1. University of Zurich, 16-18 February 2011
DCM: Advanced topics
Rosalyn Moran
Wellcome Trust Centre for Neuroimaging
Institute of Neurology
University College London
With thanks to the FIL Methods Group
for slides and images
SPM Course 2011
University of Zurich, February 2011

2. Dynamic Causal Modeling (DCM)
Hemodynamic forward model: neural activityBOLD
Electromagnetic
forward model: neural activityEEG MEG
LFP
Neural state equation:
fMRI
EEG/MEG
simple neuronal model
complicated forward model
complicated neuronal model
simple forward model
inputs

3. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI
Stochastic DCM
Embedding computational models in DCMs
Integrating tractography and DCM

4. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI
Stochastic DCM
Embedding computational models in DCMs
Integrating tractography and DCM

5. 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?

6. 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

7. The negative free energy approximation
The negative free energy F is a lower bound on the log model evidence:

8. The complexity term in F
In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies. Under gaussian assumptions:
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 !
However, comparing these expression to Equation 4, shows that both the AIC and BIC will fail in various situations. An obvious example is redundant parameterisation; the true complexity will not change when we add a parameter whose effect is identical to another parameter in measurement space. While the free-energy bound would take this redundancy into account, keeping the complexity identical, the AIC and BIC approximations would indicate that complexity has increased. In practice, many models show partial dependencies amongst parameters, meaning that AIC and BIC routinely over-estimate the effect that adding or removing parameters has on model complexity
Penny et al. submitted

9. Bayes factors
For a given dataset, to compare two models, we compare their evidences.
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:
or their log evidences
Kass & Raftery 1995, J. Am. Stat. Assoc.

10. 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
Posterior model probability in lower plot is a normalised probability:
p(m_i|y) = p(y|m_i)/sum(p(y|m_i))
Note that under flat model priors p(m_i|y) = p(y|m_i) V1 V5
stim
PPC M4
attention
M4 better than M3
BF  23
F = 3.144

11. 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 or

12. Random effects BMS for group studies
Dirichlet parameters
= “occurrences” of models in the population
Dirichlet distribution of model probabilities
Multinomial distribution of model labels
Model inversion by Variational Bayes (VB)
Want: the density from which models are sampled to generate subject-specific data.
we seek the conditional estimates of the multinomial parameters, i.e. the model probabilities , that generate switches or indicator variables, , prescribing the model for the i-th subject; where . Since the model probabilities r follow a Dirichlet distribution , the conditional expectations encode the expected probability that the k-th model will be selected for any randomly selected subject. Also, the cumulative probability density of can be used to quantify our belief that a given model m1 has a higher probability of having generated the observed data across the group than a second model m2; below, we will call this the "exceedance probability". In the following, we describe a hierarchical Bayesian model that can be inverted to obtain the Dirichlet parameters .
Using the Dirichlet density for model comparison
After the above optimization of the Dirichlet parameters, , the Dirichlet density can be used for model comparisons at the group level. There are several ways to report the results of this comparison. The simplest option is to report the Dirichlet parameter estimates (c.f. Eq. 20). Another possibility is to use to compute the expected multinomial parameters and thus the expected likelihood of obtaining a particular model, i.e. for any randomly selected subject [1]
(20)
Either the Dirichlet parameter estimates  or the expected multinomial parameters can be used, with equivalent results, to rank models at the group level.
A third option, which is perhaps the most useful one when two models (or model subsets, see below) are compared against each other, is to use the cumulative probability density of to quantify our belief that one model is more likely than a second one, given the group data. For example, when comparing two models, m1 and m2, the belief that m1 has the higher probability of having generated the observed data across the group corresponds to the "exceedance probability"
[1] Note that for the special case of "drawing" a single "sample" (model), the multinomial distribution of models reduces to . Therefore, for any given subject, represents the conditional expectation that the k-th model generated her/his data.
This enables us to estimate the Dirichlet parameters given observations or samples from a multinomial distribution
We consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new method rests on a variational approach to estimating the parameters of a Dirichlet distribution, which define a multinomial distribution of how likely it is that a specific model generated the data from a specific subject. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects
the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. That is, its probability density function returns the belief that the probabilities of K rival events are xi given that each event has been observed αi − 1 times.
The support of the Dirichlet distribution is a K-dimensional vector of real numbers in the range (0,1), all of which sum to 1. These can be viewed as the probabilities of a K-way categorical event. Another way to express this is that the domain of the Dirichlet distribution is itself a probability distribution, specifically a K-dimensional discrete distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution.
The binomial distribution is the probability distribution of the number of "successes" in n independent Bernoulli trials, with the same probability of "success" on each trial. In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number k of possible outcomes, with probabilities p1, ..., pk (so that pi ≥ 0 for i = 1, ..., k and ), and there are n independent trials. Then let the random variables Xi indicate the number of times outcome number i was observed over the n trials. The vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk).
rm as the frequency with which
model m is used in the population.
Measured data y
estimate the parameters  of the posterior
Stephan et al. 2009, NeuroImage

