Dynamic Causal Modelling
for event-related responses
J. Daunizeau
Motivation, Brain and Behaviour group, ICM, Paris, France
Wellcome Trust Centre for Neuroimaging, London, UK
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Introduction
structural, functional and effective connectivity
structural connectivity
functional connectivity
effective connectivity
O. Sporns 2007, Scholarpedia
•
structural connectivity
= presence of axonal connections
•
functional connectivity
= statistical dependencies between regional time series
•
effective connectivity
= causal (directed) influences between neuronal populations
! connections are recruited in a context-dependent fashion
Introduction
from functional segregation to functional integration
localizing brain activity:
functional segregation
u1
effective connectivity analysis:
functional integration
A
B
A
u1
u2
u1
B
u2
u1 X u2
« Where, in the brain, did
my experimental manipulation
have an effect? »
« How did my experimental manipulation
propagate through the network? »
Introduction
why do we use dynamical system theory?


u 
 x 
y
1
1
21
1
2
32
13
3
13u
1
2
2
3
3
2
time
t  t
3
 3u
t  0
t
u
t  t
 x t  0

?
t
Introduction
DCM: evolution and observation mappings
Hemodynamic
observation model:
temporal convolution
Electromagnetic
observation model:
spatial convolution
neural states dynamics
x  f ( x, u , )
fMRI
EEG/MEG
• simple neuronal model
• slow time scale
• complicated neuronal model
• fast time scale
inputs
Introduction
DCM: a parametric statistical approach
• DCM: model structure
 y  g  x,    

 x  f  x, u, 
24
2
likelihood
 p  y  , , m 
4
3
1
u
• DCM: Bayesian inference
parameter estimate:
model evidence:


priors on parameters
ˆ    p  y  ,  , m  p  m  p  m  d d

p  y m    p  y  ,  , m  p  m  p  m  d d

Introduction
DCM for EEG-MEG: auditory mismatch negativity
sequence of auditory stimuli
…
…
S
S
S
D
S
S
S
S
D
S
S-D: reorganisation
of the connectivity structure
standard condition (S)
rIFG
rIFG
deviant condition (D)
lA1
lSTG
rA1
lSTG
rSTG
lA1
rA1
rSTG
t =200 ms
Daunizeau, Kiebel et al., Neuroimage, 2009
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Neural ensembles dynamics
multi-scale perspective
macro-scale
meso-scale
Golgi
micro-scale
Nissl
EI
external granular
layer
external pyramidal
layer
EP
II
mean-field firing rate
internal granular
layer
internal pyramidal
layer
synaptic dynamics
Neural ensembles dynamics
from micro- to meso-scale
x j  t  : post-synaptic potential of j th neuron within its ensemble
1

N 
H
x
t








 j'
 H  x  t     p  x  t   dx
N  1 j ' j

 S    mean-field firing rate

mean firing rate (Hz)
ensemble density
p(x)
 

S(x)

H(x)
S(x)
membrane depolarization (mV)
mean membrane depolarization (mV)
Neural ensembles dynamics
synaptic kinematics
post-synaptic potential
membrane
depolarization (mV)
EPSP
 1  2

2
2
2   i / e S ( )  2 i / e 2   i / e 1
IPSP
Ki
1
K e 1
time (ms)
Neural ensembles dynamics
intrinsic connections within the cortical column
 7  8
Golgi
external granular
layer
8   3 e2 S ( 0 )  2 e 8   e2 7
Nissl
inhibitory
interneurons
external pyramidal
layer
internal granular
layer
internal pyramidal
layer
spiny
stellate cells
4
3
1  4
4   1 e2 S ( 0 )  2 e  4   e2 1
1
intrinsic
connections
pyramidal cells
 0  5   6
 2  5
5   2 e2 S ( 1 )  2 e 5   e2  2
3  x6
6   4 i2 S ( 7 )  2 i 6   i2 3
2
Neural ensembles dynamics
 ( i )  t    ( i )  r, t 
lateral (homogeneous)
density of connexions
from meso- to macro-scale
r1
r2
local wave propagation equation (neural field):
 2

