Dynamic Causal Modelling: Tutorial

Dynamic Causal Modelling: Tutorial

Rosalyn Moran, Wellcome Trust Centre for Neuroimaging University College London

MPS-UCL Symposium on Computational Psychiatry September 16 ‒ 22, 2012, Ringberg Castle

Structural, functional & effective connectivity

¤  anatomical/structural connectivity= presence of axonal connections

¤  functional connectivity = statistical dependencies between regional time series

¤  effective connectivity = causal (directed) influences between neurons or neuronal populations

Sporns 2007, Scholarpedia

Methods for effective connectivity analysis

•  Regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997

•  Structural Equation Models (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000

•  Volterra kernels Friston & Büchel 2000

•  Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003

•  Dynamic Causal Models (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008; ERPs David et al 2006; LFPs, Moran et al 2009

Dynamic causal modelling (DCM)

DCMs are generative models of brain responses, which provide posterior estimates of neurobiologically interpretable parameters such as the effective strength of connections among neuronal populations and their context dependent modulation

A Generic Framework

),,( θuxFdtdx

=

Neural state equation:

Electromagnetic forward model:

neural activity→EEGMEG LFP

1 and 2 state models

13 state model

fMRI EEG/MEG

Hemodynamicforward model:neural activity→BOLD

Its Key Components

Dynamic: Dynamic (differential) equations describe hidden neuronal dynamics at a level of detail constrained by the measurement Causal: In a control theory sense, input perturbations disturb equilibrium neuronal dynamics and propagate through connected networks to other brain regions Biophysical Observer: Realistic neuronal to measurement model accounts for regional HRF differences and captures key electrophysiological features eg. Power spectra Bayesian Inversion: Priors on biological parameters to constrain them within physiologically plausible ranges Model evidence objective function maximised during parameter estimation Mean and covariance estimates for a parameter set

Overview

•  Dynamic causal models for fMRI

-  Work through example: Attention to Motion

- Neural level & Hemodynamic level

-  Parameter estimation, priors & inference

•  Dynamic causal models for Steady State Responses (rat LFPs

-  Work through example: Isoflurane effects on connectivity

-  Neural mass model

Attention to Motion

Paradigm

4 conditions - fixation only baseline -  observe static dots + photic - observe moving dots + motion -  attend to moving dots

GLM Results

Büchel & Friston 1997, Cereb. Cortex Büchel et al. 1998, Brain

V5+

SPC

Attention ‒ No attention

The GLM analysis showed that activity in area V5 was not only enhanced by moving stimuli, but also by attention to motion. In the following we will model this effect in V5 using DCM, testing competing hypotheses regarding context-dependent modulation or “enabling” of V5 afferents.

Model 1/Hypothesis 1 Bottom-up Model

Model 2/ Hypothesis 2 Top-down Model

Effects of attention on motion responses in V5

Overview








Neural Dynamics and Static Observer

A(1,1)

A(2,1)

A(1,2)

A(2,2)

x1

!x = (A+uB)x +Cuy = g(x,H )+!! ~ N(0," )H{2}

y2

H{1}

y1

x2

u1

C(1)

A

Neural Dynamics and Static Observer

A(1,1)

A(2,1)

A(1,2)

A(2,2)

x1

!x = (A+uB)x +Cuy = g(x,H )+!! ~ N(0," )

u2

+ B(1,2)

H{2}

y2

H{1}

y1

x2

u1

C(1)

Vanilla DCM: Neural Dynamics : a bilinear approximation

CuxBuAdtdx m

i

ii +⎟

⎠

⎞⎜⎝

⎛+= ∑

=1

)(

Bilinear state equation:

...)0,(),(2

0 +∂∂

∂+

∂∂

+∂∂

+≈= uxuxfu

ufx

xfxfuxf

dtdx

Simply a two-dimensional taylor expansion (around x0=0, u0=0):