13. Random effects BMS for group studies
“the occurences”
“the expected likelihood”
“the exceedance probability”
Alpha, the occurences,
<r> the expected likelihood of obtaining a particular model for any randomly selected subject
. For example, when comparing two models, m1 and m2, the belief that m1 has the higher probability of having generated the observed data across the group corresponds to the "exceedance probability
Stephan et al. 2009, NeuroImage

14. group analysis (random effects), n=16, p<0.05 corrected
Task-driven lateralisation
Does the word contain the letter A or not?
letter decisions > spatial decisions •
group analysis (random effects), n=16, p<0.05 corrected
analysis with SPM2
time
Is the red letter left or right from the midline of the word?
spatial decisions > letter decisions
Stephan et al. 2003, Science

15. + Theories on inter-hemispheric integration during lateralised tasks
Information transfer (for left-lateralised task) T
|RVF  + T
|LVF
LVF
RVF
Predictions:
modulation by task conditional on visual field
asymmetric connection strengths

16. Ventral stream & letter decisions
Left MOG
-38,-90,-4
Left FG
-44,-52,-18
Right FG
38,-52,-20
Right MOG
-38,-94,0
LD|LVF
LD>SD, p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
p<0.01 uncorrected
MOG
left FG
left FG
right
MOG
right LG
left LG
right
Left LG
-12,-70,-6
Left LG
-14,-68,-2
RVF
stim.
LVF
stim.
LD>SD masked incl. with RVF>LVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
LD>SD masked incl. with LVF>RVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Stephan et al. 2007, J. Neurosci.

17. Ventral stream & letter decisions
Left MOG
-38,-90,-4
Left FG
-44,-52,-18
Right FG
38,-52,-20
Right MOG
-38,-94,0
LD>SD, p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
p<0.01 uncorrected
MOG
left FG
left FG
right
MOG
right LG
left LG
right
Left LG
-12,-70,-6
Left LG
-14,-68,-2
LD|LVF
RVF
stim.
LVF
stim.
LD>SD masked incl. with RVF>LVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
LD>SD masked incl. with LVF>RVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Stephan et al. 2007, J. Neurosci.

18. m2 m1 m2 m1 Winner! Fixed Effects Stephan et al. 2009, NeuroImage MOGLG
RVF
stim.
LVF FG LD
LD|RVF
LD|LVF
MOG LG
RVF
stim.
LVF FG
LD|RVF
LD|LVF LD m1 m2 m1
Stephan et al. 2009, NeuroImage

19. m2 m1

20. Simulation study: sampling subjects from a heterogenous population
MOG LG
RVF
stim.
LVF FG
LD|RVF
LD|LVF LD 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
MOG LG
RVF
stim.
LVF FG LD
LD|RVF
LD|LVF m2
Stephan et al. 2009, NeuroImage

21.  <r>  m1 m2 m1 m2 m1 m2 true values: 1=220.7=15.4
2=220.3=6.6
mean estimates:
1=15.4, 2=6.6
true values:
r1 = 0.7, r2=0.3
mean estimates: 
<r> m1 m2 m1 m2
true values:
1 = 1, 2=0
mean estimates:
1 = 0.89, 2=0.11  m1 m2

22. Families of Models Partition the cortical dynamics of intelligible
Speech
activity among three key multimodal
regions: the left posterior and anterior superior temporal sulcus
(subsequently referred to as regions P and A respectively) and pars
orbitalis of the inferior frontal gyrus (region F). The aim of the
study was to see how connections among regions depended on
whether the auditory input was intelligible speech or time-reversed
speech.
Partition

23. Families of Models * * e.g. Modulatory connections
BMA: weight posterior parameter
densities with model probabilities
Penny et al., 2010

