dynamic causal model for steady state responses

DYNAMIC CAUSAL MODEL FOR STEADY STATE RESPONSESRosalyn Moran

Wellcome Trust Centre for Neuroimaging

DCM for Steady State Responses

Under linearity and stationarity assumptions, the model’s biophysical parameters (e.g. post-synaptic receptor density and

time constants) prescribe the cross-spectral density of responses measured directly (e.g. local field potentials) or indirectly through

some lead-field (e.g. electroencephalographic and magnetoencephalographic data).

Overview

1. Data Features

2. The Generative Model in DCMs for Steady-State Responses – a family

of neural mass models

3. Bayesian Inversion: Parameter Estimates and Model Comparison

4. Example. DCM for Steady State Responses: Anaesthetic Depth in Rodents: Validating a few Basics

Questions of Consciousness using Anaesthesia in Humans

Overview

1. Data Features

2. The Generative Model in DCMs for Steady-State Responses - a family of neural mass models




Steady State

Statistically:

A “Wide Sense Stationary” signal has 1st and 2nd moments that do not vary with respect to time

Dynamically:

A system in steady state has settled to some equilibrium after a transient

Data Feature:

Quasi-stationary signals that underlie Spectral Densities in the Frequency Domain

Steady State

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Po

wer

(u

V2 )

Source 2

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Po

wer

(u

V2 )

Source 1

Cross Spectral Density: The Data E

EG

- M

EG

– L

FP

Tim

e S

eri

es

Cro

ss

Sp

ec

tral D

en

sity

1

1

2

2 3

3

4

4

1

2

3

4

A few LFP channels or EEG/MEG spatial modes

Overview

1. Data Features





Dynamic Causal Modelling: Generic Framework

simple neuronal model

Slow time scale

fMRI

complicated neuronal model

Fast time scale

EEG/MEG

),,( uxFdt

dx

Neural state equation:

Hemodynamicforward model:neural activityBOLD

Time Domain Data

Electromagneticforward model:

neural activityEEGMEGLFP

Time Domain ERP DataPhase Domain Data

Time Frequency DataSteady State Frequency Data

Dynamic Causal Modelling: Generic Framework


Slow time scale

fMRI


Fast time scale

EEG/MEG

),,( uxFdt

dx




Steady State Frequency Data


Time Domain Data

Frequency (Hz)

Pow

er (m

V2 )

“theta”

Dynamic Causal Modelling: Framework


fMRIfMRI


EEG/MEGEEG/MEG),,( uxF

dt

dx





Genera

tive

ModelB

ayesi

an

Invers

ion

Empirical Data

Model Structure/ Model Parameters

neuronal (source) model

State equationsExtrinsic Connections ,,uxFx

spiny stellate cells

inhibitory interneurons

PyramidalCells

Intrinsic Connections

Internal Parameters

EEG/MEG/LFPsignal

EEG/MEG/LFPsignal

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 statesin terms of their ensemble density. Neural mass models consider only point densities and describe the interaction of the means in the ensemble

Neural Mass Model

4g3g

1g2g

12

4914

41

2))(( xxuaxsHx

xx

eeee kkgk --+-==

&&

Excitatory spiny cells in granular layers

4g3g

1g2g

Intrinsicconnections

5g


Excitatory pyramidal cells in agranular layers

Inhibitory cells in agranular layers

),( uxfx

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee

Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model

4g 3g

1g2g

12

4914

41

2))(( xxuaxsHxxx

eeee kkgk --+-==

&&


4g 3g

1g2g


5g




),( uxfx

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

Synaptic ‘alpha’ kernelSynaptic ‘alpha’ kernel

Sigmoid functionSigmoid function

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee

Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model

Neural Mass Model

4g3g

1g2g

12

4914

41

2))(( xxuaxsHx

xx

eeee kkgk --+-==

&

&Excitatory spiny cells in granular layers

4g3g

1g2g


5g




),( uxfx

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

Synaptic ‘alpha’ kernel

Synaptic ‘alpha’ kernel

Sigmoid functionSigmoid function

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee

Extrinsic

Connections:

Forward

Backward

Lateral

ieH /

ie /

hrv

: Receptor Density

Neural Mass Model

4g 3g

1g2g

12

4914

41

2))(( xxuaxsHxxx

eeee kkgk --+-==

&&


4g 3g

1g2g


5g




),( uxfx

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee


Sigmoid function

Sigmoid function

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee

Extrinsic

Connections:

