DynamicCausalModelling(DCM)forfMRI：动态因果模型DCM的功能磁共振成像

activity z(t)
vasodilatory signal
s z s γ( f 1)
f
s
flow induction
f s
f
changes in volume v
changes in dHb
τv f v1/α
τq f E( f,) q v1/α q/v
v
q
BOLD signal
y(t) v, q
z Az Cu
{A,C}
z1 a11 a12 a13 0 z1 0 c12
z2
a21
a22
0
a24
z2
c21
zz43
a031
0 a42
a33 a43
a34 a44
z3 z4
0
0
0 0 0
u1 u2
12
Extension:
FG
bilinear
h { ,g ,t ,a , } n {A, B1...Bm ,C} { h , n}
changes in volume v τv f v1/α
v
changes in dHb τq f E( f,) q v1/αq/v
q
ηθ|y
y (x)
modelled
y h(u, ) X e
2
Overview
Dynamic Causal Modelling of fMRI
Definitions & motivation
The neuronal model(bilinear dynamics) The Haemodynamic model
Estimation: Bayesian framework
observation model
BOLD response
19
Haemodynamics: 2 nodes with input
Dashed Line: Real BOLD response
a11
a22
a21
Activity in z1 is coupled to z2 via coefficient ຫໍສະໝຸດ Baidu21
• System parameters :
specify exact form of system
z (t) 1
overall
system state
z(t) represented
by state variables
zn (t)
z1 z1 f1(z1...zn ,u,1)
a11 a21
0 a22
z1 z2
u2
0 b221
0 0
z1 z2
c 0u1
modulatory input u2 activity
a b221 0
through the coupling 621
Neurodynamics: reciprocal connections
u1
a11
A F z z z
Input u(t)
modulation of connectivity
direct inputs
c1
b23
B j 2F z zu j u j z
C F z u u
integration
activity z1(t)
y
a12 activity
z2(t)
y
activity z3(t)
0 0
a24 a34 a44
u3
0 0 0
b132 0 0 0
0 0 0 0
0
0
b334
0
z1
z2
z3 z4
0
c21
0
0
c12 0 0 0
0
0 0
u1 u2
0u3
13
Bilinear state equation in DCM/fMRI
state changes
z3 left
dynamic
system
z1
LG left
FG
right z4
LG
right z2
RVF CONTEXT LVF
u2
u3
u1
m
z ( A u j B j )z Cu j 1
z1 z2 zz43
aaa0132111
a12 a22 0 a42
a13 0 a33 a43
Dynamic Causal Modelling (DCM) for fMRI
Andre Marreiros
Wellcome Trust Centre for Neuroimaging
University College London
1
Thanks to...
Stefan Kiebel Lee Harrison Klaas Stephan Karl Friston
connectivity
modulation of system direct m external connectivity state inputs inputs
z1
a11
zn an1
a1n
m
u j b1j1
ann j1 bnj1
b1jn
z1
c11
bnjn zn cn1
z1 z2
a11 a21
0 a22
z1 z2
c 0u1
a21 0
activity in z2 is coupled to z1 via
coefficient a21
5
Neurodynamics: positive modulation
a11
u1
u2
a21
z1
a22
z2
z1 z2
16
Diagram
Dynamic Causal Modelling of fMRI
Network dynamics
Haemodynamic response
Priors
State space Model
fMRI data y
Model inversion using
Expectation-maximization
z
change of
state vector in time
zn zn fn (z1...zn ,u,n )
z F (z,u, )
11
Example: linear dynamic system
FG
z3 left
z1
LG left
FG
right z4
LG right
z2
LG = lingual gyrus FG = fusiform gyrus
u2
a12
z1
a21
a22
z2
z1 z2
a11 a21
a12 a22
z1 z2
u2
0 b221
0 0
z1 z2
c 0u1
a21 0
a12 0
b221 0
reciprocal connection disclosed by7 u2
Haemodynamics: reciprocal connections
Constraints on
•Connections •Hemodynamic parameters
p( )
prior
posterior
p( | y) p( y | ) p( )
Bayesian estimation
18
Overview:
stimulus function u
parameter estimation
DCM latest Extensions
3
Principles of organisation
Functional specialization
Functional integration
4
Neurodynamics: 2 nodes with input
u1
a11
u2
z1
a21
z2
a22
c1m u1 cnm um
m
z ( A u j B j )z Cu j 1
14
Conceptual overview
Neuronal state equation z F (z, u, n )
The bilinear model z (A ujB j )z Cu
effective connectivity
left LG
LG
LG
left
right
RVF
LVF
LG = lingual gyrus FG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF) visual field.
blue: observed BOLD signal red: modelled BOLD signal (DCM)
z (A
u j B j )z Cu
neuronal state equation
• Specify model (neuronal and hemodynamic level)
• Make it an observation model by adding measurement error e and confounds X (e.g. drift).
a21 0
a12 0
b221 0
green: neuronal activity red: bold response
9
Example: modelled BOLD signal
Underlying model
(modulatory inputs not shown)
FG
FG
left
right
a11
a12
a21
a22
Bold Response
Simulated response
Bold Response
z1
z2
a11 a21
a12 a22
z1 z2
u2
0 b221
0 0
z1 z2
c 0u1
a21 0
a12 0
b221 0
green: neuronal activity red: bold response
Visual input in the - left (LVF) - right (RVF) visual field.
RVF
u2
state changes
LVF
u1
effective connectivity
system input external state parameters inputs
neuronal states
y BOLD
z
λ
hemodynamic model
y
Friston e1t 5al. 2003,
NeuroImage
The hemodynamic “Balloon” model
• 5 hemodynamic parameters:
h { ,g ,t ,a, }
important for model fitting, but of no interest for statistical inference • Empirically determined a priori distributions. • Computed separately for each area
activity - dependentvasodilatory signal
s z s γ( f 1)
f
s
s
hidden states
x {z, s, f , v, q}
state equation
x F(x,u, )
flow - induction (rCBF)
f s
f
parameters
• Bayesian parameter estimation using Bayesian version of an expectation-maximization algorithm.
• Result: (Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y.
8
Haemodynamics: reciprocal connections
a11
a12
a21
a22
Bold with Noise added
Bold with Noise added
z1
z2
a11 a21
a12 a22
z1 z2
u2
0 b221
0 0
z1 z2
c 0u1
Model comparison
Posterior distribution of parameters
17
Estimation: Bayesian framework
Models of
•Hemodynamics in a single region •Neuronal interactions
likelihood term p( y | )
right LG
10
Use differential equations to describe mechanistic model of a system
• System dynamics = change of state vector in time
• Causal effects in the system: – interactions between elements – external inputs u