Bayesian Inference in fmri Will Penny

Transcription

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

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4 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

5 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

6 Likelihood and Prior Bayes Rule for Gaussians p y p (1) (1) (1) ( ) N(, ) (1) (2) (2) ( ) N(, ) Posterior Likelihood Posterior Prior p y N m P P m (1) ( ) (, ) P (1) (2) (1) (2) (1) (2) P ( 2) m (1) Relative Precision Weighting

7 Global Shrinkage Priors p 1 N 0, I Y X Observation noise prior precision of GLM coeff GLM Y observations K.J. Friston and W.D. Penny. Posterior probability maps and SPMs. NeuroImage, 19(3): , 2003.

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

9 PPM activation threshold s th Display only voxels that exceed e.g. 95% p p p th q( s ) th Probability mass p n Mean (Cbeta_*.img) Posterior density q(β n ) PPM (spmp_*.img) probability of getting an effect, given the data Std dev (SDbeta_*.img) q( n) N( n, n) mean: size of effect covariance: uncertainty

10 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): , 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, Global Shrinkage: K.J. Friston and W.D. Penny. Posterior probability maps and SPMs. NeuroImage, 19(3): , 2003.

11 Stationary Smoothness Priors Y X p 1 N 0, L amri Smooth Y Observation noise prior precision of GLM coeff prior precision of AR coeff GLM A AR coeff (correlated noise) ML Posterior Y observations

12 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

13 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.

14 Hemodynamic Response Functions Canonical Temporal Temporal derivative, df/dt

15 Hemodynamic Response Functions Canonical Temporal Dispersion Dispersion derivative, df/dw

16 Hemodynamic Response Functions Canonical Temporal Dispersion These three functions together comprise an Informed Basis Set

17 Finite Impulse Response (FIR) Fourier (F) Gamma Informed (Inf)

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

19 Bayesian Model Comparison Prior Posterior p(m y) = m p y m p(m) p y m p(m ) Log Evidence = log p(y m)

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

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

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

23 K Friston. Bayesian Estimation of Dynamical Systems: An application to fmri, Neuroimage 16, , 2002 x Hemodynamic variables [ s, f, v, q ] Dynamics x g( x, z, h ) y b( x ) Hemodynamic parameters Seconds

24 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:

25 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

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

27 Gaussian Population Receptive Fields

28 Gaussian Population Receptive Fields

29 Mexican-Hat Population Receptive Fields

30 Mexican-Hat Population Receptive Fields

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

32 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

33 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? trials 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,

34 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?

35 M Rosa, S.Bestmann, L. Harrison, and W Penny. Bayesian model selection maps for group studies. Neuroimage, Jan ; 49(1): BMS maps model 1 subject 1 subject N q( 0.5) r k r k PPM q( r k ) model K Compute log-evidence map for each model/subject r k Probability that model k generated data k model k EPM

36 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,

37 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

38 K Friston et al. (2008) Bayesian decoding of brain images. Neuroimage, 39: Relate Behavioural Descriptors, X, to fmri data Y via voxel weights 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?

39 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?

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

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

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

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

44 Multivariate Decoding Voxels

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

46 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

47 Overview Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Decoding Dynamic Causal Modelling

48 Single region z a z cu u 1 c a 11 u 1 z 1 u 2 z 1 z 2

49 Multiple regions z1 a11 0 z1 c u1 z a a z 0 u u 1 c a 11 u 1 z 1 u 2 a 21 z 1 z 2 z 2 a 22

50 Modulatory inputs z1 a11 0 z1 0 0 z1 c u1 u2 z 2 a21 a 22 z 2 b21 0 z 2 0 u 2 u 1 u 2 c a 11 u 1 z 1 b 21 u 2 a 21 z 1 z 2 z 2 a 22

51 Reciprocal connections z1 a11 a12 z1 0 0 z1 c u1 u2 z 2 a21 a 22 z 2 b21 0 z 2 0 u 2 u 1 u 2 c a 11 u 1 a 12 z 1 b 21 u 2 a 21 z 1 z 2 z 2 a 22

52 Neurodynamics Change in Neuronal Activity i i i z Az u B z Cu Intrinsic Connectivity Matrix Neuronal Activity Modulatory Connectivity Matrices Inputs Input Connectivity Matrix

53 Rowe et al. 2010, Dynamic causal modelling of effective connectivity from fmri: Are results reproducible and sensitive to Parkinson s disease and its treatment? NeuroImage, 52: Age-matched controls PD patients on medication PD patients off medication Selection of action modulates connections between PFC and SMA DA-dependent functional disconnection of the SMA

54 Brodersen et al. 2011, Generative Embedding for Model-Based Classification of fmri data. PLoS Comput. Biol. 7(6):e measurements from an individual subject step 1 model inversion C A B subject-specific inverted DCM step 2 kernel construction A B A C B B subject representation in the generative score space B C A step 4 interpretation step 3 support vector classification C B jointly discriminative model parameters separating hyperplane fitted to discriminate between groups

55 Model-based decoding of disease status: mildly aphasic patients (N=11) vs. controls (N=26) Connectional fingerprints from a 6-region DCM of auditory areas during speech perception PT PT HG (A1) HG (A1) MGB MGB S S

56 Model-based decoding of disease status: aphasic patients (N=11) vs. controls (N=26) Classification accuracy PT PT HG (A1) HG (A1) MGB MGB auditory stimuli

57 Voxel (64,-24,4) mm L.HG L.HG Multivariate searchlight classification analysis Generative embedding using DCM Voxel-based feature space Generative score space generative embedding patients controls Voxel (-42,-26,10) mm Voxel (-56,-20,10) mm L.MGB L.MGB R.HG L.HG

58 Summary Posterior Probability Maps Hemodynamic Response Functions Population Receptive Fields Computational fmri Multivariate Bayes Dynamic Causal Modelling

59 Savage-Dickey Ratios Bayesian equivalent of inference using F-tests implemented using Savage-Dickey approximations to the log Bayes Factor. W. Penny and G. Ridgway (2013). Efficient Posterior Probability Mapping using Savage-Dickey Ratios. PLoS One 8(3), e59655

60 Faces versus scrambled faces RFX analysis on 18 subjects. Data from Rik Henson.

61 Faces versus scrambled faces: Evidence for Null Probability of Null Hypothesis Using command line call to spm_bms_test_null.m

62 One parameter Likelihood and Prior p y p (1) (1) (1) ( ) N(, ) (1) (2) (2) ( ) N(, ) Posterior Likelihood Posterior Prior p y N m P P m (1) ( ) (, ) P (1) (2) (1) (2) (1) (2) P ( 2) m (1) Relative Precision Weighting

63 Two parameters

64 Bayes Rule for Gaussians Likelihood: Prior: Bayes rule:

65 Model comparison y=f(x) y = f(x) Model evidence: Occam s razor : x model evidence p(y m) space of all data sets