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

650

2)(

2))()()((

iii

i

ee

LB

e

e

xxxSHx

xx

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