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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. - PowerPoint PPT Presentation
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DCM: Advanced issues
Klaas Enno Stephan
Laboratory for Social & Neural Systems Research Institute for Empirical Research in EconomicsUniversity of Zurich
Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London
Methods & models for fMRI data analysis, University of Zurich27 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?
For which model m does p(y|m) become maximal?
Which model represents thebest balance between model fit and model complexity?
Pitt & Miyung (2002) TICS
dmpmypmyp )|(),|()|( Model evidence:
Bayesian model selection (BMS)
)|(
)|(),|(),|(
myp
mpmypmyp
Bayes’ rule:
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
dmpmypmyp )|(),|()|(
Model evidence p(y|m)Gharamani, 2004
p(y
|m
)
all possible datasets y
a specific y
Balance between fit and complexity
Generalisability of the model
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).
pmypAIC ),|(log
Logarithm is a monotonic function
Maximizing log model evidence= Maximizing model evidence
)(),|(log
)()( )|(log
mcomplexitymyp
mcomplexitymaccuracymyp
In SPM2 & SPM5, interface offers 2 approximations:
Np
mypBIC log2
),|(log
Akaike Information Criterion:
Bayesian Information Criterion:
Log model evidence = balance between fit and complexity
Penny et al. 2004, NeuroImage
Approximations to the model evidence in DCM
No. of parameters
No. ofdata points
AIC favours more complex models,BIC favours simpler models.
Bayes factors
)|(
)|(
2
112 myp
mypB
positive value, [0;[
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is made possible by Bayes factors:
To compare two models, we can just compare their log evidences.
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
Kass & Raftery classification:
Kass & Raftery 1995, J. Am. Stat. Assoc.
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:
mypqKLF
mypqKLmpqKLmyp
myp
,|,
,|,|,),|(log
)|(log
mypqKLmypF ,|,)|(log
The complexity term in F
• In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies.
• 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 !
y
Tyy CCC
mpqKL
|1
|| 2
1ln
2
1ln
2
1
)|(),(
V1 V5stim
PPCM2
attention
V1 V5stim
PPCM1
attention
V1 V5stim
PPCM3attention
V1 V5stim
PPCM4attention
BF 2966F = 7.995
M2 better than M1
BF 12F = 2.450
M3 better than M2
BF 23F = 3.144
M4 better than M3
M1 M2 M3 M4
BMS in SPM8: an example
Fixed effects BMS at group level
Group Bayes factor (GBF) for 1...K subjects:
Average Bayes factor (ABF):
Problems:- blind with regard to group heterogeneity- sensitive to outliers
k
kijij BFGBF )(
( )kKij ij
k
ABF BF
)|(~ 111 mypy)|(~ 111 mypy
)|(~ 222 mypy)|(~ 111 mypy
)|(~ pmpm kk
);(~ rDirr
)|(~ pmpm kk )|(~ pmpm kk),1;(~1 rmMultm
Random effects BMS for group studies: a variational Bayesian approach
Dirichlet parameters= “occurrences” of models in the population
Dirichlet distribution of model probabilities
Multinomial distribution of model labels
Measured data
Stephan et al. 2009, NeuroImage
Is the red letter left or right from the midline of the word?
group analysis (random effects),n=16, p<0.05
whole-brain corrected
group analysis (random effects),n=16, p<0.05
whole-brain corrected
Task-driven lateralisation
letter decisions > spatial decisions
time
•••
Does the word contain the letter A or not?
spatial decisions > letter decisions
Stephan et al. 2003, Science
MOGleft
LGleft
LGright
RVFstim.
LVFstim.
