Dynamic Causal Modelling for Steady State Responses...Dynamic Causal Modelling for SSR A framework which uses Bayesian techniques to fit differential equations to steady – state

Dynamic Causal Modelling forSteady State Responses

Dimitris Pinotsis

The Wellcome Trust Centre forNeuroimaging

A1 A2

NeuroimagingUCL Institute of Neurology, London, UK

SPM Course London May 2012

Dynamic Causal Modelling for SSR

A framework which uses Bayesian techniques to fitdifferential equations to steady – state data. It allowsfor comparison between competing models of brainarchitecture and furnishes estimates for parametersthat are not measured directly by exploiting

1. What is DCM for SSR?

that are not measured directly by exploitingelectrophysiological data.

Although it is based on sophisticated models fromcomputational neuroscience, its

application is straightforward and

does not require mathematical

training.

-Brain activity retains similar statistical features(e.g.variance) and frequency contentacross measurement period-Cannot describe nonlinear couplingbetween frequenciesnext talk

2. When should I use DCM for SSR?

Advantages:

Summarize activity in a compact way -no need to fit long time series (computationally expensive)Describe brain function in terms of acharacteristic frequency associated with the task understudy

Extract DataFeatures

Specifymodel

Maximise themodel evidence

(~-F)

Reveal hiddenbrain

architectures

Experimentaldata

DCM Chain

3. Which steps do we take when usingDCM for SSR?

architecturesdata

Get estimates ofhidden

parameters

DCM for SSR(Moran et al., Neuroimage, 2009)

Anaesthesia:Anaesthetic Depth in Rodents(Moran et al., Plos One, 2011) Dopamine in working memory(Moran et al., Current Biol., 2011)

4.Where has DCM for SSR been applied ?

(Moran et al., Current Biol., 2011) Beta oscillations in PD(Moran et al., Plos CB, 2011)(Marreiros – yesterday’s talk) Sleep and Coma(Boly et al., Science, 2011,J Neuro, 2012)Extension:DCM for Neural Fields(Pinotsis et al., Neuroimage, 2011,2012)

Overview

1. Data Features

2. Generative Model

3. Bayesian Inversion: Parameter Estimates and ModelComparisonComparison

4. Example: Glutamate and GABA in Rodent AuditoryCortex

5. DCM for Current Source Density

6. DCM for Neural Fields


Specifymodel


(~-F)

Reveal hiddenbrain

architectures

Experimentaldata

DCM Chain

architecturesdata


parameters

Cross Spectral DensityM

EG

–L

FP

Tim

eS

eri

es

Frequency (Hz)

Po

we

r(m

V2)

EE

G-

ME

G

1

2

3

4

Frequency (Hz)

Po

we

r(m

V

Summarizes brain responsein terms of power at eachfrequency

From Time Series to Cross Spectral DensitiesM

EG

–L

FP

Tim

eS

eri

es

Cro

ss

Sp

ec

tralD

en

sity

1

EE

G-

ME

G

Cro

ss

Sp

ec

tralD

en

sity

1

2

2 3

3

4

4

1

2

3

4

A few LFP channels or EEG/MEG spatial modes

Vector Auto-regression 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

}{:)( ccpA Resulting in a matrices for c Channels

From Time Series to Cross Spectral Densities

))(()( pAfg ij

ijijijij HHg )()()(

iwpijp

iwijiwijijeee

H

......

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

2. Generative Model

3. Bayesian Inversion: Parameter Estimates and Model3. Bayesian Inversion: Parameter Estimates and ModelComparison

4. Example:Glutamate and GABA in Rodent AuditoryCortex

5. DCM for Cross Spectral Density



Specifymodel


(~-F)

Reveal hiddenbrain

architectures

Experimentaldata

DCM Chain

architecturesdata


parameters

Model=equations

andparameters

A Brain Region as an Input - Output System

equations: determine the dynamics(eg. limit cycles, transients, steady –state)

1 2 3 4 5, ,{ , , , , , , , , , , , , }e i e i a F B LH H g A A A

(eg. limit cycles, transients, steady –state)parameters: fine tune the dynamics(e.g. faster, shorter)

