Klaas Enno Stephan
Zurich SPM Course 2014
14 February 2014

Overview
• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models into DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions

Generative models
  m ) p

Advantages:
1. force us to think mechanistically: how were the data caused?
2. allow one to generate synthetic data.
3. Bayesian perspective → inversion & model evidence

Dynamic Causal Modeling (DCM) fMRI
Hemodynamic forward model: neural activity  BOLD simple neuronal model complicated forward model
Electromagnetic forward model: neural activity  EEG
MEG
LFP
Neural state equation: dx dt

F ( x , u ,

) EEG/MEG complicated neuronal model simple forward model inputs

Bayesian system identification
Neural dynamics
Observer function u ( t ) dx dt
  y
    p ( y |

, m ) p (

, m )


N ( g (

),

(

))
N (


,


)
Design experimental inputs
Define likelihood model
Specify priors
Inference on model structure
Inference on parameters p ( y | m )
  p ( y |

, m ) p (

) d
 p (

| y , m )
 p ( y |

, m ) p (

, m ) p ( y | m )
Invert model
Make inferences

VB in a nutshell (mean-field approximation)
 Neg. free-energy approx. to model evidence.
 Mean field approx.
 Maximise neg. free energy wrt. q = minimise divergence, by maximising variational energies ln

|

F KL q
      
F
 ln
    q

 y

       m
 p
 y
       q
   exp
   exp ln
    q q
   exp
   exp ln
    q
 Iterative updating of sufficient statistics of approx. posteriors by gradient ascent.

Generative models p
 
| ,
      
| m

• any DCM = a particular generative model of how the data (may) have been caused
• modelling = comparing competing hypotheses about the mechanisms underlying observed data
 model space: a priori definition of hypothesis set is crucial
 model selection: determine the most plausible hypothesis (model), given the data
 inference on parameters: e.g., evaluate consistency of how model mechanisms are implemented across subjects
• model selection  model validation!
 model validation requires external criteria (external to the measured data)

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

Bayesian model selection (BMS)
Model evidence:
( | )
     log ( | )
 p y
 m

KL q
   
| m




KL q
   
| ,


 accounts for both accuracy and complexity of the model a measure of generalizability y
Gharamani, 2004 all possible datasets
Various approximations, e.g.:
- negative free energy, AIC, BIC
McKay 1992, Neural Comput.
Penny et al. 2004a, NeuroImage

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 log p ( y | m )
 accuracy ( m )
 complexity ( m )
 log p ( y |

, m )
 complexity ( m )
No. of parameters
In SPM2 & SPM5, interface offers 2 approximations:
Akaike Information Criterion: AIC
 log p ( y |

, m )
 p
Bayesian Information Criterion: BIC
 log p ( y |

, m )
 p log
2
N
No. of data points
Penny et al. 2004a, NeuroImage

The (negative) free energy approximation
• Under Gaussian assumptions about the posterior (Laplace approximation):
   

   
| m
 

 

   
| ,



F

 log ( | )
 

   
| ,



  

   
| m


 accuracy complexity

The complexity term in
F
• In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies.
KL
 q (

), p (

| m )


1
2 ln C


1
2 ln C

| y

1
2



| y
 

T
 


| y
 


• 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: Since SPM8, only
F is used for model selection !

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 yes no optimal model structure assumed to be identical across subjects?
yes
FFX BMS no
RFX BMS
BMA
FFX BMS RFX BMS
Stephan et al. 2010, NeuroImage
FFX analysis of parameter estimates
(e.g. BPA)
RFX analysis of parameter estimates
(e.g. t-test, ANOVA)

Random effects BMS for heterogeneous groups

Dirichlet parameters

= “occurrences” of models in the population r ~ Dir ( r ;

) Dirichlet distribution of model probabilities r m k
~ m k p (
~ m k m k p (
~ m
1
| m p k p )
(
| m k p )
|
~ Mult ( p ) m ; 1 , r ) y
1
~ y
1 p
~
( y
2 y
1 p
~
(
| y
1 y m
1 p
1
(
~
|
) y m
1
2 p (
|
) m y
1
2
|
) m
1
)
Multinomial distribution of model labels m
Measured data y
Model inversion by Variational
Bayes or MCMC
Stephan et al. 2009a, NeuroImage
Penny et al. 2010, PLoS Comput. Biol.

