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Contents Overview of DCM –Effective connectivity, DCM framework, generative models Model specification –Neural model, haemodynamic model Model estimation –Model inversion, parameter inference Example
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Dynamic Causal ModellingIntroduction
SPM Course (fMRI), October 2015
Peter ZeidmanWellcome Trust Centre for NeuroimagingUniversity College London
is a framework
for creating, estimating and comparing generative models
of neuroimaging timeseries
Dynamic Causal Modelling
We use these models to investigate effective connectivity of neuronal populations
Contents
• Overview of DCM– Effective connectivity, DCM framework, generative
models
• Model specification– Neural model, haemodynamic model
• Model estimation– Model inversion, parameter inference
• Example
Contents
• Overview of DCM– Effective connectivity, DCM framework, generative
models
• Model specification– Neural model, haemodynamic model
• Model estimation– Model inversion, parameter inference
• Example
The system of interest
Stimulus from Buchel and Friston, 1997Brain by Dierk Schaefer, Flickr, CC 2.0
Experimental Stimulus (Hidden) Neural Activity Observations (BOLD)
time
Vector y
BO
LD
?off
on
time
Vector u
Connectivity
• Structural ConnectivityPhysical connections of the brain
• Functional ConnectivityDependencies between BOLD observations
• Effectivity ConnectivityCausal relationships between brain regions
"Connectome" by jgmarcelino. CC 2.0 via Wikimedia CommonsFigure 1, Hong et al. 2013 PLOS ONE.KE Stefan, SPM Course 2011
DCM Framework
Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0
Experimental Stimulus (u)
Observations (y)
z = f(z,u,θn).
How brain activity z
changes over time
y = g(z, θh)
What we would see in the scanner, y, given the
neural model?
Neural Model Observation Model
DCM Framework
Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0
Experimental Stimulus (u)
Observations (y)Neural Model Observation Model
Generative model p(u,y)
DCM Framework
Experimental Stimulus (u)
Observations (y)Neural Model Observation Model
Model Inversion(Variational EM)
1. Gives parameter estimates:Given our observations y, and stimuli u, what parameters θ make the
model best fit the data?
2. Gives an approximation to the log model evidence:
Free energy = accuracy - complexity
DCM Framework
Experimental Stimulus (u)
Observations (y)Neural Model Observation Model
Experimental Stimulus (u)
Observations (y)Neural Model Observation Model
Model 1:
Model 2:
Model comparison: Which model best explains my observed data?
DCM Framework1. We embody each of our hypotheses in a generative
model.The generative model separates neural activity from haemodynamics
2. We perform model estimation (inversion)This identifies parameters θ = {θn,θh} which make the model best fit the data and the free energy (log model evidence)
3. We inspect the estimated parameters and / or we compare models to see which best explains the data.
Contents
• Overview of DCM– Effective connectivity, DCM framework, generative
models
• Model specification– Neural model, haemodynamic model
• Model estimation– Model inversion, parameter inference
• Example
The Neural Model“How does brain activity, z, change over time?”
Driving input u1
V1z1a
c
u1
z1
z2
�̇�1=𝑎𝑧+𝑐 𝑢1 Inhibitory self-connection (Hz).Rate constant: controls rate of decay in region 1. More negative = faster decay.
The Neural Model“How does brain activity, z, change over time?”
V5
V1
Driving input u1
z1
z2
a11
a22
a21
c11
�̇�1=𝑎11 𝑧1+𝑐11𝑢1
Change of activity in V1:
�̇� 2=𝑎22𝑧 2+𝑎21𝑧1
Change of activity in V5:
Self decay V1 input
The Neural Model“How does brain activity, z, change over time?”
V5
V1z1
z2
a11
a22
c11�̇�=𝐴𝑧+𝐶𝑢1
[ �̇�1�̇�2]=[𝑎11 0𝑎21 𝑎22 ] [𝑧1𝑧 2]+[𝑐110 ]𝑢1
Columns are outgoing connectionsRows are incoming connections
Driving input u1
a21
The Neural Model“How does brain activity, z, change over time?”
V5
V1z1
z2
a11
a22
c11
u1
z1
z2
�̇�=𝐴𝑧+𝐶𝑢1
Driving input u1
a21
“How does brain activity, z, change over time?”
The Neural Model
V5
V1z1
z2
a11
a22
c11
Driving input u1
a21u2
u1
z1
z2
b21
Attention u2
Could model be used to model a main effect and interaction
The Neural Model“How does brain activity, z, change over time?”
V5
V1z1
z2
a11
a22
c11
Driving input u1
a21
�̇�1=𝑎11 𝑧1+𝑐11𝑢1
Change of activity in V1:
b21
Attention u2
𝑧 2=𝑎22𝑧 2+𝑎21𝑧1+¿ ¿
Change of activity in V5:
Self decay V1 input Modulatory input
The Neural Model“How does brain activity, z, change over time?”
V5
V1z1
z2
a11
a22
c11
Driving input u1
a21
b21
Attention u2
�̇�=(𝐴+∑𝑗=1
𝑚
𝑢 𝑗𝐵𝑗) 𝑧+𝐶𝑢
For m inputs:
[ �̇�1�̇�2]=([𝑎11 0𝑎21 𝑎22 ]+𝑢2[ 0 0
𝑏21 0 ])[𝑧 1𝑧 2]+[𝑐11 00 0 ][𝑢1𝑢2]
A: Structure B: ModulatoryInput
C: DrivingInput
Change in activity per
region
External input 2(attention)
Currentactivity
per region
All external input
DCM Framework
Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0
Experimental Stimulus (u)
Observations (y)
z = f(z,u,θn).