Maximum PSP, time constants, intrinsic and extrinsic connectivity etc

Pink line =

Steady state responses

Pink line =Container (bowl)If there is no external perturbation, the ball will stay at the centreIf there is, the container will be tilted and the ball will oscillate around the centre asshownNow, imagine that the ball has a bell inside. If there is an external perturbation thebell will start ringing.CAUSE of CONTAINER TILTING NEURAL NOISEBALL BRAIN REGIONRINGINGS RESPONSES (cross spectral densities)CONTAINER MODEL (equations, parameters cf. shape/friction)STEADY STATE PERTURBATIONS means that“ the ball always stays very close to the centre” (while the bowl is tilted)

A Brain Region as an Input - Output System

ERP

(Model)Brain region

Neuronalinnovations

Predictedresponses

( , )Yg ( , )Ug k

Frequency (Hz)P

ow

er

(mV

2)

SSRSSR

Frequency Domain Generative Model(Perturbations about a fixed point)

Time DifferentialEquations

)( Buxfx

State SpaceCharacterisation

BuAxx

Transfer FunctionFrequency Domain

)(

)(

xly

Buxfx

Cxy

BuAxx

BAsICsH )()(

Linearise

mV

Stationary Time Series


Po

we

r(m

V2)

Po

we

r(m

V2)

Spectrum channel/mode 1 Cross-spectrum modes 1& 2Transfer FunctionFrequency Domain

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

Peaks Parameters

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

Po

we

r(m

VP

ow

er

(mV

2)

Po

we

r(m

V

Spectrum mode 2Transfer FunctionFrequency Domain

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

Cross-Transfer FunctionFrequency Domain

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

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

ERP vs Steady State Responses

Outputs ThroughLead fieldc3c1

outputneuronal

Freq DomainOutput

/)( 21fH

c2

+

Time Domain

ERPOutput

)(ty

Time Domain

Freq Domainoutputs1(t)

outputs2(t) output

s3(t)

neuronalstates

drivinginput u(t)

Freq DomainCortical Input

bf

aU

1)(

Time Domain

Pulse Input

inhibitoryinterneurons

IntrinsicConnections

EEG/MEG/LFPsignal

Neural Mass Model

Tens of thousands of neuronsapproximated by their averageresponse. Neural mass modelsdescribe the interaction of theseaverages between populations andsources

neuronal (source) model

State equations

Extrinsic Connections

,,uxFx

spiny stellatecells

interneurons

PyramidalCellsInternal

Parameters

4g

3g

=&

Excitatory spiny cells in granular layers

4g

3g

Intrinsicconnections

5g


Inhibitory cells in agranular layers

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

xx

Neural Mass Model: Equations for Voltage and Current

1g

2g

1

2

4914

41

2))(( xxuaxsHx

xx

eeeekkgk --+-=

=

&

&

1g

2g

Excitatory pyramidal cells in agranular layers

),( uxfx

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

4g

3g

=&


4g

3g


5g



11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

xx

Synaptic ‘alpha’ kernel

Neural Mass Model: Two Transformations

1g

2g

1

2

4914

41

2))(( xxuaxsHx

xx

eeeekkgk --+-=

=

&

&

1g

2g


),( uxfx

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

Sigmoid function

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee

ExtrinsicConnections:

Forward

Backward

Lateral

Neural Mass Model: 1st Transformation

4g

3g

=&


4g

3g


5g



11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

xx

Synaptic ‘alpha’ kernel

ieH /

ie/

hrv

: Receptor Density

1g

2g

1

2

4914

41

2))(( xxuaxsHx

xx

eeeekkgk --+-=

=

&

&

1g

2g


),( uxfx

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


Forward

Backward

Lateral

hrv

Input: presynaptic rateOutput: Average synapticdepolarization

Neural Mass Model: 2nd Transformation

4g

3g


4g

3g


5g



11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSAAHx

xx

iiii

eeLB

ee

Input: Avg synapticdepolirizationOutput: firing rate

1g

2g

1

2

4914

41

2))(( xxuaxsHx

xx

eeeekkgk --+-=

=

&

&

1g

2g


),( uxfx

12

4914

41

2))()(( xxCuxSAAHx

xx

eeLF

ee

Sigmoid function

659

32

61246

63

22

51295

52

2)(

2))()()((

xxx

xxxSHx

x

xxxSxSAAHx

xx

iiii

eeLB

ee


Forward

Backward

Lateral

)(vSr

: Firing Rate

Overview

1. Data Features

2. Generative Model






Specifymodel


(~-F)

