Dynamic Causal Model for Steady State Responses

Dynamic Causal Model for Steady State Responses

Rosalyn 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

1. Bayesian Inversion: Parameter Estimates and Model Comparison

2. Example. DCM for Steady State Responses: Glutamate with Microdialysis validation

Predicting Anaesthetic Depth

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

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Frequency (Hz)

Pow

er (

uV2)

Pow

er (

uV2)

Source 2

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

Cross Spectral Density: The data from a time series

Vector Auto-regression a p-order model:

Linear prediction formulas that attempt to predict an output y[n] of a system based on

the previous outputs

npnpnnn eyyyy ....2211

))(()( pAfg ij

ijijijij HHg )()()(

iwpijp

iwijiwijij eeeH

......

1)(

221

f 2

}{:)(....1 ccpAp Resulting in a matrices for c Channels

Cross Spectral Density for channels

i,j at frequencies

..)(

..)()(

12

1211

g

gg

Overview

1. Data Features





Dynamic Causal Modelling: Generic Framework

simple neuronal model

Slow time scale


Slow time scale

fMRIfMRI

complicated neuronal model

Fast time scale


Fast time scale

EEG/MEGEEG/MEG

),,( uxFdt

dx

Neural state equation:Neural state equation:

Hemodynamicforward model:neural activityBOLD

Time Domain Data


Time Domain Data

Electromagneticforward model:

neural activityEEGMEGLFP

Time Domain ERP DataPhase Domain Data

Time Frequency DataSteady State Frequency Data



Time Domain ERP DataPhase Domain Data

Time Frequency DataSteady State Frequency Data

Dynamic Causal Modelling: Generic Framework


Slow time scale


Slow time scale

fMRIfMRI


Fast time scale


Fast time scale

EEG/MEGEEG/MEG

),,( uxFdt

dx

Neural state equation:Neural state equation:



Steady State Frequency Data



Steady State Frequency Data


Time Domain Data


Time Domain Data

Frequency (Hz)

Pow

er (m

V2 ) “theta”

Dynamic Causal Modelling: Framework


fMRIfMRI


EEG/MEGEEG/MEG),,( uxF

dt

dx

Neural state equation:




Generative M

odel

Bay

esia

n In

vers

ion

Empirical Data

Model Structure/ Model Parameters

Dynamic Causal Modelling: Framework


fMRIfMRI


EEG/MEGEEG/MEG),,( uxF

dt

dx

Neural state equation:




Generative M

odel

Bay

esia

n In

vers

ion

Empirical Data


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

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2))(( xxuaxsHx

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eeee

Excitatory spiny cells in granular layers

43

12

Intrinsicconnections

5


Excitatory pyramidal cells in agranular layers

Inhibitory cells in agranular layers

),( uxfx

11812

102

1112511

1110

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2)(

2)()(

xxx

xxxSHx

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xxxSAAHx

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2))()(( xxCuxSAAHx

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eeLF

ee

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2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

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iiii

eeLB

ee

Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model

Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007)

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),( uxfx

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2)(

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Synaptic ‘alpha’ kernelSynaptic ‘alpha’ kernel

Sigmoid functionSigmoid function

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Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model


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2))(( xxuaxsHx

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),( uxfx

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2)(

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2))()(( xxCuxSAAHx

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2)(

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Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model


hrv

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2))(( xxuaxsHx

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),( uxfx

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2)(

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Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model


ieH /

ie /

hrv

: Receptor Density

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2))(( xxuaxsHx

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12


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),( uxfx

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Extrinsic

Connections:

Forward

Backward

Lateral

Neural Mass Model


ieH /

ie /

hrv

: Receptor Density

)(vSr

: Firing Rate

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2))(( xxuaxsHx

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),( uxfx

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Neural Mass Model


hrv

: Receptor Density

)(vSr

: Firing Rate

ieH /

ie /

Extrinsic Connections:

Forward

Backward

Lateral

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

12

34

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


Dynamic Causal Modelling: Steady State Responses

..),:()( ,/ ieieHfH


Dynamic Causal Modelling: Steady State Responses



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

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

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

ERP or Steady State Responses

Time Domain

Freq Domain

Time Domain

Freq Domain

Outputs Through Lead fieldc3c1

outputs1(t)

outputs2(t) output

s3(t)

neuronalstates

drivinginput u(t)