Forward

Backward

Lateral

ieH /

ie /

hrv

: Receptor Density

)(vSr

: Firing Rate

Neural Mass Model

4g3g

1g2g

12

4914

41

2))(( xxuaxsHx

xx

eeee kkgk --+-==

&&



5g




),( uxfx

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee


Sigmoid function

Sigmoid function

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee

hrv

: Receptor Density

)(vSr

: Firing Rate

ieH /

ie /

Extrinsic Connections:

Forward

Backward

Lateral

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

1g2g

3g4g

Frequency Domain Generative Model(Perturbations about a fixed point)

Time Differential Equations

)(

)(

xly

Buxfx

State Space Characterisation

Cxy

BuAxx

Transfer FunctionFrequency Domain

BAsICsH )()(

Linearise

mV

Frequency Domain Generative Model(Perturbations about a fixed point)

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

Pow

er (m

V2 )Po

wer

(mV2 )

Pow

er (m

V2 )

Spectrum channel/mode 1

Spectrum mode 2

Cross-spectrum modes 1& 2


..),:()(2 ,/ ieieHfH


..),:()(12 ,/ ieieHfH


..),:()(1 ,/ ieieHfH

Dynamic Causal Modelling: Framework


fMRIfMRI


EEG/MEGEEG/MEG),,( uxF

dt

dx





Genera

tive

ModelB

ayesi

an

Invers

ion

Empirical Data

Model Structure/ Model Parameters

Overview

1. Data Features





Bayesi

an

Invers

ion

)|(

)|(),|(),|(

myp

mpmypmyp

Bayes’ rules:

)|(

)|(

2

1

myp

mypBF

Model comparison via Bayes factor:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

Inference on models

Model 1Model 2

Free Energy: )),()(()(ln mypqDmypF max

Inference on parameters

Model 1

-2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

),()( mypq

%1.99)|0( yconnp

Fixed effects BMS at group level

Group Bayes factor (GBF) for 1...K subjects:

Average Bayes factor (ABF):

Problems:- blind with regard to group heterogeneity- sensitive to outliers

kn

nijij BFGBF

...1

)(

kn

nj

kn

niij FFGBF

..1

)(

..1

)( lnlnlnor

k

kn

nijij BFABF

...1

)(

Overview

1. Data Features





Depth of Anaesthesia

A1 A2

-0.06

0

0.06

0.12

mV

LFP

-0.06

0

0.06

0.12

mV

-0.06

0

0.06

0.12

mV

-0.06

0

0.06

0.12

mV

Trials:1: 1.4 Mg Isoflourane2: 1.8 Mg Isoflourane3: 2.4 Mg Isoflourane4: 2.8 Mg Isoflourane

(White Noise and Silent Auditory Stimulation)

30sec

A1

A2

Forward (Excitatory Connection)

Backward (Modulatory Connection)

A1

A2Forward (Excitatory Connection)

FB Model (1)

BF Model (2)

Models

Backward (Modulatory Connection)

Model 1 Model 20

5

10

15

20

25

30

35

Ln G

BF

Model Fits: Model 1

Results

A1

A2

He: maxEPSP

Hi: maxIPSP

IsofluraneIsoflurane

IsofluraneIsoflurane

Overview

1. Data Features





Boly et al. in prep

WakeMild SedationDeep Sedation

Anterior Cingulate

Posterior Cingulate

Increased gamma power in Mild & Deep Sedation vs WakeIncreased low frequency power in Deep Sedation

Murphy & Bruno, [..], Tononi, Boly, submitted

DCM


Anterior Cingulate

Posterior Cingulate

Is Loss of Consciousness associating with decreased thalamocortical connectivity?

Models


Mildly Sedated

Model Parameters and States of Consciousness

Loss of Consciousness

Increase in thalamic excitability

Breakdown in Backward Connections

Thalamic excitability mirrored the changes in fast rhythms that accompany propofol infusion but did not change with LOC. On the other hand, backward cortico-cortical connectivity was preserved during mild sedation, but showed a significant reduction with loss of consciousness - thus following the expression of slow power changes in EEG

Model Parameters and States of Consciousness

Summary

DCM is a generic framework for asking mechanistic questions of neuroimaging data

Neural mass models parameterise intrinsic and extrinsic ensemble connections and synaptic measures

DCM for SSR is a compact characterisation of multi- channel LFP or EEG data in the Frequency Domain

Bayesian inversion provides parameter estimates and allows model comparison for competing hypothesised architectures

Empirical results suggest valid physiological predictions

Thank You

FIL Methods Group

dynamic causal model for steady state responses

Documents

steadystate responses

model comparison example

time constants

parameter estimates

magnetoencephalographic

anaesthetic depth

spectral densities

neuroimaging dcm