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Dynamic Causal Model for evoked responses
in MEG/EEG
Rosalyn Moran
Overview
Dynamic Causal Modelling – Motivation
Dynamic Causal Modelling – Generative model
Bayesian model inversion/selection
Example
Overview
Dynamic Causal Modelling – Motivation
Dynamic Causal Modelling – Generative model
Bayesian model inversion/selection
Example
pseudo-random auditory sequence
80% standard tones – 500 Hz
20% deviant tones – 550 Hz
time
standards deviants
Mismatch negativity (MMN) – DCM Motivation
time (ms)
μV
Paradigm
Raw data(128 sensors)
Preprocessing (SPM8)
Evoked responses(here: single sensor)
Dynamic Causal Modelling- Motivation
time
sens
ors
sens
ors
standard
deviant
time (ms)
amplitude (μV)
sens
ors
sens
ors
standard
deviant
time
Conventional approach: Reduce evoked response to a few
variables.
Alternative approach that tellsus about communication among
brain sources?
Dynamic Causal Modelling- Motivation
Dynamic Causal Modelling- Motivation
),,( uxfx
)|(),|(
mypmyp
???Build a generative model for spatiotemporal dataand fit to evoked responses.
Assume that both ERs are generated by temporal dynamics of a network of a few sources
Describe temporal dynamics by differential equations
Each source projects to the sensors, following physical laws
Solve for the model parameters using Bayesian model inversion
DynamicCausal
Modelling
A1 A1
DCM uses priors for source locations
time (ms)
μV
Raw data(128 sensors)
Preprocessing (SPM8)
Evoked responses(here: single sensor)
Source LocalisationDCM
MNI coordinates
from the literature
Overview
Dynamic Causal Modelling – Motivation
Dynamic Causal Modelling – Generative model
Bayesian model inversion/selection
Example
),,( uxFx Neural state equation:
Electric/magneticforward model:
neural activityEEGMEGLFP
(linear)
Neural model:1 state variable per regionbilinear state equationno propagation delays
Neural model:8 state variables per region
nonlinear state equationpropagation delays
fMRI ERPs
inputs
Hemodynamicforward model:neural activityBOLD(nonlinear)
The Generative model
The Generative model
),,( uxfx
Source dynamics f
states x parameters θ
Input u
Evoked response
data y
),( xgy
Spatial forward model g
One Source
One Source
GranularLayer:Excitatory Cells
Infragranular layer:PyramidalCells
SupragranularLayer:Inhibitory Cells
macro-scale meso-scale micro-scale
The state of a neuron comprises a number of attributes, membrane potentials, conductances
etc. Modelling these states can become intractable. Mean field approximations summarise the states in terms of their
ensemble density. Neural mass models consider only point densities and describe the
interaction of the means in the ensemble
Dynamics
AP generation zone
synapses
AP generation zone
eH
e
1
GranularLayer:Excitatory Cells
Infragranular layer:PyramidalCells
SupragranularLayer:Inhibitory Cells
Neural mass equations and connectivity
Extrinsicforward
connectionsspiny
stellate cells
inhibitory interneurons
pyramidal cells
4 3
214
014
41
2))()((ee
LF
e
e xxCuxSIAAHx
xx
1 2)( 0xSAF
)( 0xSAL
)( 0xSABExtrinsic backward connections
Intrinsic connections
neuronal (source) model
Extrinsic lateral connections
State equations
,,uxfx
0x
278
038
87
2))()((ee
LB
e
e xxxSIAAHx
xx
236
746
63
225
1205
52
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2)(
2))()()((
iii
i
ee
LB
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e
xxxSHx
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xxxSxSAAHx
xxxxx
Overview
Dynamic Causal Modelling – Motivation
Dynamic Causal Modelling – Generative model
Bayesian model inversion/selection
Example
Model Selection & Hypothesis Testing
data y
)|( 1mypModel 1
Model 2
...
