Transcript DCM

DCM: Advanced issues
Klaas Enno Stephan
Laboratory for Social & Neural Systems
Research
Institute for Empirical Research in Economics
University of Zurich
Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London
Methods & models for fMRI data analysis, University of Zurich
27 May 2009
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
Model comparison and selection
Given competing hypotheses
on structure & functional
mechanisms of a system, which
model is the best?
Which model represents the
best balance between model
fit and model complexity?
For which model m does p(y|m)
become maximal?
Pitt & Miyung (2002) TICS
Bayesian model selection (BMS)
Bayes’ rule:
p( y |  , m) p( | m)
p( | y, m) 
p ( y | m)
Model evidence:
p( y | m)   p( y |  , m)  p( | m) d
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability)
of the model
integral usually not analytically solvable, approximations
necessary
Model evidence p(y|m)
Gharamani, 2004
Generalisability of the
model
p( y | m)   p( y |  , m)  p( | m) d
p(y|m)
Balance between fit and
complexity
a specific y
all possible
datasets y
Model evidence: probability of generating data y from parameters  that are
randomly sampled from the prior p(m).
Maximum likelihood: probability of the data y for the specific parameter vector  that
maximises p(y|,m).
Approximations to the model evidence in DCM
Maximizing log model evidence
= Maximizing model evidence
Logarithm is a
monotonic function
Log model evidence = balance between fit and complexity
log p( y | m)  accuracy(m)  com plexity(m)
 log p( y |  , m)  com plexity(m)
No. of
parameters
In SPM2 & SPM5, interface offers 2 approximations:
Akaike Information Criterion:
Bayesian Information Criterion:
AIC  log p( y |  , m)  p
p
BIC  log p( y |  , m)  log N
2
AIC favours more complex models,
BIC favours simpler models.
No. of
data points
Penny et al. 2004, NeuroImage
Bayes factors
To compare two models, we can just compare their log evidences.
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is made
possible by Bayes factors:
positive value, [0;[
p( y | m1 )
B12 
p( y | m2 )
Kass & Raftery classification:
Kass & Raftery 1995, J. Am. Stat. Assoc.
B12
p(m1|y)
Evidence
1 to 3
50-75%
weak
3 to 20
75-95%
positive
20 to 150
95-99%
strong
 150
 99%
Very strong
The negative free energy approximation
• Under Gaussian assumptions about the posterior (Laplace
approximation), the negative free energy F is a lower bound on
the log model evidence:
log p( y | m)
 log p( y |  , m)  KLq , p | m  KLq , p | y, m
 F  KLq , p | y, m
 F  log p( y | m)  KLq , p | y, m
The complexity term in F
• In contrast to AIC & BIC, the complexity term of the negative free
energy F accounts for parameter interdependencies.
KLq( ), p( | m)
1
1
1
T
 ln C  ln C | y   | y    C1  | y   
2
2
2
• The complexity term of F is higher
– the more independent the prior parameters ( effective DFs)
– the more dependent the posterior parameters
– the more the posterior mean deviates from the prior mean
• NB: SPM8 only uses F for model selection !
BMS in SPM8: an example
attention
M1
stim
M1
M2
M3
M4
M3
stim
PPC
V1
attention
V1
V5
PPC
M2
M2 better than M1
BF 2966
F = 7.995
PPC
attention
stim
V1
V5
M3 better than M2
BF  12
F = 2.450
V5
M4
attention
PPC
M4 better than M3
BF  23
F = 3.144
stim
V1
V5
Fixed effects BMS at group level
Group Bayes factor (GBF) for 1...K subjects:
GBFij   BF
(k )
ij
k
Average Bayes factor (ABF):
ABFij  K  BF
(k )
ij
k
Problems:
- blind with regard to group heterogeneity
- sensitive to outliers
Random effects BMS for group studies:
a variational Bayesian approach