Reveal hiddenbrain

architectures

Experimentaldata

Bayesian Inversion

DCM Chain

architecturesdata


parameters

Frequency (Hz)

Po

wer

Time Domain

Freq Domain

c3c1

NMM

NMM

NMM

Freq DomainOutput

Freq DomainCortical Input

)( fH

bf

aU

1)(

c2

+

• predictedresponses

• parameters• Hyper

parameters

Generativemodel

• variance

Priors

Roadmap

•Modelevidence

•Posteriorestimates ofparameters

BayesianInference

Baye

sia

nIn

vers

ion

Inference on models Inference on parameters

DATA PREDICTIONS

Variational Bayes Algorithm

• Iterative procedure• Minimize free energy and optimize parameters•Maximum accuracy over complexity constraints

Model Fit and Comparison

Baye

sia

nIn

vers

ion

Model 1

Model 2

Model 1

),()( mypq

Model comparison via Bayes factor:

)|(

)|(

2

1

myp

mypBF

Overview

1. Data Features

2. Generative Model





Pharmacological Manipulation of Glutamate andGABA

Questions of Study:

Are our estimates of excitation and inhibition veridical,

e.g. ?

Can we obtain hierarchical relationships between brainregions?

,e iH H

regions?

AIM:

NOT to explain mechanisms of isoflurane BUT

to exploit isoflurane to induce known changes in synaptictransmission and THEN

Use LFP recordings and DCM for SSR to infer changes

Moran, Tetsuya, Jung, Endepols, Graf, Dolan, Friston, Stephan, Tittgemeyer,PLoSONE, 2011

Use animal LFP recordings from primary auditory cortex (A1) & posteriorauditory field (PAF)

Manipulate neurotransmitter processing via anaesthetic agent Isoflurane

4 levels of anaesthesia: each successively decreasing glutamate andincreasing GABA (Larsen et al Brain Research 1994; Lingamaneni et alAnesthesiology 2001; Caraiscos et al J Neurosci 2004 ; de Sousa et alAnesthesiology 2000

White noise stimulus & Silence

1.4 % 1.8 % 2.4 % 2.8 %

Pharmacological Manipulation of Glutamate and GABA

-0.06

0

0.06

0.12

mV

A2

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

A1

1.4 %Isoflurane

1.8 %Isoflurane

2.4 %Isoflurane

2.8 %Isoflurane


Data

Cross-spectra white noise

Po

wer

A1

Po

wer

A1

-P

AF

0.04

0.06

0.04

0.06

1.4 %1.8 %2.4 %2.8 %

Frequency (Hz) Frequency (Hz)

Frequency (Hz)

Po

wer

A1

Po

wer

PA

FP

ow

er

A1

0 5 10 15 20 25 300

0.02

0.04

0 5 10 15 20 25 300

0.02

0.04

0 5 10 15 20 25 30

0

0.02

0.04

0.06

Predicted Observed


Model

PAF

backward

Or?

5gInhibitory cells in supra(infra) granular

layers4 4v g

ExtrinsicConnections:Forward

M2

AI

PAFforward

backward

M1

AI

PAF

forward

4g 3g

2g

12

4914

41

2))(( xxuaxsHxxx

eeee kkgk --+-==

&&


3g

1g 2g

Intrinsic connections

Excitatory spiny cells

in granular layers

Pyramidal cells in infra(supra) granularlayers

layers4 4

2 24 6 3 6 4 4

5 5

25 5 7 5 5

7 4 5

( ) ( ) 2

( ) 2

B regione e e e e e

i i i i

v g

g H A S v H S v g v

v g

g H S v g v

v g g

1 1

2 21 6 1 6 1 1( ) ( ) 2F region

e e e e e e

v g

g H A S v H S v g v

6

2 2

2 22 2 1 2 2

3 3

23 4 7 3 3

6 2 3

( ) ( ) 2

( ) 2

B regione e e e e e

i i i i

v g

i H A S v H S v g v

v g

g H S v g v

v g g

ForwardBackwardLateral

AI

PAF

lateral

lateral

Or?

