An electrophysiological validation of stochastic DCM for fMRI The goal of Dynamic Causal Modelling (DCM) of neuroimaging data is to study experimentally induced changes in effective connectivity among brain regions. In this work, we assess the predictive validity of stochastic DCM of fMRI data (Daunizeau et al., 2012), in terms of its ability to explain changes in the frequency spectrum of concurrently acquired EEG signal. First, we use a neural field model to show how dense lateral connections induce a separation of time scales, whereby fast (and high spatial frequency) modes are enslaved by slow (low spatial frequency) modes. This slaving effect is such that the frequency spectrum of fast modes (which dominate EEG signals) is controlled by the amplitude of slow modes (which dominate fMRI signals). This means we expect the frequency modulation of EEG to follow temporal variations of slow neural states driving fMRI BOLD signal changes (as in Kilner et al., 2005). Second, we use conjoint empirical EEG-fMRI data – acquired in epilepsy patients – to demonstrate the electrophysiological underpinning of (spontaneous) neural fluctuations inferred from stochastic DCM for fMRI.

J. Daunizeau Brain and Spine Institute, Paris, France Wellcome Trust Centre for Neuroimaging, London, UK

Overview 1

Dynamic Causal Modelling (DCM): introduction

2

Stochastic DCM for fMRI

3

Separation of time scales in neural fields

4

Predicting EEG frequency modulation from sDCM for fMRI

5

Conclusion

Overview 1

Dynamic Causal Modelling (DCM): introduction

2

Stochastic DCM: motivation

3

Separation of time scales in neural fields

4

Predicting EEG frequency modulation from sDCM for fMRI

5

Conclusion

DCM for fMRI: introduction structural, functional and effective connectivity structural connectivity

functional connectivity

effective connectivity

O. Sporns 2007, Scholarpedia

•

structural connectivity = presence of axonal connections

•

functional connectivity = statistical dependencies between regional time series

•

effective connectivity = causal (directed) influences between neuronal populations ! connections are recruited in a context-dependent fashion

DCM for fMRI: introduction functional integration and the neural code

activation studies: functional segregation

u1

effective connectivity studies: functional integration

1

2

1

u1

u2

u1

2

u2

u1 X u2

“where did the experimental manipulation have a specific effect?”

“how did the experimental manipulation propagate through the network?”

DCM for fMRI: introduction neural states dynamics a24

b12

d24

gating effect

2 4

u2 modulatory effect

1 3 c1 u1

driving input

f f 2 f 2 f x2 x  f ( x, u )  f  x0 ,0   x  u  ux  2  ... x u xu x 2 0

nonlinear state equation: m n  (i ) ( j)  x   A   ui B   x j D  x  Cu i 1 j 1   Stephan et al., 2009

DCM for fMRI: introduction the neuro-vascular coupling u t

m n   x   A   ui B (i )   x j D ( j )  x  Cu i 1 j 1  

experimentally controlled stimulus

neural states dynamics

vasodilatory signal

s  x   s   ( f  1)

f

s

s flow induction (rCBF)

f s

 ( h)  { ,  , ,  , E0 }  ( n)  { A, B(i ) , C, D( j ) }

Balloon model changes in volume

 v  f  v1/  v

 ( q, v ) 

 ( h )  {r0 ,  }

hemodynamic states dynamics

f

v

changes in dHb 1/   q  f E ( f,E0 ) E q0 v q / v

q

S    q  V0  k1 1  q   k2 1    k3 1  v   S0  v  

k1  4.30 E0TE k2   r0 E0TE k3  1  

BOLD signal change observation

DCM for fMRI: introduction the variational Bayesian approach to model inversion



ln p  y m   ln p  , y m   S  q   DKL q   ; p  y, m  q



free energy : functional of q

mean-field: approximate marginal posterior distributions:

q   , q   1

2

   , 

p 1 ,2 y, m  2

p 1 or 2 y, m  1

q 1 or 2 

DCM for fMRI: introduction example: audio-visual associative learning auditory cue

visual outcome

or

P(outcome|cue)

or

Put

response 0

200

400

600

800

2000

time (ms)