Inference about DCM parameters:
Bayesian 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 chosen threshold γ : c T η
θ|y
) is above a p
 
N c
T

 y
  c
T
C
 y c
• By default, γ is chosen as zero ("does the effect exist?").

Inference about DCM parameters:
Random effects group analysis (classical)
• In analogy to “random effects” analyses in SPM, 2 nd level analyses can be applied to DCM parameters:
Separate fitting of identical models for each subject
Selection of model parameters of interest one-sample t-test: parameter > 0 ?
paired t-test: parameter 1 > parameter 2 ?
rmANOVA: e.g. in case of multiple sessions per subject

Bayesian Model Averaging (BMA)
• uses the entire model space considered (or an optimal family of models)
• averages parameter estimates, weighted by posterior model probabilities
• particularly useful alternative when
– none of the models (subspaces) considered clearly outperforms all others
– when comparing groups for which the optimal model differs
 p
  n
| y
1..
N
 m
 p
  n
| n
,
 
|
1..
N

NB: p ( m | y
1..
N
) can be obtained by either FFX or RFX BMS
Penny et al. 2010, PLoS Comput. Biol.

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 yes no optimal model structure assumed to be identical across subjects?
yes
FFX BMS no
RFX BMS
BMA
FFX BMS RFX BMS
Stephan et al. 2010, NeuroImage
FFX analysis of parameter estimates
(e.g. BPA)
RFX analysis of parameter estimates
(e.g. t-test, ANOVA)

Overview
• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models into DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions

The evolution of DCM in SPM
• DCM is not one specific model, but a framework for Bayesian inversion of dynamic system models
• The default implementation in SPM is evolving over time
– improvements of numerical routines (e.g., for inversion)
– change in parameterization (e.g., self-connections, hemodynamic states in log space)
– change in priors to accommodate new variants (e.g., stochastic DCMs, endogenous DCMs etc.)
To enable replication of your results, you should ideally state which SPM version (release number) you are using when publishing papers.
The release number is stored in the DCM.mat file.

 y activity x
1
( t ) driving input u
1
( t ) t y y

 activity x
2
( t ) activity x
3
( t ) neuronal states
BOLD modulatory input u
2
( t )
The classical DCM: a deterministic, one-state, bilinear model t y
λ x hemodynamic model integration
Neural state equation
 x

( A
  u j
B
( j )
) x

Cu endogenous connectivity modulation of connectivity direct inputs
B
(
A j )
C



 x 
 x

 u j
 x 
 u
 x 
 x

Factorial structure of model specification in DCM10
• Three dimensions of model specification:
– bilinear vs. nonlinear
– single-state vs. two-state (per region)
– deterministic vs. stochastic
• Specification via GUI.

driving input bilinear DCM modulation non-linear DCM driving input modulation
Two-dimensional Taylor series (around x
0
=0, u
0
=0):
 f f
 f f f f ( ( x , , u ) ) f f ( ( , , 0 ) )
 x
 u f f
 x
 u
Bilinear state equation: dx dt



