Dynamic Causal Modelling (DCM) for fMRI
Rosalyn Moran
Virginia Tech Carilion Research Institute
With thanks to the FIL Methods Group
for slides and images
SPM Course, UCL
May 2013
Dynamic 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 250
160 published papers on DCM
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
Principles of Organisation
Functional specialization
Functional integration
Functional vs Effective
Connectivity
Functional connectivity is defined in terms of statistical dependencies, it is an
operational concept that underlies the detection of (inference about) a functional
connection, without any commitment to how that connection was caused
- Assessing mutual information & testing for significant departures from zero
- Simple assessment: patterns of correlations
- Undirected or Directed Functional Connectivity eg. Granger Connectivity
Effective connectivity is defined at the level of hidden neuronal states generating
measurements. Effective connectivity is always directed and rests on an explicit
(parameterised) model of causal influences — usually expressed in terms of difference
(discrete time) or differential (continuous time) equations.
- Eg. DCM
- causality is inherent in the form of the model ie. fluctuations in hidden neuronal states
cause changes in others: for example, changes in postsynaptic potentials in one area
are caused by inputs from other areas.
Dynamic Causal Modeling (DCM)
Hemodynamic
forward model:
neural activityBOLD
Electromagnetic
forward model:
neural activityEEG
MEG
LFP
Neural state equation:
dx
 F ( x , u,  )
dt
fMRI
simple neuronal model
complicated forward model
EEG/MEG
complicated neuronal model
simple forward model
inputs
Deterministic DCM
x = (A + uB)x + Cu
y
y
H{2}
y = g(x, H ) + e
e ~ N (0, s )
x2
H{1}
A(2,2)
A(2,1)
C(1)
u1
A(1,2)
x1
B(1,2)
A(1,1)
u2
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
Example:
a linear model of
interacting visual
regions
x4 = a42 x2 + a43 x3 + a44 x4
x3 = a31x1 + a33 x3 + a34 x4
x3
FG
left
FG
right
x1
LG
left
LG
right
x4
x2
RVF
LVF
u2
u1
x1 = a11x1 + a12 x2 + a13 x3 + c12u2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
LG = lingual gyrus
FG = fusiform gyrus
x2 = a21x1 + a22 x2 + a24 x4 + c21u1
Example:
a linear model of
interacting visual
regions
x3
x1
FG
left
LG
left
FG
right
LG
right
x4
LG = lingual gyrus
FG = fusiform gyrus
x2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
RVF
LVF
u2
u1
x1 = a11x1 + a12 x2 + a13 x3 + c12u2
x2 = a21x1 + a22 x2 + a24 x4 + c21u1
x3 = a31 x1 + a33 x3 + a34 x4
x4 = a42 x2 + a43 x3 + a44 x4
Example:
a linear model of
interacting visual
regions
x3
x1
FG
left
LG
left
  { A, C}
LG
right
x4
LG = lingual gyrus
FG = fusiform gyrus
x2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
RVF
LVF
u2
u1
state
changes
x = Ax +Cu
FG
right
é
ê
ê
ê
ê
ê
êë
x1 ùú éê
x2 ú ê
ú=ê
x3 ú ê
ú ê
x4 úû êë
effective
connectivity
0 ùé
úê
a21 a22 0 a24 ú ê
úê
a31 0 a33 a34 ú ê
úê
0 a42 a43 a44 úû êë
a11 a12 a13
system
state
input
parameters
x1 ùú é 0
ê
x2 ú ê c21
ú+ê
x3 ú ê 0
ú ê 0
x4 úû ë
c12
0
0
0
external
inputs
ù
úé ù
úê u1 ú
úê ú
úë u2 û
ú
û
Extension:
bilinear model
x3
FG
left
FG
right
x4
m
x = (A+ åu j B ( j ) )x + Cu
j=1
x1
é
ê
ê
ê
ê
ê
êë
ì
x1 ù ïéê
ú
x2 ú ïïê
ú = íê
x3 ú ïê
ú ïê
x4 úû ïêë
î
a31
0
0 a42
LG
right
x2
RVF
CONTEXT
LVF
u2
u3
u1
0 ùú é
ê
0 a24 ú ê
ú + u3 ê
a33 a34 ú ê
ú
a43 a44 úû êë
a11 a12 a13
a21 a22
LG
left
(3)
12
0 b
0
0
0
0
0
0
0
0
0
0
ùü
0 úï
ï
0 ú ïý
(3) ú
b34 ú ï
ï
ú
0 û ïþ
é
ê
ê
ê
ê
ê
êë
x1 ù é 0
ú ê
x2 ú ê c21
ú+ê
x3 ú ê 0
ú
x4 úû êë 0
c12
0
0
0
0
0
0
0
ù
úé u1 ù
ú
úê
úê u2 ú
úê u ú
úêë 3 úû
û
Vanilla DCM:
Deterministic Bilinear DCM
driving
input
Simply a two-dimensional
taylor expansion (around x0=0, u0=0):
dx
f
f
 f
 f ( x, u)  f ( x0 ,0) 
x u
ux  ...
dt
x
u
xu
2
¶f
A=
¶x
¶f
C=
¶u
modulation
u=0
Bilinear state equation:
x=0
¶2 f
B=
¶x¶u
m
dx 

