### Dynamic Causal Modelling Workshop SPM Course Hamburg 2009

Dynamic Causal Modelling
THEORY
Hanneke den Ouden
Donders Centre for Cognitive Neuroimaging
Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London
SPM Course FIL, London
22-24 October 2009
Principles of Organisation
Functional specialization
Functional integration
Overview
• Brain connectivity
• Dynamic causal models (DCMs)
– Basics
– Neural model
– Hemodynamic model
– Parameters & parameter estimation
– Inference & Model comparison
• Recent extentions to DCM
• Planning a DCM compatible study
Structural, functional & effective connectivity
• anatomical/structural connectivity
= presence of axonal connections
Sporns 2007, Scholarpedia
• functional connectivity
=
statistical dependencies between regional time series
• effective connectivity
=
causal (directed) influences between neurons or neuronal
populations
For understanding brain function
mechanistically, we can use DCM to create
models of causal interactions among
neuronal populations
to explain regional effects in terms of
interregional connectivity
Overview
• Brain connectivity
• Dynamic causal models (DCMs)
– Basics
– Neural model
– Hemodynamic model
– Parameters & parameter estimation
– Inference & Model comparison
• Recent extentions to DCM
• Planning a DCM compatible study
Basics of DCM:
Neuronal and BOLD level
• Cognitive system is modelled at its underlying
neuronal level (not directly accessible for
fMRI).
• The modelled neuronal dynamics (x) are
transformed into area-specific BOLD signals
(y) by a hemodynamic model (λ).
The aim of DCM is to estimate
parameters at the neuronal level such
that the modelled and measured
BOLD signals are optimally similar.
x
λ
y
DCM: Linear Model
u1
x2
x1
x  Ax  Cu
x3
  A , C 
x1  a 11 x1  a 12 x 2  c1u 1
x 2  a 21 x1  a 22 x 2  a 23 x 3
x 3  a 32 x 2  a 33 x 3
state
changes
effective
connectivity
 x1   a 11
  
x  a
 2   21
 x 3   0
a 12
a 22
a 31
system
state
input
external
parameters inputs
0   x1   c11
  
a 23 x 2  0
  
a 33   x 3   0
0
0
0
0   u1 
 
0 u2
 
0   u 3 
DCM: Bilinear Model
Neural State Equation
u1
X1
X2
u2
x1  a 11 x1  a 12 x 2  c1u 1

X3
m
u
j
B
( j)
j 1

 x  Cu


  A , B , C 
u3


x   A 




(2)
(3)
x 2  a 21  u 2 b 21 x1  a 22 x 2  a 23  u 3 b 23 x 3
x 3  a 32 x 2  a 33 x 3
state
changes
fixed effective
connectivity
 x1    a11
  
