Imaging Clinic
Tuesday 26th October: 10AM-4.30PM; Building 26, room 135; Clayton Campus
Dynamic Causal Modelling (tutorial)
Karl Friston, Wellcome Centre for Neuroimaging, UCL
Abstract
This tutorial is about the inversion of dynamic input-state-output systems. Identification of the systems parameters
proceeds in a Bayesian framework given known, deterministic inputs and observed responses of the [neuronal]
system.
We develop this approach for the analysis of effective connectivity or coupling in the brain, using experimentally
designed inputs and fMRI and EEG responses. In this context, the parameters correspond to effective connectivity
and, in particular, bilinear parameters reflect the changes in connectivity induced by inputs. The ensuing framework
allows one to characterise experiments, conceptually, as an experimental manipulation of integration among brain
regions (by contextual or trial-free inputs, like time or attentional set) that is perturbed or probed using evoked
responses (to trial-bound inputs like stimuli).
As with previous analyses of effective connectivity, the focus is on experimentally induced changes in coupling (c.f.
psychophysiologic interactions). However, unlike previous approaches to connectivity in neuroimaging, the causal
model ascribes responses to designed deterministic inputs, as opposed to treating inputs as unknown and
stochastic.
Dynamic Causal Modelling
State and observation equations
Model inversion
DCMs for fMRI
Bilinear models
Hemodynamic models
Attentional modulation
Two-state models
DCMs for EEG
Neural-mass models
Perceptual learning and MMN
Backward connections
DCMs for LFP
Steady-state responses
Functional integration and the enabling of specific pathways
Structural perturbations
neuronal network
Stimulus-free - u
e.g., attention, time
BA39
Dynamic perturbations
Stimuli-bound u
e.g., visual words
y
STG
V4
y
BA37
y
V1
y
measurement
y
Forward models and their inversion
Forward model (measurement)
y  g (x, )  
Observed data
Model inversion
p( y | x, , u, m)
Forward model (neuronal)
p( x, | y, u, m)

xi
x  f ( x, u, )  
input
u(t )
Model specification and inversion
u(t )
Neural dynamics
Design experimental inputs
x  f ( x, u, )
Define likelihood model
Observer function
y  g ( x, )  
p( y |  , m)  N ( g ( ), ( ))
p( , m)  N (  ,  )
Inference on models p ( y | m) 
 p( y |  , m) p( )d
p( y |  , m) p( , m)
Inference on parameters p ( | y, m) 
p ( y | m)
Specify priors
Invert model
Inference
Dynamic Causal Modelling
State and observation equations
Model inversion
DCMs for fMRI
Bilinear models
Hemodynamic models
Attentional modulation
Two-state models
DCMs for EEG
Neural-mass models
Perceptual learning and MMN
Backward connections
Induced responses
DCMs for LFP
Steady-state responses
The bilinear (neuronal) model
Input
Dynamic
perturbation
Structural
perturbation
u(t )
b23
c1
a12
x2
x1
x3
average
connectivity
bilinear
exogenous
connectivity causes
  { A, B, C}
y2
y1
x  f ( x, u, )
 ( A  uB) x  Cu
y3
f
A
x
2 f
B
xu
C
f
u
Hemodynamic models
for fMRI
basically, a convolution
x(t )
xi
signal
s  x   s  γ( f  1)
The plumbing
flow
f s
volume
dHb
τv  f  v1/ α
τq  f E( f )   v1/ α q v
0
8
16
y1
Output: a mixture of intra- and extravascular signal
y(t )  g ( x(t ))  V0 (k1 (1  q)  k2 (1  q v)  k3 (1  v))
24 sec
Neural population activity
0.4
0.3
0.2
0.1
0
0
u2
A toy example
10
20
30
40
50
60
70
80
90
100
10
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50
60
70
80
90
100
10
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30
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60
70
80
90
100
0.6
0.4
0.2
x3
0
0
0.3
0.2
0.1
0
0
u1
x1
x2
3
BOLD signal change (%)
2
1
–
–
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
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50
60
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90
100
0
10
20
30
40
50
60
70
80
90
100
4
  a11 a12

x    a21 a22
 0 a
32

0
 0 0 0   x1  c11 0 
u 


a23   u2 b21 0 0   x2    0 0   1 
u
 0 0 0   x3   0 c32   2 
a33 
3
2
1
0
-1
3
2
1
0
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
PPC
V5+
Buchel et al 1999
1) Hierarchical
architecture
.43
SPC
Photic
.92
V1
Motion
.73
.53
.40
.49
.62
2) Segregation of
motion
information to V5
3) Attentional
modulation of
prefrontal
connections
Attention
.35
V5
.53
IFG
sufficient to explain
regionally specific
attentional effects
Friston et al 1999
Nonlinear DCM: modulation of connections in inferotemporal cortex under binocular rivalry
rivalry
non-rivalry
0.02
FFA
PPA
MFG
-0.03
MFG
1.05
0.08
2.43
-0.31
0.51
FFA
0.04
faces
2.41
PPA
-0.80
-0.03
houses
0.30
0.02
faces
0.06
houses
time (s)
x  ( A  ui B(i )   xi D(i ) ) x  Cu
Stephan et al 2008
Modeling excitatory and inhibitory dynamics
Single-state DCM
Two-state DCM
input
u
x1E
x1E
x1
x1I
x1I
x  x  Cu
ij  ij exp(Aij  uBij )
x  x  Cu
ij  Aij  uBij
 11  1N 
    