PMd

PPA

FFA

PPA

cue-dependent surprise

Put

FFA

PMd

cue-independent surprise Den Ouden et al., 2010

Overview 1

Dynamic Causal Modelling (DCM): introduction

2

Stochastic DCM: motivation

3

Separation of time scales in neural fields

4

Predicting EEG frequency modulation from sDCM for fMRI

5

Conclusion

Stochastic DCM for fMRI the effect of state noise on network dynamics

2-regions DCM structure +

-

+

state-space landscape of the normal rate of convergence   x 

-

1

2 +

-

-

E2

B

C

u1  t   1

x2

moving frame of reference v  t  E1 v f t 

A

D

x t  x1

Daunizeau et al., 2012

Stochastic DCM for fMRI mediated influence: canonical model comparison u2

Model comparison: evidence against the full model

1

LBFAB  log 3 u3

A u2

0 -20 -40 -60 -200

ANOVA effects

2

1

3 u3

u  u1

p  y B, m 

20

ANOVA fit

u  u1

p  y A, m 

-100

0 LBFAB

100

200

1 2 1

1 2 2

residuals empirical distribution

2

0.4 0.3 0.2 0.1 0 -200

-100 0 100 ANOVA residuals

200

40 20 0 -20 -40

-60 network du DCM type

1 1 1

1 1 2

2 1 1

2 1 2

2 2 1

2 2 2

B Daunizeau et al., 2012

300

Stochastic DCM for fMRI

SPMresults: .\20070914MP_sDCM\analysis

350

Height threshold F = 11.230194 {p<0.05 (FWE)} Extent threshold k = 0 voxels

400 example: epileptogenic network 20 40

60 Design matrix

precuneus PFC thalamus

30 25 20 15 10 5 0

80

100

pC

pC

PF

PF

Th

Th

C: Th

pC

PF

pC

PF

Th

C: PC

pC

pC

Th

C: PF

PF

pC

PF

C: all

pC

PF

PF

Th

Th

Th

Th

B: fb

B: ff

B: ext

B: int

pC

pC

PF

D: fb

PF

null Th

Th

Log-evidence (relative) Log-evidence (r

Bayesian Model Selection

120 60 100 40

-0.72

80 20 60 0

-0.72 -0.02

pC 1 2 3 4 5 6 7 8 9 10 11 1213 14 15 1617 18 19 2021 22 23 2425 26 27 2829 30 31 3233 34 35 36

-0.12

Models

40

-0.01 -0.06

0.02

0.01

-0.05

20

PF

-0.01

-0.05

0.02

Th 0

1 2 3 4 5 6 7 8 9 10 11 1213 14 15 1617 18 19 2021 22 23 2425 26 27 2829 30 31 3233 34 35 36

-0.72

Models

del Posterior Probability Model Posterior Probability

1

null

null

dDCM

sDCM

Bayesian Model Selection 1

0.8

0.6 1

model 26 (sDCM, B: fb, D: fb)

A

0.5

Bayesian Model Selection 0

B 0.4 0.8

D

0.2 0.6

hemo

-0.5

-1 0 0.4

1 2 3 4 5 6 7 8 9 10 11 1213 14 15 1617 18 19 2021 22 23 2425 26 27 2829 30 31 3233 34 35 36

Models

A

B D

hemo

Stochastic DCM for fMRI estimating hidden states from fMRI BOLD signals simulated hidden-states time series

simulated observations vasodilatory signal blood inflow blood volume dHb content neural state