Freq DomainOutput

Freq DomainOutput

Freq DomainCortical InputFreq DomainCortical Input

/)( 21 fH

bf

aU

1)(

c2

+

Time DomainTime Domain

ERPOutputERP

Output

Pulse InputPulse Input

)(ty

Overview

1. Data Features





Generative M

odel

Bay

esia

n In

vers

ion

Empirical Data


Bayesian Inversion

Time Domain

Freq Domain

Time Domain

Freq Domain

c3c1

NMM

NMM

NMM

Freq DomainOutputFreq Domain

Output

Freq DomainCortical InputFreq Domain

Cortical Input

)( fH

bf

aU

1)(

c

2

+

Frequency (Hz)

Po

wer

Bay

esia

n In

vers

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

Overview

1. Data Features





N=7 N=8

4.2 ± 1.4μM 1.5 ± 0.8μM (36%)

Glutamatergic processing and microdialysis

- Microdialysis measurements of glutamate- Two groups of rats with different rearing conditions- LFP recordings from mPFC

Isolated mPFCControls mPFC

Low GlutamateRegular Glutamate

Isolated mPFCControls mPFC

mPFC

-0.06

0

0.06

0.12

mV

mPFC EEG

-0.06

0

0.06

0.12

mV

Glutamatergic processing and microdialysis Experimental data

Oscillations from 10 mins : one area (mPFC)

blue: control animalsred: isolated animals

* p<0.05, Bonferroni-corrected

Predictions about expected parameter estimates from the microdialysis measurements

chronic reduction in extracellular

glutamate levels

upregulation of AMPA

receptors

sensitisation of postsynaptic mechanisms

EPSPs

amplitude of synaptic kernels( He)

activation of voltage-sensitive Ca2+ channels → intracellular Ca2+ → Ca-dependent K+ currents → IAHP

SFA()

Van den Pool et al. 1996, NeuroscienceSanchez-Vives et al. 2000, J. Neurosci.

Increased EPSP

Increased adaptation

Glutamatergic processing and microdialysis Hypotheses

mPFC




Synaptic ‘alpha’ kernel

Synaptic ‘alpha’ kernel

eH

eH

iH

Decreased Sigmoid Firing

Extrinsicforward

connections

4

1 2u

5


Excitatory pyramidal cells in infragranular layers

Extrinsicforward

connections

4 3

u

5


Inhibitory cells in supragranular layers

[161, 210]

[29,37]

[195, 233]

(0.4)

(0.37)(0. 13)

[3.8 6.3](0.04)

eH

Control group estimates in blue,isolated animals in red,p values in parentheses.

Glutamatergic processing and microdialysis Results

Moran, Stephan, Kiebel, Rombach, O’Connor, Murphy, Reilly, Friston (2008)

[0.76,1.34] (0.0003)

Overview

1. Data Features

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




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

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

pyramidal

cellspyrami

dal cells

spiny stellate

cells

inhibitory interneurons

pyramidal cells

E13 E

31

E23

I32

IIRVE

II

EERVE

EE

VIIEELL

gVg

gVg

IVVgVVgVVgVC

)),((

)),((

)()()(

)1()2()2(12

)1(

)1()3()3(13

)1(

)1()1()1()1()1()1(

NMDANMDARVI

NMDANMDA

EERVE

EE

gVg

gVg

)),((

)),(()2()3()3(

23)2(

)2()3()3(23

)2(

VEMgNMDAEELL VVVfgVVgVVgVC ))(()()( )2()2()2()2()2()2(

NMDANMDARVI

NMDANMDA

IIRVI

II

EERVE

EE

gVg

gVg

gVg

)),((

)),((

)),((

)3()1()1(31

)3(

)3()2()2(32

)3(

)3()1()1(31

)3(

VEMgNMDAIIEELL VVVfgVVgVVgVVgVC ))(()()()( )3()3()3()3()3()3()3()3(

I12

40

-100 -50 0 500

0.2

0.4

0.6

0.8

1

Membrane Potential (mV)

f MG

Exogenous Input (I)

Documents

Dynamic Causal Model for Steady State Responses