Rosalyn Moran Virginia Tech Carilion Research Institute

Rosalyn Moran

Virginia Tech Carilion Research InstituteBradley Department of Electrical & Computer Engineering

Department of Psychiatry and Behavioral Medicine, VTC School of Medicine

Dynamic Causal Modelling For Cross-Spectral Densities

Data Features in DCM for CSDGenerative Models in the time domain

Generative Models in the frequency domainDCM Inversion procedure

Example 1: L-Dopa Modulations of theta spectra using DCM for CSDExample 2: Propofol Modulations of Delta and Gamma spectra using DCM for CSD

Outline




Outline

Dynamic Causal Modelling: Generic Framework

simple neuronal model

(slow time scale)

fMRI

detailed neuronal model

(synaptic time scales)

EEG/MEG

),,( uxFdtdx

Neural state equation:

Hemodynamicforward model:neural activity BOLD

Time Domain Data

Resting State Data

Electromagneticforward model:

neural activity EEGMEGLFP

Time Domain ERP DataPhase Domain Data

Time Frequency DataSpectral Data

Dynamic Causal Modelling: Generic Framework

simple neuronal model

(slow time scale)

fMRI

detailed neuronal model

(synaptic time scales)

EEG/MEG

),,( uxFdtdx

Neural state equation:

Hemodynamicforward model:neural activity BOLD

Time Domain Data

Resting State Data


neural activity EEGMEGLFP

Time Domain ERP DataPhase Domain Data

Time Frequency DataSpectral Data Frequency (Hz)

Pow

er (m

V2 )

“theta”

DCM for Steady State ResponsesUnder 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).

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

Dynamic Causal Modelling: Framework

Generative Model

Baye

sian

Inve

rsio

n

Empirical Data

Model Structure/ Model Parameters

Explanandum

Competing Hypotheses (Models)

Optimization under model constraints

Spectral Densities

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Pow

er (u

V2 )

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Pow

er (u

V2 )

Spectral Density in Source 1


Spectral Densities

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Pow

er (u

V2 )

0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Pow

er (u

V2 ) 0 5 10 15 20 25 300

5

10

15

20

25

30

Frequency (Hz)

Pow

er (u

V2 )

Cross-Spectral Density between Sources 1 & 2



Cross Spectral Density: The Data EE

G -

MEG

– L

FP T

ime

Serie

s

Cross Spectral D

ensity

1

1

2

2 3

3

4

4

1

2

3

4

A few LFP channels or EEG/MEG spatial modes

Autoregressive Model used to extract spectral representations from dataImaginary Numbers RetainedAveraged over trial types

npnpnnn eyyyy ....2211

ijijijij HHg )()()(

iwpijp

iwijiwijij eeeH

......1)( 2

21

Real and Imaginary

Data features

Cross Spectral Density: The Data

Default order 8

AR coefficients prescribe the spectral densities

Outline




A selection of intrinsic architectures in SPM

A suite of neuronal population models including neural masses, fields and

conductance-based models…expressed in terms of sets of differential equations

Neural Mass Models in DCM

neuronal (source) model

State equationsExtrinsic Connections

,,uxFx

Granular Layer

Supragranular Layer

Infragranular Layer

Intrinsic Connections

Internal Parameters

EEG/MEG/LFPsignal

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

Conductance-Based Neural Mass Models in DCM

))(()(

, gVgVVgVC

affthresholdaffaff

rev

Current in

Conductance

Potential Difference Noise Term: Since properties of tens of thousands of neurons approximated by their average response

Two governing equations: V = IR ……….. Ohms Law I = C dV/dt ……. for a capacitor

))(()(

, gVgVVgVC

affthresholdaffaff

rev

Current in

Conductance


Time constant: κ Afferent Spikes :Strength of connection x σ

Channels already open: g



))(()(

, gVgVVgVC

affthresholdaffaff

rev

Current in

Conductance


Time constant: κ Channels already open: g

σ

μ - V

Afferent Spikes :Strength of connection x σ



))(()(

, gVgVVgVC

affthresholdaffaff

rev

Intrinsic Afferents

Extrinsic Afferents


Different Neurotransmitters and Receptors?

Different Cell Types in 3/6 Layers?