FGright
FGleft
LD|RVF
LD|LVF
LD LD
0.20 0.04
0.06 0.02
0.00 0.01
0.01 0.01
0.27 0.06
0.11 0.03
MOGright
0.00 0.04
0.01 0.03
0.07 0.02
0.01 0.01
Inter-hemispheric connectivity in the visual ventral stream
LD>SD, p<0.05 cluster-level corrected(p<0.001 voxel-level cut-off)
Left MOG-38,-90,-4
Left FG-44,-52,-18
Right MOG-38,-94,0
p<0.01 uncorrected
Left LG-12,-70,-6
Left LG-14,-68,-2
LD>SD masked incl. with RVF>LVFp<0.05 cluster-level corrected(p<0.001 voxel-level cut-off)
LD>SD masked incl. with LVF>RVFp<0.05 cluster-level corrected
(p<0.001 voxel-level cut-off)
Right FG38,-52,-20
Stephan et al. 2007, J. Neurosci.
-35 -30 -25 -20 -15 -10 -5 0 5
Su
bje
cts
Log model evidence differences
MOG
LG LG
RVFstim.
LVFstim.
FGFG
LD|RVF
LD|LVF
LD LD
MOGMOG
LG LG
RVFstim.
LVFstim.
FGFG
LD
LD
LD|RVF LD|LVF
MOG
m2 m1
Stephan et al. 2009, NeuroImage
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
r1
p(r 1
|y)
p(r1>0.5 | y) = 0.997
157.0,843.0
194.2,806.11
21
21
rr
Simulation study: sampling subjects from a
heterogenous population
• Population where 70% of all subjects' data are generated by model m1 and 30% by model m2
• Random sampling of subjects from this population and generating synthetic data with observation noise
• Fitting both m1 and m2 to all data sets and performing BMS
MOG
LG LG
RVFstim.
LVFstim.
FGFG
LD|RVF
LD|LVF
LD LD
MOG
MOG
LG LG
RVFstim.
LVFstim.
FGFG
LD
LD
LD|RVF LD|LVF
MOG
m1
m2
Stephan et al. 2009, NeuroImage
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
100
200
300
400
500
600
700
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
B
0
2
4
6
8
10
12
14
16
18A
m1 m2
m1 m2 m1 m2
log GBF12
C D
<r>
true values:1=220.7=15.42=220.3=6.6
mean estimates:1=15.4, 2=6.6
true values:r1 = 0.7, r2=0.3
mean estimates:r1 = 0.7, r2=0.3
true values:1 = 1, 2=0
mean estimates:1 = 0.89, 2=0.11
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
intrinsic connectivity
direct inputs
modulation ofconnectivity
Neural state equation CuxBuAx jj )( )(
u
xC
x
x
uB
x
xA
j
j
)(
hemodynamicmodelλ
x
y
integration
BOLDyyy
activityx1(t)
activityx2(t) activity
x3(t)
neuronalstates
t
drivinginput u1(t)
modulatoryinput u2(t)
t
Stephan & Friston (2007),Handbook of Brain Connectivity
bilinear DCM
CuxDxBuAdt
dx m
i
n
j
jj
ii
1 1
)()(CuxBuA
dt
dx m
i
ii
1
)(
Bilinear state equation:
driving input
modulation
non-linear DCM
driving input
modulation
...)0,(),(2
0
uxux
fu
u
fx
x
fxfuxf
dt
dx
Two-dimensional Taylor series (around x0=0, u0=0):
Nonlinear state equation:
...2
)0,(),(2
2
22
0
x
x
fux
ux
fu
u
fx
x
fxfuxf
dt
dx
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
Neural population activity
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
fMRI signal change (%)
x1 x2
x3
CuxDxBuAdt
dx n
j
jj
m
i
ii
1
)(
1
)(
Nonlinear dynamic causal model (DCM):
Stephan et al. 2008, NeuroImage
u1
u2
Nonlinear DCM: Attention to motion
V1 IFG
V5
SPC
Motion
Photic
Attention
.82(100%)
.42(100%)
.37(90%)
.69 (100%).47
(100%)
.65 (100%)
.52 (98%)
.56(99%)
Stimuli + Task
250 radially moving dots (4.7 °/s)
Conditions:F – fixation onlyA – motion + attention (“detect changes”)N – motion without attentionS – stationary dots
Previous bilinear DCM
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?