24. definition of model space
inference on model structure or inference on model parameters?
inference on
individual models or model space partition?
inference on
parameters of an optimal model or parameters of all models?
optimal model structure assumed to be identical across subjects?
comparison of model families using FFX or RFX BMS
optimal model structure assumed to be identical across subjects?
BMA
yes no
yes no
FFX BMS
RFX BMS
FFX BMS
RFX BMS
FFX analysis of parameter estimates
(e.g. BPA)
RFX analysis of parameter estimates
(e.g. t-test, ANOVA)
Stephan et al. 2010, NeuroImage

25. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI
Stochastic DCM
Embedding computational models in DCMs
Integrating tractography and DCM

26.    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

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

28. 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

29. 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?

30. 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

31. 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

32. motion &
attention
motion &
no attention
static
dots V1 V5
PPC
observed
fitted

33. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI
Stochastic DCM
Embedding computational models in DCMs
Integrating tractography and DCM
% Generalised filtering (under the Laplace assumption)
% =====================================================================
if DCM.options.stochastic == 1
DEM = spm_LAP(DEM); % no mean-field assumption
Stochastic DCMs: Accounting for stochastic inputs to the network
and their interaction with task-specific processes may be of
particular importance for studying state-dependent processes,
e.g., short-term plasticity and trial-by-trial variations of effective
connectivity. In addition, provided that the probabilistic inversion
schemes are properly extended

34. Stochastic DCMs   neuronal states Daunizeau et al, 2009
Stochastic innovations: variance hyperparameter  
activity
x2(t)
activity
x3(t)
activity
x1(t)
neuronal
states
modulatory
input u2(t) t t
driving
input u1(t)
Daunizeau et al, 2009
Friston et al, 2008
Inversion: Generalised filtering (under the Laplace assumption)

35. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI
Stochastic DCM
Embedding computational models in DCMs
Integrating tractography and DCM

36. Learning of dynamic audio-visual associations
200
400
600
800
1000
0.2
0.4
0.6
0.8 1 CS 1 2 CS
Response
Time (ms)
200
400
600
800
2000 ± 650 or
Target Stimulus
Conditioning Stimulus TS
p(face)
Specifically, the cue could be (1) strongly predictive ( p0.9), (2)
predictive ( p 0.7), (3) nonpredictive ( p 0.5), (4) antipredictive
( p 0.3), and (5) strongly antipredictive ( p 0.1) of the visual stimulus.
trial
den Ouden et al. 2010, J. Neurosci .

37. Bayesian learning modelk
vt-1 vt rt
rt+1 ut
ut+1
volatility
probabilistic association
observed events
First, we assessed the main effect of probability, that is, in which in
brain regions the hemodynamic response reflected the probability of the
stimulus occurring, independently of which stimulus it was. We tested
for both BOLD responses that increased with the likelihood of the outcome
and responses that increased the less likely (or more surprising) the
outcome was. In other words, these contrasts tested for stimulusindependent
responses that reflected predicted or surprising outcomes,
respectively. Given the results from our previous study (den Ouden et al.,
2009), our a priori hypothesis was that the response in the putamen
would correlate positively with prediction error, i.e., negatively with the
probability of the observed outcome.
Changes over trials: Model Based Regressor
Behrens et al. 2007, Nat. Neurosci.

38. Comparison with competing learning models
400
440
480
520
560
600
0.2
0.4
0.6
0.8 1
Trial
p(F)
True
Bayes Vol
HMM fixed
HMM learn RW
Alternative learning models:
Rescorla-Wagner
HMM (2 variants)
True probabilities
BMS:
hierarchical Bayesian learner performs best
The
posterior mean of p(FCS) as estimated by the Bayesian learner (dashed line) tracks the underlying
blocked probabilities (solid line). Note that the blocked probabilities are CS2 zoomed in from B (trials
400–600, session 3). Because blocks of stable probabilities are short, however, the estimated probabilities
never quite reach their true values during a given block. Note that the estimates change
rapidly at block transitions. When an unexpected stimulus occurs, the estimates briefly move toward
p0.5. Note
den Ouden et al. 2010, J. Neurosci .

39. Stimulus-independent prediction error
Putamen
Premotor cortex
p < 0.05 (SVC)
p < 0.05
(cluster-level whole- brain corrected)
p(F)
p(H) -2
-1.5 -1
-0.5
BOLD resp. (a.u.)
p(F)
p(H) -2
-1.5 -1
-0.5
BOLD resp. (a.u.)
den Ouden et al. 2010, J. Neurosci .

