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Bayesian Inference in fMRI

Explore Bayesian approaches in fMRI using posterior probability maps, hemodynamic response functions, and population receptive fields. Understand computational fMRI, multivariate decoding, and dynamic causal modeling. Dive into the Bayesian rule for Gaussians, likelihood and prior, posterior distributions, and global shrinkage priors.

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Bayesian Inference in fMRI

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  1. Bayesian Inference in fMRI Will Penny Bayesian Approaches in Neuroscience Karolinska Institutet, Stockholm February 2016

  2. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  3. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  4. Bayes Rule for Gaussians Likelihood and Prior Posterior Likelihood Prior Posterior Relative Precision Weighting

  5. b prior precision of GLM coeff Observation noise GLM observations Global Shrinkage Priors K.J. Friston and W.D. Penny. Posterior probability maps and SPMs. NeuroImage, 19(3):1240-1249, 2003.

  6. Posterior Posterior distribution:probability of the effect given the data Posterior Probability Map: images of the probability that an activation exceeds some specified thresholdsth, given the data y Two thresholds: • activation threshold sth : percentage of whole brain mean signal (physiologically relevant size of effect) • probability pth that voxels must exceed to be displayed (e.g. 95%)

  7. Display only voxels that exceed e.g. 95% activation threshold Probability mass pn PPM (spmP_*.img) Posterior density q(βn) probability of getting an effect, given the data mean: size of effectcovariance: uncertainty PPM Mean (Cbeta_*.img) Std dev (SDbeta_*.img)

  8. Choice of Priors Stationary smoothness: W.D. Penny, N. Trujillo-Barreto, and K.J. Friston. Bayesian fMRI time series analysis with spatial priors. NeuroImage, 24(2):350-362, 2005. Nonstationary smoothness: L M Harrison, W Penny, J Daunizeau, and K J Friston. Diffusion-based spatial priors for functional magnetic resonance images. Neuroimage, 41(2):408-23, 2008. Global Shrinkage: K.J. Friston and W.D. Penny. Posterior probability maps and SPMs. NeuroImage, 19(3):1240-1249, 2003.

  9. b Smooth Y prior precision of GLM coeff prior precision of AR coeff aMRI Observation noise GLM AR coeff (correlated noise) ML Posterior observations Stationary Smoothness Priors

  10. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  11. K Friston et al. Event-Related fMRI: Characterizing differential responses, Neuroimage 7, 30-40, 1998 Canonical Two Gamma functions fitted to data from auditory cortex. “Canonical” function f(w,t) with w width and t time.

  12. Hemodynamic Response Functions Canonical Temporal derivative, df/dt Temporal

  13. Hemodynamic Response Functions Canonical Dispersion derivative, df/dw Temporal Dispersion

  14. Hemodynamic Response Functions Canonical These three functions together comprise an “Informed Basis Set” Temporal Dispersion

  15. Finite Impulse Response (FIR) Fourier (F) Gamma Informed (Inf)

  16. Hemodynamic Response Functions W.D. Penny, G Flandin and N. Trujillo-Barreto. Bayesian Comparison of Spatially Regularised General Linear Models . HBM, 28:275-293, 2007. R Henson et al. Face repetition effects in implicit and explicit memory tests as measured by fMRI. Cerebral Cortex, 12:178-186.

  17. Bayesian Model Comparison Prior Posterior Log Evidence = log p(y|m)

  18. Hemodynamic Response Functions Left Occipital Cortex: Inf-2 is the preferred model

  19. Hemodynamic Response Functions Right Occipital Cortex: Inf-3 is the preferred model

  20. Hemodynamic Response Functions Sensorimotor Cortex: Inf-3 is the preferred model

  21. K Friston. Bayesian Estimation of Dynamical Systems: An application to fMRI, Neuroimage 16, 513-530, 2002 Hemodynamic variables Dynamics Hemodynamic parameters Seconds

  22. Hemodynamic Response Functions R Buxton et al. Dynamics of Blood Flow and Oxygenation Changes During brain activation: The Balloon Model , Magnetic Resonance in Medicine, 39:855-864.