Büchel & Friston (1997)
modulation of back-ward or forward connection?
additional drivingeffect of attentionon PPC?
bilinear or nonlinearmodulation offorward connection?
V1 V5stim
PPCM2
attention
V1 V5stim
PPCM1
attention
V1 V5stim
PPCM3attention
V1 V5stim
PPCM4attention
BF = 2966
M2 better than M1
M3 better than M2
BF = 12
M4 better than M3
BF = 23
Stephan et al. 2008, NeuroImage
V1 V5stim
PPC
attention
motion
-2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
%1.99)|0( 1,5 yDp PPCVV
1.25
0.13
0.46
0.39
0.26
0.50
0.26
0.10MAP = 1.25
Stephan et al. 2008, NeuroImage
V1
V5PPC
observedfitted
motion &attention
motion &no attention
static dots
Stephan et al. 2008, NeuroImage
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
-2 -1 0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-2 -1 0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
%9.99)|0( , yDp MFGFFAPPA
%9.99)|0( , yDp MFGPPAFFA
Nonlinear DCM: Binocular rivalry
Stephan et al. 2008, NeuroImage
BR nBR
FFA
PPAMFG
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
Timing problems at long TRs/TAs
• Two potential timing problems in DCM:
1. wrong timing of inputs2. temporal shift between
regional time series because of multi-slice acquisition
• 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
1
2
slic
e a
cquis
itio
n
visualinput
Slice timing in DCM: three-level model
),,( hhxxgv
),( Tvhz
),,( uxfx n
3rd level
2nd level
1st level
sampled BOLD response
BOLD response
neuronal response
x = neuronal states u = inputsxh = hemodynamic states v = BOLD responsesn, h = neuronal and hemodynamic parameters T = sampling time points
Kiebel et al. 2007, NeuroImage
Slice timing in DCM: an example
t
1 TR 2 TR 3 TR 4 TR 5 TR
t
1 TR 2 TR 3 TR 4 TR 5 TR
Defaultsampling
Slice-specific sampling
1T
2T1T
2T1T
2T1T
2T1T
2T
1T 1T 1T 1T 1T2T 2T 2T 2T 2T
Kiebel et al. 2007, NeuroImage
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)
R2R1
R2R1
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
low probability of anatomical connection small prior variance of effective connectivity parameter
high probability of anatomical connection large prior variance of effective connectivity parameter
Integration of tractography and DCM
Stephan, Tittgemeyer, Knoesche, Moran, Friston, in revision
LG(x1)
LG(x2)
RVFstim.
LVFstim.
FG(x4)
FG(x3)
LD|LVF
LD LD
BVFstim.
LD|RVF DCM structure
