Transcript Effective connectivity & Basics of DCM

Models of Effective Connectivity &
Dynamic Causal Modelling
Hanneke den Ouden
Wellcome Trust Centre for Neuroimaging,
University College London, UK
Donders Institute for Brain, Cognition and
Behaviour, Nijmegen, the Netherlands
SPM course
Thanks to Klaas Stephan and Meike Grol for slides
Zurich, February 2009
Systems analysis in functional neuroimaging
Functional specialisation:
Functional integration:
What regions respond to a particular
experimental input?
How do regions influence each other?
 Brain Connectivity
?
?
Overview
• Brain connectivity: types & definitions
– anatomical connectivity
– functional connectivity
– effective connectivity
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
• Applications of DCM to fMRI data
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
Anatomical connectivity
• presence of axonal
connections
• neuronal communication
via synaptic contacts
• visualisation by
– tracing techniques
– diffusion tensor imaging
However,
knowing anatomical connectivity is not enough...
• Connections are recruited in a context-dependent fashion:
– Local functions depend on network activity
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However,
knowing anatomical connectivity is not enough...
• Connections are recruited in a context-dependent fashion:
– Local functions depend on network activity
• Connections show plasticity
– Synaptic plasticity =
change in the structure and transmission
properties of a synapse
– Critical for learning
– Can occur both rapidly and slowly
Need to look at functional and effective connectivity
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
• Applications of DCM to fMRI data
Different approaches to analysing functional
connectivity
Definition: statistical dependencies between regional time series
• Seed voxel correlation analysis
• Eigen-decomposition (PCA, SVD)
• Independent component analysis (ICA)
• any other technique describing statistical dependencies
amongst regional time series
Seed-voxel correlation analyses
• Very simple idea:
– hypothesis-driven choice of
a seed voxel
→ extract reference
time series
– voxel-wise correlation with
time series from all other
voxels in the brain
seed voxel
SVCA example:
Task-induced changes in functional connectivity
2 bimanual finger-tapping tasks:
During task that required more
bimanual coordination, SMA,
PPC, M1 and PM showed
increased functional connectivity
(p<0.001) with left M1
 No difference in SPMs!
Sun et al. 2003, Neuroimage
Does functional connectivity not simply
correspond to co-activation in SPMs?
No, it does not - see the
fictitious example on the
right:
regional
response A1
task T
regional response A2
Here both areas A1 and A2
are correlated identically to
task T, yet they have zero
correlation among
themselves:
r(A1,T) = r(A2,T) = 0.71
but
r(A1,A2) = 0 !
Stephan 2004, J. Anat.
Pros & Cons of functional connectivity analyses
• Pros:
– useful when we have no experimental control over
the system of interest and no model of what caused
the data (e.g. sleep, hallucinatons, etc.)
• Cons:
– interpretation of resulting patterns is difficult / arbitrary
– no mechanistic insight into the neural system of
interest
– usually suboptimal for situations where we have a
priori knowledge and experimental control about the
system of interest
For understanding brain function mechanistically,
we need models of effective connectivity, i.e.
models of causal interactions among neuronal
populations
to explain regional effects in terms of interregional
connectivity
Some models for computing effective connectivity
from fMRI data
• Structural 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. 1997
• Volterra kernels
Friston & Büchel 2000
• Time series models (e.g. MAR, Granger causality)
Harrison et al. 2003, Goebel et al. 2003
• Dynamic Causal Modelling (DCM)
bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
• Applications of DCM to fMRI data
Psycho-physiological interaction (PPI)
• bilinear model of how the influence of area A on
area B changes by the psychological context C:
AxCB
• a PPI corresponds to differences in regression
slopes for different contexts.
Psycho-physiological interaction (PPI)
Stim 1
Stim 2
Stimulus factor
Task factor
Task A
Task B
TA/S1
TB/S1
TA/S2
TB/S2
We can replace one main effect in
the GLM by the time series of an
area that shows this main effect.
Let's replace the main effect of
stimulus type by the time series of
area V1:
Friston et al. 1997, NeuroImage
GLM of a 2x2 factorial design:
y  (TA  TB )  1
main effect
of task
 ( S1  S 2 ) β 2
main effect
of stim. type
 (TA  TB ) ( S1  S 2 ) β 3
interaction
e
y  (TA  TB )  1
 V 1β 2
 (TA  TB ) V 1β 3
e
main effect
of task
V1 time series
 main effect
of stim. type
psychophysiological
interaction
Example PPI: Attentional modulation of V1→V5
Attention
V1
V5
V5 activity
SPM{Z}
time
V1 x Att.
Friston et al. 1997, NeuroImage
Büchel & Friston 1997, Cereb. Cortex
V5
V5 activity
=
attention
no attention
V1 activity
PPI: interpretation
y  (TA  TB )  1
 V 1β 2
 (TA  TB ) V 1β 3
Two possible
interpretations of
the PPI term:
e
attention
V1
attention
V5
Modulation of V1V5 by
attention
V1
V5
Modulation of the impact of attention on V5
by V1
Pros & Cons of PPIs
• Pros:
– given a single source region, we can test for its context-dependent
connectivity across the entire brain
– easy to implement
• 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
sometimes very useful, but limited causal interpretability;
in most cases, we need more powerful models
DCM!
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Priors & Parameter estimation
• Applications of DCM to fMRI data
Basic idea of DCM for fMRI
(Friston et al. 2003, NeuroImage)
• Investigate functional integration & modulation of specific cortical
pathways
• Using a bilinear state equation, a cognitive system is modelled at its
underlying neuronal level (which is not directly accessible for fMRI).
• The modelled neuronal dynamics (x) is transformed into area-
x
λ
specific BOLD signals (y) by a hemodynamic forward model (λ).
y
The aim of DCM is to estimate parameters at the
neuronal level such that the modelled and measured
BOLD signals are maximally similar.
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Priors & Parameter estimation
• Applications of DCM to fMRI data
Example:
a linear system
of dynamics in
visual cortex
x3
x1
RVF
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.
LVF
u2
u1
x1  a11 x1  a12 x2  a13 x3  c12u2
x2  a21 x1  a22 x2  a24 x4  c21u1
x3  a31 x1  a33 x3  a34 x4
x4  a42 x2  a43 x3  a44 x4
Example:
a linear system
of dynamics in
visual cortex
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
Az  Cu
FG
right
effective
connectivity
system
state
input
parameters
external
inputs
 x1   a11 a12 a13 0   x1   0 c12 
  x  c
 x  a a
 u
0
a
0
 1
24   2 
21
 2    21 22