x  a
 2    21
 x 3  
  0
a 12
a 22
a 31
0 
 0

 (2)
a 23  u 2 b 21


 0
a 33 
modulatory effective
connectivity
0
0
0
0
0


0  u3 0


 0
0 
0
0
0
system
input
external
state parameters inputs
0    x1   c11
 
(3)   
b 23  x 2  0
   
0  
  x 3   0
0
0
0
0   u1 
 
0 u2
 
0   u 3 
Basics of DCM:
Neuronal and BOLD level
• Cognitive system is modelled at its underlying
neuronal level (not directly accessible for
fMRI).
• The modelled neuronal dynamics (x) are
transformed into area-specific BOLD signals
(y) by a hemodynamic model (λ).
x
λ
y
The hemodynamic model
• 6 hemodynamic
parameters:
stimulus functions
u
t
activity
  { ,  ,  ,  ,  ,  }
h
important for model fitting,
but of no interest for
statistical inference
neural state equation
x (t )
vasodilato ry signal
s  x   s  γ ( f  1)
f
s
s
hemodynamic
state equations
flow induc tion (rCBF)
f  s
f
• Computed separately for
each area (like the
neural parameters)
 region-specific HRFs!
changes in volume
τ v  f  v
1 /α
v
changes in dHb
τ q  f E ( f,E 0 ) qE 0  v q/v
q
v
BOLD signal
Friston et al. 2000, NeuroImage
Stephan et al. 2007, NeuroImage
1 /α
y (t )   v, q 
Estimated BOLD
response
Measured vs Modelled BOLD signal
Recap
The aim of DCM is to estimate
- neural parameters {A, B, C}
- hemodynamic parameters
such that the modelled (x) and measured (y) BOLD signals are maximally
similar.
hemodynamic
model
x
λ
u1
X1
X2
X3
y
u2
u3
Overview
• Brain connectivity
• Dynamic causal models (DCMs)
– Basics
– Neural model
– Hemodynamic model
– Parameters & parameter estimation
– Inference & Model comparison
• Recent extentions to DCM
• Planning a DCM compatible study
DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
dx
dt
 ax
x ( t )  x 0 exp( at )
The coupling parameter a
determines the half life of x(t), and
thus describes the speed of the
exponential change
0.5 x 0
  ln 2 / a
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
Example: context-dependent decay
u1
stimuli
u1
context
u2
+
-
x1
+
u1
u2
u2
Z1
x1
Z2
+
x2
-
x2
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)
Estimation: Bayesian framework
Models of
Constraints on
• Haemodynamics in a single region
• Neuronal interactions
• Haemodynamic parameters
• Connections
likelihood
p( y |  )
posterior
prior
p ( | y )  p ( y |  ) p ( )
Bayesian estimation
p ( )
Conceptual overview
Neuronal states
Driving input
Modulatory input
(e.g. context/learning/drugs) (e.g. sensory stim)
b12
activity
x1(t)
y
Parameters are optimised
c1
c2
so that the predicted
matches the measured
a12
activity
x2(t)
y
BOLD Response
BOLD response
 But how confident are
we in what these
parameters tell us?
Overview
• Brain connectivity
• Dynamic causal models (DCMs)
– Basics
– Neural model
– Hemodynamic model
– Parameters & parameter estimation
– Inference & Model comparison
• Recent extentions to DCM
• Planning a DCM compatible study
Model comparison and selection
Given competing
hypotheses, which model is
the best?
log p( y | m)  accuracy (m) 
complexity(m)
Bij 
p( y | m  i)
p( y | m  j )
Pitt & Miyung (2002) TICS
Bayesian single subject analysis
• The model parameters are
distributions that have a mean
ηθ|y and covariance Cθ|y.
– Use of the cumulative normal
distribution to test the
probability that a certain
parameter is above a chosen
threshold γ:

ηθ|
y
Classical frequentist test across
Ss
• Test summary statistic: mean
ηθ|y
– One-sample t-test:
Parameter > 0?
– Paired t-test:
parameter 1 > parameter
2?
– rmANOVA: e.g. in case of
multiple sessions per subject
Neuronal
dynamics
Haemodynamics
State space
Model
Posterior densities
of parameters
Priors
Bayesian Model
inversion
fMRI data
Model
comparison
Overview
• Brain connectivity
• Dynamic causal models (DCMs)
– Basics
– Neural model
– Hemodynamic model
– Parameters & parameter estimation
– Inference & Model comparison
• Recent extentions to DCM
• Planning a DCM compatible study
Extensions to DCM
• Ext. 1: two state model
• Ext. 2: Nonlinear DCM
– excitatory & inhibitory
– Gating of connections by
other areas
u2
Two-state DCM
u1
E
x1
IE
IE
exp( A11
 uB11
)
E ,I
x1
I
x1
Nonlinear state equation
exp( Aij  uB ij )

