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Statistical Inference Rik Henson With thanks to: Karl Friston, Andrew Holmes, Stefan Kiebel, Will Penny Statistical Inference Rik Henson With thanks to:

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1 Statistical Inference Rik Henson With thanks to: Karl Friston, Andrew Holmes, Stefan Kiebel, Will Penny Statistical Inference Rik Henson With thanks to: Karl Friston, Andrew Holmes, Stefan Kiebel, Will Penny

2 OverviewOverview Motion correction Smoothing kernel Spatial normalisation Standard template fMRI time-series Statistical Parametric Map General Linear Model Design matrix Parameter Estimates

3 OverviewOverview 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity

4 If n=100,000 voxels tested with p u =0.05 of falsely rejecting H o...If n=100,000 voxels tested with p u =0.05 of falsely rejecting H o... …then approx n  p u (eg 5,000) will do so by chance (false positives, or “type I” errors) …then approx n  p u (eg 5,000) will do so by chance (false positives, or “type I” errors) Therefore need to “correct” p- values for number of comparisonsTherefore need to “correct” p- values for number of comparisons A severe correction would be a Bonferroni, where p c = p u /n…A severe correction would be a Bonferroni, where p c = p u /n… …but this is only appropriate when the n tests independent… …but this is only appropriate when the n tests independent… … SPMs are smooth, meaning that nearby voxels are correlated … SPMs are smooth, meaning that nearby voxels are correlated => Gaussian Field Theory... => Gaussian Field Theory... If n=100,000 voxels tested with p u =0.05 of falsely rejecting H o...If n=100,000 voxels tested with p u =0.05 of falsely rejecting H o... …then approx n  p u (eg 5,000) will do so by chance (false positives, or “type I” errors) …then approx n  p u (eg 5,000) will do so by chance (false positives, or “type I” errors) Therefore need to “correct” p- values for number of comparisonsTherefore need to “correct” p- values for number of comparisons A severe correction would be a Bonferroni, where p c = p u /n…A severe correction would be a Bonferroni, where p c = p u /n… …but this is only appropriate when the n tests independent… …but this is only appropriate when the n tests independent… … SPMs are smooth, meaning that nearby voxels are correlated … SPMs are smooth, meaning that nearby voxels are correlated => Gaussian Field Theory... => Gaussian Field Theory... Multiple comparisons… Gaussian 10mm FWHM (2mm pixels) p u = 0.05 SPM{t}Eg random noise

5 Gaussian Field Theory Consider SPM as lattice representation of continuous random fieldConsider SPM as lattice representation of continuous random field “Euler characteristic” - topological measure of “excursion set” (e.g, # components - # “holes”)“Euler characteristic” - topological measure of “excursion set” (e.g, # components - # “holes”) Consider SPM as lattice representation of continuous random fieldConsider SPM as lattice representation of continuous random field “Euler characteristic” - topological measure of “excursion set” (e.g, # components - # “holes”)“Euler characteristic” - topological measure of “excursion set” (e.g, # components - # “holes”) Smoothness estimated by covariance of partial derivatives of residuals (expressed as “resels” or FWHM) Smoothness estimated by covariance of partial derivatives of residuals (expressed as “resels” or FWHM) Assumes: 1) residuals are multivariate normal 2) smoothness » voxel size (practically, FWHM  3  VoxDim) Assumes: 1) residuals are multivariate normal 2) smoothness » voxel size (practically, FWHM  3  VoxDim) Not necessarily stationary: smoothness estimated locally as resels-per-voxel Not necessarily stationary: smoothness estimated locally as resels-per-voxel

6 General form for expected Euler characteristic for D dimensions:General form for expected Euler characteristic for D dimensions: E [  A u ] =  R d (  )  d ( u ) E [  (  A u )] =  R d (  )  d ( u ) General form for expected Euler characteristic for D dimensions:General form for expected Euler characteristic for D dimensions: E [  A u ] =  R d (  )  d ( u ) E [  (  A u )] =  R d (  )  d ( u ) R d (  ):d-dimensional Minkowski – function of dimension, d, space  and smoothness: R 0 (  )=  (  ) Euler characteristic of  R 1 (  )=resel diameter R 2 (  )=resel surface area R 3 (  )=resel volume  d (  ):d-dimensional EC density of Z(x) – function of dimension, d, threshold, u, and statistic, e.g. Z-statistic:  0 (u)=1-  (u)  1 (u)=(4 ln2) 1/2 exp(-u 2 /2) / (2  )  2 (u)=(4 ln2) exp(-u 2 /2) / (2  ) 3/2  3 (u)=(4 ln2) 3/2 (u 2 -1) exp(-u 2 /2) / (2  ) 2  4 (u)=(4 ln2) 2 (u 3 -3u) exp(-u 2 /2) / (2  ) 5/2 Generalised Form