 
N 1  NN 
 x1 
x    
 xN 
11EE
 IE
 11
  
 EE
 N 1
 0

11EI
11II
 1EE
N
0

0
EE
NN
0
 IE
NN
Extrinsic (betweenregion) coupling
0 

0 
 

EE
NN 
IINN 
 x1E 
 I
 x1 
x  
 E
 xN 
xI 
 N
Intrinsic (withinregion) coupling
Andre Marreiros et al
Model comparison: where is attention mediated?
 VEE1V 1 VEI1V 1
 IE
II
 V 1V 1 V 1V 1
EE
V 5V 1 0

0
 0
 0
0

0
 0
 VEE1V 1 VEI1V 1
 IE
II
 V 1V 1 V 1V 1
EE
V 5V 1 0

0
 0
 0
0

0
 0
 VEE1V 1 VEI1V 1
 IE
II
 V 1V 1 V 1V 1
EE
V 5V 1 0

0
 0
 0
0

0
 0
VEE1V 5
0
0
0
0
0
VEE5V 5 VEI5V 5 VEE5 SP
VIE5V 5 VII5V 5
0
EE
SPV
5
0
0
0
EE
SPSP
IE
SPSP
VEE1V 5
0
0
0
0
0
VEE5V 5 VEI5V 5 VEE5 SP
VIE5V 5 VII5V 5
0
EE
SPV
5
0
0
0
EE
SPSP
IE
SPSP
VEE1V 5
0
0
0
0
0
VEE5V 5 VEI5V 5 VEE5 SP
VIE5V 5 VII5V 5
EE
SPV
5
0
0
0
0
EE
SPSP
IE
SPSP
0 

0 
0 

0 
EI 
SPSP

II
SPSP

0 

0 
0 

0 
EI 
SPSP

II
SPSP

Model comparison
0 

0 
0 

0 
EI 
SPSP

II
SPSP

Andre Marreiros et al
Dynamic Causal Modelling
State and observation equations
Model inversion
DCMs for fMRI
Bilinear models
Hemodynamic models
Attentional modulation
Two-state models
DCMs for EEG
Neural-mass models
Perceptual learning and MMN
Backward connections
Induced responses
DCMs for LFP
Steady-state responses
Hierarchical connections in the brain
and laminar specificity
neuronal mass models of distributed sources
input
Inhibitory cells in supragranular layers
u
x
CV ( 2)  g L (VL  V ( 2) )  g E( 2) (VE  V ( 2) )  g I( 2) (VI  V ( 2 ) )  V
E
g E( 2)   E ( 23
 ( V(3)  VR ,  (3) )  g E( 2) )  E
I
g I( 2)   I ( 22
 ( V( 2)  VR ,  ( 2) )  g I( 2) )  I
 32I
Exogenous
input
State equations
 23E
Excitatory spiny cells in granular layers
CV (1)  g L (VL  V (1) )  g E(1) (VE  V (1) )  u  V
g E(1)   E ( 13E ( V(3)  VR , (3) )  g E(1) )   E
x  f ( x, u ,  )
u(t )
Output equation
y  g ( x, )  LV
 12I
(3)
Measured
response
 13E
Excitatory pyramidal cells in infragranular layers
CV (3)  g L (VL  V (3) )  g E(3) (VE  V (3) )  g I(3) (VI  V (3) )  V
g E(3)   E ( 31E  ( V(1)  VR ,  (1) )  g E(3) )  E
g (V (3) )
 31E
g I(3)   I ( 32I  ( V( 2)  VR ,  ( 2) )  g I(3) )  I
Comparing models (with and without backward connections)
ERPs
log-evidence
ln p( y | m)  F
IFG
A1
A1
STG
STG
STG
IFG
FB
STG
FB vs. F
IFG
STG
F
STG
without
with
0
A1
A1
A1
0
A1
0
input
input
200
400
0
200
400
Garrido et al 2007
The MMN and perceptual learning
MMN
ERP standards
ERP deviants
deviants - standards
standards
deviants
Garrido et al 2008
Model comparison:
Changes in forward and backward connections
Forward (F)
Forward and Backward (FB)
Backward (B)
IFG
IFG
IFG
STG
STG
STG
STG
STG
STG
A1
A1
A1
A1
A1
A1
IFG
A1
A1
input
STG
STG
Forward
Backward
Lateral
input
Forward
Backward
Lateral
input
Forward
Backward
Lateral
Garrido et al 2009
log evidence
Bayesian model comparison
Two subgroups
subjects
F
FB
Forward (F)
Backward (B)
Forward and Backward (FB)
Garrido et al 2008
The dynamics of plasticity:
Repetition suppression
Intrinsic connections
monotonic
phasic
200
180
160
140
120
1 2 3 4 5
1 2 3 4 5
100
80
repetition effects
60
40
20
STG
STG
0
1
2
3
4
5
Extrinsic connections
250
A1
A1
200
150
subcortical input
100
50
0
1
2
3
4
5
number of presentations
Garrido et al 2009
DCM for induced responses – a different sort of data feature
Inversion of electromagnetic model L
Aijkl
x(t )  L d (t )
d (t )
Data in channel space
gj
 g j (1 , t ) 
2


