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
Methods models for fMRI data analysis,
University of Zurich27 May 2009
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors sampling accuracy
• Integrating tractography and DCM
• 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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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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Random effects BMS for group studies a
variational Bayesian approach
Dirichlet parameters occurrences of models in
the population
Dirichlet distribution of model probabilities
Multinomial distribution of model labels
Measured data
Stephan et al. 2009, NeuroImage
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Task-driven lateralisation
Does the word contain the letter A or not?
letter decisions gt spatial decisions
group analysis (random effects),n16, plt0.05
whole-brain corrected
time
Is the red letter left or right from the midline
of the word?
spatial decisions gt letter decisions
Stephan et al. 2003, Science
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Inter-hemispheric connectivity in the visual
ventral stream
Right FG 38,-52,-20
Left MOG -38,-90,-4
Left FG -44,-52,-18
Right MOG -38,-94,0
LDLVF
0.20 ? 0.04
0.00 ? 0.01
0.07 ? 0.02
LDgtSD, plt0.05 cluster-level corrected (plt0.001
voxel-level cut-off)
plt0.01 uncorrected
0.27 ? 0.06
0.11 ? 0.03
LD
LD
0.01 ? 0.03
0.00 ? 0.04
Left LG -12,-70,-6
Left LG -14,-68,-2
0.01 ? 0.01
0.06 ? 0.02
0.01 ? 0.01
RVF stim.
LVF stim.
LDRVF
LDgtSD masked incl. with RVFgtLVF plt0.05
cluster-level corrected (plt0.001 voxel-level
cut-off)
LDgtSD masked incl. with LVFgtRVF plt0.05
cluster-level corrected (plt0.001 voxel-level
cut-off)
Stephan et al. 2007, J. Neurosci.
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m2
m1
Stephan et al. 2009, NeuroImage
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Simulation study sampling subjects from a
heterogenous population
m1

• Population where 70 of all subjects' data are
  generated by model m1 and 30 by model m2
• Random sampling of subjects from this population
  and generating synthetic data with observation
  noise
• Fitting both m1 and m2 to all data sets and
  performing BMS

m2
Stephan et al. 2009, NeuroImage
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B
A
true values ?122?0.715.4 ?222?0.36.6 mean
estimates ?115.4, ?26.6
true values r1 0.7, r20.3 mean estimates r1
0.7, r20.3
?
ltrgt
m1
m2
m1
m2
C
D
true values ?1 1, ?20 mean estimates ?1
0.89, ?20.11
?
m2
log GBF12
m1
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data

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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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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
• Timing errors sampling accuracy
• Integrating tractography and DCM
• 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
Kiebel et al. 2007, NeuroImage
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data

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Diffusion-weighted imaging
Parker Alexander, 2005, Phil. Trans. B
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Probabilistic tractography Kaden et al. 2007,
NeuroImage

• computes local fibre orientation density by
  spherical deconvolution of the diffusion-weighted
  signal
• estimates the spatial probability distribution of
  connectivity from given seed regions
• anatomical connectivity proportion of fibre
  pathways originating in a specific source region
  that intersect a target region
• If the area or volume of the source region
  approaches a point, this measure reduces to
  method by Behrens et al. (2003)

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Integration of tractography and DCM
R2
R1
low probability of anatomical connection ? small
prior variance of effective connectivity parameter
R2
R1
high probability of anatomical connection ? large
prior variance of effective connectivity parameter
Stephan, Tittgemeyer, Knoesche, Moran, Friston,
in revision
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LDLVF
LD
LD
? DCM structure
LDRVF
BVF stim.
RVF stim.
LVF stim.
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Overview

• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors sampling accuracy
• Integrating tractography and DCM
• 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 different neuronal cell types, different
  synaptic types (and their plasticity) and
  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.
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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.
(2003) 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
David et al. 2006, NeuroImage Kiebel et al.
2007, NeuroImage Moran et al. 2009, 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
Kiebel et al. 2006, NeuroImage
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DCM for steady-state responses

• models the cross-spectral density of recorded
  data
• feature extraction by means of p-order VAR model
• spectral form of neuronal innovations (i.e.
  baseline cortical activity) are estimated using a
  mixture of white and pink (1/f) components
• assumes quasi-stationary responses (i.e. changes
  in neuronal states are approximated by small
  perturbations around some fixed point)

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Frequency (Hz)
20
30
Time (s)
0
10
Moran et al. 2009, NeuroImage
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Validation study using microdialysis (in
collaboration with Conway Inst., UC Dublin)

• two groups of rats with different rearing
  conditions
• LFP recordings and microdialysis measurements
  (Glu GABA) from mPFC

Moran et al. 2008, NeuroImage
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Experimental data
FFT 10 mins time series one area (mPFC)
blue control animals red isolated animals
plt0.05, Bonferroni-corrected
Moran et al. 2008, NeuroImage
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Predictions about expected parameter estimates
from the microdialysis measurements
upregulation of AMPA receptors
?amplitude of synaptic kernels (? He)

• SFA
• (??2)

chronic reduction in extracellular glutamate
levels
? EPSPs

• ? activation of voltage-sensitive Ca2 channels
  ? ?intracellular Ca2 ? ?Ca-dependent
  K currents
• ? ?IAHP

sensitisation of postsynaptic mechanisms
Van den Pool et al. 1996, Neuroscience Sanchez-Viv
es et al. 2000, J. Neurosci.
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sensitization of post-synaptic mechanisms
Inhibitory cells in supragranular layers
3.8 6.3
29,37
(0.4)
(0.04)
Extrinsic
Extrinsic
Excitatory spiny cells in granular layers
forward
Excitatory spiny cells in granular layers
forward
connections
connections
u
u
195, 233
161, 210
(0.37)
(0. 13)
Excitatory pyramidal cells in infragranular
layers
Control group estimates in blue, isolated animals
in red, p values in parentheses.
0.76,1.34
(0.0003)
Increased neuronal adaption decreased firing rate
Moran et al. 2008, NeuroImage
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Take-home messages

• Bayesian model selection (BMS)generic approach
  to selecting an optimal model from an arbitrarily
  large number of competing models
• random effects BMS for group studiesposterior
  model probabilities and exceedance probabilities
• nonlinear DCMenables one to investigate
  synaptic gating processes via activity-dependent
  changes in connection strengths
• DCM tractography probabilities of anatomical
  connections can be used to inform the prior
  variance of DCM coupling parameters
• DCMs for electrophysiologybased on
  neurophysiologically fairly detailed neural mass
  models

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Thank you