computational models for imaging analyses zurich spm course february 6, 2015 christoph mathys
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Computational models for imaging analyses
Zurich SPM Course
February 6, 2015
Christoph Mathys
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Model-based imaging, Zurich SPM Course, Christoph Mathys
What the brain is about
• What do our imaging methods measure?
• Brain activity.
• But when does the brain become active?
• When predictions (or their precision) have to be adjusted.
•So where do the brain’s predictions come from?
• From a model.
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Model-based imaging, Zurich SPM Course, Christoph Mathys
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.
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Model-based imaging, Zurich SPM Course, Christoph Mathys
The Bayesian brain and predictive coding
Model-based prediction updating is described by Bayes’ theorem.
the Bayesian brain
This can be implemented by predictive coding.
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Hermann von Helmholtz
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Model-based imaging, Zurich SPM Course, Christoph Mathys
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.
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How to build a model
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𝑢 𝑥Sensory input Hidden states
Prediction
Inference based onprediction errors
Fundamental ingredients:
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Example of a simple learning modelRescorla-Wagner learning:
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Previous value (prediction)
Learning rate
Prediction error ()
New inputInferred value of
𝜇(𝑘)=𝜇(𝑘−1)+𝛼 (𝑢(𝑘)−𝜇(𝑘−1))
Feb 6, 2015
𝜇(𝑘−1) 𝜇(𝑘) 𝑢(𝑘)
𝛿𝑥
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Model-based imaging, Zurich SPM Course, Christoph Mathys
From perception to action
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𝜆 𝑥
Sensory input
Truehidden states
Inferredhidden states
Action
𝑢
𝑎
WorldAgent
Generative process
Inversion of perceptualgenerative model
Decision model
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Model-based imaging, Zurich SPM Course, Christoph Mathys
From perception to action
• In behavioral tasks, we observe actions ().
• How do we use them to infer beliefs ()?
• We invert (i.e., estimate) a decision model.
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𝜆 𝑥
Sensory input
Truehidden states
Inferredhidden states
Action
𝑢
𝑎
WorldAgent
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Model-based imaging, Zurich SPM Course, Christoph Mathys
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
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𝛽=0.1 𝛽=0.6
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Model-based imaging, Zurich SPM Course, Christoph Mathys
All the necessary ingredients
• Perceptual model (updates based on prediction errors)
• Value function (inferred state -> action value)
• Decision model (value -> action probability)
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Reinforcement learning example (O’Doherty et al., 2003)
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O’Doherty et al. (2003), Gläscher et al. (2010)
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Reinforcement learning example
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O’Doherty et al. (2003)
Significant effects of prediction error with fixed learning rate
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Bayesian models for the Bayesian brain
Page 14Feb 6, 2015
• Includes uncertainty about hidden states.
• I.e., beliefs have precisions.
• But how can we make them computationally tractable?
𝜆 𝑥
Sensory input
Truehidden states
Inferredhidden states
Action
𝑢
𝑎
WorldAgent
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Model-based imaging, Zurich SPM Course, Christoph Mathys
State of the world Model
Log-volatilityx3
of tendency
Gaussian random walk with constant step size ϑ
p(x3(k)) ~ N(x3
(k-1),ϑ)
Tendencyx2
towards category “1”
Gaussian random walk with step size exp(κx3+ω)
p(x2(k)) ~ N(x2
(k-1), exp(κx3+ω))
Stimulus categoryx1
(“0” or “1”)
Sigmoid trans-formation of x2
p(x1=1) = s(x2)p(x1=0) = 1-s(x2)
0
x2
1
p(x1=1)
𝑥1(𝑘−1)
𝜅 ,𝜔
𝜗
𝑥3(𝑘−1)
𝑥2(𝑘−1)
𝑥3(𝑘)
𝑥2(𝑘)
𝑥1(𝑘)
x3(k-1)
p(x3(k))
x2(k-1)
p(x2(k))
The hierarchical Gaussian filter (HGF): a computationally tractable model for individual learning under uncertainty
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Δ𝜇𝑖∝�̂� 𝑖−1
𝜋𝑖
𝛿𝑖−1
• 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.
HGF: variational inversion and update equations
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Prediction error
Precisions determine learning rate
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Example: Iglesias et al. (2013)
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Model comparison:
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Model-based imaging, Zurich SPM Course, Christoph Mathys
HGF: adaptive learning rate
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Simulation: 4.1 ,2.2 ,5.0
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Individual model-based regressors
Feb 6, 2015 Page 19
Uncertainty-weighted prediction error
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Example: Iglesias et al. (2013)
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Example: Iglesias et al. (2013)
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Example: Iglesias et al. (2013)
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Example: Iglesias et al. (2013)
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How to estimate and compare models:the HGF Toolbox
Feb 6, 2015 Page 24
• Available at
http://www.tranlsationalneuromodeling.org/tapas
• Start with README and tutorial there
• Modular, extensible
• Matlab-based
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How it’s done in SPM
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How it’s done in SPM
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How it’s done in SPM
Feb 6, 2015 Page 27
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How it’s done in SPM
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Model-based imaging, Zurich SPM Course, Christoph Mathys
How it’s done in SPM
Feb 6, 2015 Page 31
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Take home
Feb 6, 2015 Page 32
• 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.
𝜆 𝑥
Sensory input
Truehidden states
Inferredhidden states
Action
𝑢
𝑎
WorldAgent
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Model-based imaging, Zurich SPM Course, Christoph Mathys
Thank you
Feb 6, 2015 Page 33