1. Dynamic Causal Modeling of Endogenous Fluctuations
Glimpsing at Neuronal Connectivity through the Hemodynamic Veil? What We Can and Cannot Do with Biophysical Dynamic Models of fMRI Connectivity
Dynamic Causal Modeling of Endogenous Fluctuations
Karl Friston, Wellcome Centre for Neuroimaging, UCL,UK
Abstract
This talk will present recent advances in dynamic causal modeling (DCM); with a focus on (i) extending DCM to handle state-space models based on random differential equations and (ii) finessing the search over very large model spaces with the Savage-Dickey density ratio. The first advance enables DCM to infer on hidden (endogenous and smooth) fluctuations in neuronal and hemodynamic states. This permits the use of DCM in task-free designs (e.g., resting-state studies). The second advance allows one to discover networks by searching exhaustively over all combinations of connections. These developments resolve the problems faced by causal modeling based on temporal precedence and the theory of Martingales (e.g., Structural Equation Modeling and Granger causality) and circumvents the use of directed acyclic graphs (e.g., Bayes Nets and Structural Causal Modeling).

2. Outline
Dynamic causal modelling for fMRI
Modelling endogenous fluctuations
Network discovery
An empirical example

3. Functional integration and the enabling of pathways
Neuronal network of hidden states
Structural perturbations
Stimulus-free; e.g., attention, time
BA39
Dynamic perturbations
Stimuli-bound; e.g., visual words
STG V4 V1
BA37
Observation

4. Forward models and their inversion
Observed data
Forward model (measurement)
Model inversion
Forward model (neuronal)
input

5. Model specification and inversion
Design experimental inputs
Neural dynamics
Define likelihood model
Observer function
Specify priors
Invert model
Inference on parameters
Inference on models
Inference

6. The bilinear (neuronal) model for fMRI
Input
Dynamic perturbation
Structural perturbation
average
connectivity
bilinear and nonlinear
connectivity
exogenous causes

7. Output: a mixture of intra- and extravascular signal
Hemodynamic models
for fMRI
basically, a convolution
signal
The plumbing
flow
volume
dHb
sec
Output: a mixture of intra- and extravascular signal

8. A toy example u2 x3 u1 x1 x2 – – Neural population activity
0.4
0.3
0.2
0.1 10 20 30 40 50 60 70 80 90 u2
0.6
0.4
A toy example x3
0.2 10 20 30 40 50 60 70 80 90
0.3
0.2
0.1
BOLD signal change (%) 10 20 30 40 50 60 70 80 90 u1 x1 x2 2 1 – – 10 20 30 40 50 60 70 80 90 3 2 1 -1 10 20 30 40 50 60 70 80 90 2 1 10 20 30 40 50 60 70 80 90

9. An fMRI study of attention
Stimuli 250 radially moving dots at 4.7 degrees/s
Pre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)
Task: detect change in radial velocity
Scanning (no speed changes)
4 100 scan sessions;
each comprising 10 scans of 4 conditions
F A F N F A F N S
F - fixation point
A - motion stimuli with attention (detect changes)
N - motion stimuli without attention
S - no motion
V5+
PPC
Büchel et al 1999

10. Hierarchical architecture
Attentional modulation of prefrontal connections
sufficient to explain regionally specific attentional effects
Attention
.43
.53
Photic
SPC
.40
.49
.62 V1
.92
.35
IFG
.53
Segregation of motion information to V5
Motion V5
.73

11. Dynamic causal modelling for fMRI
Modelling endogenous fluctuations
Network discovery
An empirical example

12. Replacing exogenous inputs with endogenous fluctuations
Signal and noise
Hidden states
1.5
0.2
0.15 1
0.1
0.5
0.05
-0.05
-0.5
-0.1 -1
-0.15
-1.5
-0.2 50
100
150
200
250 50
100
150
200
250
time
time
Endogenous fluctuations
Network or graph generating data
Exogenous inputs
0.2
0.15
0.1
0.05
-0.05
-0.1
-0.15 50
100
150
200
250
time

13. Estimating hidden states with generalised (Bayesian) filtering1 2 3 4 5 6 7 8 9
-0.5
0.5
True and MAP connections 50
100
150
200
250
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
0.2
Hidden states
time (bins)
300
400
500
600
700
800
-0.04
-0.02
0.02
0.04
0.06
True neuronal activity
time (sec.)
MAP estimate
Extrinsic coupling parameter

14. Dynamic causal modelling for fMRI Modelling endogenous fluctuations 2 3 4 5 6 10 1
Number of models
number of nodes
Dynamic causal modelling for fMRI
Modelling endogenous fluctuations
Network discovery
An empirical example
and the problem of searching large model spaces

15. c.f., the Savage-Dickey density ratio
The concept of reduced models
Armani, Calvin Klein and Versace design houses did not refuse this year to offer very brave and reduced models of the “Thong” and “Tango”. The designers consider that a man with the body of Apollo should not obscure the wonderful parts of his body.
and their evidence
This means that we only have to invert the full model to score all reduced models;
c.f., the Savage-Dickey density ratio

16. Simulating the response of a four-node network50
100
150
200
250
-1.5 -1
-0.5
0.5 1
1.5
Signal and noise
time
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
0.2
Hidden states
Endogenous fluctuations
Network or graph generating data

17. And recovering (discovering) the true architecture5 10 15
-0.6
-0.4
-0.2
0.2
0.4
True and MAP connections 20 30 40 50 60
-600
-500
-400
-300
-200
-100
100
Log-evidence
model
log-probability 1 2 3 4 6
graph size
0.6
0.8
Model posterior
probability

18. Dynamic causal modelling for fMRI
Modelling endogenous fluctuations
Network discovery
An empirical example

19. An application to real data
200
400
600
800
1000
1200 -5 5
fef: responses
200
400
600
800
1000
1200 -5 5
ag: responses
200
400
600
800
1000
1200 -4 -2 2
ppc: responses
200
400
600
800
1000
1200 -5 5
pfc: responses
200
400
600
800
1000
1200 -2 2 4
sts: responses
200
400
600
800
1000
1200 -4 -2 2
vis: responses
With (visual motion) evoked activity
time {seconds}

20. Estimates of hidden neuronal and hemodynamic states50
100
150
200
250 -2
-1.5 -1
-0.5
0.5 1
1.5
Prediction and error
time (bins)
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
0.2
Hidden states
MAP neuronal activity
time (sec)
0.85
0.9
0.95
1.05
1.1
1.15
1.2
Hemodynamic states
visually evoked responses showing attentional modulation

21. And the results of searching model space
0.5 1
1.5 2
2.5 3
3.5
x 10 4
-400
-300
-200
-100
100
Log-posterior
model
log-probability
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
Model posterior
probability 5 10 15 20 25 30 35
-0.6
-0.4
-0.2
MAP connections (full)
MAP connections (sparse)

22. Anatomical space
Exploiting estimates of directed coupling in terms of cortical hierarchies
'vis' 'sts' 'pfc' 'ppc' 'ag' 'fef'
Functional (embedding) space

23. Thank you And thanks to CC Chen Jean Daunizeau Olivier David
Marta Garrido
Lee Harrison
Stefan Kiebel
Baojuan Li
Andre Marreiros
Rosalyn Moran
Will Penny
Klaas Stephan
And many others