Dynamic Causal Modelling for EEG/MEG:
principles
J. Daunizeau
Motivation, Brain and Behaviour group, ICM, Paris, France
Wellcome Trust Centre for Neuroimaging, London, UK
Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
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
effective connectivity analysis:
functional integration
A
B
A
B
?
u1
u1
u2
u1
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
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
• realistic observation model
• realistic neuronal model
• simple observation model
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:
ˆ  E  y, m 
priors on parameters
model evidence:

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
Introduction
DCM for fMRI: audio-visual associative learning
auditory cue
visual outcome
or
P(outcome|cue)
or
Put
response
0
200
400
600
800
2000
time (ms)
PMd
PPA
FFA
PPA
cue-dependent
surprise
Put
FFA
PMd
cue-independent
surprise
Den Ouden, Daunizeau et al., J. Neurosci., 2010
Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
Dynamical systems theory
motivation


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
Dynamical systems theory
exponentials
Dynamical systems theory
initial values and fixed points
Dynamical systems theory
time constants
Dynamical systems theory
matrix exponential
Dynamical systems theory
eigendecomposition of the Jacobian
Dynamical systems theory
dynamical modes
Dynamical systems theory
spirals
Dynamical systems theory
spirals
Dynamical systems theory
spiral state-space
Dynamical systems theory
embedding
Dynamical systems theory
kernels and convolution
Dynamical systems theory
summary
• Motivation: modelling reciprocal influences
• Link between the integral (convolution) and differential (ODE) forms
• System stability and dynamical modes can be derived from the system’s Jacobian:
o D>0: fixed points
o D>1: spirals
o D>1: limit cycles (e.g., action potentials)
o D>2: metastability (e.g., winnerless competition)
limit cycle (Vand Der Pol)
strange attractor (Lorenz)
Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
Neural ensembles dynamics
DCM for M/EEG: systems of neural populations
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
DCM for M/EEG: 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
DCM for M/EEG: synaptic dynamics
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
DCM for M/EEG: 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
DCM for M/EEG: 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
DCM for M/EEG: 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
Overview
1 DCM: introduction
2 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
5 Conclusion
Bayesian inference
forward and inverse problems
forward problem
p  y , m
likelihood
posterior distribution
p  y , m 
inverse problem
Bayesian inference
the electromagnetic forward problem
y  t    L(i ) w (0i )   j  (ij )  t     t 
i
j
Bayesian paradigm
deriving the likelihood function
- Model of data with unknown parameters:
y  f  
- But data is noisy:

e.g., GLM:
f    X
y  f    
- Assume noise/residuals is ‘small’:
f
 1

p     exp   2  2 
 2

P    4   0.05

→ Distribution of data, given fixed parameters:
2
 1
p  y    exp   2  y  f    
 2

Bayesian paradigm
likelihood, priors and the model evidence
Likelihood:
Prior:

Bayes rule:
generative model 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
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 Dynamical systems theory
3 Neural states dynamics
4 Bayesian inference
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
Conclusion
planning a compatible DCM study
•
Suitable experimental design:
– any design that is suitable for a GLM
– preferably multi-factorial (e.g. 2 x 2)
• e.g. one factor that varies the driving (sensory) input
• and one factor that varies the modulatory input
• Hypothesis and model:
– define specific a priori hypothesis
– which models are relevant to test this hypothesis?
– check existence of effect on data features of interest
– there exists formal methods for optimizing the experimental design
for the ensuing bayesian model comparison
[Daunizeau et al., PLoS Comp. Biol., 2011]
Many thanks to:
Karl J. Friston (UCL, London, UK)
Will D. Penny (UCL, London, UK)
Klaas E. Stephan (UZH, Zurich, Switzerland)
Stefan Kiebel (MPI, Leipzig, Germany)