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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.
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
Dynamic Causal ModellingState and observation equationsModel inversion
DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models
DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connections
DCMs for LFPSteady-state responses
V1
V4
BA37
STG
BA39
Structural perturbationsStimulus-free - u
e.g., attention, time
Dynamic perturbationsStimuli-bound u
e.g., visual words
Functional integration and the enabling of specific pathways
y
y y
y
y
measurement
neuronal network
Observed data
)(tu
ix
input
( , , )x f x u
),(xgy
Forward model (measurement)
Model inversion
Forward models and their inversion
Forward model (neuronal)
( | , , , )p y x u m ( , | , , )p x y u m
Model specification and inversion
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),,(
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dpmypmyp
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( , ) ( , )
N
N
p y m g
p m
Invert model
Inference
Define likelihood model
Specify priors
Neural dynamics
Observer function
Design experimental inputs)(tu
Inference on models
Inference on parameters
Dynamic Causal ModellingState and observation equationsModel inversion
DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models
DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connectionsInduced responses
DCMs for LFPSteady-state responses
1y 3y2y
u
fC
ux
fB
x
fA
CBA
2
},,{
The bilinear (neuronal) model
23b
12a
1c
averageconnectivity
exogenous causes
bilinearconnectivity
CuxuBA
uxfx
)(
),,( Input
)(tu
1x3x
2x
Dynamic perturbation
Structural perturbation
f s
1( ) / ατq f E f v q v 1/ ατv f v
( 1)s x s γ f
0 1 2 3( ) ( ( )) ( (1 ) (1 ) (1 ))y t g x t V k q k q v k v
Output: a mixture of intra- and extravascular signal
)(tx
0 8 16 24 sec
ix
Hemodynamic models for fMRI
basically, a convolution
1y
signal
flow
dHbvolume
The plumbing
Neural population activity
BOLD signal change (%)
x1 x2u1
x3
u2
– –
11 12 1 111
21 22 23 2 21 22
32 33 3 32
0 0 0 0 0
0 0 0 0
0 0 0 0 0
a a x cu
x a a a u b xu
a a x c
A toy example
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100-1
0
1
2
3
4
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
Stimuli 250 radially moving dots at 4.7 degrees/s
Pre-Scanning5 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 attentionS - no motion
Buchel et al 1999
V5+
PPC
An fMRI study of attention
V1 IFG
V5
Photic
Attention
.92
.43
.62
.40
.53.35
.73
.49
.53
3) Attentional modulation of prefrontal connections
sufficient to explain regionally specific attentional effects2) Segregation of
motion information to V5
1) Hierarchical architecture
Friston et al 1999
SPC
Motion
FFA PPA
MFG
-0.80
-0.31
faces houses faces houses
rivalry non-rivalry
1.05 0.08
0.300.51
2.43 2.41
0.04 -0.03 0.02 0.06
0.02 -0.03
FFAPPAMFG
time (s)
Stephan et al 2008
( ) ( )( )i ii ix A u B x D x Cu
Nonlinear DCM: modulation of connections in inferotemporal cortex under binocular rivalry
uinput
Single-state DCM
1x
Intrinsic (within-region) coupling
Extrinsic (between-region) coupling
NNNN
N
ijijij
x
x
x
uBA
Cuxx
1
1
111
Two-state DCM
Ex1
IN
EN
I
E
IINN
IENN
EENN
EENN
EEN
IIIE
EEN
EIEE
ijijijij
x
x
x
x
x
uBA
Cuxx
1
1
1
1111
11111
00
0
00
0
)exp(
Ix1
I
E
x
x
1
1
Modeling excitatory and inhibitory dynamics
Andre Marreiros et al
IISPSP
IESPSP
EISPSP
EESPSP
EESPV
IIVV
IEVV
EESPV
EIVV
EEVV
EEVV
IIVV
IEVV
EEVV
EIVV
EEVV
0000
000
0000
00
0000
000
5
5555
5555515
1111
511111
IISPSP
IESPSP
EISPSP
EESPSP
EESPV
IIVV
IEVV
EESPV
EIVV
EEVV
EEVV
IIVV
IEVV
EEVV
EIVV
EEVV
0000
000
0000
00
0000
000
5
5555
5555515
1111
511111
IISPSP
IESPSP
EISPSP
EESPSP
EESPV
IIVV
IEVV
EESPV
EIVV
EEVV
EEVV
IIVV
IEVV
EEVV
EIVV
EEVV
0000
000
0000
00
0000
000
5
5555
5555515
1111
511111
Model comparison: where is attention mediated?
