Bayesian Brain: Dynamic Causal Modelling (DCM)

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Bayesian Brain Dynamic Causal Modelling (DCM)
This material was modified from Uta Noppeney et
al. (Functional Imaging Lab, Wellcome Dept. of
Imaging Neuroscience, Institute of Neurology,
University College London)
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Bounded rationality
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Bounded rationality

• System 1
• Fast
• Intuitive, associative
• heuristics biases

• System 2
• Slow (lazy)
• Deliberate, reasoning
• Rational

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Bounded rationality
neocortex (system 2)
limbic system and brainstem (system 1)
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System 1 very prone to biases

• Seeing order in randomness
• Mental corner cutting
• Misinterpretation of incomplete data
• Halo effect
• False consensus effect
• Group think
• Self serving bias
• Sunk cost fallacy
• Cognitive dissonance reduction

• Confirmation bias
• Authority bias
• Small numbers fallacy
• In-group bias
• Recall bias
• Anchoring bias
• Inaccurate covariation detection
• Distortions due to plausibility

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• Bounded Rationality
• The Small Numbers Problem of Individual
  Experience
• Prone to See Patterns Even in Random Data

• Critical Thinking
• Decision Supports
• Research
• Large Ns gt individual experience
• Controls reduce bias

The Human Problem
Evidence-Based Practice
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Evidence-based deciison-making
Its hard to tell the signal from the noise. The
story the data tell us is often the one wed like
to hear, and we usually make sure it has a happy
ending. It is when we deny our role
in the process
that the odds
of failure rise. Nate Silver
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(????? ??? ??? ???? ? ??? ? 25? ???? ???.)
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We dont like uncertainty!
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Nate Silver
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538 (??? ???? ?) 435 (????) 100 (????) 3 (???
D.C)
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??(??)? ??? ??? ?? ???, ?? ?? ??? ?? ?? ?? ??
????. ?? ???(??)? ? ??? ??? ??? ?? ??? ??? ???
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• ?? ?? ??? ??? ?? ??? ?? ??? (P(A))
• ??? ?? ?? ????? ??? ????(P(B)).
• ??? ??? ???? ?? ???? ??? ???? ?? (P(BA))
• ?? ??
• ??? ??? ????? ?? ???? ??? ??? ??? ?? ???? ?? ???
  ??? (P(AB))

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?? ? (Out-of-sample)?? ??
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???? ??? ???!
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System analyses in functional neuroimaging
Functional integration Analyses of inter-regional
effects what are the interactions between the
elements of a given neuronal system?
Functional specialisation Analyses of regionally
specific effects which areas constitute a
neuronal system?
Functional connectivity the temporal
correlation between spatially remote
neurophysiological events
Effective connectivity the influence that the
elements of a neuronal system exert over another
MODEL-free
MODEL-dependent
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Approaches to functional integration

• Functional Connectivity
• Eigenimage analysis and PCA
• Nonlinear PCA
• ICA
• Effective Connectivity
• Psychophysiological Interactions
• MAR and State space Models
• Structure Equation Models
• Volterra Models
• Dynamic Causal Models

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Psychophysiological interactions
Context
X
source
target
Set
stimuli
Context-sensitive connectivity
Modulation of stimulus-specific responses
source
source
target
target
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The aim of Dynamic Causal Modeling
(DCM) Functional integration and the modulation
of specific pathways
Contextual inputs Stimulus-free - u2(t) e.g.
cognitive set/time
BA39
Perturbing inputs Stimuli-bound u1(t) e.g.
visual words
STG
V4
V1
BA37
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Neuronal model
Conceptual overview
neuronal changes
latent connectivity
induced response
induced connectivity
Input u(t)
The bilinear model
c1
b23
neuronal states
a12
activity z2(t)
activity z3(t)
Neural activity z1(t)
y
Hemodynamic model
y
y
BOLD
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Rational statistical inference(Bayes, Laplace)
Sum over space of hypotheses
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Conceptual overview

