Models of Effective Connectivity

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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 Zurich, February 2009
Thanks to Klaas Stephan and Meike Grol for slides
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Systems analysis in functional neuroimaging
Functional specialisation What regions respond
to a particular experimental input?
Functional integration How do regions influence
each other? ? Brain Connectivity
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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

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Structural, functional effective connectivity
Sporns 2007, Scholarpedia

• anatomical/structural connectivity presence of
  axonal connections
• functional connectivity statistical
  dependencies between regional time series
• effective connectivity causal (directed)
  influences between neurons or neuronal populations

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Anatomical connectivity

• presence of axonal connections
• neuronal communication via synaptic contacts
• visualisation by
• tracing techniques
• diffusion tensor imaging

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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

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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
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Overview

• Brain connectivity types definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
• Applications of DCM to fMRI data

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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

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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
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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 (plt0.001) with
  left M1
• ? No difference in SPMs!

Sun et al. 2003, Neuroimage
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Does functional connectivity not simply
correspond to co-activation in SPMs?
regional response A2
regional response A1
task T

• No, it does not - see the fictitious example on
  the right
• 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.
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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

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For understanding brain function mechanistically,
we need models of effective connectivity,
i.e.models of causal interactions among
neuronal populationsto explain regional effects
in terms of interregional connectivity
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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

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Overview

• Brain connectivity types definitions
• Functional connectivity
• Psycho-physiological interactions (PPI)
• Dynamic causal models (DCMs)
• Applications of DCM to fMRI data

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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.

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Psycho-physiological interaction (PPI)
Task factor
GLM of a 2x2 factorial design
Task B
Task A
main effect of task
TA/S1
TB/S1
Stim 1
main effect of stim. type
Stimulus factor
interaction
TA/S2
TB/S2
Stim 2
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
main effect of task
V1 time series ?? main effect of stim. type
psycho- physiological interaction
Friston et al. 1997, NeuroImage
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Example PPI Attentional modulation of V1?V5
Attention

V1 x Att.
Friston et al. 1997, NeuroImage Büchel Friston
1997, Cereb. Cortex
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PPI interpretation
Two possible interpretations of the PPI term
attention
attention
V1
V1
Modulation of V1?V5 by attention
Modulation of the impact of attention on V5 by V1
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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!
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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

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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-specific BOLD signals (y) by a
  hemodynamic forward model (?).

The aim of DCM is to estimate parameters at the
neuronal level such that the modelled and
measured BOLD signals are maximally similar.
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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

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Example a linear system of dynamics in visual
cortex
LG lingual gyrus FG fusiform gyrus Visual
input in the - left (LVF) - right
(RVF)visual field.
x4
x3
x1
x2
RVF
LVF
u1
u2
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Example a linear system of dynamics in visual
cortex
LG lingual gyrus FG fusiform gyrus Visual
input in the - left (LVF) - right
(RVF)visual field.
x4
x3
x1
x2
RVF
LVF
u2
u1
state changes
effective connectivity
externalinputs
systemstate
input parameters
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Extension bilinear dynamic system
x4
x3
x1
x2
RVF
LVF
CONTEXT
u3
u2
u1
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y
BOLD
y
y
y
?
hemodynamic model
activity x2(t)
activity x3(t)
activity x1(t)
x
neuronal states
integration
Stephan Friston (2007),Handbook of Brain
Connectivity
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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

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The hemodynamic model in DCM
u
stimulus functions

• 6 hemodynamic parameters

neural state equation
important for model fitting, but of no interest
for statistical inference
hemodynamic state equations

• Computed separately for each area (like the
  neural parameters)? region-specific HRFs!

Estimated BOLD response
Friston et al. 2000, NeuroImage Stephan et al.
2007, NeuroImage
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Example modelled BOLD signal
black observed BOLD signal red modelled BOLD
signal
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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

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Bayesian statistics
new data
prior knowledge
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.
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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
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DCM parameters rate constants
Integration of a first-order linear differential
equation gives anexponential function
Coupling parameter is inverselyproportional
to the half life ? of x(t)
The coupling parameter a thus describes the
speed ofthe exponential change in x(t)
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
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Example context-dependent decay
u1
stimuli u1
context u2
u2
-

-
x1
x1

x2

x2
-
-
Penny, Stephan, Mechelli, Friston NeuroImage
(2004)
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DCM Summary

• Select 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 parameter distributions,
  characterised by mean ??y and covariance C?y.

neuronal states
activity x1(t)
activity x2(t)
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Inference about DCM parametersBayesian
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 ??y) is above a chosen
  threshold ?
• By default, ? is chosen as zero ("does the effect
  exist?").

??
??y
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Inference about DCM parametersgroup 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 gt 0 ?
paired t-test parameter 1 gt parameter 2 ?
rmANOVA e.g. in case of multiple sessions per
subject
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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

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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?

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Multifactorial design explaining interactions
with DCM
Lets 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?
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Simulated data
A1

Stim1

Stim 1Task B
Stim 2Task B
Stim 1Task A
Stim 2Task A
A1
A2

Stim2



Task A
Task B
A2
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X1
Stim 1Task B
Stim 2Task B
Stim 1Task A
Stim 2Task A
X2
plus added noise (SNR1)
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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.

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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!