  A   ui B ( i )  x  Cu
dt 
i 1

DCM parameters = rate constants
Generic solution to the ODEs in DCM:
s
z1
dz1
dt
sz1
z1 (t )
z1 (0) exp( st ),
Decay function
A
0.10
B
If A B 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
1
0.8
0.6
0.5z1 (0)
0.4
0.2
0
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ln 2 / s
z1 (0) 1
Example:
context-dependent decay
stimuli
u1
context
u2
u1
-
+
x = Ax + u2 B (2) x + Cu1
é 2
é x ù é -1 a ù
b11 0
12 ú
ê
ê 1 ú=ê
x + u2
ê 0 b2
ê x ú ê a
ú
-1
22
ë 2 û ë 21
û
ë
-
x1
+ +
u1
u2
u2
Z1
+
x1
x2
-
Z2
-
Penny et al. 2004, NeuroImage
x2
ù é
ùé u ù
c
0
úx +ê 1
úê 1 ú
ú ê 0 0 úê u ú
ûë 2 û
û ë
bilinear DCM
non-linear DCM
modulation
driving
input
driving
input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
dx
2 f x2
f
f
2 f
 f ( x, u)  f ( x0 ,0) 
 ...
x u
ux  ... 2
dt
x 2
x
u
xu
Bilinear state equation:
m
dx 

  A   ui B ( i )  x  Cu
dt 
i 1

Nonlinear state equation:
m
n
dx 
(i )
( j) 

  A   ui B   x j D  x  Cu
dt 
i 1
j 1

Neural population activity
0.4
0.3
0.2
u2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0.6
u1
0.4
x3
0.2
0
0.3
0.2
0.1
0
x1
x2
3
fMRI signal change (%)
2
1
0
Nonlinear dynamic causal model (DCM)
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
4
3
m
n

dx 
(i )
( j)
  A   ui B   x j D  x  Cu
dt 
i 1
j 1

2
1
0
-1
3
2
1
Stephan et al. 2008, NeuroImage
0
y


y
BOLD
y

activity
x2(t)
neuronal
states
hemodynamic
model
x
integration
modulatory
input u2(t)
t
Neural state equation
endogenous
connectivity
t
λ
activity
x3(t)
activity
x1(t)
driving
input u1(t)
y
modulation of
connectivity
direct inputs
x  ( A   u j B( j ) ) x  Cu
x
x
 x

