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Computational models for imaging analyses. Zurich SPM Course February 14, 2014 Christoph Mathys. What the brain is about. What do our imaging methods measure? Brain activity. But when does the brain become active? When predictions have to be adjusted.

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computational models for imaging analyses

Computational models for imaging analyses

Zurich SPM Course

February 14, 2014

ChristophMathys

what the brain is about
What the brain is about
  • What do our imaging methods measure?
      • Brain activity.
  • But when does the brain become active?
    • When predictions have to be adjusted.
  • So where do the brain’s predictions come from?
      • From a model.

Model-based imaging, Zurich SPM Course, Christoph Mathys

what does this mean for neuroimaging
What does this mean for neuroimaging?

If brain activity reflects model updating, we need to understand what model is updated in what way to make sense of brain activity.

Model-based imaging, Zurich SPM Course, Christoph Mathys

the bayesian brain and predictive coding
The Bayesian brain and predictive coding
  • Model-based prediction updating is described by Bayes’ theorem.
  • the Bayesian brain
  • This is implemented by predictive coding.

Hermann von Helmholtz

Model-based imaging, Zurich SPM Course, Christoph Mathys

advantages of model based imaging
Advantages of model-based imaging
  • Model-based imaging permits us
    • to infer the computational (predictive) mechanisms underlying neuronal activity.
  • to localize such mechanisms.
  • to compare different models.

Model-based imaging, Zurich SPM Course, Christoph Mathys

how to build a model
How to build a model

Fundamental ingredients:

Prediction

Sensory input

Hidden states

Inference based on

prediction errors

Model-based imaging, Zurich SPM Course, Christoph Mathys

example of a simple learning model
Example of a simple learning model
  • Rescorla-Wagner learning:
  • Learning rate
  • Previous value (prediction)
  • Prediction error ()
  • New input
  • Inferred value of

Model-based imaging, Zurich SPM Course, Christoph Mathys

from perception to action
From perception to action

Agent

World

Sensory input

Generative process

Inversion of perceptual

generative model

Inferred

hidden states

True

hidden states

Action

Decision model

Model-based imaging, Zurich SPM Course, Christoph Mathys

from perception to action1
From perception to action

Agent

World

  • In behavioral tasks, we observe actions ().
  • How do we use them to infer beliefs ()?
  • We introduce a decision model.

Sensory input

True

hidden states

Inferred

hidden states

Action

Model-based imaging, Zurich SPM Course, Christoph Mathys

example of a simple decision model
Example of a simple decision model
  • Say 3 options A, B, and C have values , , and .
  • Then we can translate these values into action probabilities via a «softmax» function:
  • The parameter determines the sensitivity to value differences

Model-based imaging, Zurich SPM Course, Christoph Mathys

all the necessary ingredients
All the necessary ingredients
  • Perceptual model (updates based on prediction errors)
  • Value function (inferred state -> action value)
  • Decision model (value -> action probability)

Model-based imaging, Zurich SPM Course, Christoph Mathys

reinforcement learning example o doherty et al 2003
Reinforcement learning example (O’Doherty et al., 2003)

O’Doherty et al. (2003), Gläscher et al. (2010)

Model-based imaging, Zurich SPM Course, Christoph Mathys

reinforcement learning example
Reinforcement learning example

Significant effects of prediction error with fixed learning rate

O’Doherty et al. (2003)

Model-based imaging, Zurich SPM Course, Christoph Mathys

bayesian models for the bayesian brain
Bayesian models for the Bayesian brain

Agent

World

Sensory input

True

hidden states

Inferred

hidden states

  • Includes uncertainty about hidden states.
  • I.e., beliefs have precisions.
  • But how can we make them computationally tractable?

Action

Model-based imaging, Zurich SPM Course, Christoph Mathys

slide15
The hierarchical Gaussian filter (HGF): a computationally tractable model for individual learning under uncertainty

Model-based imaging, Zurich SPM Course, Christoph Mathys

hgf variational inversion and update equations
HGF: variational inversion and update equations
  • Inversion proceeds by introducing a mean field approximation and fitting quadratic approximations to the resulting variational energies (Mathys et al., 2011).
  • This leads to simple one-step update equations for the sufficient statistics (mean and precision) of the approximate Gaussian posteriors of the states .
  • The updates of the means have the same structure as value updates in Rescorla-Wagner learning:
  • Furthermore, the updates are precision-weighted prediction errors.
  • Prediction error
  • Precisions determine learning rate

Model-based imaging, Zurich SPM Course, Christoph Mathys

hgf adaptive learning rate
HGF: adaptive learning rate
  • Simulation:

Model-based imaging, Zurich SPM Course, Christoph Mathys

individual model based regressors
Individual model-based regressors
  • Uncertainty-weighted prediction error

Model-based imaging, Zurich SPM Course, Christoph Mathys

example iglesias et al 2013
Example: Iglesias et al. (2013)

Model-based imaging, Zurich SPM Course, Christoph Mathys

example iglesias et al 20131
Example: Iglesias et al. (2013)
  • Model comparison:

Model-based imaging, Zurich SPM Course, Christoph Mathys

example iglesias et al 20132
Example: Iglesias et al. (2013)

Model-based imaging, Zurich SPM Course, Christoph Mathys

example iglesias et al 20133
Example: Iglesias et al. (2013)

Model-based imaging, Zurich SPM Course, Christoph Mathys

example iglesias et al 20134
Example: Iglesias et al. (2013)

Model-based imaging, Zurich SPM Course, Christoph Mathys

how to estimate and compare models the hgf toolbox
How to estimate and compare models:the HGF Toolbox
  • Available at http://www.tranlsationalneuromodeling.org/tapas
  • Start with README and tutorial there
  • Modular, extensible
  • Matlab-based

Model-based imaging, Zurich SPM Course, Christoph Mathys

how it s done in spm
How it’s done in SPM

Model-based imaging, Zurich SPM Course, Christoph Mathys

how it s done in spm1
How it’s done in SPM

Model-based imaging, Zurich SPM Course, Christoph Mathys

how it s done in spm2
How it’s done in SPM

Model-based imaging, Zurich SPM Course, Christoph Mathys

how it s done in spm3
How it’s done in SPM

Model-based imaging, Zurich SPM Course, Christoph Mathys

how it s done in spm4
How it’s done in SPM

Model-based imaging, Zurich SPM Course, Christoph Mathys

take home
Take home

Agent

World

  • The brain is an organ whose job is prediction.
  • To make its predictions, it needs a model.
  • Model-based imaging infers the model at work in the brain.
  • It enables inference on mechanisms,localization of mechanisms, and model comparison.

Sensory input

True

hidden states

Inferred

hidden states

Action

Model-based imaging, Zurich SPM Course, Christoph Mathys

thank you
Thank you

Model-based imaging, Zurich SPM Course, Christoph Mathys