r ~ Dir(r; )
Dirichlet parameters
= “occurrences” of models in the population
Dirichlet distribution of model probabilities
mk ~ p(mk | p)
mk ~ p(mk | p)
mk ~ p(mk | p)
m1 ~ Mult(m;1, r )
Multinomial distribution of model labels
y1 ~ p( y1 | m1 )
y1 ~ p( y1 | m1 )
y2 ~ p( y2 | m2 )
y1 ~ p( y1 | m1 )
Measured data
Stephan et al. 2009, NeuroImage
Task-driven
lateralisation
Does the word
contain the letter A
or not?
letter decisions > spatial decisions
•
•
•
group analysis (random effects),
n=16, p<0.05
whole-brain corrected
Is the red letter left
or right from the
midline of the
word?
spatial decisions > letter decisions
Stephan et al. 2003, Science
Inter-hemispheric connectivity in the visual ventral stream
Left MOG
-38,-90,-4
Left FG
-44,-52,-18
Right FG
38,-52,-20
LD|LVF
0.20
 0.04
0.00
 0.01
Right MOG
-38,-94,0
0.07
 0.02
LD>SD, p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
p<0.01 uncorrected
MOG
left
LD
Left LG
-12,-70,-6
FG
left
FG
right
MOG
right
0.27
 0.06
0.11
 0.03
0.00
 0.04
0.01
 0.03
LG
left
0.01
 0.01
LD>SD masked incl. with RVF>LVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Stephan et al. 2007, J. Neurosci.
RVF
stim.
0.01
 0.01
LG
right
LD
Left LG
-14,-68,-2
0.06
 0.02
LD|RVF
LVF
stim.
LD>SD masked incl. with LVF>RVF
p<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
LD
MOG
FG
LD|LVF
MOG
FG
LD|RVF
MOG
LD|LVF
RVF
stim.
LD
Subjects
-30
-25
-20
LD
LG
LG
LVF
stim.
RVF LD|RVF
stim.
m2
-35
-15
MOG
FG
LD
LG
LG
FG
LVF
stim.
m1
-10
-5
0
5
Log model evidence differences
Stephan et al. 2009, NeuroImage
p(r >0.5 | y) = 0.997
1
5
4.5
4
1  11.806 ,  2  2.194
r1  0.843, r2  0.157
3.5
p(r 1|y)
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
r
0.6
1
0.7
0.8
0.9
1
LD|LVF
Simulation study: sampling subjects
from a heterogenous population
m1
MOG
• Population where 70% of all
subjects' data are generated
by model m1 and 30% by
model m2
FG
LD
LD
LG
LG
RVF LD|RVF
stim.
• Random sampling of
subjects from this population
and generating synthetic
data with observation noise
Stephan et al. 2009, NeuroImage
LVF
stim.
LD
m2
MOG
• Fitting both m1 and m2 to all
data sets and performing
BMS
MOG
FG
FG
MOG
FG
LD|RVF
LD|LVF
LG
LG
RVF
stim.
LD
LVF
stim.
A
18
true values:
1=220.7=15.4
2=220.3=6.6
16
14
12

mean estimates:
1=15.4, 2=6.6
10
8
B
1
true values:
r1 = 0.7, r2=0.3
0.9
0.8
mean estimates:
r1 = 0.7, r2=0.3
0.7
0.6
<r> 0.5
0.4
6
4
0.2
2
0.1
0
C
0.3
m1
0
m2
1
D
m1
m2
700
0.9
true values:
1 = 1, 2=0
0.8
0.7
mean estimates:
1 = 0.89, 2=0.11
0.6

0.5
600
500
400
300
0.4
0.3
200
0.2
100
0.1
0
0
m1
m2
log GBF12
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
y


y
BOLD
y

activity
x2(t)
neuronal
states
t
Neural state equation
intrinsic connectivity
modulation of
connectivity
direct inputs
Stephan & Friston (2007),
Handbook of Brain Connectivity
hemodynamic
model
x
integration
modulatory
input u2(t)
t
λ
activity
x3(t)
activity
x1(t)
driving
input u1(t)
y
x  ( A  u j B( j ) ) x  Cu
x
x
 x

u j x
A
B( j)
C
x
u
bilinear DCM
non-linear DCM
modulation
driving
input
driving
input
modulation
Two-dimensional Taylor series (around x0=0, u0=0):
dx
f
f
2 f
2 f x2
 f ( x, u)  f ( x0 ,0) 
x u
ux  ... 2
 ...
dt
x
u
xu
x 2
Bilinear state equation:
m
dx 
(i ) 
  A   ui B  x  Cu
dt 
i 1

Nonlinear state equation:
m
n
dx 
(i )
( j) 