u j x
A
B( j)
x
C
u
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
Basics of DCM:
Neuronal and BOLD level
y
• Cognitive system is modelled at its underlying
neuronal level (not directly accessible for fMRI).
• The modelled neuronal dynamics (x) are
λ
transformed into area-specific BOLD signals (y) by
a hemodynamic model (λ).
• Overcoming Regional variability of the
haemodynamic response
• ie DCM not based on temporal precedence at the
measurement level
x
Basics of DCM:
Neuronal and BOLD level
y
PLoSBIOLOGY
Identifying Neural Drivers with Functional
MRI: An Electrophysiological Validation
Olivier David 1,2*, Isabelle Guillemain 1,2, Sandrine Saillet 1,2, Sebastien Reyt 1,2, Colin Deransart 1,2,
λ
x
Christoph Segebarth 1,2, Antoine Depaulis1,2
1 INSERM, U836, Grenoble Institut des Neurosciences, Grenoble, France, 2 Université Joseph Fourier, Grenoble, France
Whether functional magnetic resonance imaging (fMRI) allows the identification of neural drivers remains an open
question of particular importance to refine physiological and neuropsychological models of the brain, and/or to
understand neurophysiopathology. Here, in a rat model of absence epilepsy showing spontaneous spike-and-wave
discharges originating from the first somatosensory cortex (S1BF), we performed simultaneous electroencephalo“Connectivity analysis applied directly on
graphic (EEG) and fMRI measurements, and subsequent intracerebral EEG (iEEG) recordings in regions strongly
activated in fMRI (S1BF, thalamus,
striatum). fMRI
connectivity
was determined from fMRI time series directly and
fMRIand
signals
failed
because
from hidden state variables using a measure of Granger causality and Dynamic Causal Modelling that relates synaptic
hemodynamics
varied
between
regions,
activity to fMRI. fMRI connectivity
was compared to directed
functional
coupling estimated
from iEEG using asymmetry
in generalised synchronisation
metrics.
The
neural
driver
of
spike-and-wave
discharges
was
estimated in S1BF from
rendering termporal precedence
iEEG, and from fMRI only when hemodynamic effects were explicitly removed. Functional connectivity analysis applied
directly on fMRI signals failed
because hemodynamics
between
regions,
rendering temporal precedence
irrelevant”
….The varied
neural
driver
was
irrelevant. This paper provides the first experimental substantiation of the theoretical possibility to improve
identified
using
DCM,
where
these
effects
interregional coupling estimation
from hidden
neural states
of fMRI.
As such,
it has important
implications for future
studies on brain connectivity
using
functional
neuroimaging.
are accounted for…
Citation: David O, Guillemain I, Saillet S, Reyt S, Deransart C, et al. (2008) Identifying neural drivers with functional MRI: an electrophysiological validation. PLoS Biol 6(12):
e315. doi:10.1371/journal.pbio.0060315
Introduction
Distinguishing effer ent fr om affer ent connect ions i n