A
 i m 

1 u i
B
( i )

 x

Cu

...
2 x f
2 x
2

...
2
Nonlinear state equation: dx dt


 A
 i m 

1 u i
B
( i )  j n 

1 x j
D
( j )

 x

Cu

u
1 x
1 u
2 x
3
Nonlinear dynamic causal model (DCM) dx dt


 A
 i m 

1 u i
B
( i )  j n 

1 x j
D
( j )

 x

Cu
Stephan et al. 2008, NeuroImage x
2
3
2
1
0
0
4
3
2
1
0
-1
0
3
2
1
0
0
0.3
0.2
0.1
0
0
0.6
0.4
0.2
0
0
0.4
0.3
0.2
0.1
0
0
10
10
10
10
10
Neural population activity
10 20 30 40 50 60 70 80 90 100
20 30 40 50
20 30 40 50 fMRI signal change (%)
20 30 40 50
20 30 40 50
20 30 40 50
60 70 80 90 100
60 70 80 90 100
60 70 80 90 100
60 70 80 90 100
60 70 80 90 100

attention
0.10
stim
0.26
PPC
V1
0.26
1.25
0.13
V5
0.46
0.50
0.39
motion
Stephan et al. 2008, NeuroImage
MAP = 1.25
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2 -1 0 1 2 3 4 p ( D
V
PPC
5 , V 1

0 | y )

99 .
1 %
5

Two-state DCM
Single-state DCM input u x
1
Two-state DCM x
1
E x
1
I x
1
E x
1
I x 
 ij


 x

Cu
A ij
 uB ij
 








11
N 1




1

N

NN



 x




 x
 x
N
1




Marreiros et al. 2008, NeuroImage
 ij
  x
   x

Cu
  ij
  exp( A ij
   uB ij
 
)
 
 



EE
11
IE
11






0
EE
N 1


0
0
EI
11
II
11
Extrinsic
(between-region) coupling





1
0
EE
N
EE
NN
 IE
NN
0
0

 EE
NN
 II
NN







Intrinsic
(within-region) coupling x







 x
1
E x x x

1
E
N
I
I
N








Stochastic DCM dx

( A
 dt v u

 j u B ) x Cv j
  
• all states are represented in generalised coordinates of motion
• random state fluctuations w ( x ) account for endogenous fluctuations, have unknown precision and smoothness
 two hyperparameters
• fluctuations w ( v ) induce uncertainty about how inputs influence neuronal activity
• can be fitted to resting state data
Li et al. 2011, NeuroImage
2
1
0
-1
-2
-3
0
1.3
1.2
1.1
1
0.9
0.8
0
0.1
0.05
0
-0.05
-0.1
0
0
-0.5
-1
0
0.5
1
Estimates of hidden causes and states
(Generalised filtering) inputs or causes - V2
200 1000 400 600 hidden states - neuronal
800 1200 excitatory signal
200 1000 1200 400 600 800 hidden states - hemodynamic flow volume dHb
200 1000 1200 400 600 predicted BOLD signal
800 observed predicted
200 400 600 time (seconds)
800 1000 1200

• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions

Prediction errors drive synaptic plasticity
PE ( t )
 x
3 x
1 a
0
R
McLaren 1989 w t
 w
0
  k t 

1
PE k w t
 a
0
   t
0 x
2
PE
  synaptic plasticity during learning = f
(prediction error)

Learning of dynamic audio-visual associations
Conditioning Stimulus Target Stimulus or or
0
CS
200
TS Response
400 600
Time (ms)
800 2000
±
650
1
0.8
0.6
0.4
0.2
0
0 200 400 trial
600 800 1000
CS
1
CS
2 den Ouden et al. 2010, J. Neurosci.

Hierarchical Bayesian learning model prior on volatility volatility probabilistic association v t-1 k r t v t r t+1 observed events u t u t+1 p
 

1 p
 v t

1
| v t
, k

~ N
 v t
, exp( k )
 p
 r t

1
| r t
, v t

~ Dir
 r t
, exp( v t
)

Behrens et al. 2007, Nat. Neurosci.