  A   ui B   x j D  x  Cu
dt 
i 1
j 1

Neural population activity
0.4
0.3
0.2
u2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0.6
u1
0.4
x3
0.2
0
0.3
0.2
0.1
0
x1
x2
3
fMRI signal change (%)
2
1
0
Nonlinear dynamic causal model (DCM):
4
m
n

dx 
(i )
( j)
  A   ui B   x j D  x  Cu
dt 
i 1
j 1

2
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
3
1
0
-1
3
2
1
Stephan et al. 2008, NeuroImage
0
Nonlinear DCM: Attention to motion
Stimuli + Task
Previous bilinear DCM
Attention
Photic
.52 (98%)
.37
(90%)
.42
(100%)
Büchel & Friston
(1997)
250 radially moving dots
(4.7 °/s)
Conditions:
F – fixation only
A – motion + attention
(“detect changes”)
N – motion without attention
S – stationary dots
V1
Motion
SPC
.56
(99%)
.69 (100%)
.47
(100%)
.82
(100%)
.65 (100%)
IFG
V5
Friston et al. (2003)
Friston et al. (2003):
attention modulates backward connections
IFG→SPC and SPC→V5.
Q: Is a nonlinear mechanism (gain control) a
better explanation of the data?
attention
 modulation of back-
M1
PPC
ward or forward
connection?
stim
 additional driving
effect of attention
on PPC?
M3
stim
M2
M2 better than M1
BF = 2966
V1
attention
PPC
Stephan et al. 2008, NeuroImage
V1
V5
BF = 12
M3 better than M2
V1
V5
M4
 bilinear or nonlinear
modulation of
forward connection?
attention
stim
V5
PPC
attention
PPC
BF = 23
M4 better than M3
stim
V1
V5
attention
MAP = 1.25
0.10
0.8
0.7
PPC
0.6
0.26
0.5
0.39
1.25
stim
0.26
V1
0.13
0.46
0.50
V5
0.4
0.3
0.2
0.1
0
-2
motion
Stephan et al. 2008, NeuroImage
-1
0
1
2
3
4
p( DVPPC
5,V 1  0 | y)  99.1%
5
motion &
attention
static
motion &
no attention dots
V1
V5
PPC
observed
fitted
Stephan et al. 2008, NeuroImage
Nonlinear DCM: Binocular rivalry
0.7
0.6
rivalry
0.5
non-rivalry
0.4
0.3
0.02
-0.03
0.2
0.1
MFG
1.05
0
-2
FFA
1
2
3
4
5
6
7
2.41
-0.31
0.51
0
MFG
p( DFFA
,PPA  0 | y)  99.9%
0.08
2.43
-1
0.30
0.6
PPA
-0.80
0.7
0.5
0.4
0.04
-0.03
0.02
0.06
0.3
0.2
faces houses faces houses
0.1
0
-2
-1
0
1
2
3
4
5
6
7
MFG
p( DPPA
,FFA  0 | y)  99.9%
Stephan et al. 2008, NeuroImage
FFA
PPA
MFG
BR
nBR
time (s)
Stephan et al. 2008, NeuroImage
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
•
•
•
•
Two potential timing problems in
DCM:
1. wrong timing of inputs
2. temporal shift between
regional time series because
of multi-slice acquisition
slice acquisition
Timing problems at long TRs/TAs
DCM is robust against timing errors up to approx. ± 1 s
–
compensatory changes of σ and θh
Possible corrections:
–
slice-timing in SPM (not for long TAs)
–
restriction of the model to neighbouring regions
–
in both cases: adjust temporal reference bin in SPM defaults
(defaults.stats.fmri.t0)
Best solution: Slice-specific sampling within DCM
2
1
visual
input
Slice timing in DCM: three-level model
3rd level
z  h(v, T )
sampled
BOLD response
2nd level
v  g ( x, x , )
BOLD response
1st
x  f ( x, , u)
neuronal response
level
h
n
h
x = neuronal states
xh = hemodynamic states
n, h = neuronal and hemodynamic parameters
Kiebel et al. 2007, NeuroImage
u = inputs
v = BOLD responses
T = sampling time points
Slice timing in DCM: an example
1 TR
3 TR
2 TR
4 TR
5 TR
Default
sampling
t
T1
T2
T1
T2
T1
T2
T1
T2
T1
T2
1 TR
2 TR
3 TR
4 TR
5 TR
Slice-specific
sampling
t
T1
Kiebel et al. 2007, NeuroImage
T2
T1
T2
T1
T2
T1
T2
T1
T2
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
Diffusion-weighted imaging
Parker & Alexander, 2005,
Phil. Trans. B
