Transcript Effective connectivity & Dynamic Causal Modelling (DCM) Meike J. Grol •Leiden Institute for Brain and Cognition (LIBC), Leiden, The Netherlands •Leiden University - Institute for Psychological Research.

Effective connectivity
& Dynamic Causal Modelling
(DCM)
Meike J. Grol
•Leiden Institute for Brain and Cognition (LIBC),
Leiden, The Netherlands
•Leiden University - Institute for Psychological
Research (LU-IPR), Leiden, The Netherlands
•Department of Radiology, Leiden University
Medical Center
•F. C. Donders Centre for Cognitive NeuroImaging,
Nijmegen, The Netherlands
Zürich SPM Course
February 29, 2008
Thanks to Klaas Enno Stephan and
Miranda van Turennout for slides
Functional Segregation
Where are the regional responses
to an experimental input?
Functional Integration
How do regions influence each other?
?
?
System analyses in functional
neuroimaging
Functional specialisation
Functional integration
Analyses of regionally specific effects:
which areas constitute a neuronal
system?
Analyses of inter-regional effects:
what are the interactions between the
elements of a given neuronal system?
Functional connectivity
= the temporal correlation between
spatially remote neurophysiological
events
MECHANISM-FREE
Effective connectivity
= the influence that the elements of a
neuronal system exert over another
MECHANISTIC MODEL
Functional connectivity: methods
• Seed-voxel correlation analyses
• Eigenimage analysis
– Principal Components Analysis (PCA)
– Singular Value Decomposition (SVD)
– Partial Least Squares (PLS)
• Independent Component Analysis (ICA)
Seed-voxel correlation analysis
(SVCA)
seed voxel
– hypothesis-driven
choice of a ‘seed voxel’
→ reference timeseries
– correlation with
timeseries of all other
voxels in the brain
SVCA example
Finger-tapping task:
all voxels (p<0.005,
uncorrected) that
showed changed
functional
connectivity with the
left ant. cerebellum
in schizophrenic
patients after
medication with
olanzapine
Stephan et al., Psychol. Med. (2001)
p<0.005, uncorrected
Functional Connectivity Analyses
• Pros:
– useful when we have no model of what caused the
data (e.g. sleep, hallucinatons, etc.)
• Cons:
– usually suboptimal for situations where we have a
priori knowledge and experimental control about the
system of interest
– resulting modes/patterns can be difficult to interpret
– no mechanistic insight into the neural system of
interest
models of effective connectivity necessary
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)
Psycho-physiological interaction
(PPI)
• bilinear model of how the influence of area A
on area B changes by the psychological
context C: A x C  B
• a PPI corresponds to differences in
regression slopes for different contexts.
PPI example:
attentional modulation
of V1→V5
V5 activity
Attention
V1
time
V5
Friston et al. 1997, NeuroImage 6:218-229
Büchel & Friston 1997, Cereb. Cortex 7:768-778
V5 activity
V5
=
V1 x Att.
SPM{Z}
attention
no attention
V1 activity
Pros & Cons of PPIs
• Pros:
– given a single source region, we can test for its contextdependent connectivity across the entire brain
• Cons:
– very simplistic model:
only allows to model contributions from a single area
– ignores time-series properties of data
– operates at the level of BOLD time series
limited causal interpretability in neural terms,
more powerful models needed
DCM!
Dynamic Causal Modelling
• Basic idea
• Neural level in DCM
• Haemodynamic level in DCM
• Priors and Parameter estimation
• Interpretation and inference of parameters
• Practical steps of a DCM study
The aim of DCM
Functional integration and the modulation of specific pathways
Contextual inputs
Stimulus-free - u2(t)
{e.g. cognitive set/time}
BA39
Perturbing inputs
Stimuli-bound u1(t)
{e.g. visual words}
y
STG
V4
y
BA37
y
V1
y
y
DCM for fMRI: the basic idea
• 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).
The aim of DCM is to estimate parameters at
the neuronal level (computed separately for each
area) such that the modelled BOLD signals are
maximally similar to the experimentally
measured BOLD signals.
z
λ
y
Dynamic Causal Modelling
• Basic idea
• Neural level in DCM
• Haemodynamic level in DCM
• Priors and Parameter estimation
• Interpretation and inference of parameters
• Practical steps of a DCM study
What is a system?
Input u(t)
System =
a set of elements which
interact in a spatially and
temporally specific fashion
State changes of a system are
dependent on:
– the current state
– external inputs
– its connectivity
– time constants & delays
connectivity parameters 
system
state z(t)
dz
 F ( z, u, )
dt
Example:
linear
dynamic
system
z3
z1
RVF
u2
FG
left
LG
left
FG
right
LG
right
z4
LG = lingual gyrus
FG = fusiform gyrus
z2
Visual input in the
- left (LVF)
- right (RVF)
visual field.
LVF
u1
Extension:
bilinear
dynamic
system
z3
FG
left
FG
right
z4
z1
LG
left
LG
right
z2
RVF
u2
Intrinsic connectivity
CONTEXT
u3
LVF
u1
Modulation of connectivity
Direct inputs
Dynamic Causal Modelling
• Basic idea
• Neural level in DCM
• Hemodynamic level in DCM
• Priors and Parameter estimation
• Interpretation and inference of parameters
• Practical steps of a DCM study
The hemodynamic “Balloon” model
• 5 hemodynamic
parameters:
activity
z (t )
  { ,  , , , }
h
vasodilato
ry signal
s  z  s  γ( f  1)
important for model fitting,
but of no interest for
statistical inference
• Empirically determined
a priori distributions.
• Computed separately for
each area (like the neural
parameters)
 region-specific HRFs!
s
f
flow induction
f  s
f
changes involume
τv  f  v
1 /α
v
changes indHb
τq  f E ( f, ) q  v1/α q/v
q
v
BOLD signal
y (t )   v, q 
Friston et al. 2000,
NeuroImage
Example: modelled BOLD signal
RVF
FG
left
FG
right
LG
left
LG
right
LVF
black: observed BOLD signal
red:
modelled BOLD signal
Dynamic Causal Modelling
• Basic idea
• Neural level in DCM
• Haemodynamic level in DCM
• Priors & Parameter estimation
• Interpretation and inference of parameters
• Practical steps of a DCM study
Bayesian estimation
Models of
Constraints on
•Responses in a single region
•Neuronal interactions
•Connections
•Biophysical parameters
p( y |  )
p( )
p( | y)  p( y |  ) p( )
Bayesian estimation
posterior
 likelihood
∙ prior
Priors in DCM
• needed for Bayesian estimation,
embody constraints on
parameter estimation
• express our prior knowledge or
“belief” about parameters of the
model
• hemodynamic parameters:
empirical priors
• Coupling parameters of selfconnections:
principled prior
• coupling parameters other
connections:
shrinkage priors
Bayes Theorem
p( | y)  p( y |  )  p( )
posterior
 likelihood
∙ prior
Shrinkage Priors
Small & variable effect
Large & variable effect
Small but clear effect
Large & clear effect
Dynamic Causal Modelling
• Basic idea
• Neural level in DCM
• Haemodynamic level in DCM
• Priors & Parameter estimation
• Interpretation and inference about parameters
• Practical steps of a DCM study
DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
dz
 az
dt
z(t )  z0 exp(at)
Coupling parameter a is inversely
proportional to the half life  of z(t):
z ( )  0.5 z0
The coupling parameter a
thus describes the speed of
the exponential change in z(t)
0.5 z0
 z0 exp(a )
a  ln 2 / 
  ln 2 / a
Example:
context-dependent decay
stimuli
context
u1
-
+
-
Z1
+
+
u2
u1
u1
u2
u2
Z1
z
Z2 1
z2
Z2
-
z  Az  u2 B 2 z  Cu1
-
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)
2


