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SPC. V1. V5. SPC. V1. V5. Dynamic Causal Models. Will Penny Olivier David, Karl Friston, Lee Harrison, Stefan Kiebel, Andrea Mechelli, Klaas Stephan. Wellcome Department of Imaging Neuroscience, ION, UCL, UK. MultiModal Brain Imaging, Copenhagen, October 25-26, 2005. - PowerPoint PPT Presentation
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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.
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
Friston et al.(2003) Neuro-Image, 19 (4), pp. 1273-1302.
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
Single region
1 11 1 1z a z cu
u2
u1
z1
z2
z1
u1
a11c
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
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
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
Neurodynamics
i ii
z Az u B z Cu
InputsChange inNeuronalActivity
NeuronalActivity
IntrinsicConnectivityMatrix
ModulatoryConnectivityMatrices
InputConnectivityMatrix
V1
V5
SPC
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
• Single word processing
Hemodynamics
( , , )
( )
g z
y b
x x h
x
Hemodynamic variables
For each region:
[ , , , ]s f v qx
Hemodynamic parameters
Seconds
Dynamics
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
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.
Contents
• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
• Single word processing
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
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
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.
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.
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
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
Contents
• Neurodynamic model
• Hemodynamic model
• Bayesian estimation
• Attention to visual motion
• Single word processing
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
Specify regions of interest
Identify regions of Interest eg. V1, V5, SPC
GLM analysis
V1
V5
SPC
Motion
Photic
Att
Model 1
V1
V5
SPC
Motion
Photic
Att
Model 1V1
V5
Estimation
SPC
Time (seconds)
Posterior Inference
B321
P(B
3 21|y
)
How muchattention (input 3) changes connection fromV1 (region 1) to V5 (region(2)
V1
V5
SPC
Motion
Photic
Att
Model 1
Motion
Photic
Att
V1
V5
SPC
Model 2
Bayes Factor B12 > 1019
VeryStrong
V1
V5
SPC
Motion
Photic
Att
Model 1
V1
V5
SPC
Motion
Photic
Att
Model 3
Bayes Factor B13=3.6
Positive
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.
Summary
• Neurodynamic model
• Hemodynamic model
• Bayesian estimation
• Attention to visual motion