Probabilistic tractography: Kaden et al. 2007, NeuroImage
• computes local fibre orientation
density by spherical deconvolution of
the diffusion-weighted signal
• estimates the spatial probability
distribution of connectivity from given
seed regions
• anatomical connectivity = proportion
of fibre pathways originating in a
specific source region that intersect
a target region
• If the area or volume of the source
region approaches a point, this
measure reduces to method by
Behrens et al. (2003)
1.6
Integration of
tractography
and DCM
1.4
1.2
1
R1
R2
0.8
0.6
0.4
0.2
0
-2
-1
0
1
2
low probability of anatomical connection
 small prior variance of effective connectivity parameter
1.6
1.4
1.2
1
R1
R2
0.8
0.6
0.4
0.2
0
Stephan, Tittgemeyer, Knoesche,
Moran, Friston, in revision
-2
-1
0
1
high probability of anatomical connection
 large prior variance of effective connectivity parameter
2
LD|LVF
FG
(x3)
probabilistic
tractography
FG
(x4)
LD
 DCM
structure
FG
left
*
 24
 1.50  102
 24  43.6%
LG
(x2)
LG
left
LG
right
12*  1.17  102
12  34.2%
LD|RVF
RVF
stim.
FG
right
13*  5.37  10 3
13  15.7%
LD
LG
(x1)
 34*  2.23  103
 34  6.5%
BVF
stim.
LVF
stim.
  6.5%
v  0.0384
2
1.8
  15.7%
1.6
 connectionspecific priors
for coupling
parameters
v  0.1070
1.4
1.2
1
0.8
0.6
  34.2%
  43.6%
v  0.5268
v  0.7746
0.4
0.2
0
-3
-2
-1
0
1
2
3
 anatomical
connectivity
m 1: a=-32,b=-32 m 2: a=-16,b=-32 m 3: a=-16,b=-28 m 4: a=-12,b=-32 m 5: a=-12,b=-28 m 6: a=-12,b=-24 m 7: a=-12,b=-20 m 8: a=-8,b=-32 m 9: a=-8,b=-28
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 10: a=-8,b=-24 m 11: a=-8,b=-20 m 12: a=-8,b=-16 m 13: a=-8,b=-12 m 14: a=-4,b=-32 m 15: a=-4,b=-28 m 16: a=-4,b=-24 m 17: a=-4,b=-20 m 18: a=-4,b=-16
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0
m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
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0
0
0
0.5
1
0
0.5
1
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0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 28: a=0,b=-16 m 29: a=0,b=-12 m 30: a=0,b=-8
m 31: a=0,b=-4
m 32: a=0,b=0
m 33: a=0,b=4
m 34: a=0,b=8
m 35: a=0,b=12 m 36: a=0,b=16
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 37: a=0,b=20 m 38: a=0,b=24 m 39: a=0,b=28 m 40: a=0,b=32 m 41: a=4,b=-32
m 42: a=4,b=0
m 43: a=4,b=4
m 44: a=4,b=8
m 45: a=4,b=12
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 46: a=4,b=16 m 47: a=4,b=20 m 48: a=4,b=24 m 49: a=4,b=28 m 50: a=4,b=32 m 51: a=8,b=12 m 52: a=8,b=16 m 53: a=8,b=20 m 54: a=8,b=24
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0.5
1
0
0.5
1
0
0.5
1
0
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1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
m 55: a=8,b=28 m 56: a=8,b=32 m 57: a=12,b=20 m 58: a=12,b=24 m 59: a=12,b=28 m 60: a=12,b=32 m 61: a=16,b=28 m 62: a=16,b=32
m 63 & m 64
1
1
1
1
1
1
1
1
1
0.5
0
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0
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1
0
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0
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1
0
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0
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0
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0
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1
0
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1
0
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0