3 2 2  (i )
2

2




c     r, t   c  (i )  r, t 
 2
t
2
 t

 (i )    ii ' S   (i ') 
i'
0th-order approximation: standing wave
Neural ensembles dynamics
extrinsic connections between brain regions
extrinsic
lateral
 L S ( 0 )
connections
 7  8
8   e2 (( B   L   3 I ) S ( 0 ))  2 e 8   e2 7
inhibitory
interneurons
spiny
stellate cells
extrinsic
forward
connections
 F S ( 0 )
4
3
1  4
4   e2 (( F   L   1 I ) S ( 0 )   u u )  2 e 4   e2 1
1
pyramidal cells
2
 0  5   6
 2  5
5   e2 (( B   L ) S ( 0 )   2 S ( 1 ))  2 e 5   e2 2
3  x6
6   i2 4 S ( 7 )  2 i 6   i2 3
extrinsic
backward  B S ( 0 )
connections
Observation mappings
the electromagnetic forward model
y  t    L(i ) w (0i )   j  (ij )  t     t 
i
j
 t 
N  0, Qy 
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Bayesian inference
forward and inverse problems
forward problem
p  y , m
likelihood
posterior distribution
p  y , m 
inverse problem
Bayesian inference
likelihood and priors
u
likelihood
p  y , m
prior
p  m 
posterior
generative model m
p  y, m  
p  y  , m  p  m 
p  y m
Bayesian inference
model comparison
Principle of parsimony :
« plurality should not be assumed without necessity »
Model evidence:
y = f(x)

p  y m    p  y  , m  p  m  d

y=f(x)
x
model evidence p(y|m)
“Occam’s razor” :
space of all data sets
Bayesian inference
the variational Bayesian approach

ln p  y m   ln p  , y m   S  q   DKL q   ; p  y, m 
q

free energy : functional of q
mean-field: approximate marginal posterior distributions:
q   , q  
1
2
p 1 , 2 y , m 
2
p 1 or 2 y, m 
1
q 1 or 2 
Bayesian inference
EM in a nutshell
Specify generative forward model
(with prior distributions of parameters)
Evoked responses
Expectation-Maximization algorithm
Iterative procedure:
1. Compute model response using
current set of parameters
2. Compare model response with data
3. Improve parameters, if possible
1. Posterior distributions of parameters
p ( | y, m)
2. Model evidence
p ( y | m)
Bayesian inference
DCM: key model parameters
21
1
2
32
13
3
13u
 3u
u
21,32 ,13 
state-state coupling
 3u
input-state coupling
13u
input-dependent modulatory effect
Bayesian inference
model comparison for group studies
ln p  y m1   ln p  y m2 
differences in log- model evidences
m1
m2
subjects
fixed effect
assume all subjects correspond to the same model
random effect
assume different subjects might correspond to different models
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Example
models for deviant response generation
Garrido et al., (2007), NeuroImage
Example
group-level model comparison
Group level
log-evidence
Bayesian Model Comparison
subjects
Forward (F)
Backward (B)
Forward and Backward (FB)
Garrido et al., (2007), NeuroImage
Example
MMN: temporal hypotheses
Models for Deviant Response Generation
Do forward and backward connections operate
as a function of time?
Peristimulus time 1
Peristimulus time 2
Garrido et al., PNAS, 2008
Example
MMN: (best) model fit
time (ms)
time (ms)
Garrido et al., PNAS, 2008
Example
MMN: group-level model comparison across time
Garrido et al., PNAS, 2008
Overview
1. DCM: introduction
2. Neural ensembles dynamics
3. Bayesian inference
4. Example
5. Conclusion
Conclusion
back to the auditory mismatch negativity
sequence of auditory stimuli
…
…
S
S
S
D
S
S
S
S
D
S
S-D: reorganisation
of the connectivity structure
standard condition (S)
rIFG
rIFG
deviant condition (D)
lA1
lSTG
t ~ 200 ms
rA1
lSTG
rSTG
lA1
rA1
rSTG
Conclusion
DCM for EEG/MEG: variants
input
 second-order mean-field DCM
1st and 2d order moments
depolarization
250
0
0
200
-20
-20
150
-40
-40
100
-60
-60
50
-80
-80
0
0
100
200
300
time (ms)
auto-spectral density
LA
-100
0
100
200
300
time (ms)
-100
0
100
200
300
time (ms)
auto-spectral density
CA1
cross-spectral density
CA1-LA
frequency (Hz)
frequency (Hz)
 DCM for steady-state responses
frequency (Hz)
 DCM for induced responses
 DCM for phase coupling
Many thanks to:
Karl J. Friston (FIL, London, UK)
Klaas E. Stephan (UZH, Zurich, Switzerland)
Stefan Kiebel (MPI, Leipzig, Germany)