1

4

y

3

x

0.5

5

2

0

1 -0.5 0 -1

10

20

30

-1

40

10

20

time

30

40

time

full lagged correlation matrix

instantaneous states correlation matrix

lagged neural states correlation matrix

1

2

1

1 0.8

10

4

0.6

20

0.4

30

2

0.2

40

0

50

-0.2

60

-0.4

70

-0.6 -0.8

80 20

40

60

80

-1

0.5

6 8

3

0

10 12

4

-0.5

14 5

16 1

2

3

4

5

5

10

15

-1

Stochastic DCM are sDCM neural state estimates noise, or what? 3 consistent ROIs across patients (N=9)

network connectivity

PFC: structure of residuals Model fit: versus y 4

-0.60

pC

-0.60 -0.02

PF pC

0.01

excluded

2

PF

fitted

Th -0.03

-0.01

y

0.01

0

0.03

-2

Th -0.60

-4

-6

macrosopic sDCM “neural” states

-6

-4

-2

0

2

0.1

PF pC Th

0.05

xˆ ( n )  t 

0 -0.05 -0.1 -0.15 0

4



50

100

150

200 250 time (scans, TR = 3s)

300

350

400

Overview 1

Dynamic Causal Modelling (DCM): introduction

2

Stochastic DCM: motivation

3

Separation of time scales in neural fields

4

Predicting EEG frequency modulation from sDCM for fMRI

5

Conclusion

Separation of time scales in neural fields preamble: Kilner’s heuristic fMRI BOLD signal: • Amplitude modulation of EEG frequency bands (inconsistent results)

Power spectrum

• Frequency modulation of EEG signals (Kilner et al., 2005)

EEG frequency

Separation of time scales in neural fields multiscale portrait of brain dynamics macro-scale

meso-scale Golgi

micro-scale

Nissl

EI

external granular layer external pyramidal layer

EP

II

mean-field firing rate

internal granular layer internal pyramidal layer

synaptic dynamics

Separation of time scales in neural fields from micro- to meso-scale  j  t  : post-synaptic potential of j th neuron within its ensemble 1  N  H  j '  t        H   t     p   t   d   t   N  1 j ' j   S   t   mean-field firing rate

 



membrane depolarization  (mV)

mean firing rate (Hz)

ensemble density

p(x)

 

S

H

mean membrane depolarization  (mV)

Separation of time scales in neural fields from meso- to macro-scale

  r, t   G  S   r, t   r 2  v 2t 2  G 0 1 1 G  r, t   t exp    H t  4  2 2  v t  

2

v 1

“dispersive” propagator (Bojak & Liley, 2010)

spatio-temporal dynamics of mean PSP:

  2 2     2   1    r , t       r , t  2 t    t  0  v    v   2   r , t  G v S  r , t        2 t 2 2  

mean PSP neural field

propagated arrival rate of mean PSP

Separation of time scales in neural fields emergence of slow and fast modes

projection of mean PSP onto Laplacian (spatial) eigenmodes:

2 w( k )  r   k w( k )  r  : k  0  0

 z (k )

 (k )  (k ) z t     w  r   r , t  dr  1 D    z ( k )  t    w( k )  r    r , t  dr   2 D

   0 0 1   z1( k )    v  ( k )   (0) 2   z2   c  z   1   k  0  z ( k )  O  z 2     2  z3( k )   very slow time scale of fundamental mode   1  2   2   this happens if: v   0    

G 0v cz   S  z1(0)  1  S  z1(0)  2 (0)





: slow modes enslaves the dynamics of the fast modes

Separation of time scales in neural fields The effect of neuronal activation on EEG frequency spectrum EEG frequency spectrum (enslaved by large-scale dynamics of slow modes)

EEG centre frequency

v 2     1   2k    i   1    i  2           v 2 1     k  kc  1   2k    i  i    2   c  z (0)    i  2       

2

     d



  

  

    d

centre EEG frequency

normalized EEG frequency spectrum normalized frequency power

centre frequency (Hz)

22

20 -1

10

18

  

 2 16

-2

10

-3

10

14

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16

32

48



64

80

frequency (rad/s)

96 112 128

10

20

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40

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z (0)

fundamental mode dV (mV)

10 0

10

20

30

fundamental mode dV (mV)

z

(0)

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50

Separation of time scales in neural fields Heuristic hemodynamic correlate of the EEG

large-scale coupling of distal neural fields (sDCM for fMRI):

x ( n )  f ( n )  x ( n ) , u, ( n )    f ( n )  x ( n ) , u, ( n )   Ax ( n )   ui B (i ) x ( n )  Cu   x (jn ) D ( j ) x ( n ) i

j

linear

gating

 