Explaining RTs by different learning models
Reaction times
450
440
430
420
410
400
390
0.1
0.3
0.5
0.7
0.9
p(outcome)
5 alternative learning models:
• categorical probabilities
• hierarchical Bayesian learner
• Rescorla-Wagner
• Hidden Markov models
(2 variants) den Ouden et al. 2010, J. Neurosci.
1
0.8
0.6
True
Bayes Vol
HMM fixed
HMM learn
RW
0.4
0.2
0
400 440 480
Trial
520 560 600
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Bayesian model selection: hierarchical Bayesian model performs best
Categorical model
Bayesian learner
HMM (fixed) HMM (learn) Rescorla-
Wagner

Stimulus-independent prediction error
Putamen Premotor cortex
0
-0.5
-1
-1.5
-2 p(F) p(H) p < 0.05
(SVC )
0
-0.5
-1
-1.5
-2 p(F) p(H) p < 0.05
(cluster-level wholebrain corrected) den Ouden et al. 2010, J. Neurosci .

Prediction error (PE) activity in the putamen
PE during active sensory learning
PE during incidental sensory learning den Ouden et al. 2009,
Cerebral Cortex p < 0.05
(SVC )
PE during reinforcement learning PE = “teaching signal” for synaptic plasticity during learning
O'Doherty et al. 2004,
Science
Could the putamen be regulating trial-by-trial changes of task-relevant connections?

Prediction errors control plasticity during adaptive cognition
• Influence of visual areas on premotor cortex:
– stronger for surprising stimuli
– weaker for expected stimuli
Hierarchical
Bayesian learning model
p = 0.010
PUT
PMd
p = 0.017
PPA FFA den Ouden et al. 2010, J. Neurosci .

• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions

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

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)

Integration of tractography and DCM
Stephan, Tittgemeyer et al.
2009, NeuroImage
R
1
R
2
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2 -1 0 1 low probability of anatomical connection
 small prior variance of effective connectivity parameter
2
R
1
R
2
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2 -1 0 1 high probability of anatomical connection
 large prior variance of effective connectivity parameter
2

Proof of concept study
FG FG probabilistic tractography

13

15.7%
FG left
 
34
6.5%
FG right
 
24
43.6%
Stephan, Tittgemeyer et al.
2009, NeuroImage
LG
 DCM
LG
 connectionspecific priors
2 for coupling
1.8
parameters
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-3
  v

34.2%
0.5268
-2 -1
LG left

12

34.2%
LG right
 anatomical connectivity

 
6.5% v

0.0384
0
 
15.7% v

0.1070
1
 
43.6% v

0.7746
2 3

Connection-specific prior variance
 as a function of anatomical connection probability

  ij m 9: a=-8,b=-28

0
1
 
0 exp( m 1: a=-32,b=-32
1
0.5
  m 2: a=-16,b=-32
1
0.5
m 3: a=-16,b=-28
1
0.5
m 4: a=-12,b=-32
1
0.5
m 5: a=-12,b=-28
1
0.5
m 6: a=-12,b=-24
1
0.5
m 7: a=-12,b=-20
1
0.5
1
0.5
m 8: a=-8,b=-32
1
0.5
0
0
1
0.5
0.5
m 10: a=-8,b=-24  1
0
0 0.5
1
0
0 0.5
1
0
0
1 m 11: a=-8,b=-20
)
0.5
0
0 ij 0.5
0.5
1 m 12: a=-8,b=-16
1
0
0
0.5
0.5
1
1
0
0
0.5
0.5
m 13: a=-8,b=-12
1
0
0 0.5
1
1
0.5
0
0 0.5
m 14: a=-4,b=-32
1
0
0 0.5
1
1
0.5
0
0 0.5
m 15: a=-4,b=-28
1
0
0 0.5
1
1
0.5
0
0 0.5
m 16: a=-4,b=-24
1
0
0 0.5
1
1
0.5
0
0 0.5
m 17: a=-4,b=-20
1
0
0 0.5
1
1
0.5
0
0 0.5
m 18: a=-4,b=-16
1
0
0 0.5
1
1 m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0 m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
1 1 1 1 1 1 1 1 1
• 64 different mappings by systematic search across hyperparameters
 and

