Dynamic Causal Models
description
Transcript of Dynamic Causal Models
![Page 1: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/1.jpg)
Dynamic Causal Models
Will Penny
Olivier David, Karl Friston, Lee Harrison, Stefan Kiebel, Andrea Mechelli,
Klaas Stephan
MultiModal Brain Imaging, Copenhagen, October 25-26, 2005
V1
V5
SPC
V1
V5
SPC
Wellcome Department of Imaging Neuroscience, ION, UCL, UK.
![Page 2: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/2.jpg)
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
Friston et al.(2003) Neuro-Image, 19 (4), pp. 1273-1302.
![Page 3: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/3.jpg)
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
![Page 4: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/4.jpg)
Single region
1 11 1 1z a z cu
u2
u1
z1
z2
z1
u1
a11c
![Page 5: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/5.jpg)
Multiple regions
1 11 1 1
2 21 22 2 2
0
0
z a z uc
z a a z u
u2
u1
z1
z2
z1
z2
u1
a11
a22
c
a21
![Page 6: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/6.jpg)
Modulatory inputs
1 11 1 1 12
2 21 22 2 21 2 2
0 0 0
0 0
z a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u1
a11
a22
c
a21
b21
![Page 7: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/7.jpg)
Reciprocal connections
1 11 12 1 1 12
2 21 22 2 21 2 2
0 0
0 0
z a a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u1
a11
a22
c
a1
2
a21
b21
![Page 8: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/8.jpg)
Neurodynamics
i ii
z Az u B z Cu
InputsChange inNeuronalActivity
NeuronalActivity
IntrinsicConnectivityMatrix
ModulatoryConnectivityMatrices
InputConnectivityMatrix
V1
V5
SPC
![Page 9: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/9.jpg)
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
• Single word processing
![Page 10: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/10.jpg)
Hemodynamics
( , , )
( )
g z
y b
x x h
x
Hemodynamic variables
For each region:
[ , , , ]s f v qx
Hemodynamic parameters
Seconds
Dynamics
![Page 11: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/11.jpg)
Why have explicit models for neurodynamics and hemodynamics ?
44332211 ucucucucazz
For 4 event types u1, u2, u3, u4 :
In a GLM for a single region, y=X+e, with 3 basis functions per event type (canonical,shifter, stretcher) there are 12 parameters to estimate. These relate hemodynamics directly to each stimulus.
In a (single region) DCM there are 4 neuronal efficacy parameters relating neuronal activity to each stimulus
And 5 hemodynamic parameters relating neuronal activity to the BOLDsignal.
A total of 9 parameters.
),( hzby
![Page 12: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/12.jpg)
DCM Priors
Hemodynamics
Rate of signal decay: 0.65Elimination rate: 0.41Transit time: 0.98Grubbs exponent: 0.32Oxygenation fraction: 0.34
E[h]Cov[h]
Neurodynamics
Stability priors ensure principal Lyapunov exponent is less than zerowith high probability.
![Page 13: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/13.jpg)
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
• Single word processing
![Page 14: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/14.jpg)
Bayesian Estimation
2( ) ( ; , )p pp N
Relative Precision Weighting
2( | ) ( ; , )ep y N y
2( | ) ( ; , )p y N
2 2 2
22 2
1 1 1
1 1
e p
pe p
y
y e
p
y
Normal densities
![Page 15: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/15.jpg)
Multiple parameters
(1)θ
(2)θ
( ) ( ; , )p pp Nθ θ μ C
( | ) ( ; , )ep Ny θ y Xθ C
( | ) ( ; , )p Nθ y θ μ C
1 1 1
1 1
Te p
Te p p
C X C X C
μ C X C y C μ
y Xθ e
One-step if Ce, Cp and p are known
GeneralLinear Model
![Page 16: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/16.jpg)
Nonlinear models( )b y θ e
( )( ) ( ) ( )
( )
( )
ii i
i
i
i
bb b
b
r y b
r
μθ μ θ μ
θμ
Jθ
θ θ μ
μ
J θ e
( ) ( ; , )
( ) ( ; , )
p p
p i p
p N
p N
θ θ μ C
θ θ μ μ C
( | ) ( ; ( ), )
( | ) ( ; , )e
e
p N b
p N
y θ y θ C
r θ r J θ C
,i iμ C
1 1 1
1
1 11 1
Ti e p
Ti i i e p p i
C J C J C
μ μ C J C r C μ μ
Gauss-Newton ascent with priors
Linearization
CurrentEstimates
{ , , , }θ A B C h
r
Friston et al.(2002) Neuro-Image, 16 (2), pp. 513-530.