• Constraints on
• Connections
• Biophysical parameters

• Models of
• Responses in a single region
• Neuronal interactions

Bayesian estimation
posterior ? likelihood prior
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Example linear dynamic system
LG lingual gyrus FG fusiform gyrus Visual
input in the - left (LVF) - right
(RVF)visual field.
z4
z3
z1
z2
RVF
LVF
u2
u1
state changes
effective connectivity
externalinputs
systemstate
input parameters
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Extension bilinear dynamic system
z4
z3
z1
z2
RVF
LVF
CONTEXT
u2
u3
u1
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Bilinear state equation in DCM
state changes
intrinsic connectivity
m externalinputs
systemstate
direct inputs
modulation of connectivity
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Neuronal model
Conceptual overview
neuronal changes
latent connectivity
induced response
induced connectivity
Input u(t)
The bilinear model
c1
b23
neuronal states
a12
activity z2(t)
activity z3(t)
activity z1(t)
y
y
y
Hemodynamic model
BOLD
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The hemodynamic Balloon model

• 5 hemodynamic parameters

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

• Empirically determinedprior distributions.
• Computed separately for each area (like the
  neural parameters).

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stimulus function u
Overviewparameter estimation
neural state equation

• Combining the neural and hemodynamic states gives
  the complete forward model.
• An observation model includes measurement error
  e and confounds X (e.g. drift).
• Bayesian parameter estimation by means of a
  Levenberg-Marquardt gradient ascent, embedded
  into an EM algorithm.
• ResultGaussian a posteriori parameter
  distributions, characterised by mean ??y and
  covariance C?y.

parameters
hidden states
state equation
observation model
modelled BOLD response
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Overviewparameter estimation

• Constraints on
• Connections
• Biophysical parameters

• Models of
• Responses in a single region
• Neuronal interactions

posterior ? likelihood prior
Bayesian estimation
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Priors in DCM
Bayes Theorem

• needed for Bayesian estimation, embody
  constraints on parameter estimation
• express our prior knowledge or belief about
  parameters of the model
• hemodynamic parametersempirical priors
• temporal scalingprincipled prior
• coupling parametersshrinkage priors

posterior ? likelihood prior
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Priors in DCM

• Principled priors
• System stabilityin the absence of input, the
  neuronal states must return to a stable mode
• Constraints on prior variance of intrinsic
  connections (A) Probability lt0.001 of obtaining
  a non-negative Lyapunov exponent (largest real
  eigenvalue of the intrinsic coupling matrix)
• Self-inhibition Priors on the decay rate
  constant s (?s1, Cs0.105) these allow for
  neural transients with a half life in the range
  of 300 ms to 2 seconds

• Shrinkage priorsfor coupling parameters (?0)?
  conservative estimates!

• Temporal scaling
• Identical in all areas by factorising A and B
  with s (a single rate constant for all regions)
  all connection strengths are relative to the
  self-connections.

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Shrinkage Priors
Small variable effect
Large variable effect
Small but clear effect
Large clear effect
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EM and gradient ascent

• Bayesian parameter estimation by means of
  expectation maximisation (EM)
• E-step gradient ascent (Fisher scoring
  Levenberg-Marquardt regularisation) to compute
• (i) the conditional mean ??y ( expansion point
  of gradient ascent),
• (ii) the conditional covariance C?y
• M-step Estimation of hyperparameters ?i for
  error covariance components Qi
• Note Gaussian assumptions about the posterior
  (Laplace approximation)

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Parameter estimation in DCM

• Bayesian parameter estimation under Gaussian
  assumptions by means of EM and gradient ascent.
• ResultGaussian a posteriori parameter
  distributions with mean ??y and covariance C?y.

• Combining the neural and hemodynamic states gives
  the complete forward model
• The observation model includes measurement error
  ? and confounds X (e.g. drift)

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The DCM cycle
Hypothesis abouta neural system
Statistical test on parameters of optimal model
Definition of DCMs as systemmodels
Bayesian modelselection of optimal DCM
Design a study thatallows to investigatethat
system
Parameter estimationfor all DCMs considered
Data acquisition
Extraction of time seriesfrom SPMs
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Planning a DCM-compatible study

• Suitable experimental design
• preferably multi-factorial (e.g. 2 x 2)
• e.g. one factor that varies the driving (sensory)
  input
• and one factor that varies the contextual input
• Hypothesis and model
• define specific a priori hypothesis
• Which alternative models?
• which parameters are relevant to test this
  hypothesis?
• TR
• as short as possible (optimal lt 2 s)