• yields anatomically informed (intuitive and counterintuitive) and uninformed priors
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0 0.5
1 m 28: a=0,b=-16
1
0
0 0.5
1 m 29: a=0,b=-12
1
0
0 0.5
1 m 30: a=0,b=-8
1
0
0 0.5
1 m 31: a=0,b=-4
1
0
0 0.5
m 32: a=0,b=0
1
1
0
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10 20 30 model
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Models with anatomically informed priors (of an intuitive form)
10 20 30 model
40
50 60
50 60
50 60

m 1: a=-32,b=-32
1 m 2: a=-16,b=-32
1 m 3: a=-16,b=-28
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1 m 5: a=-12,b=-28
1 m 6: a=-12,b=-24
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1 1 m 8: a=-8,b=-32
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m 63 & m 64
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Models with anatomically informed priors (of an intuitive form) were clearly superior to anatomically uninformed ones: Bayes Factor >10 9

• Generative models & analysis options
• Extended DCM for fMRI: nonlinear, two-state, stochastic
• Embedding computational models in DCMs
• Integrating tractography and DCM
• Applications of DCM to clinical questions

Model-based predictions for single patients model structure
BMS set of parameter estimates model-based decoding

BMS: Parkison‘s disease and treatment
Age-matched controls
Rowe et al. 2010,
NeuroImage
PD patients on medication
Selection of action modulates connections between PFC and SMA
PD patients off medication
DA-dependent functional disconnection of the SMA

Model-based decoding by generative embedding step 1 — model inversion
A step 2 — kernel construction
A → B
A → C
B → B
B → C
C
B measurements from an individual subject subject-specific inverted generative model subject representation in the generative score space
A step 4 — interpretation
B
C jointly discriminative model parameters
Brodersen et al. 2011, PLoS Comput. Biol.
separating hyperplane fitted to discriminate between groups step 3 — support vector classification

Discovering remote or “hidden” brain lesions

Discovering remote or “hidden” brain lesions

Model-based decoding of disease status: mildly aphasic patients (N=11) vs. controls (N=26)
Connectional fingerprints from a 6-region DCM of auditory areas during speech perception
PT PT
HG
(A1)
HG
(A1)
MGB
S
MGB
S
Brodersen et al. 2011, PLoS Comput. Biol.

PT
Model-based decoding of disease status: aphasic patients (N=11) vs. controls (N=26)
Classification accuracy
PT
HG
(A1)
HG
(A1)
MGB auditory stimuli
Brodersen et al. 2011, PLoS Comput. Biol.
MGB

Multivariate searchlight classification analysis
Voxel-based feature space
Generative embedding using DCM
Generative score space
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0
0.1
0
-0.1
-0.5
-0.1
-0.5
0
Voxel (-42,-26,10) mm
0.5
0
10
-0.15
-0.2
-0.25
-0.3
patients
-10 -10
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0 0
Voxel (-56,-20,10) mm
0.5
10
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-0.3
-0.35
-0.4
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-0.2
L.MGB

0
L.MGB
-0.5
0.5
0.5
0
0
-0.5
0
R.HG

L.HG

Definition of ROIs
Are regions of interest defined anatomically or functionally?
anatomically functionally
A 1 ROI definition and n model inversions unbiased estimate
Functional contrasts
Are the functional contrasts defined across all subjects or between groups?
across subjects between groups

B 1 ROI definition and n model inversions slightly optimistic estimate: voxel selection for training set and test set based on test data
C Repeat n times:
1 ROI definition and n model inversions unbiased estimate
 D 1 ROI definition and n model inversions highly optimistic estimate: voxel selection for training set and test set based on test data and test labels
 E Repeat n times:
1 ROI definition and 1 model inversion slightly optimistic estimate: voxel selection for training set based on test data and test labels
F Repeat n times:
1 ROI definition and n model inversions unbiased estimate
Brodersen et al. 2011, PLoS Comput. Biol.

