Rosalyn Moran

Virginia Tech Carilion Research Institute

Dynamic Causal Modelling for Cross Spectral Densities

Outline

• DCM & Spectral Data Features (the Basics)• DCM for CSD vs DCM for SSR

• DCM for CSD Example

Outline

• DCM & Spectral Data Features (the Basics)• DCM for CSD vs DCM for SSR

• DCM for CSD Example

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 activity BOLD

Time Domain Data

Electromagneticforward model:neural activity EEGMEGLFP

Time Domain ERP DataPhase Domain DataTime Frequency DataSteady State Frequency DataCross Spectral Densities (Frequency Domain)

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:

Electromagneticforward model:

neural activity EEGMEGLFP

CSDs

Hemodynamicforward model:neural activity BOLD

Time Domain Data

Frequency (Hz)

Pow

er (m

V2 )

“theta”

Dynamic Causal Modelling: Framework

simple neuronal model

fMRIfMRI

complicated neuronal model

EEG/MEGEEG/MEG

),,( uxFdt

dx

Neural state equation:

Electromagneticforward model:

Hemodynamicforward model:

Generative M

odel

Baye

sian

Inve

rsio

n

Empirical Data

Model Structure/ Model Parameters

Inference on models

Dynamic Causal Modelling: FrameworkBa

yesi

an In

vers

ion

)|(

)|(),|(),|(

myp

mpmypmyp

Bayes’ rules:

Model 1Model 2 Model 1

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

-2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

%1.99)|0( yconnp

Inference on parameters

)|(

)|(

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 Inference on parameters

Dynamic Causal Modelling: FrameworkBa

yesi

an In

vers

ion

)|(

)|(

2

1

myp

mypBF

Model comparison via Bayes factor:

)|(

)|(),|(),|(

myp

mpmypmyp

Bayes’ rules:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

Model 1Model 2 Model 1

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

-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

Dynamic Causal Modelling: Neural Mass Model

neuronal (source) model

State equations

Extrinsic Connections

,,uxFx

spiny stellate cells

inhibitory interneurons

PyramidalCells

Intrinsic Connections

Internal Parameters

EEG/MEG/LFPsignal

EEG/MEG/LFPsignal

Properties of tens of thousands of neurons approximated by their average response

Dynamic equations mimic physiology and produce electrophysiological responses

A Neural Mass Model (6) layer cortical regions)State equations: A dynamical systems description

of anatomy and physiology ,,uxFx

Extrinsic Connections

spiny stellate

cells

Supragranular Pyramidal

Cells + inhibitory

interneurons

Deep Pyramidal

Cells + inhibitory

interneurons

Intrinsic Connections

Internal ParametersEg.Time constants of Sodium ion channels

GABAa receptors

AMPA receptors

Neurotransmitters: Glu/GABA

Dynamics mimicked at AMPA and GABA receptors

AP generation zone

synapses

eH

e

Cortico-cortical connection

GABAa receptors

AMPA receptors

Neurotransmitters: Glu/GABA

AP generation zone

1 Intrinsic Connection

Cortico-cortical connection

GranularLayer:Excitatory Cells

SupragranularLayer:Inhibitory Cells

3

InfragranularLayer:Pyramidal Cells

Parameters quantify contributions at AMPA and GABA receptors

synapses

eH

e

Cortico-cortical connection

GABAa receptors

AMPA receptors

Neurotransmitters: Glu/GABA

AP generation zone

1 Intrinsic ConnectionGranularLayer:Excitatory Cells

SupragranularLayer:Inhibitory Cells

3

InfragranularLayer:Pyramidal Cells

Extrinsicforward

connections

4 3

214

014

41

2))()((

ee

LF

e

e xxCuxSIAA

Hx

xx

1 2)( 0xSAF

)( 0xSAL

)( 0xSABExtrinsic backward connections

Intrinsic connections

Extrinsic lateral connections

,,uxfx

0x

278

038

87

2))()((

ee

LB

e

e xxxSIAA

Hx

xx

236

746

63

225

1205

52

650

2)(

2))()()((

iii

i

ee

LB

e

e

xxxS

Hx

xx

xxxSxSAA

Hx

xx

xxx

spiny stellate

cells

inhibitory interneurons

pyramidal cells

State equations in a 6 layer cortical model

Time Differential Equations

)(

)(

xly

Buxfx

State Space Characterisation

Cxy

BuAxx

Transfer FunctionFrequency Domain

BAsICsH )()(

Linearise

mV

State equations to Spectra

Moran, Kiebel, Stephan, Reilly, Daunizeau, Friston (2007) A neural mass model of spectral responses in electrophysiology. NeuroImage

Predicted response (Pyramidal Cell Depolarization)

Given an empirical recording: estimate parameters of the model

4g 3g

2g1

24914

41

2))(( xxuaxsHxxx

eeee kkgk

&&

Excitatory spiny cells in granular layers

3g

1g 2g

5g

AMPA receptor density

GABAa receptor density

Glutamaterelease

0 2 4 6 8 10 12 14 16 180.7

0.8

0.9

1

1.1

1.2

1.3

1.4

800

2

4

6

8

10

12

14

16

Frequency (Hz)

Fre

qu

ency

(H

z)

No

rmal

ised

Po

wer

(a

.u.)

