations necessary
Model evidence: probability of generating data y from parameters that arerandomly sampled from the prior p(m).
Maximum likelihood: probability of the data y for the specific parameter vector that maximises p(y|,m).
Logarithm is amonotonic function
Maximizing log model evidence
= Maximizing model evidence
)(),|(log
)()()|(log
mcomplexitymyp
mcomplexitymaccuracymyp
Log model evidence = balance between fit and complexity
[Penny et al. 2004, NeuroImage][Penny et al. 2010, PLoS Comp Biol]
Approximations to the model evidence in DCM
The negative variotional free energy (F) approximation
Under Gaussian assumptions about the posterior (Laplace approximation),the negative free energy F is a lower bound on the log model evidence:
mypqKLmypF ,|,)|(log Kullback-Leibler (KL) divergence
The complexity term in F
• The negative free energy F accounts for parameterinterdependencies.
• The complexity term of F is higher
– the more independent the prior parameters ( effective DFs)
– the more dependent the posterior parameters
– the more the posterior mean deviates from the prior mean
• NB: SPM8 only uses F for model selection !
y
T
yy CCC
mpqKL
|1
||2
1
2
1
2
1
)|(),([Penny et al. 2004 Neuroimage][Stephan et al. 2009 Neuroimage]
Bayes factors
)|(
)|(
2
112
myp
mypBF
positive value, [0;[
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is madepossible by Bayes factors:
To compare two models, we can just compare their logevidences.
Very strong 99% 150
strong95-99%20 to 150
positive75-95%3 to 20
weak50-75%1 to 3
Evidencep(m1|y)BF12
Kass & Raftery classification:[Kass & Raftery 1995, J. Am. Stat. Assoc.]
Bayesian Model Selection in group studies.
Fixed effects BMS at group level
Group Bayes factor (GBF) for 1...K subjects:
Average Bayes factor (ABF):
k
kijij BFGBF )(
( )kKij ij
k
ABF BF
)|(
)|(
j
iij
myp
mypBF
Problems:► blind with regard to group heterogeneity;► sensitive to outliers.
)|(~ 111 mypy)|(~ 111 mypy
)|(~ 222 mypy)|(~ 111 mypy
)|(~ pmpm kk
);(~ rDirr
)|(~ pmpm kk )|(~ pmpm kk),1;(~1 rmMultm
Random effects BMS for group studies
Dirichlet parameters= “occurrences” of models in the populations
Dirichlet distribution of model probabilities
Multinomial distribution of subject-specificmodels
Measured data
[Stephan et al. 2009, Neuroimage]
-5 -4 -3 -2 -1 0 1 2 3 4 5
Sim
ula
ted
data
sets
Log model evidence differences
x1 x2u1
x3
u2
x1 x2u1
x3
u2
incorrect model (m2) correct model (m1)
m2 m1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
r1
p(r
1|y
)
p(r1>0.5 | y) = 1.000
1 2
/
0.934, 0.066
k k kk
r
r r
Exceedance probability
1 220.537, 1.463 Estimates of Dirichlet parameters
Post. expectations of modelprobabilities
%100
|211
yrrp
[Stephan et al. 2009, Neuroimage]
-35 -30 -25 -20 -15 -10 -5 0 5
Su
bje
cts
Log model evidence differences
MOG
LG LG
RVFstim.
LVFstim.
FGFG
LD|RVF
LD|LVF
LD LD
MOGMOG
LG LG
RVFstim.
LVFstim.
FGFG
LD
LD
LD|RVF LD|LVF
MOG
m2 m1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
r1
p(r
1|y
)
p(r1>0.5 | y) = 0.997
157.0,843.0
194.2,806.11
21
21
rr
[Stephan et al. 2009, Neuroimage]
Interface in SPM8
k
kkkr
Post. expectations ofmodel probabilities
yrrp
kKj
jkk |
:\}...1{
Exceedance probability
Levels of inference: Group/population level
-- Family level ---- System/model level --
-- Parameter/connection level --
♣ Family level:- Useful when no clear winning model // models have common characteristics .Models assigned to subsets (families) with shared parametersInference: a class of models that best explains the data.
FFX: subjects assumed to use identical systems.RFX: optimal models vary across subjects.
Variational Bayes: fast/accurate Nmod < N_sub.Gibbs sampling: optimal N_mod >> N_sub.
♣ System level:- Useful when a clear winning model can be identified (BMS).Inference: the best combination of inputs+connections that explains the data.
♣ Connection level:- Useful if interested in connectivity parameters (e.g. modulations).Inference: Bayesian parameters averaging (BPA) or t-test on DCM parameters.
Inference: BMA on the winning family (or the whole model space).
[Penny et al. 2010, PLoS Comp Biol]
Fig. 1. This schematic summarizes the typical sequence of analysis in DCM, depending on the question of interest. Abbreviations: FFX=fixedeffects, RFX=random effects, BMS=Bayesian model selection, BPA=Bayesian parameter averaging, BMA=Bayesian model averaging,ANOVA=analysis of variance.
Stephan et al. (2010). Ten Simple Rules for DCM. NeuroImage
It is helpful to constrain your DCM model space.
number of ROIs limited to 8 in SPM8 (GUI).
(e.g., 5 ROIs, fully connected, 1 Billion alternatives for modulations!).
Define sets of models that are plausible, in a systematic way, given priorknowledge about the system (e.g. anatomical, TMS, previous studies).
Bad models will affect your BMS results (BMS = a “relative” space)!
BMS has nothing to say about the “true” models.find the most plausible (useful) model, given a set of alternatives.Best model = best balance between accuracy and complexity.
BMS cannot be applied to models fitted to different data!(Only models with the same ROIs can be compared using BMS).
Extensions in DCM for fMRI (SPM8):
• Bayesian Model Selection BMS [Penny et al. 2004 Neuroimage].
• Slice specific sampling [Kiebel et al. 2007 Neuroimage].
• Refined hemodynamic model [Stephan et al. 2007 Neuroimage].
• The two-state DCM [Marreiros et al. 2008 Neuroimage].
• The non-linear DCM [Stephan et al. 2008 Neuroimage].
• Random-effects BMS (VB) [Stephan et al. 2009 Neuroimage].
• Random-effects BMS (Gibbs) [Penny et al. 2010 PLoS Comp Biol].
• Stochastic DCM [Daunizeau et al. 2009 Physica D].
• Anatomical-based priors for DCM [Stephan et al. 2009 Neuroimage].
• Family level inference BMS [Penny et al. 2010 PLoS Comp Biol].
• Bayesian model averaging BMA [Penny et al. 2010 PLoS Comp Biol].
Design a study thatallows to investigatethat system
Extraction ofROI time
series
Parameter estimationfor all DCMs considered
Bayesian modelselection ofoptimal DCM
Inference on thefamily, model,
connection level
Hypothesis abouta neural system
Data analysis (SPM)
Define plausibleDCMs
The DCM cycle
Theoritical reviews:
Stephan et al. (2010). Ten Simple Rules for DCM. NeuroImage
Daunizeau et al. (2010). DCM: a critical review of the biophysical and statistical foundations. NeuroImage
Friston (2009). Causal modelling and brain connectivity in fMRI. PLoS Biol
Applications: (recent examples of DCM-fMRI at the FIL)
- Word reading via the putamen:Seghier and Price (2010) Cerebral Cortex.
- Intelligible speech perception:Leff et al. (2008) J Neurosci.
- Associative learning and prediction error:den Ouden et al. (2009) Cerebral Cortex.