M3

150

200

250

Lo

gG

rou

pB

ayes

Facto

r


Model

AI

PAF

forward

backward

PAbackward

M1

M1

M2

M3

0

50

100

150

White Noise

Silence

Lo

gG

rou

p

AI

PAF

forward

AI

PAF

lateral

lateral

M2

M3

DCM recovers known neuronatomy

Decreased Glutamate Release

Increased gabaergic transmission

1.4 %Isoflurane

1.8 %Isoflurane

2.4 %Isoflurane

2.8 %Isoflurane


Physiological Parameters

AI

PAF

Excitatory spiny cellsin granular layers

Excitatory pyramidal cellsin agranular layers

Inhibitory cells inagranular layers

Synaptic ‘alpha’ kerneleH

eH

iH


Parametric Effectof Isoflurane

-1.5

-1

-0.5

0.5

1

**

**

**

Do

se

Sp

ec

ific

Ga

in

0

Excitatory spiny cellsin granular layers

Excitatory pyramidal cellsin agranular layers

Inhibitory cells inagranular layers

eH

eH

iH

1.4 1.8 2.4 2.8-1.5

Isoflurane Dose

DCM recovers known drug-induced changes

Overview

1. Data Features

2. Generative Model

3. Bayesian Inversion: Parameter Estimates and Model3. Bayesian Inversion: Parameter Estimates and ModelComparison




Specifymodel



(~-F)

Test models orMAP parameters

Find yourexperimental

data

DCM Chain

1. Interface Additions

2. New CSD routines, similar to SSR

3. SPM_NLSI_GN accommodates imaginary numbers, slopes,curvatures

4. A host of new results features, in channel and source space!

dataAlso report lags,

coherence,covariance &phase delays

Friston, Bastos, Litvak, Stephan, Fries, Moran, Neuroimage, 2012

Interface Additions


Time DifferentialEquations

)( Buxfx

State SpaceCharacterisation

BuAxx

Transfer FunctionFrequency Domain

Complex NumberComplex Number

)(

)(

xly

Buxfx

Cxy

BuAxx

BAsICsH )()(

Linearise

mV

Im

r

φb

What is a complex number ?

The Fourier transform of a signal is a continuouscomplex function

Re

z=a + ib

i2 = -1

= r(cosφ + isinφ)

H(ω) = F(f)= ∫∞

f(t)e-2πωit dt-∞

= reiφ

a

1.5

2

2.5

3

3.5

real

prediction and response: E-Step: 32

-0.2

0

0.2

0.4

0.6

0.8

1

imag

inar

y

prediction and response: E-Step: 32

DCM for CSD: data fits have two parts:real and imaginary

0 10 20 30 40 500

0.5

1

1.5

Frequency (Hz)0 10 20 30 40 50

-1

-0.8

-0.6

-0.4

-0.2

Frequency (Hz)im

agin

ary

| H2(ω) . H2*(ω) |

DCM for SSR

Predictionsand Data

| H1(ω) . H1*(ω) |

| H1(ω) . H2*(ω) |

4g 3g2g

12491441

2))(( xxuaxsHxxx eeee kkgk --+-==&&

3g

1g 2g

5g


Pyramidal cells

Inhibitory cells4g 3g

2g124914

412))(( xxuaxsHxxx eeee kkgk --+-==&

&3g

1g 2g

5g


Pyramidal cells

Inhibitory cells

H2(ω) . H2*(ω)

Predictions

DCM for Cross Spectral Density

H1(ω) . H1*(ω)

H1(ω) . H2*(ω)

4g 3g2g

12491441


3g

1g 2g

5g


Pyramidal cells


2g124914


&3g

1g 2g

5g


Pyramidal cells

Inhibitory cells

H2(ω) . H2*(ω)