7 Levels of Inference Three levels of inference:Three levels of inference: –extreme voxel values  voxel-level inference –big suprathreshold clusters  cluster-level inference –many suprathreshold clusters  set-level inference Three levels of inference:Three levels of inference: –extreme voxel values  voxel-level inference –big suprathreshold clusters  cluster-level inference –many suprathreshold clusters  set-level inference n=82 n=32 n=1 2 Parameters: “Height” threshold, u - t > 3.09 “Extent” threshold, k - 12 voxels Dimension, D - 3 Volume, S - 32 3 voxels Smoothness, FWHM - 4.7 voxels Omnibus: P(c  7, t  u) = 0.031 voxel-level: P(t  4.37) =.048 set-level: P(c  3, n  k, t  u) = 0.019 cluster-level: P(n  82, t  u) = 0.029

8 (Spatial) Specificity vs. Sensitivity

9 Small-volume correction If have an a priori region of interest, no need to correct for whole- brain!If have an a priori region of interest, no need to correct for whole- brain! But can use GFT to correct for a Small Volume (SVC)But can use GFT to correct for a Small Volume (SVC) Volume can be based on:Volume can be based on: – An anatomically-defined region – A geometric approximation to the above (eg rhomboid/sphere) – A functionally-defined mask (based on an ORTHOGONAL contrast!) Extent of correction can be APPROXIMATED by a Bonferonni correction for the number of resels…Extent of correction can be APPROXIMATED by a Bonferonni correction for the number of resels…..but correction also depends on shape (surface area) as well as size (volume) of region (may want to smooth volume if rough)..but correction also depends on shape (surface area) as well as size (volume) of region (may want to smooth volume if rough) If have an a priori region of interest, no need to correct for whole- brain!If have an a priori region of interest, no need to correct for whole- brain! But can use GFT to correct for a Small Volume (SVC)But can use GFT to correct for a Small Volume (SVC) Volume can be based on:Volume can be based on: – An anatomically-defined region – A geometric approximation to the above (eg rhomboid/sphere) – A functionally-defined mask (based on an ORTHOGONAL contrast!) Extent of correction can be APPROXIMATED by a Bonferonni correction for the number of resels…Extent of correction can be APPROXIMATED by a Bonferonni correction for the number of resels…..but correction also depends on shape (surface area) as well as size (volume) of region (may want to smooth volume if rough)..but correction also depends on shape (surface area) as well as size (volume) of region (may want to smooth volume if rough)

10 Example SPM window

11 OverviewOverview 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity

12 Fixed vs. Random Effects Subject 1 Subjects can be Fixed or Random variablesSubjects can be Fixed or Random variables If subjects are a Fixed variable in a single design matrix (SPM “sessions”), the error term conflates within- and between-subject varianceIf subjects are a Fixed variable in a single design matrix (SPM “sessions”), the error term conflates within- and between-subject variance Subjects can be Fixed or Random variablesSubjects can be Fixed or Random variables If subjects are a Fixed variable in a single design matrix (SPM “sessions”), the error term conflates within- and between-subject varianceIf subjects are a Fixed variable in a single design matrix (SPM “sessions”), the error term conflates within- and between-subject variance Subject 2 Subject 3 Subject 4 Subject 6 Multi-subject Fixed Effect model error df ~ 300 Subject 5 –In PET, this is not such a problem because the within-subject (between-scan) variance can be as great as the between-subject variance; but in fMRI the between-scan variance is normally much smaller than the between-subject variance If one wishes to make an inference from a subject sample to the population, one needs to treat subjects as a Random variable, and needs a proper mixture of within- and between-subject variance If one wishes to make an inference from a subject sample to the population, one needs to treat subjects as a Random variable, and needs a proper mixture of within- and between-subject variance In SPM, this is achieved by a two-stage procedure: In SPM, this is achieved by a two-stage procedure: 1) (Contrasts of) parameters are estimated from a (Fixed Effect) model for each subject 2) Images of these contrasts become the data for a second design matrix (usually simple t-test or ANOVA)

13 WHEN special case of n independent observations per subject: var(  pop ) =  2 b   +  2 w / Nn Two-stage “Summary Statistic” approach       p < 0.001 (uncorrected) SPM{t} 1 st -level (within-subject)2 nd -level (between-subject) contrast images of c  i 11 ^ 22 ^ 33 ^ 44 ^ 55 ^ 66 ^ 11 ^  ^  ^  ^  ^  ^   w   within-subject error ^ N=6 subjects (error df =5) One-sample t-test ^  pop 