distributed networks is critical to construct formal theories
integrated neuroscience, these formal ideas have initiated a
search for neural networks using sophisticated signal analysis
techniques to estimate the connect ivity between distant
regions [4,12–18]. At the brain level, connectivity analyses
The hemodynamic model
• 6 hemodynamic
parameters:
stimulus functions
u
t
activity
x(t )
  { ,  , ,  ,  ,  }
h
neural state equation
vasodilatory signal
important for model fitting, but
of no interest for statistical
inference
s = x - k s - g ( f -1)
f
s
s
hemodynamic state
equations
flow induction (rCBF)
f  s
f
• Computed separately for
each area  region-specific
HRFs!
changes in volume
τv  f  v1 /α
v
v
changes in dHb
τq  f E ( f,E0 ) qE0  v1/α q/v
q
BOLD signal
Friston et al. 2000, NeuroImage
Stephan et al. 2007, NeuroImage
y(t) = l ( v, q)
Estimated BOLD
response
The hemodynamic model
• 6 hemodynamic
parameters:
stimulus functions
u
t
activity
x(t )
  { ,  , ,  ,  ,  }
h
neural state equation
vasodilatory signal
important for model fitting, but
of no interest for statistical
inference
s = x - k s - g ( f -1)
f
s
s
hemodynamic state
equations
flow induction (rCBF)
f  s
f
• Computed separately for
each area  region-specific
HRFs!
changes in volume
τv  f  v1 /α
v
v
changes in dHb
τq  f E ( f,E0 ) qE0  v1/α q/v
q
BOLD signal
Friston et al. 2000, NeuroImage
Stephan et al. 2007, NeuroImage
y(t) = l ( v, q)
Estimated BOLD
response
How interdependent are neural and hemodynamic
parameter estimates?
1
A
0.8
5
0.6
10
B
0.4
15
C
0.2
20
0
25
-0.2
h
ε
30
-0.4
35
-0.6
-0.8
40
5
10
15
20
25
30
35
40
-1
Stephan et al. 2007, NeuroImage
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
DCM is a Bayesian approach
new data
prior knowledge
p( y |  )
p ( )
p( | y )  p( y |  ) p( )
posterior
 likelihood
∙ prior
Bayes 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.
stimulus function u
Overview:
parameter estimation
•
•
•
•
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 inversion:
parameter estimation by
means of variational EM
under Laplace approximation
Result:
Gaussian a posteriori
parameter distributions,
characterised by
mean ηθ|y and
covariance Cθ|y.
neural state
equation
x  ( A   u j B j ) x  Cu
activity - dependent vasodilatory signal
s  z  s  γ( f  1)
s
s
f
parameters
flow - induction (rCBF)
hidden states
z ={x,s, f ,v,q}
state equation
z = F(x,u,q )
f  s
 h  { ,  , ,  , }
f
 n  { A, B1...B m , C}
  { h , n }
changes in volume
τv  f  v1 /α
v
changes in dHb
τq  f E ( f, ) q  v1/α q/v
q
v
ηθ|y
y   (x )
y  h(u ,  )  X  e
modelled
BOLD response
observation model
VB in a nutshell (mean-field approximation)
 Neg. free-energy
approx. to model
evidence.
 Mean field approx.
 Maximise neg. free
energy wrt. q =
minimise divergence,
by maximising
variational energies
ln p  y | m   F  KL  q  ,   , p  ,  | y  
F  ln p  y,  ,   q  KL  q  ,   , p  ,  | m  
p  ,  | y   q  ,    q   q   
q    exp  I   exp  ln p  y,  ,  

