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

A dynamic causal model (DCM) of steady-state responses in electrophysiological data is summarised in terms of their cross-spectral density.

Where

These spectral data-features are generated by a biologically plausible, neural-mass model of coupled electromagnetic sources; where each source comprises three sub-populations.

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 indirectlythrough some lead-field (e.g., electroencephalographic and magnetoencephalographicdata).

Inversion of the ensuing DCM provides conditional probabilities on the synaptic parameters of intrinsic and extrinsic connections in the underlying neuronal network

Overview

1. Data Features

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

3. Bayesian Inversion: Parameter Estimates and Model Comparison

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

Predicting Anaesthetic Depth

Overview

1. Data Features





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 E

EG

- M

EG

– L

FP

Tim

e S

eri

es

1

2

3

4

Cro

ss

Sp

ec

tral D

en

sity

1

1

2

2 3

3

4

4

Cross Spectral Density

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





),,( uxFx Neural state equation:

Electric/magneticforward model:

neural activityEEGMEGLFP

(linear)

Neural model:1 state variable per regionbilinear state equationno propagation delays

Neural model:8 state variables per region

nonlinear state equationpropagation delays

fMRIfMRI ERPsERPs

inputs

Hemodynamicforward model:neural activityBOLD(nonlinear)

DCM

),,( uxFx Neural state equation:



(linear)

Neural model:1 state variable per regionbilinear state equationno propagation delays

Neural model:8 state variables per region

nonlinear state equationpropagation delays

fMRIfMRI ERPs

ERPs

inputs

Hemodynamicforward model:neural activityBOLD(nonlinear)

Neural model:8-10 state variables per region

propagation delayslinearised model

modulation transfer function

SSRsSSRs



(linear)

DCM for SSRs

Neural Mass Model

Extrinsic Connections

neuronal (source) model

State equations ,,uxFx

spiny stellate cells

inhibitory interneurons

pyramidal cells

Intrinsic Connections

Internal Parameters

MEG/EEG/LFPsignal

MEG/EEG/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

1. Synaptic Input Sigmoid Response Function

2. Synaptic Impulse Response Function Membrane Potential v

Firi

ng R

ate

Am

plitu

de

(E/IP

SP

)

Time msec (E/IPSP)

)exp()(

)()(

////

/

ieieieie

ie

ttHth

rthtv

)exp(1

1

))(exp(1

1

2121

vr

spiny stellate

cells

inhibitory interneurons

pyramidal cells

A F,L,B

BLFieie AHH ,,

2/1,54321 ,,,,,,,,,

Neural Mass Model

Backward connections

Lateral connections

Forward connections

inputs

4

4

3

3

1

1

2

2

u 12

4914

41

2))(( xxuaxsHx

xx

eeee

12

4914

41

2))(( xxuaxsHx

xx

eeee

5

5

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in infragranular layers

Inhibitory cells in supragranular layers

output

4

4

3

3

1

1

2

2

u

Intrinsicconnections

5

5




),( uxfx

11812

102

1112511

1110

72

8938

87

2)(

2)()(

xxx

xxxSHx

xx

xxxSIBHx

xx

iiii

eeee

12

4914

41

2))()(( xxCuxSIFHx

xx

eeee

659

32

61246

63

22

51295

52

2)(

2))()((

xxx

xxxSHx

x

xxxSxBSHx

xx

iiii

eeee

constant input

ERP or Steady State Responses

Time Domain

Freq Domain

Time Domain

Freq Domain

Outputs Through Lead fieldc

3

c1

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

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

• Transfer Function and Convolution Kernels

• First Order Volterra Series Expansion:

Exact Linear Impulse Response

Cross Spectral Density

dtetdtet tjkj

tjki

kij

)()()(

)(

)()(

tu

ts

k

iki

1

0

110 )()()(

dtuty

• By Definition, the Cross Spectral Density is given by

c3

c1

outputs1(t)

outputs2(t) output

s3(t)

c2

u1

Overview

1. Data Features

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




Bayesian Inversion

Time Domain

Freq Domain

Time Domain

Freq Domain

c3c1

NMM

NMM

NMM

Freq DomainOutput

Freq DomainOutput

Freq DomainCortical Input

Freq DomainCortical Input

)( fH

bf

aU

1)(

c2

+Frequency (Hz)