 x3   a31 0 a33 a34   x3   0 0  u2 
   
  

0
a
a
a
x
x
0
0
42
43
44   4 


 4 
Extension:
bilinear
dynamic
system
x3
FG
left
FG
right
x4
m
x  ( A   u j B ( j ) ) x  Cu
j 1
x1
LG
left
LG
right
x2
RVF
CONTEXT
LVF
u2
u3
u1
0 b12(3)
 x1    a11 a12 a13 0 


 x   a a
0
a
0 0
24 

 2     21 22
 u3
0 0
 x3    a31 0 a33 a34 


  
0
a
a
a
x
42
43
44 
 4  
0 0
0 

0 0 
0 b34(3) 

0 0 
0







 x1   0 c12
 x  c
0
 2    21
 x3   0 0
  
 x4   0 0
0
 u1 

0  
 u2
0  
  u3 
0
y
y
BOLD
y
λ
activity
x2(t)
neuronal
states
t
Neural state equation
endogenous
connectivity
modulation of
connectivity
direct inputs
Stephan & Friston (2007),
Handbook of Brain Connectivity
x
integration
modulatory
input u2(t)
t
hemodynamic
model
activity
x3(t)
activity
x1(t)
driving
input u1(t)
y
x  ( A  u j B( j ) ) x  Cu
x
x
 x