How brain activity z
changes over time
y = g(z, θh)
What we would see in the scanner, y, given the
neural model?
Neural Model Observation Model
The Haemodynamic Model
Contents
• Overview of DCM– Effective connectivity, DCM framework, generative
models
• Model specification– Neural model, haemodynamic model
• Model estimation– Model inversion, parameter inference
• Example
DCM Framework
Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0
Experimental Stimulus (u)
Observations (y)
z = f(z,u,θn).
How brain activity z
changes over time
y = g(z, θh)
What we would see in the scanner, y, given the
neural model?
Neural Model Observation Model
Bayesian Models
posterior likelihood ∙ prior
new data prior knowledge
parameter estimates
pypyp ||
Priors• All parameters have prior distributions,
• Between-regions neuronal coupling parameters have shrinkage priors.
• Haemodynamic parameters have empirical priors.
-1 -0.5 0 0.5 1Connection strength (Hz)
Prior on between-region coupling
N(0,1/64)
Prior means stored in DCM.M.pE, covariance in DCM.M.pC
Model Estimation
• Inverting the model (via Variational EM) gives:
• Posterior probability distribution for each parameter,
• Posterior estimate of the noise precision
• Approximation of the model evidence,
Posterior mean stored in DCM.Ep
Posterior variance stored in DCM.Vp.
Noise precision stored in DCM.Ce
Free energy stored in DCM.F
Bayesian Model Reduction
Option 1:
Individually fit each model to the data (then inspect or compare)
Option 2:
Fit only the full model (model 1) then use ‘post-hoc model reduction’ (Bayesian Model Reduction) to estimate the others
Model 1 Model 2 Model 3
Contents
• Overview of DCM– Effective connectivity, DCM framework, generative
models
• Model specification– Neural model, haemodynamic model
• Model estimation– Model inversion, parameter inference
• Example
PREPARING DATA
0
2
4
6
8
10
12
Choosing Regions of Interest
We generally start with SPM results
+
ROI Options
1. A sphere with given radius
Positioned at the group peak
Allowed to vary for each subject, within a radius of the group peak
or
2. An anatomical mask
Pre-processing
1. Regress out nuisance effects (anything not specified in the ‘effects of interest f-contrast’)
2. Remove confounds such as low frequency drift
3. Summarise the ROI by performing PCA and retaining the first component
200 400 600 800 1000-4
-3
-2
-1
0
1
2
3
1st eigenvariate: test
time \{seconds\}
230 voxels in VOI from mask VOI_test_mask.niiVariance: 81.66%
New in SPM12: VOI_xx_eigen.nii(When using the batch only)
EXAMPLE
Reading > fixation (29 controls)Lesion (Patient AH)
1. Extracted regions of interest. Spheres placed at the peak SPM coordinates from two contrasts:
A. Reading in patient > controls B. Reading in controls
2. Asked which region should receive the driving input
Seghier et al., Neuropsychologia, 2012
Key:ControlsPatient
Bayesian Model Averaging
Seghier et al., Neuropsychologia, 2012
TROUBLESHOOTING
0 200 400 600 800 1000 1200-2
-1
0
1
2
3
4
responses and predictionsvariance explained 87%
time {seconds}
1 2 3 4 5 6 7 8 9-1
-0.5
0
0.5
1
intrinsic and extrinsic connectionslargest connection strength 0.58
parameter}
posterior correlationsestimable parameters 13
10 20 30 40 50
10
20
30
40
50
spm_dcm_fmri_check(DCM) spm_dcm_explore (DCM)
From Jean Daunizeau’s website
Further ReadingThe original DCM paper Friston et al. 2003, NeuroImage
Descriptive / tutorial papers
Role of General Systems Theory Stephan 2004, J Anatomy
DCM: Ten simple rules for the clinician Kahan et al. 2013, NeuroImage
Ten Simple Rules for DCM Stephan et al. 2010, NeuroImage
DCM Extensions
Two-state DCM Marreiros et al. 2008, NeuroImage
Non-linear DCM Stephan et al. 2008, NeuroImage
Stochastic DCM Li et al. 2011, NeuroImageFriston et al. 2011, NeuroImageDaunizeau et al. 2012, Front Comput Neurosci
Post-hoc DCM Friston and Penny, 2011, NeuroImageRosa and Friston, 2012, J Neuro Methods
A DCM for Resting State fMRI Friston et al., 2014, NeuroImage
EXTRAS
Approximates: The log model evidence:Posterior over parameters:
The log model evidence is decomposed:
The difference between the true and approximate posterior
Free energy (Laplace approximation)
Accuracy Complexity-
Variational Bayes
The Free Energy
Accuracy Complexity-
Complexity
posterior-prior parameter means
Prior precisions
Occam’s factor
Volume of posterior parameters
Volume of prior parameters
(Terms for hyperparameters not shown)
Distance between prior and posterior means
DCM parameters = rate constants
dx axdt
0( ) exp( )x t x at
The coupling parameter ‘a’ thus describes the speed ofthe exponential change in x(t)
0
0
( ) 0.5exp( )
x xx a
Integration of a first-order linear differential equation gives anexponential function:
/2lna
00.5x
a/2ln
Coupling parameter a is inverselyproportional to the half life of x(t):
Main effects → driving inputsInteractions → modulatory inputs
FFA
Amy
Face
ValenceFrom a factorial design:
A factorial design translates easily to DCMA (fictitious!) example of a 2x2 design:
Factor 1:Stimulus (face or inverted face)
Factor 2:Valence (neutral or angry)
Main effect of face: FFA
Interaction of Stimulus x Valence: Amygdala
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