Po

wer

)(p)),|(||)(()|(ln mypqDmypF

Model Evidence

Approximate Posterior

)|(

),(),|(),|(

)(),|()|(

mGp

mpmGpmGp

dpmGpmGp

),(),(

))(),((),|(

Nmp

NmGp

Invert model

Make inferences

Define likelihood model

Specify priors

Neural Parameters

Observer function

Inference on models

Inference on parameters

Inversion

2/121

2 )|)((|)(

)()(

iii

iii

jH

G

LBFaieie AAAgHH ,,,,,,,,,,, ,,54321

Overview

1. Data Features





Glutamate & microdialysis

Schizophrenic: Rearing Models

Low GlutamateRegular Glutamate

Isolated mPFCControls mPFC

Low GlutamateRegular Glutamate

Isolated mPFCN=8

Controls mPFCN=7

mPFC-0.06

0

0.06

0.12

mV

mPFC EEG

-0.06

0

0.06

0.12

mV

Hypotheses

• Main findings from microdialysis:

– reduction in prefrontal glutamate levels of isolated group

→ sensitization of post-synaptic mechanisms (e.g. upregulation)

• Model parameters should reflect

amplitude of synaptic kernels

coupling parameters of glutamatergic connections

neuronal adaptation (i.e., 2)

Results

connections

Extrinsicforward

connections

4

1 2u

Intrinsic 5



Extrinsicforward

connections

4 3

u

Intrinsic 5




[161, 210]

[29,37]

[195, 233]

(0.4)

(0.37)(0. 13)

[3.8,6.3]

[4.6,3.9]

[0.76,1.34] (0.0003)

(0.17)

(0.04)eH

et

2

Control group estimates in blue Isolated animals in red with p values in parentheses.

In our simulation excitatory parameters were inferred with inhibitory connectivity (and impulse response) prior parameter variances set to zero.

sensitization of post-synaptic mechanisms

Increased neuronal adaption: decrease firing rate

Two-tailed paired t-test

Moran et al., NeuroImage, 2007

Model Fits

Overview

1. Data Features





Case Study: 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

(1 per condition)

30sec

A1

A2

Forward (Excitatory Connection)

Backward (Inhibitory Connection)

A1

A2


A1

A2


Lateral (Mixed Connection)

FB Model (1)

F Model (2)

L Model (3)

Lateral (Mixed Connection)

Models

1 2 3 40

50

100

150

200

250

300

A2 to A1: Modulatory

trial

stre

ngth

(%)

1 2 3 40

20

40

60

80

100

A1 to A2: Excitatory

trial

stre

ngth

(%)

Results

1 2 30

100

200

300

400

500

600

700

Log-

evid

ence

(rel

ative

)

Models

Bayesian Model Selection

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Post

erio

r Mod

el P

roba

bility

Bayesian Model Selection

Models

Data identity has not been verified

A1

A2

Forward

Backward

FB Model (1)

Pathological Beta Rhythms in Parkinson’s

Traditional theory of negative motor symptoms induced by an

unbalance in the striatal outputs of direct ( ) /indirect ( ) pathways

Newer theory focused on pathological synchrony: STN

Beta oscillations correlate to disease state

20 Hz

Chronic loss Dopamine innervations in the Striatum

Pathological Beta Rhythms

DDD

Neuronal states:LFP model subsets

STN

Str

GPe

D

Ctx

GPiTh

GABA Glut

Pathological Beta Rhythms

STN

Str

GPe

Ctx

GPiTh

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

GPe to STN

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Str to GPe

Control

PD

Effects of Chronic

Dopamine Loss

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

Documents

Dynamic Causal Model for Steady State Responses