A= !f!x u=0

C = !f!u x=0

B = !2 f!x!u

driving input

modulation

dx axdt

= 0( ) exp( )x t x at=The coupling parameter a thus describes the speed of the exponential change

Integration of a first-order linear differential equation gives an exponential function:

If AàB is 0.10 s-1 this means that, per unit time, the increase in activity in B corresponds to 10% of the activity in A If AàB is -0.10 s-1 this means that, per unit time, the decrease in activity in B corresponds to 10% of the activity in A

X1 X2 X3

Connectivity Parameters: Rate Constants

Context Dependent Decay

-

x2

stimuli u1

context u2

x1

+

+

-

-

- + u 1

Z 1

u 2

Z 2

u1

u2

x2

x1

Penny et al. 2004, NeuroImage

+

!x = Ax +u2B(2)x +Cu1

!x1!x2

!

"

##

$

%

&&=

'1 a12a21 '1

!

"

##

$

%

&&x +u2

b112 0

0 b222

!

"

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&&x + c1 0

0 0

!

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u1u2

!

"

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&&

Neuronal activity to BOLD

•  Cognitive system is modelled at its underlying neuronal level (not directly accessible for fMRI).

•  The modelled neuronal

dynamics (x) are transformed into area-

specific BOLD signals (y) by a hemodynamic model (λ).

λ

x

y

},,,,,{ ερατγκθ =h

important for model fitting, but usually of no interest for statistical inference

•  6 parameters:

•  Computed separately for each area → region-specific HRFs

sftionflow induc

= (rCBF)

s

v

vq q/vvEf,EEfqτ /α

dHbchanges in1

00 )( −=/αvfvτ volumechanges in

1−=

f

q

vasodilatory signal

!s = x !!s!" ( f !1)s

f

Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage

stimulus functions u t

neural state equations

Baloon Model

Estimated BOLD response

BOLD signal y(t) = ! v,q( )

Activityx(t)

Hemodynamic Model

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

A

B

C

θh

ε

Stephan et al. 2007, NeuroImage

Hemodynamic and Neural Parameter Correlations

Overview








Model evidence:

Approximation: Free Energy

∑∑ −=kk

mypmypBF )(ln)(ln 212,1

Fixed Effects Model selection via log Group Bayes factor:

accounts for both accuracy and complexity of the model

allows for inference about structure (generalisability) of the model

)|( imyp

)],|(),([)|(ln λθθ GpqKLmypF i −=

( | , )p r y α

Random Effects Model selection via Model probability:

)( 1 Kkqkr ααα ++= …

Inference on Models

•  Gaussian assumptions about the posterior distributions of the parameters

•  posterior probability that a certain parameter (or contrast of parameters) is above a chosen threshold γ:

•  By default, γ is chosen as zero – the prior ("does the effect exist?").

Inference on Single Subject Parameters

Likelihood distributions from different subjects are independent

NiiN

iN

yy

N

iyyyy

N

iyyy

CC

CC

,...,|1

|1|,...,|

1

1|

1,...,|

11

1

θθθθ

θθ

ηη ⎟⎠

⎞⎜⎝

⎛=

=

∑

∑

=

−

=

−−

Under Gaussian assumptions this is easy to compute:

group posterior covariance

individual posterior covariances

group posterior mean

individual posterior covariances and means

Inference on Multi Subject Parameters: FFX Bayesian Parameter Averaging

•  In analogy to “random effects” analyses in SPM, 2nd level analyses can be applied to DCM parameters:

Separate fitting of identical models for each subject

Selection of parameters of interest

one-sample t-test: parameter > 0 ?

paired t-test: parameter 1 > parameter 2 ?

rmANOVA: e.g. in case of multiple

sessions per subject

Inference on Multi Subject Parameters: RFX Summary Statistic Approach

Roadmap

Model 1/Hypothesis 1 Bottom-up Model

Model 2/ Hypothesis 2 Top-down Model

Effects of attention on motion responses in V5

Overview





•  Dynamic causal models for Steady State Responses (rat LFPs)



Dynamic Causal Models and Physiological Inference: AValidation Study Using Isoflurane Anaesthesia inRodentsRosalyn J. Moran1*, Fabienne Jung3, Tetsuya Kumagai3, Heike Endepols3, Rudolf Graf3, Raymond J.

