1. 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 Zurich 2009

2.    BOLD y y y y λ x neuronal states hemodynamic model activity
x2(t)
activity
x3(t)
activity
x1(t) x
neuronal
states
modulatory
input u2(t) t
integration
intrinsic connectivity
direct inputs
modulation of
connectivity
Neural state equation t
driving
input u1(t)
Stephan & Friston (2007), Handbook of Brain Connectivity

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

4. Model comparison and selection
Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?
Pitt & Miyung (2002) TICS
Which model represents the best balance between model fit and model complexity?
For which model m does p(y|m) become maximal?

5. 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

6. Model evidence p(y|m) Balance between fit and complexity
Generalisability of the model
Gharamani, 2004
p(y|m)
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).

7. Approximations to the model evidence in DCM
Logarithm is a monotonic function
Maximizing log model evidence
= Maximizing model evidence
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

8. 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;[
B12
p(m1|y)
Evidence
1 to 3
50-75%
weak
3 to 20
75-95%
positive
20 to 150
95-99%
strong
 150
 99%
Very strong
Kass & Raftery classification:
Kass & Raftery 1995, J. Am. Stat. Assoc.

9. SPM2/SPM5: Two models with identical numbers of parameters AIC:
BF = 3.3
BMS result:
BF = 3.3
BIC:
BF = 3.3

10. SPM2/SPM5: Two models with different numbers of parameters AIC: &
compatible AIC/BIC based decisions about models
AIC:
BF = 0.1
BMS result:
BF = 0.7
BIC:
BF = 0.7

11. SPM2/SPM5: AIC: 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

12. 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:

13. 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 !

14. BMS in SPM8: an example M1 M2 M3 M4 attention PPC PPC BF 2966
M2 better than M1
attention
stim V1 V5
stim V1 V5 M1 M2 M3 M4 V1 V5
stim
PPC M3
attention
M3 better than M2
BF  12
F = 2.450 V1 V5
stim
PPC M4
attention
M4 better than M3
BF  23
F = 3.144

15. 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

16. 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

17. 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

18. m2 m1 incorrect model (m2) correct model (m1) x1 x2 u1 x3 u2 x1 x2 u1

19. Estimates of Dirichlet parameters
Post. expectations of model probabilities
Exceedance probability
Simulations
Stephan et al. 2009, in revision

20. m2 m1 Stephan et al. 2009, in revision MOG LG RVF stim. LVF FG LD
LD|RVF
LD|LVF
MOG LG
RVF
stim.
LVF FG
LD|RVF
LD|LVF LD m2 m1
Stephan et al. 2009, in revision

21. Stephan et al. 2009, in revision

22. Interface in SPM8 Post. expectations of model probabilities
Exceedance probability

23. Further reading on BMS of DCMs
Theoretical papers:
Penny et al. (2004) Comparing dynamic causal models. NeuroImage 22:
Stephan et al. (2007) Comparing hemodynamic models with DCM. NeuroImage 38:
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:
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:

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

25. bilinear DCM nonlinear DCM
driving
input
modulation
driving
input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
Bilinear state equation:
Nonlinear state equation:

26. u2 x1 x2 x3 u1 Nonlinear dynamic causal model (DCM):
Neural population activity
fMRI signal change (%) u2 x1 x2 x3 u1
Nonlinear dynamic causal model (DCM):
Stephan et al. 2008, NeuroImage

27. Nonlinear DCM: Attention to motion
Stimuli + Task
Previous bilinear DCM V1
IFG V5
SPC
Motion
Photic
Attention
.82
(100%)
.42
.37
(90%)
.69 (100%)
.47
.65 (100%)
.52 (98%)
.56
(99%)
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?

28. M1 M2   M3  M4 modulation of back- ward or forward connection?
attention M1 M2 
modulation of back-
ward or forward
connection?
PPC
PPC
BF = 2966
M2 better than M1
attention
stim V1 V5
stim V1 V5
M3 better than M2
BF = 12 
additional driving
effect of attention
on PPC? V1 V5
stim
PPC M3
attention
M4 better than M3
BF = 23 
bilinear or nonlinear
modulation of
forward connection? V1 V5
stim
PPC M4
attention
Stephan et al. 2008, NeuroImage

29. attention PPC stim V1 V5 motion MAP = 1.25
0.10
PPC
0.26
0.39
1.25
0.26
stim V1
0.13 V5
0.46
0.50
motion
Stephan et al. 2008, NeuroImage

30. motion & attention motion & no attention static dots V1 V5 PPC
observed
fitted
Stephan et al. 2008, NeuroImage

31. Nonlinear DCM: Binocular rivalry
FFA
PPA
MFG
-0.80
-0.31
faces
houses
rivalry
non-rivalry
1.05
0.08
0.30
0.51
2.43
2.41
0.04
-0.03
0.02
0.06
Stephan et al. 2008, NeuroImage

32. FFA
PPA
MFG BR
nBR
time (s)
Stephan et al. 2008, NeuroImage

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

34. The hemodynamic model in DCMu
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 determined a 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

35. Region-specific HRFs RVF LVFLG
left
right
RVF
LVF FG
E0=0.5
E0=0.9
black: measured BOLD signal red: predicted BOLD signal

36. Recent changes in the hemodynamic model (Stephan et al
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

37. How interdependent are our neural and hemodynamic parameter estimates?B C h ε
Stephan et al. 2007, NeuroImage

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

39. 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 2
slice acquisition 1
visual input
DCM is robust against timing errors up to approx. ± 1 s
compensatory changes of σ and θh
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

40. Slice timing in DCM: three-level model
sampled
BOLD response
3rd level
2nd level
BOLD response
1st level
neuronal response
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

41. Slice timing in DCM: an example
1 TR
2 TR
3 TR
4 TR
5 TR
Default
sampling t
1 TR
2 TR
3 TR
4 TR
5 TR
Slice-specific sampling t

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

43. DCM: generative model for fMRI and ERPs
Hemodynamic forward model: neural activityBOLD
(nonlinear)
Electric/magnetic
forward model: neural activityEEG MEG
LFP
(linear)
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

44. 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

45. Neural mass model of a cortical macrocolumnx t r i n s c p u
Excitatory
Interneurons
He, e
mean firing rate  mean postsynaptic potential (PSP) 1 2
Pyramidal
Cells
He, e
MEG/EEG
signal 3 4
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

46. g g g g g g g g g Synaptic ‘alpha’ kernel Sigmoid function5 g
Intrinsic
connections
Synaptic ‘alpha’ kernel
Inhibitory cells in agranular layers 4 g 4 g 3 g 3 g
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers 1 2 4 9 ) ( x u a s H e k g - + =
&
Exogenous input u 1 g 1 g 2 g 2 g
Sigmoid function
Excitatory pyramidal cells in agranular layers
Extrinsic Connections:
Forward
Backward
Lateral
Moran et al. 2008, NeuroImage

47. Electromagnetic forward model for M/EEG
Depolarisation of
pyramidal cells
Forward model:
lead field & gain matrix
Scalp data
Forward model

48. Thank you