1. Dynamic Causal Modelling (DCM) for fMRI
Andre Marreiros
Wellcome Trust Centre for Neuroimaging
University College London

2. Thanks to... Stefan Kiebel Lee Harrison Klaas Stephan Karl Friston
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

3. Overview Dynamic Causal Modelling of fMRI Definitions & motivation
The neuronal model (bilinear dynamics)
The Haemodynamic 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
Estimation: Bayesian framework
DCM latest Extensions

4. Principles of organisation
Functional specialization
Functional integration
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

5. Neurodynamics: 2 nodes with inputz1 z2
activity in is coupled to via
coefficient

6. Neurodynamics: positive modulationz1 z2
modulatory input u2 activity through the coupling

7. Neurodynamics: reciprocal connectionsz1 z2
reciprocal
connection
disclosed by u2

8. Haemodynamics: reciprocal connections
Simulated response
Bold
Response
a12
Bold
Response
a22
green: neuronal activity
red: bold response

9. Haemodynamics: reciprocal connections
Bold
with
Noise added
a12
Bold
with
Noise added
a22
green: neuronal activity
red: bold response

10. Example: modelled BOLD signal
Underlying model (modulatory inputs not shown)
left LG FG
left FG
right LG
left LG
right
right LG
RVF
LVF
LG = lingual gyrus Visual input in the FG = fusiform gyrus - left (LVF) - right (RVF) visual field.
blue: observed BOLD signal
red: modelled BOLD signal (DCM)

11. Use differential equations to describe mechanistic model of a system
System dynamics = change of state vector in time
Causal effects in the system:
interactions between elements
external inputs u
System parameters  : specify exact form of system
overall system state represented by state variables
change of state vector in time

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

13. Extension: bilinear dynamic systemFG
left FG
right z3 z4 LG
left LG
right z1 z2
RVF
CONTEXT
LVF u2 u3 u1

14. Bilinear state equation in DCM/fMRI
state changes
modulation of
connectivity
system state
direct
inputs
m external inputs
connectivity

15. Neuronal state equation
Conceptual overview
Neuronal state equation
The bilinear model
effective connectivity
modulation of
connectivity
Input
u(t)
direct inputs c1
b23
integration
neuronal
states λ z y
a12
activity
z2(t)
activity
z3(t)
activity
z1(t)
hemodynamic
model y y y
BOLD
Friston et al. 2003, NeuroImage

16. The hemodynamic “Balloon” model
5 hemodynamic parameters: } , { r a t g k q = h
important for model fitting, but of no interest for statistical inference
Empirically determined a priori distributions.
Computed separately for each area

17. Expectation-maximization Posterior distribution
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

18. Estimation: Bayesian framework
Models of
Hemodynamics in a single region
Neuronal interactions
Constraints on
Connections
Hemodynamic parameters
prior
likelihood term
posterior
Bayesian estimation

19. modelled BOLD response
stimulus function u
Overview: parameter estimation
neuronal state
equation
Specify model (neuronal and hemodynamic level)
Make it an observation model by adding measurement error e and confounds X (e.g. drift).
Bayesian parameter estimation using Bayesian version of an expectation-maximization algorithm.
Result: (Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y.
parameters
hidden states
state equation
ηθ|y
observation model
modelled BOLD response

20. Haemodynamics: 2 nodes with input
Dashed Line: Real BOLD response
a11
a22
Activity in z1 is coupled to
z2 via coefficient a21

21. Inference about DCM parameters: single-subject analysis
Bayesian parameter estimation in DCM: Gaussian assumptions about the posterior distributions of the parameters
Use of the cumulative normal distribution to test the probability by which a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ: 
ηθ|y

22. Model comparison and selection
Pitt & Miyung (2002), TICS
Given competing hypotheses, which model is the best?

23. Comparison of three simple models
Model 1: attentional modulation of V1→V5
Model 2: attentional modulation of SPC→V5
Model 3: attentional modulation of V1→V5 and SPC→V5
Attention
Attention
Photic
Photic
SPC
Photic
SPC
SPC
0.85
0.55
0.03
0.70
0.86
0.85
0.75
0.70
1.36
0.84 V1
1.42
0.89 V1
1.36
0.85 V1
-0.02
0.57
0.56
-0.02 V5
0.57
-0.02 V5 V5
Motion
Motion
Motion
0.23
0.23
Attention
Attention
Bayesian model selection: Model 1 better than model 2,
model 1 and model 3 equal
→ Decision for model 1: in this experiment, attention primarily modulates V1→V5

24. 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)

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

26. Extension III: Nonlinear DCM for fMRI
Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas.
The D matrices encode which of the n neural units gate which connections in the system.
attention
0.19
(100%)
Can V5 activity during attention to motion be explained by allowing activity in SPC to modulate the V1-to-V5 connection?
The posterior density of indicates that this gating existed with 97.4% confidence.
SPC
0.03
(100%)
0.01
(97.4%)
1.65
(100%) V1 V5
0.04
(100%)
motion

27. DCM uses a Bayesian framework to estimate these
Conclusions
Dynamic Causal Modelling (DCM) of fMRI is mechanistic model that is informed by anatomical and physiological principles.
DCM uses a deterministic differential equation to model neuro-dynamics (represented by matrices A,B and C)
DCM uses a Bayesian framework to estimate these
DCM combines state-equations for dynamics with observation model (fMRI: BOLD response, M/EEG: lead field).
t-contrast: tests for single dimension in parameter space
F-contrast: tests for multiple dimensions
inference at first or second level (fixed or random effects)
over conditions or groups: main effect, difference, interaction:
average over time window
parametric modulation with extrinsic variable
power comparison in time-frequency domain
DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. ERPs)