Dolan1, Karl J. Friston1, Klaas E. Stephan1,2, Marc Tittgemeyer3

1Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, London, United Kingdom, 2 Laboratory for Social and Neural Systems

Research, Department of Economics, University of Zurich, Zurich, Switzerland, 3Max Planck Institute for Neurological Research, Cologne, Germany

Abstract

Generative models of neuroimaging and electrophysiological data present new opportunities for accessing hidden or latentbrain states. Dynamic causal modeling (DCM) uses Bayesian model inversion and selection to infer the synaptic mechanismsunderlying empirically observed brain responses. DCM for electrophysiological data, in particular, aims to estimate therelative strength of synaptic transmission at different cell types and via specific neurotransmitters. Here, we report a DCMvalidation study concerning inference on excitatory and inhibitory synaptic transmission, using different doses of a volatileanaesthetic agent (isoflurane) to parametrically modify excitatory and inhibitory synaptic processing while recording localfield potentials (LFPs) from primary auditory cortex (A1) and the posterior auditory field (PAF) in the auditory belt region inrodents. We test whether DCM can infer, from the LFP measurements, the expected drug-induced changes in synaptictransmission mediated via fast ionotropic receptors; i.e., excitatory (glutamatergic) AMPA and inhibitory GABAA receptors.Cross- and auto-spectra from the two regions were used to optimise three DCMs based on biologically plausible neuralmass models and specific network architectures. Consistent with known extrinsic connectivity patterns in sensoryhierarchies, we found that a model comprising forward connections from A1 to PAF and backward connections from PAF toA1 outperformed a model with forward connections from PAF to A1 and backward connections from A1 to PAF and amodel with reciprocal lateral connections. The parameter estimates from the most plausible model indicated that theamplitude of fast glutamatergic excitatory postsynaptic potentials (EPSPs) and inhibitory postsynaptic potentials (IPSPs)behaved as predicted by previous neurophysiological studies. Specifically, with increasing levels of anaesthesia,glutamatergic EPSPs decreased linearly, whereas fast GABAergic IPSPs displayed a nonlinear (saturating) increase. Theconsistency of our model-based in vivo results with experimental in vitro results lends further validity to the capacity of DCMto infer on synaptic processes using macroscopic neurophysiological data.

Citation: Moran RJ, Jung F, Kumagai T, Endepols H, Graf R, et al. (2011) Dynamic Causal Models and Physiological Inference: A Validation Study Using IsofluraneAnaesthesia in Rodents. PLoS ONE 6(8): e22790. doi:10.1371/journal.pone.0022790

Editor: Vladimir N. Uversky, University of South Florida College of Medicine, United States of America

Received March 1, 2011; Accepted July 6, 2011; Published August 2, 2011

Copyright: ! 2011 Moran et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was supported by the Max Planck Society (MT, FJ, TK, HE, RG, RJM), the NEUROCHOICE project of SystemsX.ch (KES), and the Wellcome Trust(RJD, KJF, RJM). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: [email protected]

Introduction

Neural mass models have been used to simulate the electro-physiological response of cortical regions and have recently servedas generative models for empirical M/EEG and LFP data[1,2,3,4,5,6,7,8,9]. These models furnish mathematical descrip-tions of detailed physiological processes including thalamic burstfiring [1], spike frequency adaptation [10], neuronal noise [11],nonlinear channel conductances [12] and neuromodulation [13].Of particular interest to empirical neuroscience is the inversion orfitting of these generative models to real experimental data, wheremechanistic hypotheses regarding the genesis of data features canbe tested. Dynamic causal modelling (DCM) provides a generalframework in which neuronal ensemble models are inverted or‘fitted’ to data. A particular ensemble model, known as an alpha-kernel model [14] is often used within DCMs of M/EEG and LFPdata. The form of the dynamics is constrained by parameters that