![Page 17: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/17.jpg)
Model Comparison I
{ , , , }θ A B C h
( | , ) ( | )( | , )
( | )
p m p mp m
p m
y θ θθ y
y
V1
V5
SPC
( | ) ( | , ) ( | )p m p m p m dy y θ θ θ
Model, m Parameters:
PriorPosterior Likelihood
Evidence
Laplace, AIC, BIC approximations
Model fit + complexityPenny et al. (2004) NeuroImage, 22(3), pp. 1157-1172.
![Page 18: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/18.jpg)
Model Comparison II
{ , , , }θ A B C h
( | , ) ( | )( | , )
( | )
p m p mp m
p m
y θ θθ y
y
V1
V5
SPC
Model, mParameters:
PriorPosterior Likelihood
( | ) ( )( | )
( )
p m p mp m
p
yy
y
PriorPosterior Evidence
Parameter Parameter
Model Model
![Page 19: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/19.jpg)
Model Comparison III
V1
V5
SPC
( | ) ( | , ) ( | )p m i p m i p m i d y y θ θ θ
( | ) ( | , ) ( | )p m j p m j p m j d y y θ θ θ
Model, m=i
V1
V5
SPC
Model, m=j
Model Evidences:
Bayes factor:
( | )
( | )ij
p m iB
p m j
y
y
1 to 3: Weak3 to 20: Positive20 to 100: Strong>100: Very Strong
![Page 20: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/20.jpg)
Contents
• Neurodynamic model
• Hemodynamic model
• Bayesian estimation
• Attention to visual motion
• Single word processing
![Page 21: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/21.jpg)
Attention to Visual MotionAttention to Visual Motion
STIMULISTIMULI
250 radially moving dots at 4.7 degrees/s250 radially moving dots at 4.7 degrees/s
PRE-SCANNINGPRE-SCANNING
5 x 30s trials with 5 speed changes (reducing to 1%)5 x 30s trials with 5 speed changes (reducing to 1%)
Task - detect change in radial velocityTask - detect change in radial velocity
SCANNINGSCANNING (no speed changes)(no speed changes)
6 normal subjects, 4 100 scan sessions;6 normal subjects, 4 100 scan sessions;
each session comprising 10 scans of 4 different conditioneach session comprising 10 scans of 4 different condition
1. Photic2. Motion3. Attention
Experimental Factors
Buchel et al. 1997
![Page 22: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/22.jpg)
Specify regions of interest
Identify regions of Interest eg. V1, V5, SPC
GLM analysis
V1
V5
SPC
Motion
Photic
Att
Model 1
![Page 23: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/23.jpg)
V1
V5
SPC
Motion
Photic
Att
Model 1V1
V5
Estimation
SPC
Time (seconds)
![Page 24: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/24.jpg)
Posterior Inference
B321
P(B
3 21|y
)
How muchattention (input 3) changes connection fromV1 (region 1) to V5 (region(2)
![Page 25: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/25.jpg)
V1
V5
SPC
Motion
Photic
Att
Model 1
Motion
Photic
Att
V1
V5
SPC
Model 2
Bayes Factor B12 > 1019
VeryStrong
![Page 26: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/26.jpg)
V1
V5
SPC
Motion
Photic
Att
Model 1
V1
V5
SPC
Motion
Photic
Att
Model 3
Bayes Factor B13=3.6
Positive
![Page 27: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/27.jpg)
V1
V5
SPC
Motion
Photic
Att
Model 1
Motion
Photic
Att
Att
V1
V5
SPC
Model 4
Bayes Factor B14=2.8
Weak
Penny et al. (2004) NeuroImage, Special Issue.
![Page 28: Dynamic Causal Models](https://reader036.fdocuments.us/reader036/viewer/2022062314/56813af4550346895da37326/html5/thumbnails/28.jpg)
Summary
• Neurodynamic model
• Hemodynamic model
• Bayesian estimation
• Attention to visual motion