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0
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1
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1
log group Bayes factor
600
400
200
log group Bayes factor
0
0
10
20
30
model
40
50
60
0
10
20
30
model
40
50
60
10
20
30
model
40
50
60
700
695
690
685
680
post. model prob.
0.6
0.5
0.4
0.3
0.2
0.1
0
0
m 1: a=-32,b=-32 m 2: a=-16,b=-32 m 3: a=-16,b=-28 m 4: a=-12,b=-32 m 5: a=-12,b=-28 m 6: a=-12,b=-24 m 7: a=-12,b=-20 m 8: a=-8,b=-32 m 9: a=-8,b=-28
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
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0
0
0
0
0
0
0
0
0
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
0
0.5
1
m 10: a=-8,b=-24 m 11: a=-8,b=-20 m 12: a=-8,b=-16 m 13: a=-8,b=-12 m 14: a=-4,b=-32 m 15: a=-4,b=-28 m 16: a=-4,b=-24 m 17: a=-4,b=-20 m 18: a=-4,b=-16
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
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0
0
0
0
0
0
0
0
0
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
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1
0
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1
0
0.5
1
m 19: a=-4,b=-12 m 20: a=-4,b=-8 m 21: a=-4,b=-4 m 22: a=-4,b=0
m 23: a=-4,b=4 m 24: a=0,b=-32 m 25: a=0,b=-28 m 26: a=0,b=-24 m 27: a=0,b=-20
1
1
1
1
1
1
1
1
1
0.5
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0.5
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0
0
0
0
0
0
0
0
0
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
0
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1
0
0.5
1
m 28: a=0,b=-16 m 29: a=0,b=-12 m 30: a=0,b=-8
m 31: a=0,b=-4
m 32: a=0,b=0
m 33: a=0,b=4
m 34: a=0,b=8
m 35: a=0,b=12 m 36: a=0,b=16
1
1
1
1
1
1
1
1
1
0.5
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1
m 37: a=0,b=20 m 38: a=0,b=24 m 39: a=0,b=28 m 40: a=0,b=32 m 41: a=4,b=-32
m 42: a=4,b=0
m 43: a=4,b=4
m 44: a=4,b=8
m 45: a=4,b=12
1
1
1
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1
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0.5
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m 46: a=4,b=16 m 47: a=4,b=20 m 48: a=4,b=24 m 49: a=4,b=28 m 50: a=4,b=32 m 51: a=8,b=12 m 52: a=8,b=16 m 53: a=8,b=20 m 54: a=8,b=24
1
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m 55: a=8,b=28 m 56: a=8,b=32 m 57: a=12,b=20 m 58: a=12,b=24 m 59: a=12,b=28 m 60: a=12,b=32 m 61: a=16,b=28 m 62: a=16,b=32
m 63 & m 64
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1
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
DCM: generative model for fMRI and ERPs
Hemodynamic
forward model:
neural activityBOLD
(nonlinear)
Electric/magnetic
forward model:
neural activityEEG
MEG
LFP
(linear)
Neural state equation:
x  F ( x, u, )
fMRI
Neural model:
1 state variable per region
bilinear state equation
no propagation delays
ERPs
Neural model:
8 state variables per region
nonlinear state equation
propagation delays
inputs
DCMs for M/EEG and LFPs
•
can be fitted both to frequency spectra and
ERPs
•
models different neuronal cell types, different
synaptic types (and their plasticity) and spikefrequency adaptation (SFA)
•
ongoing model validation by LFP recordings
in rats, combined with pharmacological
manipulations
Example of
single-neuron
SFA
Tombaugh et al. 2005, J.Neurosci.
standards
deviants
A1
A2
Neural mass model of a cortical macrocolumn
E
x
t
r
i
n
s
i
c
i
n
p
u
t
s
Excitatory
Interneurons
mean firing
rate

