DCM

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DCM Advanced issues
Klaas Enno Stephan Laboratory for Social
Neural Systems Research Institute for Empirical
Research in Economics University of
Zurich Functional Imaging Laboratory
(FIL) Wellcome Trust Centre for
Neuroimaging University College London
SPM Course Zurich2009
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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

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• The hemodynamic model in DCM
• Timing errors sampling accuracy
• DCMs for electrophysiological data

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Model comparison and selection
Given competing hypotheses on structure
functional mechanisms of a system, which model is
the best?
Which model represents thebest balance between
model fit and model complexity?
For which model m does p(ym) become maximal?
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Bayesian model selection (BMS)
Bayes rule
Model evidence
accounts for both accuracy and complexity of the
model
allows for inference about structure
(generalisability) of the model
integral usually not analytically solvable,
approximations necessary
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Model evidence p(ym)
Gharamani, 2004
Balance between fit and complexity Generalisabili
ty of the model
p(ym)
a specific y
all possible datasets y
Model evidence probability of generating data y
from parameters ? that are randomly sampled from
the prior p(m). Maximum likelihood probability
of the data y for the specific parameter vector ?
that maximises p(y?,m).
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Approximations to the model evidence in DCM
Maximizing log model evidence Maximizing model
evidence
Logarithm is a monotonic function
Log model evidence balance between fit and
complexity
No. of parameters
In SPM2 SPM5, interface offers 2 approximations
No. of data points
Akaike Information Criterion
Bayesian Information Criterion
AIC favours more complex models, BIC favours
simpler models.
Penny et al. 2004, NeuroImage
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Bayes factors
To compare two models, we can just compare their
log evidences.
But the log evidence is just some number not
very intuitive!
A more intuitive interpretation of model
comparisons is made possible by Bayes factors
positive value, 0??
Kass Raftery classification
Kass Raftery 1995, J. Am. Stat. Assoc.
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SPM2/SPM5 Two models with identical numbers of
parameters
AIC BF 3.3
BMS result BF 3.3
BIC BF 3.3
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SPM2/SPM5 Two models with different numbers of
parameters compatible AIC/BIC based decisions
about models
AIC BF 0.1
BMS result BF 0.7
BIC BF 0.7
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SPM2/SPM5 Two models with different numbers of
parameters incompatible AIC/BIC based decisions
about models
AIC BF 0.3
BMS result AIC and BIC disagree about which
model is superior - no decision can be made.
BIC BF 2.2
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The negative free energy approximation

• Under Gaussian assumptions about the posterior
  (Laplace approximation), the negative free energy
  F is a lower bound on the log model evidence

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The complexity term in F

• In contrast to AIC BIC, the complexity term of
  the negative free energy F accounts for parameter
  interdependencies.
• The complexity term of F is higher
• the more independent the prior parameters (?
  effective DFs)
• the more dependent the posterior parameters
• the more the posterior mean deviates from the
  prior mean
• NB SPM8 only uses F for model selection !

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BMS in SPM8 an example
attention
M1
M2
PPC
PPC
attention
V1
V5
stim
V1
V5
stim
M1
M2
M3
M4
M3 better than M2
BF ? 12 ?F 2.450
M4 better than M3
BF ? 23 ?F 3.144
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Fixed effects BMS at group level

• Group Bayes factor (GBF) for 1...K subjects
• Average Bayes factor (ABF)
• Problems
• blind with regard to group heterogeneity
• sensitive to outliers

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A suboptimal solution...

• Positive Evidence Ratio (PER)
• i.e. number of subjects in which there is
  positive evidence for model i divided by number
  of subjects in which there is positive evidence
  for model j

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Random effects BMS for group studies
Dirichlet parameters occurrences of models in
the populations
Dirichlet distribution of model probabilities
Multinomial distribution of subject-specific
models
Measured data
Stephan et al. 2009, in revision
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incorrect model (m2)
correct model (m1)
m2
m1
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Estimates of Dirichlet parameters
Post. expectations of model probabilities
Exceedance probability
Stephan et al. 2009, in revision
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m2
m1
Stephan et al. 2009, in revision
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Stephan et al. 2009, in revision
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Interface in SPM8
Post. expectations of model probabilities
Exceedance probability
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Further reading on BMS of DCMs

• Theoretical papers
• Penny et al. (2004) Comparing dynamic causal
  models. NeuroImage 22 1157-1172.
• Stephan et al. (2007) Comparing hemodynamic
  models with DCM. NeuroImage 38 387-401.
• Stephan et al. (2009) Bayesian model selection
  for group studies. NeuroImage, in revision.
• Examples of application
• Grol et al. (2007) Parieto-frontal connectivity
  during visually-guided grasping. J. Neurosci. 27
  11877-11887.
• Kumar et al. (2007) Hierarchical processing of
  auditory objects in humans. PLoS Computat. Biol.
  3 e100.
• Stephan et al. (2007) Inter-hemispheric
  integration of visual processing during
  task-driven lateralization. J. Neurosci. 27
  3512-3522.