DCM for Cross Spectral Density

Predictionsand Data

H1(ω) . H1*(ω)

H1(ω) . H2*(ω)

4g 3g2g

12491441


3g

1g 2g

5g


Pyramidal cells


2g124914


&3g

1g 2g

5g


Pyramidal cells

Inhibitory cells

Spectra Abs(H (ω) . H *(ω)) , Abs(H (ω) . H *(ω)) …

4g 3g2g

12491441


3g

1g 2g

5g


Pyramidal cells

Inhibitory cells

4g 3g2g

12491441


3g

1g 2g

5g


Pyramidal cells

Inhibitory cells

H1(ω)H2(ω)

Spectra

Coherence

Delay at particular frequencies

Covariance (lags over time, collapsed across frequencies )

Abs(H1(ω) . H1*(ω)) , Abs(H1(ω) . H2*(ω)) …

|(H1(ω).H2*(ω))|2 / { (H1(ω).H1*(ω)) + (H2(ω).H2*(ω))}

arg(H1(ω).H2*(ω))/2πf

Real( F-1(H1(ω).H2*(ω)))

Can also optimize complex-valued quantities Understand how biophysical parameters affect conventionallinear systems measures

Overview

1. Data Features

2. Generative Model





Specifymodel



(~-F)

Test models orMAP parameters

Find yourexperimental

data

DCM Chain

New NFM routines

data

And also reportspatial extent of

intrinsic connectionsand conduction speed

Pinotsis, Moran, Friston, Neuroimage 2012

Main difference with previous models:Brain activity is deployed on a cortical patch as opposed tobeing centred around a point

P

2

1

2

( , ) expx

L x

Lead field modified to enable a mapping of spatially distributedactivity (coloured waves) to a time series at P

Lateral interactions in the visual cortex

Novelty of DCM for NFs: Can get estimates of parametersrelating to spatial properties of sources when there is NOSPATIAL INFO in the data

intrinsic connection strength

spatial decay rate connection extent

c xK x = ae

a

c

Inhibitory cells in supragranular layers

23 23,a c32 32,a c

Exogenous input

Hidden states

22 2 2 2

2 2 22 23 2 2 23 2 23 23 3 3

2

2 ( ) ( ( ) ( ))

i i i i

xx

v v v m

sc s c s c v s v

Jansen and Rit Neural Field Models

( )U t

( , ) ( , )V x t f V U

Augment old equations with wave equations describingpropagation of afferent spike rate between points on the cortex

Exogenousinput

13 13,a c

( )U t Excitatory spiny cells in granular layers

Excitatory pyramidal cells in supra- and infragranular layers

Endogenousoutput 3( )v t

31 31,a c

23 23,a c32 32,a cObserved signals

23 3 3 3

2 2 23 31 3 3 31 3 31 31 1 2 1 2

2

2 ( ) ( ( ) ( )) ( ) ( )

e e e e

xx

v v v m

sc s c s c v v s v v

21 1 1 1

2 2 21 13 1 1 13 1 13 13 3 3

2 ( )

2 ( ) ( ( ) ( ))

e e e e

xx

v v v m U

sc s c s c v s v

( ) ( , ) ( , )y t L x V x t dx

conduction velocitys

Inference on Model Parameters

Summary

• DCM is a generic framework for asking mechanistic questions based onneuroimaging data (e.g. drug-induced changes in balance of synaptictransmission)

• Neural mass models parameterise intrinsic and extrinsic ensemble connectionsand synaptic measures (time constants, effective connectivity,…)

• DCM for SSR and CSD provide a compact characterisation of multi- channel LFPor EEG data in the frequency domainor EEG data in the frequency domain

• Bayesian inversion provides parameter estimates and allows model comparisonfor competing hypothesised architectures

• Neural field models incorporate propagation of activity on a cortical patch, so onecan distinguish between spatial effects and other factors such as cortico-thalamicinteractions or intrinsic cell properties

• Neural field models yield estimates of parameters related to topographicproperties of the sources such as spatial decay rate of synaptic connections andintrinsic conduction speed, even when using spatially unresolved data

Thanks to FIL Methods GroupKarl Friston

Rick Adams

Harriet Brown

Jean Daunizeau

Marta Garrido

Guillaume Flandin

Stefan Kiebel

VladimirVladimir LitvakLitvak

…and thank you !