q ( )
q     exp  I    exp  ln p  y,  ,  


q ( ) 
 Iterative updating of sufficient statistics of approx. posteriors by
gradient ascent.


Bayesian Inversion
Specify generative forward model
(with prior distributions of parameters)
Regional responses
Variational Expectation-Maximization algorithm
Iterative procedure:
1.
2.
3.
Compute model response using
current set of parameters
Compare model response with data
Improve parameters, if possible
1. Posterior distributions of parameters
p ( | y , m)
2. Model evidence
p ( y | m)
Inference about DCM parameters:
Bayesian single-subject analysis
• Gaussian assumptions about the posterior distributions of the
parameters
• posterior probability that a certain parameter (or contrast of
parameters) is above a chosen threshold γ:
• By default, γ is chosen as zero – the prior ("does the effect exist?").
Inference about DCM parameters:
Bayesian parameter averaging (FFX group analysis)
Under Gaussian assumptions this is
easy to compute:
Likelihood distributions from different
subjects are independent
individual
posterior
covariances
group
posterior
covariance
N
C|1y1 ,..., y N   C|1yi
i 1
 | y ,..., y
1
group
posterior
mean
N
 N 1

   C | yi | yi C | y1 ,..., y N
 i 1

individual posterior
covariances and means
Inference 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 subject
Selection of 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
Inference about Model Architecture, Bayesian Model
Selection
Model evidence:
p( y | mi )
Approximation: Free Energy
F  ln p( y | mi )  KL[q( ), p( | G,  )]
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
Fixed Effects Model selection via
Random Effects Model selection
log Group Bayes factor:
via Model probability:
BF1, 2   ln p( y m1 )  ln p( y m2 )
k
k
p (r | y,  )
rk
q
  k ( 1     K )
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
Bayesian Model Selection
DCM – Attention to Motion
Results
Paradigm
4 conditions
- fixation only
- observe static dots
- observe moving dots
- attend to moving dots
baseline
SPC
+ photic
V3A
+ motion
V5+
Attention – No attention
Büchel & Friston 1997, Cereb. Cortex
Büchel et al. 1998, Brain
What connection in the network
mediates attention ?
Bayesian Model Selection
m1
m2
Modulation
By attention
Modulation
By attention
PPC
External
stim
V1
m3
V5
m4
Modulation
By attention
PPC
stim
V1
V5
PPC
stim
V1
V5
ln p y m
0.10
PPC
0.26
V1
0.39
0.26
1.25
stim
PPC
stim
V1
V5
estimated
effective synaptic strengths
for best model (m4)
attention
models marginal likelihood
0.13
0.46
[Stephan et al., Neuroimage, 2008]
Modulation
By attention
V5
Parameter Inference
attention
MAP = 1.25
0.10
0.8
0.7
PPC
0.6
0.26
0.5
0.39
1.25
stim
0.26
V1
0.13
0.46
0.50
V5
0.4
0.3
0.2
0.1
0
-2
motion
Stephan et al. 2008, NeuroImage
-1
0
1
2
3
4
p( DVPPC
5,V 1  0 | y )  99.1%
5
Data Fits
motion &
attention
static
motion &
no attention dots
V1
V5
PPC
observed
fitted
Overview
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Parameter estimation, priors & inference
• Applications of DCM to fMRI data
- Attention to Motion
- The Status Quo Bias
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
Difficulty
High
Low
Decision
Accept
Reject
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
Difficulty
High
Low
Decision
Accept
Reject
Main effect of difficulty
in medial frontal and right inferior frontal cortex
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
Difficulty
High
Low
Decision
Accept
Reject
Interaction of decision and difficulty
in region of subthalamic nucleus:
Greater activity in STN when default is rejected in difficult trials
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
DCM: “aim was to establish a possible mechanistic explanation for the interaction
effect seen in the STN.
Whether rejecting the default option is reflected in a modulation of connection
strength from rIFC to STN, from MFC to STN, or both “…
MFC
rIFC
STN
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
Difficulty
Difficulty
MFC
rIFC
Reject
Difficulty
MFC
rIFC
rIFC
Reject
Reject
STN
STN
STN
Difficulty
Difficulty
Difficulty
MFC
MFC
STN
Reject
rIFC
Reject
STN
Difficulty
rIFC
Reject
STN
Reject
Reject
Difficulty
Difficulty
MFC
MFC
Difficulty
MFC
rIFC
rIFC
STN
Difficulty
MFC
rIFC
Reject
STN
Reject
Difficulty
MFC
rIFC
Reject
STN
Reject
Example:
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
Difficulty
Difficulty
MFC
rIFC
Reject
Difficulty
MFC
Reject
STN
STN
Difficulty
Difficulty
Difficulty
MFC
MFC
STN
Reject
rIFC
Reject
STN
Difficulty
rIFC
Reject
STN
Reject
Reject
Difficulty
Difficulty
MFC
MFC
Difficulty
MFC
rIFC
rIFC
STN
rIFC
rIFC
Reject
STN
Difficulty
MFC
rIFC
Reject
STN
Reject
Difficulty
MFC
rIFC
Reject
STN
Reject
Overcoming status quo bias in the human brain
Fleming et al PNAS 2010
The summary statistic approach
Effects across subjects consistently greater than zero
P < 0.01 *
P < 0.001 **
The evolution of DCM in SPM
• DCM is not one specific model, but a framework for Bayesian inversion of
dynamic system models
• The default implementation in SPM is evolving over time
– better numerical routines for inversion
– change 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.
GLM vs. DCM
DCM 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.
Thank you