Spiny stellate cells

Pyramidal cells

Inhibitory interneuron

))(()(

, gVgVVgVC

affthresholdaffaff

rev

Current Conductance Reversal Pot – Potential Diff

Afferent Firing No. open channelsTime ConstantConductance

Unit noise

Firing Variance

Exogenous input

E13

)(tI

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in extragranular layers

Inhibitory cells in extragranular layers

Measured response

)( )3(Vg

E31

E23I

32

EERVE

EE

VEELL

gVg

IVVgVVgVC

)),((

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

13)1(

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

IIRVI

II

EERVE

EE

VIIEELL

gVg

gVg

VVgVVgVVgVC

)),((

)),((

)()()(

)2()2()2(22

)2(

)2()3()3(23

)2(

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

IIRVI

II

EERVE

EE

VIIEELL

gVg

gVg

VVgVVgVVgVC

)),((

)),((

)()()(

)3()2()2(32

)3(

)3()1()1(31

)3(

)3()3()3()3()3()3(

I22


vivHi

iv

dthvt

tetHthhv

ieieaffieie

tt

2//// 2)(

)(0;0

0;.)(;


Pyramidal cells


MaximumPost Synaptic Potential

Parameterised Sigmoid

Inverse TimeConstant

Synaptic Kernel

H

Intrinsic connectivity

Convolution-Based Neural Mass Models in DCM

Extrinsic Forward Input

Extrinsic Backward Input


vivHi

iv

dthvt

tetHthhv

ieieaffieie

tt

2//// 2)(

)(0;0

0;.)(;


Pyramidal cells


12

1611

11

2))(( viIvHi

iv

eeee

5

Exogenous input

1)(tI

Excitatory spiny cells being granular layers

Excitatory pyramidal cells in extragranular layers

Inhibitory cells in extragranular layers

Measured response

)( 6vg

2

34547

52

5755

55

42

4634

44

2)(

2)(

iivvivHi

ivvivHi

iv

iiii

eeee

326

32

3743

33

22

2122

22

2)(

2)(

iivvivHi

ivvivHi

iv

iiii

eeee

MaximumPost Synaptic Potential

Parameterised Sigmoid

Inverse TimeConstant

Synaptic Kernel

H

Intrinsic connectivity

Convolution-Based Neural Mass Models in DCM




Spiny stellate

Pyramidal cells


Extrinsic Output




GABA Receptors

AMPA Receptors

NMDA Receptors

))(()(

, gVgVVgVC

affthresholdaffaff

rev

4 population CanonicalMicro-Circuit (CMC)

Spiny stellate

Superficial pyramidal


Deep pyramidal

4-subpopulationCanonical Microcircuit

BackwardExtrinsic Output

ForwardExtrinsic Output




Temporal Derivatives

Outline




Time Differential Equations

)()(

xlyBuxfx

State Space Characterisation

CxyBuAxx

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

u: spectral innovationsWhite and colored noise


CxyBuAxx

Generative Model of Spectra

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

0100100000002000000000

0100000000000000001100000002000000000010000000

000002000000000002000000000200000000100000000000010000000000001000

52

32

.42

22

12

gH

gH

gHgH

gH

A

iiii

eeee

iiii

eeee

eeee

00000000

000

eeH

B

000100000000

TC

Populated According to the neural mass model equations

The Output State(Pyramidal Cells)

The Input State(Stellate Cells)


CxyBuAxx

0100100000002000000000

0100000000000000001100000002000000000010000000

000002000000000002000000000200000000100000000000010000000000001000

52

32

.42

22

12

gH

gH

gHgH

gH

A

iiii

eeee

iiii

eeee

eeee

00000000

000

eeH

B

000100000000

TC

Modulation Transfer FunctionAn analytic mixture of state space parameters

Output Spectrum (Y) = Modulation Transfer Function x Spectrum of Innovations


Freq

uenc

yNMDA connectivty

Posterior Cingulate Cortex

4 5 6 7 8

4

6

8

10

12

14

162 4 6 8 10 12 14 16

0

0.5

1

1.5

2

2.5

3

3.5

4

Frequency

Log

Powe

r

Posterior Cingulate Cortex

Freq

uenc

y

NMDA connectivty

Anterior Cingulate Cortex

4 5 6 7 8

4

6

8

10

12

14

16

2 4 6 8 10 12 14 160

2

4

6

8

10

12

Frequency

Log

Powe

r

Anterior Cingulate Cortex

)),(( )2()2()2()2(NMDARVNMDAINMDA gVg


Observer Model in the Frequency Domain

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

+ White Noise in Electrodes

Interconnected Neural mass models

Lead Field

Sensor LevelSpectral Responses

Summary: Neural Mass Models in DCM

Outline




Dynamic Causal Modelling: Inversion & Inference

fMRI EEG/MEG

Neural

state equation:


Hemodynamicforward model:

Generative Model

Baye

sian

Inve

rsio

n

Empirical Data

Model Structure/ Model Parameters

Inference on models

Dynamic Causal Modelling: Inversion & InferenceBa

yesia

n In

vers

ion

)|()|(),|(),|(

mypmpmypmyp 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

mypmypBF

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: Inversion & InferenceBa

yesia

n In

vers

ion

)|()|(

2

1

mypmypBF

Model comparison via Bayes factor:

)|()|(),|(),|(

mypmpmypmyp 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

A Neural Mass Model

Inversion in the real & complex domain

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5

Frequency (Hz)

real

prediction and response: E-Step: 32

0 10 20 30 40 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

imag

inar

y

prediction and response: E-Step: 32

0 10 20 30 40 50 60 70 80-2

-1.5

-1

-0.5

0

0.5

1

1.5

parameter

conditional [minus prior] expectation

Outline




Dopaminergic modulation in Humans

Aim: Infer plausible synaptic effects of dopamine in humans via non-invasive imaging

Approach: Double blind cross-over (within subject) design, with participants on placebo or

levodopa

Use MEG to measure effects of increased dopaminergic transmission

Study a simple paradigm with “known” dopaminergic effects (from the animal literature): working memory maintenance

Apply DCM to one region (a region with sustained activity throughout maintenance prefrontal)

Moran, Symmonds, Stephan, Friston, Dolan (2011) An In Vivo Assay of Synaptic Function Mediating Human Cognition, Current Biology

• Animal unit recordings have shown selective persistent activity of dorsolateral prefrontal neurons during the delay period of a delayed-response visuospatial WM task (Goldman-Rakic et al, 1996)

• The neuronal basis for sustained activity in prefrontal neurons involves recurrent excitation among pyramidal neurons and is modulated by dopamine (Gao, Krimer, Goldman-Rakic, 2001)

• Dose dependant inverted U

Working Memory

Dopamine in Working Memory

• DA terminals converge on pyramidal cells and inhibitory interneurons in PFC (Sesack et al, 1998)

• DA modulation occurs through several pre and post synaptic mechanisms (Seamans & Yang, 2004)

- Increase in NMDA mediated responses in pyramidal cells – postsynaptic D1 mechanism

- Decrease in AMPA EPSPs in pyramidal cells – presynaptic D1 mechanism

- Increase in spontaneous IPSP Amplitude and Frequency in GABAergic interneurons

- Decrease in extrinsic input current

Gao et al, 2001

Wang et al, 1999

Seamans et al, 2001

Memory

Probe Image

Target Image

. . . .4 sec

. . 300 ms

. . . . 2 sec

. . 300 ms

Memory

e.g. match e.g. no match

WM Paradigm in MEG on and off levodopa

Maintenance Period

Load titrated to 70% accuracy(predrug)

Behavioural Results

Memory

Probe Image

Target Image

match

68

69

70

71

72

73

74

75

76

77

Placebo L-Dopa

Titration

*

% A

ccur

acy

Activity at sensors during maintenance

• Localised main effect and interaction in right prefrontal cortex

• Significant effects of memory in different frequency bands (channels over time)

• Sustained effect throughout maintenance in delta - theta - alpha bands

Broad Band Low Frequency Activity

P A P AP A

Tim

e (s

)0

4 sensors

Interaction: Memory and Dopamine

c

Time (msec)

Freq

uenc

y (H

z)

0 2 4 6 8 10 12 14 16 180.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Frequency (Hz)

Nor

mal

ised

Pow

er (

a.u.

)

L-Dopa

Placebo

Sustained Activity during memory maintenance:Sensor Space

DCM Architecture

AMPA receptors

NMDA receptors

GABAa receptors

Receptor Types

Pyramidal Cell (Population 3)

Inhibitory Interneurons (Population 2)

Spiny Stellates (Population 1)

Cell Populations

3,2

2,1

1,3

2,33,3

3,1

γ : The strengths of presynaptic inputs to and postsynaptic conductances of transmitter-receptor pairs

i.e. a coupling measure that absorbs a number of biophysical processes, e.g.:Receptor DensityTransmitter Reuptake

Synaptic Hypotheses

-100 -50 0 500

0.2

0.4

0.6

0.8

1

Membrane Potential (mV)

pyramidal

cellspyrami

dal cells

spiny stellate

cells

inhibitory interneurons

pyramidal cells

Ext

rinsi

c C

ortic

al In

put (

u)