u j x
A
B( j)
C
x
u
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Priors & Parameter estimation
• Applications of DCM to fMRI data
The hemodynamic model in DCM
• 6 hemodynamic
parameters:
stimulus functions
u
t
activity
x (t )
 h  { ,  , , ,  ,  }
neural state equation
vasodilato
ry signal
important for model fitting, but
of no interest for statistical
inference
s  x  s  γ( f  1)
f
s
s
hemodynamic state
equations
flow induction(rCBF)
f  s
f
• Computed separately for
each area (like the neural
parameters)
 region-specific HRFs!
Balloon model
changes involume
τv  f  v1/α
v
changes indHb
τq  f E ( f,E0 ) qE0  v1/α q/v
q
v
BOLD signal
Friston et al. 2000, NeuroImage
Stephan et al. 2007, NeuroImage
y (t )   v, q 
Estimated BOLD
response
Example: modelled BOLD signal
RVF
FG
left
FG
right
LG
left
LG
right
LVF
black: observed BOLD signal
red:
modelled BOLD signal
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
– Basic idea
– Neural level
– Hemodynamic level
– Priors & Parameter estimation
• Applications of DCM to fMRI data
Bayesian statistics
new data
prior knowledge
p( y |  )
p ( )
p( | y)  p( y |  ) p( )
posterior
 likelihood
∙ prior
Bayes theorem allows us to express
our prior knowledge or “belief” about
parameters of the model
The posterior probability of the
parameters given the data is an
optimal combination of prior knowledge
and new data, weighted by their
relative precision.
Priors in DCM
• embody constraints on parameter estimation
– hemodynamic parameters: empirical priors
– coupling parameters of self-connections: principled priors
– coupling parameters other connections: shrinkage priors
Small & variable effect
Large & variable effect
Small but clear effect
Large & clear effect
DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
dx
 ax
dt
x(t )  x0 exp(at )
Coupling parameter a is inversely
proportional to the half life  of x(t):
x( )  0.5x0
 x0 exp(a )
The coupling parameter a
thus describes the speed of
the exponential change in x(t)
0.5x0
a  ln 2 / 
  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
stimuli
u1
context
u2
+
-
x1
+
u1
u1
u2
u2
Z1
x
Z2 1
x2
+
x2
-
x  Ax  u2 B (2) x  Cu1
-
Penny, Stephan, Mechelli, Friston
NeuroImage (2004)
 x1   
 x    a 21
 2 
2

a12 
b11
x

u
2
 
 0
0 
 c1 0  u1 
x
 u 
2
0
0

b 22     2 
DCM Summary
Select areas you want to model
•
Extract timeseries of these areas
(x(t))
•
Specify at neuronal level
Modulatory input
(e.g. context/learning/drugs)
b12
– what drives areas (c)
– how areas interact (a)
– what modulates interactions (b)
•
neuronal
states
State-space model with 2 levels:
activity
x1(t)
– Hidden neural dynamics
– Predicted BOLD response
•
Estimate model parameters:
Gaussian a posteriori parameter
distributions, characterised by
mean ηθ|y and
covariance Cθ|y.
BOLD
ηθ|y
y
Driving input
(e.g. sensory stim)
c1
a12
c2
activity
x2(t)
y
Inference about DCM parameters:
Bayesian single-subject analysis
• Gaussian assumptions about the posterior distributions of the
parameters
• Use of the cumulative normal distribution to test the probability that
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
• By default, γ is chosen as zero ("does the effect exist?").
Inference about DCM parameters:
group analysis (classical)
• 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 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
Overview
• Brain connectivity: types & definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
• Applications of DCM to fMRI data
– Design of experiments and models
– Some empirical examples and simulations
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?
Multifactorial design:
explaining interactions with DCM
Stim 1
Stim 2
Stimulus factor
Task factor
Stim1/
Task A
Stim2/
Task A
Task A
Task B
TA/S1
TB/S1
X1
X2
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.
Stim1
How do we model this using DCM?
Stim2
Task A
Task B
GLM
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
X1
Stim 1
Task A
Stim 2
Task A
Stim 1
Task B
Stim 2
Task B
X2
plus added noise (SNR=1)
Final point: GLM vs. DCM
DCM tries to model the same phenomena as a GLM, just in a
different way:
It is a model, based on connectivity and its modulation, for
explaining experimentally controlled variance in local responses.
If there is no evidence for an experimental effect (no activation
detected by a GLM) → inclusion of this region in a DCM is not
meaningful.
Thank you
Stay tuned to find out how to
… select the best model comparing various DCMs
… test whether one region influences the connection
between other regions
… do DCM on your M/EEG & LFP data
… and lots more!