Model n
)|( 2myp
)|( nmyp
),|( 1myp
),|( 2myp
),|( nmyp
Model selection:
)|( imyp
best?
STG STG
A1 A1
STG
A1 A1
Model Selection & Hypothesis Testing
data yModel
selection:
)|( 1mypModel 1
Model 2
...
Model n
)|( 2myp
)|( nmyp
),|( 1myp
),|( 2myp
),|( nmyp
)|( imyp
STG STG
A1 A1
STG STG
A1 A1
)()|()|( pypyp posterior likelihood ∙ prior
)|( yp )(p
In DCM for ERPs priors include time constants, PSP, delays etc.
The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.
new data prior knowledge
Bayesian Statistics
)|(),(),|(),|(
)(),|()|(
mypmpmypmyp
dpmypmyp
),(),( Nmp
Invert model
Make inferences
Define likelihood model
Specify priors
Neural Parameters: Dynamic Model
Observer function:Forward Spatial Model
Inference on models
Inference on parameters
0)( xLy
LBFaieie AAAgHH ,,,,,,,,,,,, ,54321
Bayesian Inversion
Bayesian Inversion
Evoked responsesSpecify generative forward model
(with prior distributions of parameters)
Expectation-Maximization algorithm
Iterative procedure: 1. Compute model response using current set of parameters
2. Compare model response with data3. Improve parameters, if possible
1. Posterior distributions of parameters
2. Model evidence )|( myp
),|( myp
Model evidence:Approximation: Free Energy
kk
mypmypBF )(ln)(ln 212,1
Fixed Effects Model selection via log Group Bayes factor:
accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model
Bayesian Model Selection
)|( imyp
)],|(),([)|(ln GpqKLmypF i
( | , )p r y
Random Effects Model selectionvia Model probability:
)( 1 Kkqkr
Overview
Dynamic Causal Modelling – Motivation
Dynamic Causal Modelling – Generative model
Bayesian model inversion/selection
Example
pseudo-random auditory sequence
80% standard tones – 500 Hz
20% deviant tones – 550 Hz
time
standards deviants
Mismatch negativity (MMN) – DCM Motivation
time (ms)
μV
Paradigm
Raw data(128 sensors)
Preprocessing (SPM8)
Evoked responses(here: single sensor)
Garrido et al., (2007), NeuroImage
Model for mismatch negativity
Garrido et al., (2007), NeuroImage
Models for Deviant Response Generation
Bayesian Model Comparison
Forward (F)
Backward (B)
Forward and Backward (FB)
subjects
log
-evi
denc
e
Group level
Group model comparison
Garrido et al., (2007), NeuroImage
Temporal Hypotheses
Garrido et al., PNAS, 2008
Peristimulus time 1
Peristimulus time 2
Do forward and backward connections operate as a function of time? Models for Deviant Response Generation
Grand mean ERPs
Garrido M. I. et.al. PNAS 2007;104:20961-20966
©2007 by National Academy of Sciences
Model Fit
Garrido et al., PNAS, 2008
time (ms) time (ms)
Bayesian model comparison across subjects
Garrido M. I. et.al. PNAS 2007;104:20961-20966
©2007 by National Academy of Sciences
Bayesian model comparison across subjects
• First :
Forward and Backward Connections are required to produce a deviant, “mismatch” response
• Then this was refined to show:
Forward Connections are sufficient to generate early components of the mismatch ERP but Forward and Backward connections are required to generate late components of the ERP
Summary
DCM enables testing hypotheses about how brain sources communicate.
DCM is based on a neurobiologically plausible generative model of evoked responses.
Differences between conditions are modelled as modulation of connectivity.
Inference: Bayesian model selection Posterior Connectivity Estimates
mPFC
VTA
LFP
DCM for Induced Responses
DCM for Phase Coupling
Conductance Based Mean Field Models
DCM for Steady State Responses
0 20 400
5
0 20 400
5
0 20 400
5