1. Dynamic Causal Modelling (DCM): Theory
Burkhard Pleger
Thanks to
Klaas Enno Stephan Functional Imaging Lab
Wellcome Dept. of Imaging Neuroscience
Institute of Neurology
University College London

2. Overview DCMs as generic models of dynamic systems
Neural and hemodynamic levels in DCM
Parameter estimation
Priors in DCM
Bayesian parameter estimation
Interpretation of parameters
Bayesian model selection

3. System analyses in functional neuroimaging
Functional specialisation
Analyses of regionally specific effects: which areas constitute a neuronal system?
Functional integration
Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system?
Functional connectivity
= the temporal correlation between spatially remote neurophysiological events
Effective connectivity
= the influence that the elements of a neuronal system exert over another
MECHANISM-FREE
MECHANISTIC MODEL

4. Models of effective connectivity
Structural Equation Modelling (SEM)
Psycho-physiological interactions (PPI)
Multivariate autoregressive models (MAR) & Granger causality techniques
Kalman filtering
Volterra series
Dynamic Causal Modelling (DCM)
Friston et al., NeuroImage 2003

5. Models of effective connectivity = system models
Models of effective connectivity = system models. But what precisely is a system?
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

6. 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

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

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

9. Overview DCMs as generic models of dynamic systems
Neural and hemodynamic levels in DCM
Parameter estimation
Priors in DCM
Bayesian parameter estimation
Interpretation of parameters
Bayesian model selection

10. 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.

11. Conceptual overview Input u(t) neuronal z states λ y y y BOLD y c1 b23
Neural 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

12. The hemodynamic “Balloon” model
5 hemodynamic parameters:
Empirically determined a priori distributions.
Computed separately for each area (like the neural parameters).

13. 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)

14. Overview DCMs as generic models of dynamic systems
Neural and hemodynamic levels in DCM
Parameter estimation
Priors in DCM
Bayesian parameter estimation
Interpretation of parameters
Bayesian model selection

15. modelled BOLD response
stimulus function u
Overview: parameter estimation
neural state
equation
Combining the neural and hemodynamic states gives the complete forward model.
An observation model includes measurement error e and confounds X (e.g. drift).
Bayesian parameter estimation
Result: Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.
parameters
hidden states
state equation
ηθ|y
observation model
modelled BOLD response

16. Priors in DCM Bayes Theorem
needed for Bayesian estimation, embody constraints on parameter estimation
express our prior knowledge or “belief” about parameters of the model
hemodynamic parameters: empirical priors
temporal scaling: principled prior, self-inhibition
coupling parameters: shrinkage priors
Bayes Theorem
posterior  likelihood ∙ prior

17. Priors in DCM, self-inhibition
self-inhibition: ensured by priors on the decay rate constant σ (ησ=1, Cσ=0.105) → these allow for neural transients with a half life in the range of 300 ms to 2 seconds NB: a single rate constant for all regions!
Identical temporal scaling in all areas by factorising A and B with σ: all connection strengths are relative to the self-connections.

18. Priors in DCM, coupling
system stability: in the absence of input, the neuronal states must return to a stable mode
→ constraints on prior variance of intrinsic connections (A)
shrinkage priors for coupling parameters (η=0) → conservative estimates!

19. Shrinkage Priors Small & variable effect Large & variable effect
Small but clear effect
Large & clear effect