2 feedback of fast modes onto slow modes → neural noise : O z  

fMRI: slow rate of energy dissipation

x( n )  z1(0) z1(0)  z1(0) z3(0)

EEG: fast PSP, with centre frequency

      1

 '   (0) 3

z

2

x(n)  O  x2 

Overview 1

Dynamic Causal Modelling (DCM): introduction

2

Stochastic DCM: motivation

3

Separation of time scales in neural fields

4

Predicting EEG frequency modulation from sDCM for fMRI

5

Conclusion

Stochastic DCM validation derivation of the EEG frequency modulation 8

EEG setup of the recording session

neural field model: fast modes empirical EEG data: subject MP

frequency power (A.U.)

log-power (A.U.)

EEG log- frequency power (A.U.)

6

4

2

0

-2

-4

-6

-8 20

40 60 80 frequency (Hz)

100

120

frequency (Hz)

slow EEG frequency modulation

   t 

...to be compared with sDCM “neural” states estimates: GSW GSW

frequency (Hz)

time (scans)

H1 :    xˆ ( n )

X 0    e

H0 :   X 0   e

Stochastic DCM validation single channel comparison to sDCM “neural” states estimates best channel: structure of residuals

best channel: model fit

36

34

34

32 30

100

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observed predicted 100

200

26

400

28

30

best channel: adjusted fit

36

20

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confounds

best channel fit: adjusted EEG frequency modulation

best channel: adjusted correlation

1

0.5

0.5 adjusted

EEG centre frequency (Hz)

34

best channel: adjusted residuals

adjusted predicted

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250

GSW

best channel fit: adjusted EEG frequency modulation

0 -0.5

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predicted

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GSW

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observed

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best channel fit: EEG frequency modulation

26

matrix of confounds

best channel fit: EEG frequency modulation

100

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400

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predicted

GSW

GSW GSW

GSW

Stochastic DCM validation subject-level and group-level Bayesian model comparison LBF  log p  H1    log p  H 0  

→ Bayes factor for each channel/subject:

best subject: BMC across channels

group distribution of log Bayes factors

Bayesian model comparison accross channels 40

250 200

log-Bayes factor

log- Bayes factor

30 20 10 0 -10 0

posterior on model frequencies

150 100 50 0

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10

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20

EEG channels

25

30

-50

1

2

3

4 5 6 subjects

Exceedance probability (N=9):



7

8

9



P f H1  0.5   0.99

→ sDCM neural states estimates predict concurrent EEG frequency modulation above and beyond (>100) confounds

Overview 1

Dynamic Causal Modelling (DCM): introduction

2

Stochastic DCM: motivation

3

Separation of time scales in neural fields

4

Predicting EEG frequency modulation from sDCM for fMRI

5

Conclusion

References Daunizeau et al., 2012: Stochastic Dynamic Causal Modelling of fMRI data: should we care about neural noise? Neuroimage,62: 464-481.

Den Ouden et al., 2010: Striatal prediction error modulates cortical coupling. J. Neurosci, 30: 3210-3219. Bojak et al., 2010: Axonal velocity distributions in neural field equations. PLoS Comp. Biol. 6: e1000653. Stephan et al., 2009: Bayesian model selection for group studies. Neuroimage, 46: 1004-1017. Daunizeau et al., 2009: Dynamic Causal Modelling of distributed electromagnetic responses. Neuroimage, 47: 590-601. Stephan et al., 2008: Nonlinear dynamic causal models for fMRI. Neuroimage, 42: 649-662.

Kilner et al., 2005: hemodynamic correlate of EEG: a heuristic. Neuroimage, 28: 280-286.

many thanks to:

Karl J. Friston (London, UK) Klaas E. Stephan (Zurich, Switzerland) Louis Lemieux (Chalfont, UK) Anna Vaudano (Roma, Italy) Dimitris Pinotsis (London, UK)