encode the strength of transmission at different types of synapses.Clearly, it is important to provide construct validity for thesemodel parameters and ensure that they have a physiologicalinterpretability. In this paper, we address this issue using LFPsignals, acquired by invasive recordings in rat auditory cortex,under different levels of anaesthesia. This work is one from a seriesof ongoing validation studies of the models employed in DCM forelectrophysiological data [15] using invasive recordings. Here, wefocus on the ability of DCM to infer on specific aspects of synaptictransmission, i.e., whether it obtains plausible estimates ofexperimentally induced changes in transmission at excitatoryglutamatergic synapses vs. inhibitory GABAergic synapses.Pharmacological interventions can manipulate aspects of

synaptic processing and can thus be used to validate modelpredictions: Here, we use isoflurane, a volatile anaesthetic agentthat is used commonly in animal laboratory studies [16]. While,compared to other pharmacological agents, it induces a diverse

PLoS ONE | www.plosone.org 1 August 2011 | Volume 6 | Issue 8 | e22790

Anaesthesia Depth

A1 PAF

Trials: 1: 1.4 Mg Isoflourane 2: 1.8 Mg Isoflourane 3: 2.4 Mg Isoflourane 4: 2.8 Mg Isoflourane 5: awake (1 per condition)

- 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

30sec

Steady-State Spectral Responses

Connectivity effected by Isoflurane: Extrinsic or Intrinsic? (Bayesian Model Comparison) How so? (Posterior Parameter Estimates)

A1

PAF

Extrinsic Forward Connection

Intrinsic Connection

Extrinsic Backward Connection

Intrinsic Connection A1 PAF

Hypothesised mechanisms of action

Overview





•  Dynamic causal models for Steady State Responses (rat LFPs)



Neural Mass Model

4γ 4γ 3γ 3γ

1γ 1γ 2γ 2γu 1

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41

2))(( xxuaxsHx

xx

eeee κκγκ −−+−=

=

12

4914

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2))(( xxuaxsHx

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eeee κκγκ −−+−=

=

5γ 5γ

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in infragranular layers

Inhibitory cells in supragranular layers

input

4γ 4γ 3γ 3γ

1γ 1γ 2γ 2γu

Intrinsicconnections

5γ 5γ

Excitatory spiny cells in granular layers

Excitatory pyramidal cells in infragranular layers

Inhibitory cells in supragranular layers

),( uxfx =

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2)()(

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−−=

=

−−+=

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κκγκ

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2))()(( xxCuxSIFHxxx

eeee κκγκ −−++=

=

Synaptic ‘alpha’ kernelSynaptic ‘alpha’ kernel

Sigmoid functionSigmoid function

659

32

61246

63

22

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2)(

2))()((

xxxxxxSHx

xxxxSxBSHx

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−=

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κκγκ

µ

κκγκ

Backward connections

Lateral connections

Forward connections

Time (msec)

Freq

uenc

y (H

z)

0 2 4 6 8 10 12 14 16 18 0.7 0.8 0.9 1

1.1 1.2 1.3 1.4

Frequency (Hz) Nor

mal

ised

Pow

er

L-Dopa Placebo

Time to Frequency Domain

Linearise around a stable fixed point or LC

DCM for SSR

DCM for CSD

Connectivity effected by Isoflurane: Extrinsic or Intrinsic? (Bayesian Model Comparison) How so? (Posterior Parameter Estimates)

A1

PAF

Extrinsic Forward Connection

Intrinsic Connection

Extrinsic Backward Connection

Intrinsic Connection A1 PAF

Hypothesised mechanisms of action

Thank You

Ray Dolan Karl Friston Klaas Enno Stephan Methods Group Emo9on Group

Acknowledgments

Marc Ti@gemeyer Fabienne Jung Max Planck Ins,tute for Neurological Research, Koln

Dynamic Causal Modelling: Tutorial

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