LGleft
LGright
FGright
FGleft
* 313
13
5.37 10
15.7%
* 334
34
2.23 10
6.5%
* 224
24
1.50 10
43.6%
* 212
12
1.17 10
34.2%
anatomical connectivity
probabilistictractography
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
6.5%
0.0384v
15.7%
0.1070v
34.2%
0.5268v
43.6%
0.7746v
connection-specific priors for coupling parameters
0 0.5 10
0.5
1m1: a=-32,b=-32
0 0.5 10
0.5
1m2: a=-16,b=-32
0 0.5 10
0.5
1m3: a=-16,b=-28
0 0.5 10
0.5
1m4: a=-12,b=-32
0 0.5 10
0.5
1m5: a=-12,b=-28
0 0.5 10
0.5
1m6: a=-12,b=-24
0 0.5 10
0.5
1m7: a=-12,b=-20
0 0.5 10
0.5
1m8: a=-8,b=-32
0 0.5 10
0.5
1m9: a=-8,b=-28
0 0.5 10
0.5
1m10: a=-8,b=-24
0 0.5 10
0.5
1m11: a=-8,b=-20
0 0.5 10
0.5
1m12: a=-8,b=-16
0 0.5 10
0.5
1m13: a=-8,b=-12
0 0.5 10
0.5
1m14: a=-4,b=-32
0 0.5 10
0.5
1m15: a=-4,b=-28
0 0.5 10
0.5
1m16: a=-4,b=-24
0 0.5 10
0.5
1m17: a=-4,b=-20
0 0.5 10
0.5
1m18: a=-4,b=-16
0 0.5 10
0.5
1m19: a=-4,b=-12
0 0.5 10
0.5
1m20: a=-4,b=-8
0 0.5 10
0.5
1m21: a=-4,b=-4
0 0.5 10
0.5
1m22: a=-4,b=0
0 0.5 10
0.5
1m23: a=-4,b=4
0 0.5 10
0.5
1m24: a=0,b=-32
0 0.5 10
0.5
1m25: a=0,b=-28
0 0.5 10
0.5
1m26: a=0,b=-24
0 0.5 10
0.5
1m27: a=0,b=-20
0 0.5 10
0.5
1m28: a=0,b=-16
0 0.5 10
0.5
1m29: a=0,b=-12
0 0.5 10
0.5
1m30: a=0,b=-8
0 0.5 10
0.5
1m31: a=0,b=-4
0 0.5 10
0.5
1m32: a=0,b=0
0 0.5 10
0.5
1m33: a=0,b=4
0 0.5 10
0.5
1m34: a=0,b=8
0 0.5 10
0.5
1m35: a=0,b=12
0 0.5 10
0.5
1m36: a=0,b=16
0 0.5 10
0.5
1m37: a=0,b=20
0 0.5 10
0.5
1m38: a=0,b=24
0 0.5 10
0.5
1m39: a=0,b=28
0 0.5 10
0.5
1m40: a=0,b=32
0 0.5 10
0.5
1m41: a=4,b=-32
0 0.5 10
0.5
1m42: a=4,b=0
0 0.5 10
0.5
1m43: a=4,b=4
0 0.5 10
0.5
1m44: a=4,b=8
0 0.5 10
0.5
1m45: a=4,b=12
0 0.5 10
0.5
1m46: a=4,b=16
0 0.5 10
0.5
1m47: a=4,b=20
0 0.5 10
0.5
1m48: a=4,b=24
0 0.5 10
0.5
1m49: a=4,b=28
0 0.5 10
0.5
1m50: a=4,b=32
0 0.5 10
0.5
1m51: a=8,b=12
0 0.5 10
0.5
1m52: a=8,b=16
0 0.5 10
0.5
1m53: a=8,b=20
0 0.5 10
0.5
1m54: a=8,b=24
0 0.5 10
0.5
1m55: a=8,b=28
0 0.5 10
0.5
1m56: a=8,b=32
0 0.5 10
0.5
1m57: a=12,b=20
0 0.5 10
0.5
1m58: a=12,b=24
0 0.5 10
0.5
1m59: a=12,b=28
0 0.5 10
0.5
1m60: a=12,b=32
0 0.5 10
0.5
1m61: a=16,b=28
0 0.5 10
0.5
1m62: a=16,b=32
0 0.5 10
0.5
1m63 & m64
0 10 20 30 40 50 600
200
400
600
model
log
gro
up
Bay
es f
acto
r
0 10 20 30 40 50 60
680
685
690
695
700
model
log
gro
up
Bay
es f
acto
r
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
model
po
st.
mo
del
pro
b.