Model comparison
Andre Marreiros et al
Hierarchical connections in the brain and laminar specificity
Dynamic Causal ModellingState and observation equationsModel inversion
DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models
DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connectionsInduced responses
DCMs for LFPSteady-state responses
neuronal mass models of distributed sources
State equations
( , , ) x f x u
Output equation
(3)( , ) y g x LV
Exogenous input
E13
( )u t
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in infragranular layers
Inhibitory cells in supragranular layers
Measured response
)( )3(Vg
(1) (1) (1) (1)
(1) (3) (3) (1)13
( ) ( )
( ( , ) )
L L E E V
EE E V R E E
CV g V V g V V u
g V g
E31
IIRVI
II
EERVE
EE
VIIEELL
gVg
gVg
VVgVVgVVgVC
)),((
)),((
)()()(
)2()2()2(22
)2(
)2()3()3(23
)2(
)2()2()2()2()2()2(
IIRVI
II
EERVE
EE
VIIEELL
gVg
gVg
VVgVVgVVgVC
)),((
)),((
)()()(
)3()2()2(32
)3(
)3()1()1(31
)3(
)3()3()3()3()3()3(
E23I
32 12I
uinput
x
ERPs
Comparing models (with and without backward connections)
A1 A1
STG
input
STG
IFG
FB
A1 A1
STG
input
STG
IFG
F
0 200 400
0
0 200 400
0
FB vs. F
without with
A1A1
STGSTG
IFG
Garrido et al 2007
log-evidence
ln ( | ) Fp y m
The MMN and perceptual learning
MMN
standards deviants
ERP standardsERP deviantsdeviants - standards
Garrido et al 2008
Model comparison:Changes in forward and backward connections
A1 A1
STG STG
ForwardBackward
Lateral
input
A1 A1
STG STG
ForwardBackward
Lateral
input
A1 A1
STG
ForwardBackward
Lateral
input
-
STG
IFGIFGIFG
Forward (F) Backward (B) Forward and Backward (FB)
Garrido et al 2009
A1A1
STGSTG
IFGA1 A1
STG STG
ForwardBackward
Lateral
input
A1 A1
STG STG
ForwardBackward
Lateral
input
A1 A1
STG
ForwardBackward
Lateral
input
-
STG
IFGIFGIFG
Forward (F) Backward (B) Forward and Backward (FB)
FFB
log
evid
ence
Bayesian model comparison
subjects
Forward (F)
Backward (B)
Forward and Backward (FB)
Two subgroups
Garrido et al 2008
1 2 3 4 5 1 2 3 4 5
A1 A1
STG
subcortical input
STG
repetition effects
monotonic phasic
1 2 3 4 50
20
40
60
80
100
120
140
160
180
200
1 2 3 4 50
50
100
150
200
250
Intrinsic connections
Extrinsic connections
number of presentations
The dynamics of plasticity:Repetition suppression
Garrido et al 2009
K frequency modes in j-th source
KKij
Kij
Kijij
ij
AA
AA
A
1
111
Nonlinear (between-frequency) coupling
Linear (within-frequency) coupling
Extrinsic (between-source) coupling
Neuronal model for spectral features
)()()(1
1
1111
tu
C
C
tg
AA
AA
g
g
tg
JJJJ
J
J
Data in channel space
12
( ) ( )
( , )
( ) ( ( ))
( , )
j
j j
j K
x t L d t
g t
g t FT x t
g t
)(td
Inversion of electromagnetic model L
)(tu
klijA
jg
input
Intrinsic (within-source) coupling
),( tgi
DCM for induced responses – a different sort of data feature
CC Chen et al 2008
LV RV
RFLF
input
LV RV
RFLF
input
Frequency-specific coupling during face-processing
CC Chen et al 2008
From 32 Hz (gamma) to 10 Hz (alpha) t = 4.72; p = 0.002
4 12 20 28 36 44
44
36
28
20
12
4
SPM t df 72; FWHM 7.8 x 6.5 Hz
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Right hemisphereLeft hemisphere
Forward Backward Forward BackwardFr
eque
ncy
(Hz)
LV RV
RFLF
input
FLBL FNBL FLBN FNBN
-59890
-16308 -16306 -11895
-70000
-60000
-50000
-40000
-30000
-20000
-10000
0
Functional asymmetries in forward and backward connections
CC Chen et al 2008
Dynamic Causal ModellingState and observation equationsModel inversion
DCMs for fMRIBilinear modelsHemodynamic modelsAttentional modulationTwo-state models
DCMs for EEGNeural-mass modelsPerceptual learning and MMNBackward connections
DCMs for LFPSteady-state responses
Glutamatergic stellate cells
GABAergic cells
Glutamatergic Projection cells
Data
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
0 20 400
5
Cortex
GPe
StriatumSTN
Cortex GPeStriatum STN
DCMs for steady-state responses:characterizing coupling parameters Cross-spectral data features
6-OHDA lesion model of Parkinsonism
Moran et al
1. Cortex
2. Striatum
3. External globus pallidus (GPe)
4. Subthalamic Nucleus (STN)
6. Thalamus
5. Entopeduncular Nucleus (EPN)
Changes in the basal ganglia-cortical circuits
Moran et al
Control 6-OHDA Lesioned
1
2
3
4
6
4.25 ± 0.17
1.44 ± 0.18
5.24 ± 0.16
6. 91 ± 0.190.90
± 0
.21
1.43 ± 0.38
0.29 ± 0.31
0.85 ± 0.36
5
0.72 ± 0.44
1
2
3
4
5
3.43 ± 0.16
3.07 ± 0.17
5.00 ± 0.15
2.33 ± 0.21 1.0
4 ±
0.20
1.18 ± 0.33
1.03 ± 0.356
0.74 ± 0.28
MAP estimates
EPN
to T
hala
mus
Thal
amus
to C
tx
Ctx
to S
triat
um
Ctx
to S
TN
Stria
tum
to G
Pe
Stria
tum
to E
PN
STN
to E
PN
STN
to G
Pe
GPe
to S
TN
0
1
2
3
4
5
6
7
8
**
Thank you
And thanks to
CC ChenJean Daunizeau Marta GarridoLee HarrisonStefan Kiebel
Andre MarreirosRosalyn Moran
Will PennyKlaas Stephan
And many others