NMDANMDARVNMDANMDA

AMPAAMPARVAMPAAMPA

VEMgNMDAEAMPALL

gVg

gVg

VVVfgVVgVVgVC

)),((

)),((

))(()()(

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

)2(

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

)2(

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

GABAaGABAaRVGABAaGABAa

NMDANMDARVRVNMDANMDA

AMPAAMPARVRVAMPAAMPA

VIGABAaEMgNMDAEAMPALL

gVg

gVVg

gVVg

VVgVVVfgVVgVVgVC

)),((

))],(),(([

))],(),(([

)())(()()(

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

)3(

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

)1()1(1,3

)3(

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

)1()1(1,3

)3(

)3()3()3()3()3()3()3()3()3(

GABAaGABAaRVGABAaGABAa

AMPAAMPARVAMPAAMPA

VIGABAaEAMPALL

gVg

gVg

VVgVVgVVgVC

)),((

)),((

)()()(

)1()2()2(2,1

)1(

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

)1(

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

3,31,3

3,22,3

3,1

2,1

L-Dopa relative to Placebo, Memory – No Memory Trials 1. Decrease in AMPA coupling (decreased γ1,3) 2. Increased sensitivity by NMDA receptors (increased α) 3. Increase in GABA coupling (increased γ3,2) 4. Decreased exogenous input (decreased u)

Parameter Estimates

L-Dopa : Memory – No Memory:

Interaction of Parameter and Task on L-Dopa ( p = 0.009)

L-Dopa : Memory – No Memory

MA

P Pa

ram

eter

est

imat

es

γ1,3 α γ 3,2 u u

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-8

-7

-6

-5

-4

-3

-2

-1

0x 10-4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16 *

*

L-Dopa relative to Placebo, Memory – No Memory Trials 1. Decrease in AMPA coupling (decreased γ1,3) 2. Increased sensitivity by NMDA receptors (increased α) 3. Increase in GABA coupling (increased γ3,2) 4. Decreased exogenous input (decreased u)


Individual Behaviour

L-Dopa : Memory – No Memory

MA

P Pa

ram

eter

est

imat

es

γ1,3 α γ 3,2 u-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-8

-7

-6

-5

-4

-3

-2

-1

0x 10-4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

*

*

• Decrease in AMPA coupling (decreased γ1,3)• Increased sensitivity by NMDA receptors

(increased α)

Performance Increase

AM

PA c

onne

ctiv

ity γ

1,3

-10 -5 0 5 10 15 20-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

R = -0.51p < 0.05

Performance Increase

NM

DA

Non

linea

rity

α

-10 -5 0 5 10 15 20-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

R = 0.59p < 0.05


Outline




Connectivity changes underlying spectral EEG changes during propofol-induced loss of consciousness.

WakeMild Sedation: Responsive to commandDeep Sedation: Loss of Consciousness

Boly, Moran, Murphy, Boveroux, Bruno, Noirhomme, Ledoux, Bonhomme, Brichant, Tononi, Laureys, Friston, J Neuroscience, 2012

Propofol-induced loss of consciousness


Anterior Cingulate/mPFC

Precuneus/Posterior Cingulate


Increased gamma power in Propofol vs WakeIncreased low frequency power when consiousness is lost

Murphy et al. 2011


Anterior Cingulate/mPFC

Precuneus/Posterior Cingulate

Bayesian Model Selection

WakeMild SedationDeep Sedation


ACC PCCACC PCC ACC PCC

Thalamus Thalami

WakeMild SedationDeep Sedation


ACC PCCACC PCC ACC PCC

Thalamus Thalami

Wake


Parameters of Winning Model ACC PCC

Thalamus

Wake

Mild Sedation:Increase in thalamic excitability


ACC PCC

Thalamus

ACC PCC

Thalamus

Wake

Mild Sedation:Increase in thalamic excitability


ACC PCC

Thalamus

ACC PCC

Thalamus

Loss of Consciousness:Breakdown in Cortical Backward Connections

ACC PCC

Thalamus


Loss of Consciousness

:Breakdown in Cortical Backward Connections

ACC PCC

Thalamus

Boly, Moran, Murphy,Boveroux, Bruno, Noirhomme, Ledoux, Bonhomme, Brichant, Tononi, Laureys, Friston, J Neuroscience, 2012

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

Rosalyn Moran Virginia Tech Carilion Research Institute

Documents

spectral representations

hemodynamicforward model

time constants

time domaingenerative

magnetoencephalographic

postsynaptic receptor

csdgenerative models

csd outline data features