0 0.5 10
0.5
1m1: a=-32,b=-32
0 0.5 10
0.5
1m2: a=-16,b=-32
0 0.5 10
0.5
1m3: a=-16,b=-28
0 0.5 10
0.5
1m4: a=-12,b=-32
0 0.5 10
0.5
1m5: a=-12,b=-28
0 0.5 10
0.5
1m6: a=-12,b=-24
0 0.5 10
0.5
1m7: a=-12,b=-20
0 0.5 10
0.5
1m8: a=-8,b=-32
0 0.5 10
0.5
1m9: a=-8,b=-28
0 0.5 10
0.5
1m10: a=-8,b=-24
0 0.5 10
0.5
1m11: a=-8,b=-20
0 0.5 10
0.5
1m12: a=-8,b=-16
0 0.5 10
0.5
1m13: a=-8,b=-12
0 0.5 10
0.5
1m14: a=-4,b=-32
0 0.5 10
0.5
1m15: a=-4,b=-28
0 0.5 10
0.5
1m16: a=-4,b=-24
0 0.5 10
0.5
1m17: a=-4,b=-20
0 0.5 10
0.5
1m18: a=-4,b=-16
0 0.5 10
0.5
1m19: a=-4,b=-12
0 0.5 10
0.5
1m20: a=-4,b=-8
0 0.5 10
0.5
1m21: a=-4,b=-4
0 0.5 10
0.5
1m22: a=-4,b=0
0 0.5 10
0.5
1m23: a=-4,b=4
0 0.5 10
0.5
1m24: a=0,b=-32
0 0.5 10
0.5
1m25: a=0,b=-28
0 0.5 10
0.5
1m26: a=0,b=-24
0 0.5 10
0.5
1m27: a=0,b=-20
0 0.5 10
0.5
1m28: a=0,b=-16
0 0.5 10
0.5
1m29: a=0,b=-12
0 0.5 10
0.5
1m30: a=0,b=-8
0 0.5 10
0.5
1m31: a=0,b=-4
0 0.5 10
0.5
1m32: a=0,b=0
0 0.5 10
0.5
1m33: a=0,b=4
0 0.5 10
0.5
1m34: a=0,b=8
0 0.5 10
0.5
1m35: a=0,b=12
0 0.5 10
0.5
1m36: a=0,b=16
0 0.5 10
0.5
1m37: a=0,b=20
0 0.5 10
0.5
1m38: a=0,b=24
0 0.5 10
0.5
1m39: a=0,b=28
0 0.5 10
0.5
1m40: a=0,b=32
0 0.5 10
0.5
1m41: a=4,b=-32
0 0.5 10
0.5
1m42: a=4,b=0
0 0.5 10
0.5
1m43: a=4,b=4
0 0.5 10
0.5
1m44: a=4,b=8
0 0.5 10
0.5
1m45: a=4,b=12
0 0.5 10
0.5
1m46: a=4,b=16
0 0.5 10
0.5
1m47: a=4,b=20
0 0.5 10
0.5
1m48: a=4,b=24
0 0.5 10
0.5
1m49: a=4,b=28
0 0.5 10
0.5
1m50: a=4,b=32
0 0.5 10
0.5
1m51: a=8,b=12
0 0.5 10
0.5
1m52: a=8,b=16
0 0.5 10
0.5
1m53: a=8,b=20
0 0.5 10
0.5
1m54: a=8,b=24
0 0.5 10
0.5
1m55: a=8,b=28
0 0.5 10
0.5
1m56: a=8,b=32
0 0.5 10
0.5
1m57: a=12,b=20
0 0.5 10
0.5
1m58: a=12,b=24
0 0.5 10
0.5
1m59: a=12,b=28
0 0.5 10
0.5
1m60: a=12,b=32
0 0.5 10
0.5
1m61: a=16,b=28
0 0.5 10
0.5
1m62: a=16,b=32
0 0.5 10
0.5
1m63 & m64
Overview
• Bayesian model selection (BMS)
• Nonlinear DCM for fMRI
• Timing errors & sampling accuracy
• Integrating tractography and DCM
• DCMs for electrophysiological data
),,( uxFx Neural state equation:
Electric/magneticforward model:
neural activityEEGMEGLFP
(linear)
DCM: generative model for fMRI and ERPs
Neural model:1 state variable per regionbilinear state equationno propagation delays
Neural model:8 state variables per region
nonlinear state equationpropagation delays
fMRIfMRI ERPsERPs
inputs
Hemodynamicforward model:neural activityBOLD(nonlinear)
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 spike-frequency adaptation (SFA)
• ongoing model validation by LFP recordings in rats, combined with pharmacological manipulations
standards deviants
A1
A2
Tombaugh et al. 2005, J.Neurosci.