A
dt 
dx
m
uB
i
i 1
n
(i)

x
j 1
j
D
( j)

 x  Cu


Planning a DCM-compatible study
• Suitable experimental design:
– any design that is suitable for a GLM
– preferably multi-factorial (e.g. 2 x 2)
• e.g. one factor that varies the driving (sensory) input
• and one factor that varies the contextual input
• Hypothesis and model:
– Define specific a priori hypothesis
– Which parameters are relevant to test this hypothesis?
– If you want to verify that intended model is suitable to test this
hypothesis, then use simulations
– Define criteria for inference
– What are the alternative models to test?
So, DCM….
•
enables one to infer hidden neuronal processes from fMRI data
•
tries to model the same phenomena as a GLM
– explaining experimentally controlled variance in local responses
– based on connectivity and its modulation
•
allows one to test mechanistic hypotheses about observed effects
•
is informed by anatomical and physiological principles.
•
uses a Bayesian framework to estimate model parameters
•
is a generic approach to modeling experimentally perturbed dynamic
systems.
– provides an observation model for neuroimaging data, e.g. fMRI, M/EEG
– DCM is not model or modality specific (Models will change and the method
extended to other modalities e.g. ERPs)
Some useful references
• The first DCM paper: Dynamic Causal Modelling (2003). Friston et
al. NeuroImage 19:1273-1302.
• Physiological validation of DCM for fMRI: Identifying neural drivers
with functional MRI: an electrophysiological validation (2008). David et
al. PLoS Biol. 6 2683–2697
• Hemodynamic model: Comparing hemodynamic models with DCM
(2007). Stephan et al. NeuroImage 38:387-401
• Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008).
Stephan et al. NeuroImage 42:649-662
• Two-state model: Dynamic causal modelling for fMRI: A two-state
model (2008). Marreiros et al. NeuroImage 39:269-278
• Group Bayesian model comparison: Bayesian model selection for
group studies (2009). Stephan et al. NeuroImage 46:1004-10174
• Watch out for: 10 Simple Rules for DCM, Stephan et al (in prep).
Time to do a DCM!
Dynamic Causal Modelling
PRACTICAL
Andre Marreiros
Hanneke den Ouden
Donders Centre for Cognitive Neuroimaging
Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London
SPM Course FIL, London
22-24 October 2009
Attention to Motion in the visual system
DCM – Attention
to
Motion
Stimuli 250 radially moving dots at 4.7 degrees/s
Pre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)
Scanning (no speed changes)
F A F N F A F N S ….
F - fixation
S - observe static dots
N - observe moving dots
A - attend moving dots
Parameters
- blocks of 10 scans
- 360 scans total
- TR = 3.22 seconds
+ photic
+ motion
+ attention
Attention to Motion in the visual system
Results
SPC
V3A
V5+
Attention – No attention
Büchel & Friston 1997, Cereb. Cortex
Büchel et al. 1998, Brain
- fixation only
- observe static dots
- observe moving dots
+ photic
+ motion
+ attention
 V1
 V5
 V5 + parietal cortex
DCM: comparison of 2 models
Model 1
attentional modulation
of V1→V5: forward
Photic
SPC
V1
Model 2
attentional modulation
of SPC→V5: backward
Photic
Attention
V1
V5
Motion
Attention
SPC
Motion
V5
Bayesian model selection: Which model is optimal?
Attention to Motion in the visual system
Ingredients for a DCM
Specific hypothesis/question
Model:
based on hypothesis
Timeseries:
from the SPM
Inputs:
from design matrix
Model 1
Model 2
attentional modulation
of V1→V5: forward
Photic
SPC
attentional modulation
of SPC→V5: backward
Photic
V1
Attention
SPC
V1
V5
Motion
Attention
V5
Motion
Attention to Motion in the visual system
DCM – GUI basic steps
1 – Extract the time series (from all regions of interest)
2 – Specify the model
3 – Estimate the model
4 – Review the estimated model
5 – Repeat steps 2 and 3 for all models in model space
6 – Compare models