L-Dopa

Placebo

Frequency (Hz)

No

rmal

ised

Po

wer

(a

.u.)

0 6 11 16

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Measurement

AMPA time constant

GABArelease

Bayesian Inversion

Increased activity at GABAreceptors in supragranular layers

Superficial layers

Granular layers

Deep layers

GABAa TC

Moran, Stephan, Seidenbecher, Pape, Dolan, Friston (2009) Dynamic Causal Model of Steady State Responses. NeuroImage Friston, Bastos, Litvak, Stephan, Fries, Moran (2012) DCM for complex data: cross-spectra, coherence and phase-delays. NeuroImage

Neuromodulators: Acetylcholine/Dopamine

E13 E

31

E23

I32

EERVE

EE

VEELL

gVg

IVVgVVgVC

)),((

)()()1()3()3(

13)1(

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

NMDANMDARVI

INMDA gVg )),(( )2()2()2(23

)2(

fMg

= ))(exp(1

1)3(V

NMDANMDARVI

INMDA

IIRVI

II

EERVE

EE

gVg

gVg

gVg

)),((

)),((

)),((

)2()2()2(31

)2(

)2()2()2(22

)2(

)2()3()3(23

)2(

Superficial layers

Granular layers

Deep layers

IIRVI

II

EERVE

EE

VIINMDAMGNMDA

EELL

gVg

gVg

VVgVVfg

VVgVVgVC

)),((

)),((

)()(

)()(

)2()2()2(22

)2(

)2()3()3(23

)2(

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

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

Sodium Channel

Chloride Channel

Potassium Channel

Depolarization dependentCalcium Channel

NMDA mediated switch

Neurotransmitters: Glu/GABA

A conductance model offers more biological plausibility

Moran, Stephan, Dolan, Friston (2011) Consistent Spectral Predictors for Dynamic Causal Models of Steady State Responses. NeuroImage

0 2 4 6 8 10 12 14 16 180.7

0.8

0.9

1

1.1

1.2

1.3

1.4

800

2

4

6

8

Frequency (Hz)

Fre

qu

ency

(H

z)

No

rmal

ised

Po

wer

(a

.u.)

Frequency (Hz)N

orm

alis

ed P

ow

er

(a.u

.)

AMPA/NMDA Ratio higher in Prefrontal Regions than Parietal Regions

0 6 11 160.7

0.8

0.9

1

1.1

1.2

1.31.4

Roadmap

Specify model

Extract Data Features

Maximise the model evidence

(~-F)

Test models or MAP parameters

Find your experimental data

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50

60 70 80 90 100

Prediction Prediction

Prediction

Summary:DCM for Steady State Responses

Cortical Macrocolumns and free parameters

dx/dt = Ax + B

| H1(ω) . H1*(ω) |

| H1(ω) . H2*(ω) |

| H2(ω) . H2*(ω) |

Generative Model

Frequency (Hz)

Prediction Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Prediction Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Prediction Response

Pow

er (m

V2 )

0 10 20 30 40 50

60 70 80 90 100

Cortical Macrocolumns and free parameters

dx/dt = Ax + B

| H1(ω) . H1*(ω) |

| H1(ω) . H2*(ω) |

)exp()(

)()()(

////

/

ieieieie

ie

ttHth

tuthtv

| H2(ω) . H2*(ω) |

Model Inversion

Summary:DCM for Steady State Responses

Outline

• DCM & Spectral Data Features (the Basics)• DCM for CSD vs DCM for SSR

• DCM for CSD Example

Time to Frequency DomainLinearise around a stable fixed point or LC

DCM for SSR

DCM for CSD

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50

60 70 80 90 100

Prediction Prediction

Prediction

DCM for Cross Spectral Densities

Cortical Macrocolumns and free parameters

dx/dt = Ax + B H1(ω) . H2*(ω)

)exp()(

)()()(

////

/

ieieieie

ie

ttHth

tuthtv

Generative Model

Spectra and Phase lagCoherence

Cross Correlations

H1(ω) . H1*(ω)

H2(ω) . H2*(ω)

