1. 5th March 2008 Andreina Mendez Stephanie Burnett
DCM Theory
5th March 2008
Andreina Mendez
Stephanie Burnett

2. Recap Last session: PPIs (psycho-physiological interactions)
Functional vs. effective connectivity
Functional connectivity: temporal correlation between spatially remote neurophysiological events
Effective connectivity: the influence that the elements of a neuronal system exert over each other
Standard fMRI analysis
PPIs, SEM, DCM

3. Introduction: DCM and its place in the methods family tree
Standard fMRI analysis
The BOLD signal (related to brain activity in some implicit way) in some set of brain is correlated, and is also correlated with your task
Task
BOLD signal
Of course you don’t do that, you cite converging evidence and blah blah, but still…
“This is a frontoparietal
network collection of brain regions involved in activated while processing coffee”

4. Introduction: DCM and its place in the methods family tree
PPIs
Represent how the (experimental) context modulates connectivity between a brain region of interest, and anywhere else
E.g. (Whatever gives rise to the) signal in one brain region (V1) will lead to a signal in V5, and the strength of this signal in V5 depends on attention V1 V5
attention
DCM models how neuronal activity
causes the BOLD signal (forward model)
i.e. your conclusions are about neural processes
DCM models bidirectional and modulatory
interactions, between multiple brain regions

5. Introduction: DCM and its place in the methods family tree
Your experimental task causes neuronal activity in an input brain region, and this generates a BOLD signal.
The neuronal activity in this input region, due to your task, then causes/modulates neuronal activity in other brain regions (with resultant patterns of BOLD signals across the brain)
“This sounds more like something I’d enjoy writing up!”

6. DCM basics
DCM models interactions between neuronal populations (fMRI/MEG/EEG)
The aim is to estimate, and make inferences about,
1. The coupling among brain areas
2. How that coupling is influenced by changes in experimental context

7. DCM basics
DCM starts with a realistic model of how brain regions interact
Adds a forward model of how neuronal activity causes the signals you observe (e.g. BOLD)
…and estimates the parameters in your model (effective connectivity), given your observed data
Neural and hemodynamic models
(more later on)

8. DCM basics DCM treats the brain as a nonlinear dynamic system
The system has inputs, state variables, and outputs
Your experiment is a designed perturbation of the system
A primary visual area (V1) receives
‘photic’ input and has reciprocal connections with the motionsensitive
area V5. ‘Motion’ input modulates the forward
connection from V1 to V5. Area V5 has reciprocal connections
with superior parietal cortex (SPC). ‘Attention’ input modulates
the top-down connection from SPC to V5.

9. DCM basics
Inputs
In functional connectivity models (e.g. standard fMRI analysis), conceptually your input can enter anywhere
In effective connectivity models (e.g. DCM), input only enters at certain places

10. DCM basics Inputs can exert their influence in two ways
1. Direct influence
e.g. visual input to V1
2. Vicarious influence
e.g. attentional modulation of the coupling between SPC and V5

11. DCM basics State variables
neuronal activities, and other neuro/biophysical variables needed to form the outputs (later on)
Neuronal priors
Hemodynamic priors

12. DCM basics
Output
the BOLD signal you’ve measured in the brain regions specified in your model

13. Models of effective connectivity = system models.
System = set of elements which interact in a spatially and temporally specific fashion.
System dynamics = change of state vector in time
Causal effects in the system:
interactions between elements
external inputs u
System parameters  : specify the nature of the interactions
general state equation for non- autonomous systems
overall system state represented by state variables
change of state vector in time
Mechanistic or system models

14. Example: linear dynamic systemFG
left FG
right z3 z4
LG = lingual gyrus
FG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF) visual field. LG
left LG
right z1 z2
RVF
LVF u2 u1
state changes
effective
connectivity
system state
input
parameters
external inputs

15. Bilinear state equation in DCM
state changes
intrinsic
connectivity
modulation of
connectivity
system state
direct
inputs
m external inputs

16. Bilinear state equation in DCM
state changes
intrinsic
connectivity
modulation of
connectivity
system state
direct
inputs
m external inputs

17. Extension: bilinear dynamic systemLG
left
right
RVF
LVF FG z1 z2 z4 z3 u2 u1
CONTEXT u3

18. DCM for fMRI: the basic idea
Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI).
The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ). λ z y
The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.

19. The hemodynamic “Balloon” model
5 hemodynamic parameters:
Empirically determined a priori distributions.
Computed separately for each area (like the neural parameters).
Buxton et al. 1998
Hemodynamic parameters, e.g. rho is the resting oxygen extraction fraction.