mean
postsynaptic
potential (PSP)
He, e
g1
g2
Pyramidal
Cells
MEG/EEG
signal
He, e
g3
g4
Inhibitory
Interneurons
Parameters:
Hi, e
Excitatory connection
mean PSP

mean firing rate
Inhibitory connection
 e, i : synaptic time constant (excitatory and inhibitory)
 He, Hi: synaptic efficacy (excitatory and inhibitory)
 g1,…,g4: intrinsic connection strengths
Jansen & Rit (1995) Biol. Cybern.
 propagation delays
David et al. (2003) NeuroImage
Intrinsic
connections
g5
Synaptic ‘alpha’ kernel
Inhibitory cells in agranular layers
x7  x8
x8  k e H e ( A B  A L  g 3 ) S ( x9 )  2k e x8  k e2 x7
x10  x11
x11  k i H ig 5 S ( x12 )  2k i x11  k i2 x10
x12  x8  x11
g4
g3
Excitatory
Excitatory
spiny
spiny
cellscells
in granular
in granular
layers
layers
x1 x1x4 x4
x4 x 4k 
) 2kk xx  k x
kee((
(ux)9 ) 2kCu
HAe (g1 sA
( x ga1))S
eH
e x4
F
L
9
g1
2
ee 14
x  f ( x, u)
2
e 1
g2
Exogenous input
u
Sigmoid function
Excitatory pyramidal cells in agranular layers
x2  x5
Extrinsic
Connections:
x5  k e H e (( A B  AL ) S ( x9 )  g 2 S ( x1 ))  2k e x5  k e2 x2
 3  x6
Forward
x6  k i H ig 4 S ( x12 )  2k i x6  k i2 x3
x9  x5  x6
Backward
Lateral
David et al. 2006, NeuroImage Kiebel et al. 2007, NeuroImage Moran et al. 2009, NeuroImage
Electromagnetic forward model for M/EEG
Depolarisation of
pyramidal cells
Forward model:
lead field & gain matrix
x0  f ( x, u, )
LK
Scalp data
y  g ( x, )  LKx0
Forward model
Kiebel et al. 2006, NeuroImage
DCM for steady-state responses
• models the cross-spectral
density of recorded data
• spectral form of neuronal
innovations (i.e. baseline
cortical activity) are
estimated using a mixture of
white and pink (1/f)
components
• assumes quasi-stationary
responses (i.e. changes in
neuronal states are
approximated by small
perturbations around some
fixed point)
Frequency (Hz)
• feature extraction by means
of p-order VAR model
10
20
30
0
Time (s)
10
Moran et al. 2009, NeuroImage
Validation study using microdialysis
(in collaboration with Conway Inst., UC Dublin)
- two groups of rats with different rearing conditions
- LFP recordings and microdialysis measurements (Glu & GABA) from mPFC
Controls mPFC
Isolated mPFC
Regular Glutamate
Low Glutamate
mPFC EEG
0.12
0.06
mV
0
-0.06
mPFC
VTA
Moran et al. 2008, NeuroImage
Experimental data
FFT 10 mins time series: one area (mPFC)
blue: control animals
red: isolated animals
* p<0.05, Bonferroni-corrected
Moran et al. 2008, NeuroImage
Predictions about expected parameter estimates
from the microdialysis measurements
amplitude of
synaptic kernels
( He)
upregulation of
AMPA receptors
SFA
(2)
chronic reduction
in extracellular
glutamate levels
 EPSPs
sensitisation of
postsynaptic
mechanisms
Van den Pool et al. 1996, Neuroscience
Sanchez-Vives et al. 2000, J. Neurosci.
 activation of voltage-sensitive
Ca2+ channels
→ intracellular Ca2+
→ Ca-dependent K+ currents
→ IAHP
g5
sensitization of postsynaptic mechanisms
Inhibitory cells in supragranular layers
g 3 [29,37]
g4
g4
Extrinsic
forward
connections
u
H e[3.8 6.3]
(0.4)
(0.04)
Excitatory spiny cells in granular layers
Excitatory spiny cells in granular layers
g 1 [161, 210]
(0. 13)
g 2 [195, 233]
(0.37)
Excitatory pyramidal cells in infragranular layers
2
Control group estimates in blue,
isolated animals in red,
p values in parentheses.
[0.76,1.34]
(0.0003)
Increased neuronal adaption:
decreased firing rate
Moran et al. 2008, NeuroImage
Take-home messages
• Bayesian model selection (BMS):
generic approach to selecting an optimal model from an arbitrarily large
number of competing models
• random effects BMS for group studies:
posterior model probabilities and exceedance probabilities
• nonlinear DCM:
enables one to investigate synaptic gating processes via activity-dependent
changes in connection strengths
• DCM & tractography:
probabilities of anatomical connections can be used to inform the prior
variance of DCM coupling parameters
• DCMs for electrophysiology:
based on neurophysiologically fairly detailed neural mass models
Thank you