Example of single-neuron SFA
Neural mass model of a cortical macrocolumn
ExcitatoryInterneurons
He, e
PyramidalCells
He, e
InhibitoryInterneurons
Hi, e
Extrinsic inputs
Excitatory connection
Inhibitory connection
e, i : synaptic time constant (excitatory and inhibitory) He, Hi: synaptic efficacy (excitatory and inhibitory) 1,…,: intrinsic connection strengths propagation delays
21
43
MEG/EEGsignal
MEG/EEGsignal
Parameters:
Parameters:
Jansen & Rit (1995) Biol. Cybern.David et al. (2003) NeuroImage
mean firing rate
mean
postsynaptic potential (PSP)
mean PSP
mean firing rate
43
12
12
4914
41
2))(( xxuaxsHx
xx
eeee
Excitatory spiny cells in granular layers
Exogenous input u
43
12
Intrinsicconnections
5
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in agranular layers
Inhibitory cells in agranular layers
),( uxfx
11812
102
1112511
1110
72
8938
87
2)(
2)()(
xxx
xxxSHx
xx
xxxSAAHx
xx
iiii
eeLB
ee
12
4914
41
2))()(( xxCuxSAAHx
xx
eeLF
ee
Synaptic ‘alpha’ kernelSynaptic ‘alpha’ kernel
Sigmoid functionSigmoid function
659
32
61246
63
22
51295
52
2)(
2))()()((
xxx
xxxSHx
x
xxxSxSAAHx
xx
iiii
eeLB
ee
Extrinsic
Connections:
Forward
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
matrixScalp data
),,(0 uxfx LK 0),( LKxxgy
Forward model
Kiebel et al. 2006, NeuroImage
DCM for steady-state responses
• models the cross-spectral density of recorded data
• feature extraction by means of p-order VAR model
• 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)
10
20
30
Fre
qu
en
cy
(H
z)
Time (s)
0 10
Moran et al. 2009, NeuroImage
Validation study using microdialysis (in collaboration with Conway Inst., UC Dublin)
Low GlutamateRegular Glutamate
Isolated mPFCControls mPFC
Low GlutamateRegular Glutamate
Isolated mPFCControls mPFC
mPFC
VTA
-0.06
0
0.06
0.12
mV
mPFC EEG
-0.06
0
0.06
0.12
mV
- two groups of rats with different rearing conditions
- LFP recordings and microdialysis measurements (Glu & GABA) from mPFC
Moran et al. 2008, NeuroImage
Experimental data
FFT 10 mins time series: one area (mPFC)
blue: control animalsred: isolated animals
* p<0.05, Bonferroni-corrected
Moran et al. 2008, NeuroImage
Predictions about expected parameter estimates from the microdialysis
measurements
chronic reduction in extracellular
glutamate levels
upregulation of AMPA receptors
sensitisation of postsynaptic mechanisms
EPSPs
amplitude of synaptic kernels
( He)
activation of voltage-sensitive Ca2+ channels → intracellular Ca2+ → Ca-dependent K+ currents → IAHP
SFA(2)
Van den Pool et al. 1996, NeuroscienceSanchez-Vives et al. 2000, J. Neurosci.
Extrinsicforward
connections
4
1 2u
5
Excitatory spiny cells in granular layers
Excitatory pyramidal cells in infragranular layers
Extrinsicforward
connections
4 3
u
5
Excitatory spiny cells in granular layers
Inhibitory cells in supragranular layers
[161, 210]
[29,37]
[195, 233]
(0.4)
(0.37)(0. 13)
[3.8 6.3]
[0.76,1.34] (0.0003)
(0.04)eH
2
Control group estimates in blue,isolated animals in red,p values in parentheses.
sensitization of post-synaptic mechanisms
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