20. Priors on biophysical parameters

21. Conceptual overview Input u(t) neuronal z states λ y y y y BOLD
Neural state equation
The bilinear model
effective connectivity
modulation of
connectivity
Input
u(t)
direct inputs c1
integration
neuronal
states λ z y
b23
Schematic summary of the conceptual basis of DCM. The dynamics in a system of
interacting neuronal populations (blue boxes), which are not directly observable by fMRI, is
modeled using a bilinear state equation (green box). Integrating the state equation gives predicted
neural dynamics (z) that enter a model of the hemodynamic response (λ) to give predicted BOLD
responses (y) (green boxes). The parameters at both neural and hemodynamic levels are adjusted
such that the diferences between predicted and measured BOLD series are minimized. Critically,
the neural dynamics are determined by experimental manipulations. These enter the model in
the form of external or driving inputs. Driving inputs (u1; e.g. sensory stimuli) elicit local
responses directly that are propagated through the system according to the intrinsic connections.
The strengths of these connections can be changed by modulatory inputs (u2; e.g. changes in
cognitive set, attention, or learning).
a12
activity
z2(t)
activity
z1(t)
activity
z3(t)
hemodynamic
model y y y
BOLD
Friston et al. 2003, NeuroImage

22. Estimating model parameters
Bayes Theorem
posterior  likelihood ∙ prior ) ( | q p y ×
DCMs are biologically plausible (i.e. complicated) - they have lots of free parameters
A Bayesian framework is a good way to embody the constraints on these parameters

23. Use Bayes’ theorem to estimate model parameters
posterior  likelihood ∙ prior ) ( | q p y ×
Priors – empirical (haemodynamic parameters) and non-empirical (eg. shrinkage priors, temporal scaling)
Likelihood derived from error and confounds (eg. drift)
Calculate the Posterior probability for each effect, and the probability that it exceeds a set threshold
You need shrinkage priors to estimate the posterior – they make the system stable. That’s why the prior distribution is so narrow.
Inferences about the strength (= speed) of connections between the brain regions in your model

24. Interpretation of parameters
EM algorithm – works out the parameters in a model
Bayesian model selection to test between alternative models
Single subject analysis
Use the cumulative normal distribution to test the probability with which a certain parameter is above a chosen threshold γ: 
ηθ|y

25. Model comparison and selection
A good model of your data will balance model fit with complexity (overfitting models noise)
You find this by taking evidence ratios (the “Bayes factor”)
The “Bayes factor” is a summary of the evidence in favour of one model as opposed to another

26. Bayesian Model Selection
Bayes’ theorem:
Model evidence:
The log model evidence can be represented as:
Bayes factor:
Basically this is how you maximise model evidence. The best model has the highest free energy.
Penny et al. 2004, NeuroImage

27. Interpretation of parameters
- Group analysis:
Like “random effects” analysis in SPM, 2nd level analysis can be applied to DCM parameters:
Separate fitting of identical models for each subject
Selection of bilinear parameters of interest
One sample t-test:
Parameter > 0?
Paired t-test:
Parameter 1 > parameter 2?
rm ANOVA:
For multiple sessions
per subject

28. New stuff in DCM
1. DCM now accounts for the slice timing problem

29. Extension I: Slice timing model
Potential timing problem in DCM: temporal shift between regional time series because of multi-slice acquisition 2
slice acquisition 1
visual input
Solution:
Modelling of (known) slice timing of each area.
Slice timing extension now allows for any slice timing differences.
Long TRs (> 2 sec) no longer a limitation.
(Kiebel et al., 2007)

30. New stuff in DCM
1. DCM now accounts for the slice timing problem (SPM5)
2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.)

31. Extension II: Two-state model
input
Single-state DCM
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
Two-state DCM

32. New stuff in DCM
1. DCM now accounts for the slice timing problem (SPM5)
2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.)
3. Biological plausibility: more complex balloon model (SPM5)
4. Non-linear version of DCM as well as bilinear (SPM8)

33. Expectation-maximization Posterior distribution
fMRI DCM Diagram
Dynamic Causal Modelling of fMRI
Network
dynamics
Haemodynamic
response
Priors
Model
comparison
State space
Model
In the same way like SPM for fMRI
Use SPM2‘s methods!
However, model needs to be adapted to make proper inference
Comparison with fMRI analysis to aid illustration
Also, conventional model (to be shown) in the same framework
Model inversion
using
Expectation-maximization
Posterior distribution
of parameters
fMRI
data y

34. SUMMARY DCM good because Next week: practical issues in DCM
Causal – models effective connectivity, not functional connectivity
Neuronally plausible
Forward model of how neuronal activity causes BOLD signal
Inputs only enter at certain places; can model vicarious (modulatory) input; can have reciprocal connections and loops
Next week: practical issues in DCM

35. REFERENCES
Karl J. Friston. Dynamic Causal Modelling. Human brain function. Chapter 22. Second Edition.
K.J Friston, L. Harrison and W. Penny. Dynamic Causal Modelling. Neuroimage 2003; 19:
SPM Manual
Last year’s presentation

36. THANK YOU
Special thanks to:
- Andre Marreiros
- Maria Joao