z

a
12
 1 

b
11

z

u
2
 z  a 21  

 2 
 0
0  c1 0  u1 
2 z  
 u 
0
0
b22    2 
Inference about DCM parameters:
Bayesian single-subject analysis
• Bayesian parameter estimation in DCM: Gaussian assumptions
about the posterior distributions of the parameters
• 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:
 cT  
 y

p  N 
 cT C y c






ηθ|y
• 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
Inference about DCM parameters:
group analyses
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.
“Random effects” analyses:
2nd level analyses can be applied to DCM parameters.
Separate fitting of identical models for each subject
Selection of bilinear parameters of interest
one-sample t-test:
parameter > 0 ?
paired t-test:
RmANOVA: e.g. in case of
parameter 1 > parameter 2 ?
multiple sessions per subject
Dynamic Causal Modelling
• Basic idea
• Neural level in DCM
• Haemodynamic level in DCM
• Priors & Parameter estimation
• Interpretation and inference about parameters
• Practical steps of a DCM study
Planning a DCM-compatible study
• Suitable experimental design:
– 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?
–Ensure that intended model is suitable to test this hypothesis → simulations
–Define criteria for inference
–What are the alternative models to test?
Multifactorial design:
explaining interactions with DCM
Stim2/
Task A
Task A
Task B
Stim 1
Stim1/
Task A
TA/S1
TB/S1
X1
X2
Stim 2
Stimulus factor
Task factor
TA/S2
TB/S2
Stim 1/
Task B
Stim 2/
Task B
X1
X2
Let’s assume that an SPM analysis
shows a main effect of stimulus in
X1 and a stimulus  task interaction
in X2.
How do we model this using DCM?
GLM
Stim1
Stim2
Task A
Task B
DCM
Simulated data
A1
+++
Stim1
+
A1
Stim2
+
+++
+++
Task A
Stim 1
Task A
A2
+
Task B
A2
Stim 2
Task A
Stim 1
Task B
Stim 2
Task B