Andre Marreiros

Jeremie Mattout

Rosalyn MoranRosalyn Moran

Ashwini Oswal

Will Penny

Ged Ridway

Klaas Stephan

Yen Yu

Field Power Spectra

2

1

2

( , ) expx

L x

( ) ( , ) ( , ) ( )Y Y Ng g g

Exogenous input

Hidden states

Observed signals

( ) ( , ) ( , )y t L x V x t dx

( , ) ( , )V x t f V U

( )U t

'( , ) , , ,

( , ) ( , )

Re( ) ~ (0, ( , )) Im( ) ~ (

Y m U mj

N UN N U U

j j j j jg L T g T L

g g k

0, ( , ))

Mass Power Spectra

( , ) ( )L x L

( ) ( , ) ( , ) ( )

( , ) ( ) , ,

Y Y N

k

g g

g L T g T L

g

Exogenous input

Hidden states

Observed signals

( )U t

( )y t L V

( ) ( , )V t f V U

'

1

( , ) ( ) , ,

( , ) ( , )

, ( , )

( , )

Re( ) ~ (0, ( , )) Im

kY m U mk

N UN N U U

k k j tm m

km

k

g L T g T L

g g k

T t e dt

g ft e

x u

( ) ~ (0, ( , ))

Maximum postsynaptic depolarization8, 32 (mV)

Postsynaptic time constants1/4, 1/28 (ms-1)

Amplitude of intrinsic connectivity kernels2000, 8000, 2000, 1000

Intrinsic connectivity decay constant0.32 (mm-1)0.32 (mm-1)

Sigmoid parameters(post synaptic firing rate function)0.54, 0,0.135

Conduction velocity3 m/s

Radius of cortical source50 (mm)

Exogenous input

Neural Mass Neural Field

( )U t

( )U t

Neural Mass vs Neural Field

( , )Yg Difference in predicted spectra because of difference in underlying model:

Hidden states

Observed signals

, ( , )D Q D x x t t Q x t dx dt

2 2V B V BV ABF V GU

( )y t L V

2 2V B V BV ABD F V GU

( ) ( , ) ( , )y t L x V x t dx

Inverse conduction speedS

pect

rald

ensi

ty

0 20 400

10

0 20 400

5

10

0 20 400

5

10

0 20 400

10

0 20 400

5

0 20 400

2

4

10

Connectivity decayconstant

Changing model parameters

New peaks appear:- as intrinsic speed decreases- as connectivity extent increases

Spe

ctra

lden

sity

Frequency (Hz)

0 20 400

0 20 400

10

20

0 20 400

50

0 20 400

0 20 400

20

0 20 400

10

Pinotsis, Moran, Friston, Neuroimage 2012

Roadmap

•predictedresponses

•parameters•hyperparameters

Generativemodel

•variancePriors

•Model Evidence•Posteriors

BayesianInference

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Y Y N

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( , ) ( , )p m N

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p G m p G m p d

p G m p mp G m

p G m

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Re( ) ~ (0, ( , )) Im( ) ~ (0, ( , ))

NN Ng

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Measured data Specify generative forward model(with prior distributions of parameters)

Variational Laplace Algorithm

Iterative procedure:

1. Compute model response usingcurrent set of parameters andhyperparameters

log ( ) ( ( ) ( , ))

log ( , ) ( ( ) ( ))q

F p y m D q p y m

p y m D q p m

Bayesian Model Inversion

Maximize a free energy bound to model evidence :

Iterative procedure: hyperparameters2. Compare model response with data3. Improve parameters and

hyperparameters

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2

1

myp

mypBF

Model comparison via Bayes factor:

Maximum accuracy over complexity constraints

),()( mypq

Documents

Dynamic Causal Modelling for Steady State Responses...Dynamic Causal Modelling for SSR A framework which uses Bayesian techniques to fit differential equations to steady – state