g j (t )  FT ( x j (t ))  

 g j (K , t ) 


input
K frequency modes in j-th source
u(t )
gi ( , t )
Linear (within-frequency) coupling
Intrinsic (within-source) coupling
 g1   A11
g (t )        
 g J   AJ 1



 Aij11

Aij   
 AijK 1

A1J 
 C1 
  g (t )    u (t )
C J 
AJJ 
Extrinsic (between-source) coupling



Aij1K 

 
AijKK 
Nonlinear (between-frequency) coupling
Neuronal model for spectral features
CC Chen et al 2008
Frequency-specific coupling during face-processing

LF
LF
RF
LV
RV
RF



LV
RV
input
input
CC Chen et al 2008
Functional asymmetries in forward and backward connections
SPM t df 72; FWHM 7.8 x 6.5 Hz
4
-16306
28
36
44
-11895
12
44
-30000
20
-10000
-16308
20
28
Frequency (Hz)
0
-20000
12
4
FNBN
36
FLBL FNBL FLBN
-40000
From 32 Hz (gamma) to 10 Hz (alpha)
-50000
-60000
t = 4.72; p = 0.002
-59890
-70000
LF
RF
LV
RV
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.02
-0.02
-0.04
-0.04
-0.06
input
Forward Backward
-0.06
-0.08
-0.08
-0.1
-0.1
Left hemisphere
Forward Backward
Right hemisphere
CC Chen et al 2008
Dynamic Causal Modelling
State and observation equations
Model inversion
DCMs for fMRI
Bilinear models
Hemodynamic models
Attentional modulation
Two-state models
DCMs for EEG
Neural-mass models
Perceptual learning and MMN
Backward connections
DCMs for LFP
Steady-state responses
DCMs for steady-state responses:
characterizing coupling parameters
Cross-spectral data features
6-OHDA lesion model of Parkinsonism
1. Cortex
Striatum
Cortex
0
5
0
20
40
0
5
0
20
40
5
0
5
0
20
40
5
0
20
40
0
0
0
20
40
0
20
40
0
20
40
5
0
20
40
5
0
Striatum
2. Striatum
0
Cortex
5
STN
GPe
5
GPe
3. External
globus
pallidus
(GPe)
0
0
20
40
0
6. Thalamus
5
STN
Glutamatergic stellate cells
4. Subthalamic
Nucleus
(STN)
5. Entopeduncular
Nucleus
(EPN)
0
0
20
40
GABAergic cells
Glutamatergic Projection cells
Data
Moran et al
Changes in the basal ganglia-cortical circuits
1.44 ± 0.18
3.07 ± 0.17
1
1
1.03 ± 0.35
2
0.85 ± 0.36
2
5.24 ± 0.16
MAP estimates
3.43 ± 0.16
4.25 ± 0.17
5.00 ± 0.15
0.29 ± 0.31
8
*
*
7
6
5
0.74 ± 0.28
4
3
GPe to STN
STN to GPe
STN to EPN
Striatum to EPN
5
Striatum to GPe
1.18 ± 0.33
Ctx to STN
5
Ctx to Striatum
1.04 ± 0.20
0.90 ± 0.21
1.43 ± 0.38
1
0
2.33 ± 0.21
6. 91 ± 0.19
0.72 ± 0.44
2
Thalamus to Ctx
6
6
EPN to Thalamus
3
3
4
4
Control
6-OHDA Lesioned
Moran et al
Thank you
And thanks to
CC Chen
Jean Daunizeau
Marta Garrido
Lee Harrison
Stefan Kiebel
Andre Marreiros
Rosalyn Moran
Will Penny
Klaas Stephan
And many others