  23. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  24. Population Receptive Fields S. Kumar and W. Penny (2014). Estimating Neural Response Functions from fMRI.Frontiers in Neuroinformatics, 8th May, doi: 10.3389/fninf.2014.00048. K Friston et al. (2007) Variational free energy and the Laplace approximation. Neuroimage, 34, 220–234.

  25. Gaussian Population Receptive Fields

  26. Gaussian Population Receptive Fields

  27. Mexican-Hat Population Receptive Fields

  28. Mexican-Hat Population Receptive Fields

  29. Which Parametric Function is a Better Descriptor ? Mexican-Hat Gaussian

  30. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  31. 1 2 3 4 … 1 2 40 trials Computational fMRI Subjects pressed 1 of 4 buttons depending on the category of visual stimulus. The 4 categories of stimuli occurred with different frequencies over a session. Brain responses are then hypothesised to be proportional to the surprise, S, associated with each stimulus where S=log(1/p). But over what time scale is the probability p estimated ? And do different brain regions use different time scales ? L. Harrison, S Bestmann, M. Rosa, W. Penny and G. Green (2011). Time scales of representation in the human brain: weighing past information to predict future events. Frontiers in Human Neuroscience, 5, 00037.

  32. Long Time Scale (LTS) Short Time Scale (STS) Enter surprise as a Parametric Modulator in first level GLM analysis. Which surprise variable (STS or LTS) underlies the best model of fMRI responses?

  33. BMS maps subject 1 model 1 subject N PPM model K EPM Probability that model k generated data model k Compute log-evidence map for each model/subject M Rosa, S.Bestmann, L. Harrison, and W Penny. Bayesian model selection maps for group studies. Neuroimage, Jan 1 2010; 49(1):217-24.

  34. Exceedance Probability Maps L. Harrison, S Bestmann, M. Rosa, W. Penny and G. Green (2011). Time scales of representation in the human brain: weighing past information to predict future events. Frontiers in Human Neuroscience, 5, 00037.

  35. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  36. K Friston et al. (2008) Bayesian decoding of brain images. Neuroimage, 39:181-205. Relate Behavioural Descriptors, X, to fMRI data Y via voxel weights b As the number of voxels in a region will likely exceed the number of time points in the fMRI time series, and only some combination of them will be useful for prediction we need to select ‘features’ Which type of feature will be useful for decoding (1) voxels, (2) clusters, (3) singular vectors, (4) support vectors ?

  37. Multivariate Decoding 1.Voxels 2.Clusters Which type of feature will be useful for decoding (1) voxels, (2) clusters, (3) singular vectors, (4) support vectors ?

  38. A Morcom and K Friston (2012) Decoding episodic memory in ageing: A Bayesian Analysis of activity patterns predicting memory. Neuroimage 59, 1772-1782. Clustered Clusters Voxels Distributed

  39. A Morcom and K Friston (2012) Decoding episodic memory in ageing: A Bayesian analysis of activity patterns predicting memory. Neuroimage 59, 1772-1782. The more clustered the representation the better the memory

  40. Multivariate Decoding Q. With what sort of neural code is motion represented with in V5 ? • Voxels Voxels Clusters

  41. Multivariate Decoding Q. Which brain region can motion best be decoded from: V5 or PFC ? A. V5.

  42. Multivariate Decoding Voxels

  43. A Maas et al (2014) Laminar activity in the hippocampus and entorhinal cortex related to novelty and episodic encoding Nature Communications, 5:5547.

  44. A Maas et al (2014) Laminar activity in the hippocampus and entorhinal cortex related to novelty and episodic encoding Nature Communications, 5:5547. Novelty Subsequent Memory

  45. Overview • Posterior Probability Maps • Hemodynamic Response Functions • Population Receptive Fields • Computational fMRI • Multivariate Decoding • Dynamic Causal Modelling

  46. Single region u1 c u1 a11 z1 u2 z1 z2

  47. u1 c a11 z1 a21 z2 a22 Multiple regions u1 u2 z1 z2

  48. Modulatory inputs u1 u2 c u1 a11 z1 u2 b21 z1 a21 z2 z2 a22

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