14 Limitations of 2-stage approach Summary statistic approach is a special case, valid only when each subject’s design matrix is identical (“balanced designs”)Summary statistic approach is a special case, valid only when each subject’s design matrix is identical (“balanced designs”) In practice, the approach is reasonably robust to unbalanced designs (Penny, 2004)In practice, the approach is reasonably robust to unbalanced designs (Penny, 2004) More generally, exact solutions to any hierarchical GLM can be obtained using ReMLMore generally, exact solutions to any hierarchical GLM can be obtained using ReML This is computationally expensive to perform at every voxel (so not implemented in SPM2)This is computationally expensive to perform at every voxel (so not implemented in SPM2) Plus modelling of nonsphericity at 2 nd -level can minimise potential bias of unbalanced designs…Plus modelling of nonsphericity at 2 nd -level can minimise potential bias of unbalanced designs… Summary statistic approach is a special case, valid only when each subject’s design matrix is identical (“balanced designs”)Summary statistic approach is a special case, valid only when each subject’s design matrix is identical (“balanced designs”) In practice, the approach is reasonably robust to unbalanced designs (Penny, 2004)In practice, the approach is reasonably robust to unbalanced designs (Penny, 2004) More generally, exact solutions to any hierarchical GLM can be obtained using ReMLMore generally, exact solutions to any hierarchical GLM can be obtained using ReML This is computationally expensive to perform at every voxel (so not implemented in SPM2)This is computationally expensive to perform at every voxel (so not implemented in SPM2) Plus modelling of nonsphericity at 2 nd -level can minimise potential bias of unbalanced designs…Plus modelling of nonsphericity at 2 nd -level can minimise potential bias of unbalanced designs… New in SPM2

15 Nonsphericity again! When tests at 2 nd -level are more complicated than 1/2-sample t-tests, errors can be non i.i.dWhen tests at 2 nd -level are more complicated than 1/2-sample t-tests, errors can be non i.i.d For example, two groups (e.g, patients and controls) may have different variances (non-identically distributed; inhomogeniety of variance)For example, two groups (e.g, patients and controls) may have different variances (non-identically distributed; inhomogeniety of variance) When tests at 2 nd -level are more complicated than 1/2-sample t-tests, errors can be non i.i.dWhen tests at 2 nd -level are more complicated than 1/2-sample t-tests, errors can be non i.i.d For example, two groups (e.g, patients and controls) may have different variances (non-identically distributed; inhomogeniety of variance)For example, two groups (e.g, patients and controls) may have different variances (non-identically distributed; inhomogeniety of variance) New in SPM2 Inhomogeneous variance (3 groups of 4 subjects) Q Repeated measures (3 groups of 4 subjects) Q 1 2 3 Or when taking more than one parameter per subject (repeated measures, e.g, multiple basis functions in event-related fMRI), errors may be non-independent Or when taking more than one parameter per subject (repeated measures, e.g, multiple basis functions in event-related fMRI), errors may be non-independent (If nonsphericity correction selected, inhomogeniety assumed, and further option for repeated measures) Same method of variance component estimation with ReML (that used for autocorrelation) is used Same method of variance component estimation with ReML (that used for autocorrelation) is used (Greenhouse-Geisser correction for repeated- measures ANOVAs is a special case approximation)

16 Hierarchical Models Two-stage approach is special case of Hierarchical GLMTwo-stage approach is special case of Hierarchical GLM In a Bayesian framework, parameters of one level can be made priors on distribution of parameters at lower level: “Parametric Empirical Bayes” (Friston et al, 2002)In a Bayesian framework, parameters of one level can be made priors on distribution of parameters at lower level: “Parametric Empirical Bayes” (Friston et al, 2002) The parameters and hyperparameters at each level can be estimated using EM algorithm (generalisation of ReML)The parameters and hyperparameters at each level can be estimated using EM algorithm (generalisation of ReML) Note parameters and hyperparameters at final level do not differ from classical frameworkNote parameters and hyperparameters at final level do not differ from classical framework Second-level could be subjects; it could also be voxels…Second-level could be subjects; it could also be voxels… Two-stage approach is special case of Hierarchical GLMTwo-stage approach is special case of Hierarchical GLM In a Bayesian framework, parameters of one level can be made priors on distribution of parameters at lower level: “Parametric Empirical Bayes” (Friston et al, 2002)In a Bayesian framework, parameters of one level can be made priors on distribution of parameters at lower level: “Parametric Empirical Bayes” (Friston et al, 2002) The parameters and hyperparameters at each level can be estimated using EM algorithm (generalisation of ReML)The parameters and hyperparameters at each level can be estimated using EM algorithm (generalisation of ReML) Note parameters and hyperparameters at final level do not differ from classical frameworkNote parameters and hyperparameters at final level do not differ from classical framework Second-level could be subjects; it could also be voxels…Second-level could be subjects; it could also be voxels… y  = X (1)  (1) +  (1)  (1)  = X (2)  (2) +  (2) …  (n-1)  = X (n)  (n) +  (n) C  (i) =  k (i) Q k (i) New in SPM2