Generative embedding for detecting patient subgroups
Brodersen et al. 2014, NeuroImage: Clinical

Generative embedding of variational
Gaussian Mixture Models
Supervised:
SVM classification
Unsupervised:
GMM clustering
71 % number of clusters number of clusters
• 42 controls vs. 41 schizophrenic patients
• fMRI data from working memory task (Deserno et al. 2012, J. Neurosci)
Brodersen et al. 2014, NeuroImage: Clinical

Detecting subgroups of patients in schizophrenia
Optimal cluster solution
• three distinct subgroups (total N=41)
• subgroups differ (p < 0.05) wrt. negative symptoms on the positive and negative symptom scale (PANSS)
Brodersen et al. 2014, NeuroImage: Clinical

TAPAS www.translationalneuromodeling.org/tapas
• PhysIO: physiological noise correction
• Hierarchical Gaussian
Filter (HGF)
• Variational Bayesian linear regression
• Mixed effects inference for classification studies
Brodersen et al. 2011, PLoS CB
Kasper et al., in prep.
Mathys et al. 2011,
Front. Hum. Neurosci.
Brodersen et al. 2012, J Mach Learning Res

Methods papers: DCM for fMRI and BMS – part 1
• Brodersen KH, Schofield TM, Leff AP, Ong CS, Lomakina EI, Buhmann JM, Stephan KE (2011) Generative embedding for model-based classification of fMRI data. PLoS Computational Biology 7: e1002079.
• Brodersen KH, Deserno L, Schlagenhauf F, Lin Z, Penny WD, Buhmann JM, Stephan KE (2014) Dissecting psychiatric spectrum disorders by generative embedding. NeuroImage: Clinical 4: 98-111
• Daunizeau J, David, O, Stephan KE (2011) Dynamic Causal Modelling: A critical review of the biophysical and statistical foundations. NeuroImage 58: 312-322.
• Daunizeau J, Stephan KE, Friston KJ (2012) Stochastic Dynamic Causal Modelling of fMRI data: Should we care about neural noise? NeuroImage 62: 464-481.
• Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. NeuroImage 19:1273-1302.
• Friston K, Stephan KE, Li B, Daunizeau J (2010) Generalised filtering. Mathematical Problems in Engineering 2010:
621670.
• Friston KJ, Li B, Daunizeau J, Stephan KE (2011) Network discovery with DCM. NeuroImage 56: 1202 –1221.
• Friston K, Penny W (2011) Post hoc Bayesian model selection. Neuroimage 56: 2089-2099.
• Kiebel SJ, Kloppel S, Weiskopf N, Friston KJ (2007) Dynamic causal modeling: a generative model of slice timing in fMRI. NeuroImage 34:1487-1496.
• Li B, Daunizeau J, Stephan KE, Penny WD, Friston KJ (2011). Stochastic DCM and generalised filtering. NeuroImage
58: 442-457
• Marreiros AC, Kiebel SJ, Friston KJ (2008) Dynamic causal modelling for fMRI: a two-state model. NeuroImage
39:269-278.
• Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Comparing dynamic causal models. NeuroImage 22:1157-
1172.
• 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:S264-274.

Methods papers: DCM for fMRI and BMS – part 2
• Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing Families of Dynamic
Causal Models. PLoS Computational Biology 6: e1000709.
• Penny WD (2012) Comparing dynamic causal models using AIC, BIC and free energy. Neuroimage 59: 319-330.
• Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI responses. Curr Opin Neurobiol
14:629-635.
• Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM.
NeuroImage 38:387-401.
• 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 32:129-144.
• Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM.
NeuroImage 38:387-401.
• Stephan KE, Kasper L, Harrison LM, Daunizeau J, den Ouden HE, Breakspear M, Friston KJ (2008) Nonlinear dynamic causal models for fMRI. NeuroImage 42:649-662.
• Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009a) Bayesian model selection for group studies.
NeuroImage 46:1004-1017.
• Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009b) Tractography-based priors for dynamic causal models. NeuroImage 47: 1628-1638.
• Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, Friston KJ (2010) Ten simple rules for Dynamic
Causal Modelling. NeuroImage 49: 3099-3109.