Dynamic Causal Models
Will Penny
Olivier David, Karl Friston, Lee Harrison, Andrea Mechelli, Klaas Stephan
Mathematics in Brain Imaging, IPAM, UCLA, USA, July 22 2004.
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
• Single word processingFriston et al.(2003) Neuro-Image, 19 (4), pp. 1273-1302.
Contents• Neurodynamic model
• Hemodynamic model
• Model estimation and comparison
• Attention to visual motion
• Single word processing
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
00
z a z ucz 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 00 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 00 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
a12
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
Hemodynamicvariables
For each region:
[ , , , ]s f v q=x
Hemodynamicparameters
Seconds
Dynamics
Hemodynamic saturation
Sub-linear and super-linearresponses to pairs of stimuli
Equivalent input-outputfunctions
Neuronal impulse
Neuronal impulses
Why have explicit models for neurodynamicsand hemodynamics ?
44332211 ucucucucazz ++++=
For 4 event types u1, u2, u3, u4 :
In a GLM for a single region, y=Xβ+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 PriorsHemodynamics
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 Estimation2( ) ( ; , )p pp Nθ θ µ σ=
Relative Precision Weighting
2( | ) ( ; , )ep y N y θ σθ =
2( | ) ( ; , )p y Nθ θ µ σ=
2 2 2
22 2
1 1 1
1 1
e p
pe p
y
σ σ σ
µ σ µσ σ
= +
= +
y eθ= +
pµ
yµ
Normal densities
Multiple parameters
(1)θ
(2)θ
( ) ( ; , )p pp N=θ θ µ C
( | ) ( ; , )ep N=y θ y Xθ C
( | ) ( ; , )p N=θ 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 µp are known
GeneralLinear Model
Nonlinear models( )b= +y θ e
( )( ) ( ) ( )
( )
( )
ii i
i
i
i
bb b
b
r y br
∂= + −
∂∂
=∂
∆ = −= −= ∆ +
µθ µ θ µθ
µJθ
θ θ µµ
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 bp N
=
∆ = ∆
y θ y θ Cr 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
θ∆
r
Friston et al.(2002) Neuro-Image, 16 (2), pp. 513-530.
Model Comparison I
{ , , , }=θ A B C h
( | , ) ( | )( | , )( | )
p m p mp mp m
=y θ θθ y
y
V1
V5
SPC
( | ) ( | , ) ( | )p m p m p m d= ∫y 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 mp m
=y θ θθ y
y
V1
V5
SPC
Model, mParameters:
PriorPosterior Likelihood
( | ) ( )( | )( )
p m p mp mp
=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:
( | )( | )ijp m iBp m j
==
=yy
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
STIMULISTIMULI250 250 radiallyradially moving dots at 4.7 degrees/smoving dots at 4.7 degrees/s
PREPRE--SCANNINGSCANNING5 x 30s trials with 5 speed changes (reducing to 1%)5 x 30s trials with 5 speed changes (reducing to 1%)Task Task -- detect change in radial velocitydetect 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 1−10
−5
0
5
10
−4
−2
0
2
4
0 200 400 600 800 1000 1200−4
−2
0
2
4
V1
V5
Estimation
SPC
Time (seconds)
Posterior Inference
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
6
B321
P(B
3 21|y
)
γ
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.
Contents• Neurodynamic model
• Hemodynamic model
• Bayesian estimation
• Attention to visual motion
• Single word processing
Single word processing
SPM{F}
“dog”“radio”“gate”….
10,15,30,60 and 90words perminute
A2
WA
A1
0.92
0.380.47
-0.62
-0.51
0.37
Input 1:Word Presentation
Input 1
Estimated Model
Intrinsic connections: Full connectivityassumed – only significant connectionsshown.
Modulatory connections: modulation of all self-connections, only A1 and WA significant
Hemodynamic saturation in A1
Seconds
y
Summed individual responses
Responseto pair
IndividualStimuli
Pair ofStimuli
Neuronal Saturation in A1
Seconds
zt modulationof A1 selfconnection
Withor
without
A1
-0.62
Identifiability
−−
−=
11
1
3231
2321
1312
aaaaaa
A τ
Estimation of relative intrinsic connections and modulatoryconnections is robust to errors inestimation of hemodynamics due toeg. slice timing problems
Indeterminacy in neurodynamic and hemodynamic time constants is soaked up in τ
u2
z1
z2
u1
-τc
a12
a21
b21
-τ
Summary
• Neurodynamic model• Hemodynamic model• Bayesian estimation• Attention to visual motion• Single word processing
Neuronal Transients and BOLD
300ms 500ms
More enduring transients produce bigger BOLD signals
SecondsSeconds
BOLD is sensitive to frequencycontent of transients
Seconds
SecondsRelative timings of transients areamplified in BOLD
Neurodynamics Hemodynamics