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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• The hemodynamic model in DCM
• Timing errors sampling accuracy
• DCMs for electrophysiological data

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bilinear DCM
driving input
modulation
Two-dimensional Taylor series (around x00, u00)
Nonlinear state equation
Bilinear state equation
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u2
u1
Nonlinear dynamic causal model (DCM)
Stephan et al. 2008, NeuroImage
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Nonlinear DCM Attention to motion
Stimuli Task
Previous bilinear DCM
Büchel Friston (1997)
250 radially moving dots (4.7 /s)
Friston et al. (2003)
Conditions F fixation only A motion
attention (detect changes) N motion
without attention S stationary dots
Friston et al. (2003)attention modulates
backward connections IFG?SPC and SPC?V5. Q Is a
nonlinear mechanism (gain control) a better
explanation of the data?
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attention
M1
M2
?
modulation of back- ward or forward connection?
PPC
PPC
attention
V1
stim
V1
V5
stim
V5
?
additional driving effect of attention on PPC?
?
bilinear or nonlinear modulation of forward
connection?
Stephan et al. 2008, NeuroImage
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attention
MAP 1.25
0.10
PPC
0.26
0.39
1.25
0.26
V1
stim
0.13
V5
0.46
0.50
motion
Stephan et al. 2008, NeuroImage
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motion attention
static dots
motion no attention
V1
V5
PPC
observed
fitted
Stephan et al. 2008, NeuroImage
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Nonlinear DCM Binocular rivalry
Stephan et al. 2008, NeuroImage
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BR
nBR
time (s)
Stephan et al. 2008, NeuroImage
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• The hemodynamic model in DCM
• Timing errors sampling accuracy
• DCMs for electrophysiological data

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The hemodynamic model in DCM
u
stimulus functions
neural state equation

• 6 hemodynamic parameters

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

• Empirically determineda priori distributions.
• Area-specific estimates (like neural
  parameters)? region-specific HRFs!

BOLD signal change equation
Friston et al. 2000, NeuroImage Stephan et al.
2007, NeuroImage
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Region-specific HRFs
E00.1
E00.5
E00.9
black measured BOLD signal red predicted BOLD
signal
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Recent changes in the hemodynamic model(Stephan
et al. 2007, NeuroImage)

• new output non-linearity, based on new exp. data
  and mathematical derivations

BMS indicates that new model performs better than
original Buxton model

• field-dependency of output coefficients is
  handled better, e.g. by estimating
  intra-/extravascular BOLD signal ratio??

less problematic to apply DCM to high-field fMRI
data
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How interdependent are our neural and hemodynamic
parameter estimates?
A
B
C
?h
e
Stephan et al. 2007, NeuroImage
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• The hemodynamic model in DCM
• Timing errors sampling accuracy
• DCMs for electrophysiological data

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Timing problems at long TRs/TAs

• Two potential timing problems in DCM
• wrong timing of inputs
• temporal shift between regional time series
  because of multi-slice acquisition

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slice acquisition
1
visualinput

• DCM is robust against timing errors up to approx.
  1 s
• compensatory changes of s and ?h
• Possible corrections
• slice-timing in SPM (not for long TAs)
• restriction of the model to neighbouring regions
• in both cases adjust temporal reference bin in
  SPM defaults (defaults.stats.fmri.t0)
• Best solution Slice-specific sampling within DCM

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Slice timing in DCM three-level model
sampled BOLD response
3rd level
2nd level
BOLD response
neuronal response
1st level
x neuronal states u inputs xh
hemodynamic states v BOLD responses ?n, ?h
neuronal and hemodynamic parameters T sampling
time points
Kiebel et al. 2007, NeuroImage
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Slice timing in DCM an example
3 TR
1 TR
2 TR
4 TR
5 TR
Default sampling
t
3 TR
1 TR
2 TR
4 TR
5 TR
Slice-specific sampling
t
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• The hemodynamic model in DCM
• Timing errors sampling accuracy
• DCMs for electrophysiological data

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DCM generative model for fMRI and ERPs
Electric/magnetic forward modelneural
activity?EEGMEG LFP (linear)
Hemodynamicforward modelneural
activity?BOLD (nonlinear)
Neural state equation
fMRI
ERPs
Neural model 1 state variable per
region bilinear state equation no propagation
delays
Neural model 8 state variables per
region nonlinear state equation propagation delays
inputs
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DCMs for M/EEG and LFPs

• can be fitted both to frequency spectra and ERPs
• models synaptic plasticity and of spike-frequency
  adaptation (SFA)
• ongoing model validation by LFP recordings in
  rats, combined with pharmacological manipulations

standards
deviants
A1
A2
Example of single-neuron SFA
Tombaugh et al. 2005, J.Neurosci.
Moran et al. 2008, NeuroImage
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Neural mass model of a cortical macrocolumn
E x t r i n s i c i n p u t s
Excitatory Interneurons He, ??e
mean firing rate ? mean postsynaptic potential
(PSP)
?2
?1
Pyramidal Cells He, ?e
MEG/EEG signal
?4
?3
mean PSP? mean firing rate
Inhibitory Interneurons Hi, ?e
Excitatory connection
Inhibitory connection

• te, ti synaptic time constant (excitatory
  and inhibitory)
• He, Hi synaptic efficacy (excitatory and
  inhibitory)
• g1,,g4 intrinsic connection strengths
• propagation delays

Parameters
Jansen Rit (1995) Biol. Cybern. David et al.
(2006) NeuroImage
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Intrinsic connections
Synaptic alpha kernel
Inhibitory cells in agranular layers
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers
Exogenous input u
Sigmoid function
Excitatory pyramidal cells in agranular layers
Extrinsic Connections Forward Backward Lateral
Moran et al. 2008, NeuroImage
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Electromagnetic forward model for M/EEG
Forward model lead field gain matrix
Depolarisation of pyramidal cells
Scalp data
Forward model
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Thank you