Cortical Macrocolumns and free parameters

dx/dt = Ax + B

)exp()(

)()()(

////

/

ieieieie

ie

ttHth

tuthtv

Model Inversion using

full complex signal

Spectra and Phase lagCoherence

Cross Correlations

H1(ω) . H2*(ω)

H2(ω) . H2*(ω)

H1(ω) . H1*(ω)

DCM for Cross Spectral Densities Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz)

Response

Pow

er (m

V2 )

0 10 20 30 40 50

60 70 80 90 100

Prediction Prediction

Prediction

Accommodating Imaginary Numbers

F

ln2

1ln2

1

2

1

2ln2

ln2

1

2

1

)),()(()ln(

11

T

T n

ypqDyF

E:

)( 11

11

T

T

G

GG

M:

11 )()(

))((2

1)(

))((2

1)(

II

PPGGPtrI

GGPtrI

jiTT

ij

TTi

Real and imaginary errors

Real and imaginary derivatives wrt fx, G

1. Interface Additions2. New CSD routines, similar to SSR3. SPM_NLSI_GN accommodates imag numbers, slopes, curvatures 4. A host of new results features, in channel and source space!

Roadmap

Specify model

Extract Data Features

Maximise the model evidence

(~-F)

Test models or MAP parameters

Find your experimental data And also report

phase lags coherence &

delaysIn channel or source space

PFC

Hipp

Conditional Estimates: Spectral Power

10 20 30 400

2

4

6

8

10

12

14

16

18

mode 1 to 1

frequency Hz10 20 30 40

0

2

4

6

8

10

12

14

16

18

mode 2 to 1

frequency Hz

10 20 30 400

5

10

15

Spectral density over modes(in channel-space)

frequency (Hz)

abs(

CSD

)

10 20 30 400

2

4

6

8

10

12

14

16

18

mode 2 to 2

frequency Hz

predicted: trial 1observed: trial 1

predicted: mode 1observed: mode 1predicted: mode 2observed: mode 2

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

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

Pow

er

PFC

Hipp

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Channels: 2 to 1

frequency Hz

predicted: trial 1

observed: trial 1

0 10 20 30 40 501

1

1

1

1

1

1

1

Coh: pfc to hipp

frequency Hz

Conditional Estimates: Coherence

|(H1(ω).H2*(ω))|2

______________________

{(H1(ω).H1*(ω)) + (H2(ω).H2*(ω))}

PFC

Hipp

F-1(H1(ω).H1*(ω)) F-1(H1(ω).H2

*(ω))

F-1(H2(ω).H1*(ω)) F-1(H2(ω).H2

*(ω))

-100 -50 0 50 100

-0.05

0

0.05

0.1

0.15

0.2mode 1 to 1

lag (ms)-100 -50 0 50 100

-0.05

0

0.05

0.1

0.15

0.2mode 2 to 1

lag (ms)

-100 -50 0 50 100

-0.05

0

0.05

0.1

0.15

0.2

Auto-covariance(in channel-space)

Lag (ms)

auto

-cov

aria

nce

-100 -50 0 50 100

-0.05

0

0.05

0.1

0.15

0.2

mode 2 to 2

lag (ms)

trial 1

channel 1channel 2

Conditional Estimates: Covariance

PFC

Hipp

arg(H1(ω).H2*(ω))

____________ ω

0 10 20 30 40 50-25

-20

-15

-10

-5

0

5Delay (ms) 2 to 1

frequency Hz

predicted: trial 1observed: trial 1

0 10 20 30 40 50-25

-20

-15

-10

-5

0

5

10

Delay (ms) PfC to Hipp

Frequency Hz

trial 1

Conditional Estimates: Delays

Outline

• DCM & Spectral Data Features (the Basics)• DCM for CSD vs DCM for SSR

• DCM for CSD Examples

Pharmacological Manipulation of Glutamate and GABA

- 4 levels of anaesthesia: each successively decreasing glutamate and increasing GABA

(Larsen et al Brain Research 1994; Lingamaneni et al Anesthesiology 2001; Caraiscos et al J Neurosci 2004 ; de Sousa et al Anesthesiology 2000 )

- LFP recordings from primary auditory cortex (A1) & posterior auditory field (PAF)

- White noise stimulus & Silence

-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

SummaryDCM for CSD:

Suitable for long time series with trial-specific spectral features eg pronounced beta

Fits complex spectral data features

Offers similar connectivity estimates to DCM for ERPs

With estimates of frequency specific delays and coherence

Can be used with all biophysical, Neural Mass Models (CMC, LFP etc.)

Thank You

The FIL Methods Group

Karl FristonDimitris PinotsisMarco LeiteVladimir LitvakJean DaunizeauStephan KiebelWill PennyKlaas StephanAndre BastosPascal Fries

Acknowledgments