40. Prediction error (PE) activity in the putamen
PE during
reinforcement learning
O'Doherty et al , Science
PE during
incidental
sensory learning
den Ouden et al , Cerebral Cortex
According to the free energy principle (and other learning theories):
synaptic plasticity during learning = PE dependent changes in connectivity

41. Prediction error in PMd: cause or effect?
Model 1
Model 2
den Ouden et al. 2010, J. Neurosci .

42. Prediction error gates visuo-motor connections
Modulation of visuo-motor connections by striatal PE activity
Influence of visual areas on premotor cortex:
stronger for surprising stimuli
weaker for expected stimuli
p(H)
p(F)
PUT
d = 0.010 0.003
p = 0.010
d = 0.011 0.004
p = 0.017
PMd
PPA
FFA
den Ouden et al. 2010, J. Neurosci .

43. Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI
Stochastic DCM
Embedding computational models in DCMs
Integrating tractography and DCM

44. Diffusion-tensor imaging
Parker & Alexander, 2005,
Phil. Trans. B

45. Probabilistic tractography: Kaden et al. 2007, NeuroImage
computes local fibre orientation density by 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)

46. Integration of tractography and DCM
low probability of anatomical connection
 small prior variance of effective connectivity parameter R1 R2
high probability of anatomical connection
 large prior variance of effective connectivity parameter
Stephan, Tittgemeyer et al. 2009, NeuroImage

47.  anatomical connectivity  DCM structure
LD|LVF
probabilistic
tractography LG
left
right FG
 anatomical connectivity FG
(x3) FG
(x4) LD LD LG
(x1) LG
(x2)
 DCM structure
LD|RVF
RVF
stim.
BVF
stim.
LVF
stim.
 connection-specific priors for coupling parameters
Stephan, Tittgemeyer et al. 2009, NeuroImage

48. Connection-specific prior variance  as a function of anatomical connection probability 
64 different mappings by systematic search across hyper-parameters  and 
yields anatomically informed (intuitive and counterintuitive) and uninformed priors

50. Stephan, Tittgemeyer et al. 2009, NeuroImage

51. Methods papers on DCM for fMRI and BMS – part 1
Daunizeau J., Friston K. J., Kiebel S. J. ; Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models, Physica D (2009) 238:
Chumbley JR, Friston KJ, Fearn T, Kiebel SJ (2007) A Metropolis-Hastings algorithm for dynamic causal models. Neuroimage 38:
Daunizeau J, David, O, Stephan KE (2010) Dynamic Causal Modelling: A critical review of the biophysical and statistical foundations. NeuroImage, in press.
Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. NeuroImage 19:
Kasess CH, Stephan KE, Weissenbacher A, Pezawas L, Moser E, Windischberger C (2010) Multi-Subject Analyses with Dynamic Causal Modeling. NeuroImage 49:
Kiebel SJ, Kloppel S, Weiskopf N, Friston KJ (2007) Dynamic causal modeling: a generative model of slice timing in fMRI. NeuroImage 34:
Marreiros AC, Kiebel SJ, Friston KJ (2008) Dynamic causal modelling for fMRI: a two-state model. NeuroImage 39:
Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Comparing dynamic causal models. NeuroImage 22:
Penny WD, Stephan KE, Mechelli A, Friston KJ (2004b) Modelling functional integration: a comparison of structural equation and dynamic causal models. NeuroImage 23 Suppl 1:S
Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing Families of Dynamic Causal Models. PLoS Computational Biology, in press.

52. Methods papers on DCM for fMRI and BMS – part 2
Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI responses. Curr Opin Neurobiol 14:
Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM. NeuroImage 38:
Stephan KE, Harrison LM, Kiebel SJ, David O, Penny WD, Friston KJ (2007) Dynamic causal models of neural system dynamics: current state and future extensions. J Biosci 32:
Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM. Neuroimage 38:
Stephan KE, Kasper L, Harrison LM, Daunizeau J, den Ouden HE, Breakspear M, Friston KJ (2008) Nonlinear dynamic causal models for fMRI. NeuroImage 42:
Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009) Bayesian model selection for group studies. NeuroImage 46:
Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009) Tractography-based priors for dynamic causal models. NeuroImage 47:
Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, Friston KJ (2010) Ten simple rules for Dynamic Causal Modelling. NeuroImage 49:

53. Thank you