17 Parametric Empirical Bayes & PPMs Bayes rule:Bayes rule: p( p(  |y) = p(y|  ) p(  ) Bayes rule:Bayes rule: p( p(  |y) = p(y|  ) p(  ) New in SPM2 Posterior Likelihood Prior (PPM) (SPM) What are the priors? –In “classical” SPM, no (flat) priors –In “full” Bayes, priors might be from theoretical arguments, or from independent data –In “empirical” Bayes, priors derive from same data, assuming a hierarchical model for generation of that data

18 Parametric Empirical Bayes & PPMs Bayes rule:Bayes rule: p( p(  |y) = p(y|  ) p(  ) Bayes rule:Bayes rule: p( p(  |y) = p(y|  ) p(  ) New in SPM2 Posterior Likelihood Prior (PPM) (SPM) Classical T-test Bayesian test For PPMs in SPM2, priors come from distribution over voxels If remove mean over voxels, prior mean can be set to zero (a “shrinkage” prior) One can threshold posteriors for a given probability of a parameter estimate greater than some value  … …to give a posterior probability map (PPM)

19 Parametric Empirical Bayes & PPMs New in SPM2 Activations greater than certain amount Voxels with non-zero activations Can infer no responses Cannot “prove the null hypothesis” No fallacy of inference Fallacy of inference (large df) Inference independent of search volume Correct for search volume Computationally expensive Computationally faster Activations greater than certain amount Voxels with non-zero activations Can infer no responses Cannot “prove the null hypothesis” No fallacy of inference Fallacy of inference (large df) Inference independent of search volume Correct for search volume Computationally expensive Computationally faster

20 OverviewOverview 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity

21 A taxonomy of design Categorical designs Categorical designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Parametric designs Linear - Cognitive components and dimensions Nonlinear- Polynomial expansions Factorial designs Factorial designs Categorical- Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric- Linear and nonlinear interactions - Psychophysiological Interactions

22 A taxonomy of design Categorical designs Categorical designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Parametric designs Linear - Cognitive components and dimensions Nonlinear- Polynomial expansions Factorial designs Factorial designs Categorical- Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric- Linear and nonlinear interactions - Psychophysiological Interactions

23 A categorical analysis Experimental design Word generationG Word repetitionR R G R G R G R G R G R G G - R = Intrinsic word generation …under assumption of pure insertion, ie, that G and R do not differ in other ways

24 A taxonomy of design Categorical designs Categorical designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Parametric designs Linear - Cognitive components and dimensions Nonlinear- Polynomial expansions Factorial designs Factorial designs Categorical- Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric- Linear and nonlinear interactions - Psychophysiological Interactions

25 Cognitive Conjunctions One way to minimise problem of pure insertion is to isolate same process in several different ways (ie, multiple subtractions of different conditions)One way to minimise problem of pure insertion is to isolate same process in several different ways (ie, multiple subtractions of different conditions) A1A2 B2B1 Task (1/2) Viewing Naming Stimuli (A/B) Objects Colours Visual ProcessingV Object Recognition R Phonological RetrievalP Object viewingR,V Colour viewingV Object namingP,R,V Colour namingP,V (Object - Colour viewing) [1 -1 0 0] & (Object - Colour naming) [0 0 1 -1] [ R,V - V ] & [ P,R,V - P,V ] = R & R = R (assuming RxP = 0; see later) Price et al, 1997 Common object recognition response (R)

26 Cognitive Conjunctions Original (SPM97) definition of conjunctions entailed sum of two simple effects (A1-A2 + B1-B2) plus exclusive masking with interaction (A1-A2) - (B1-B2)Original (SPM97) definition of conjunctions entailed sum of two simple effects (A1-A2 + B1-B2) plus exclusive masking with interaction (A1-A2) - (B1-B2) Ie, “effects significant and of similar size”Ie, “effects significant and of similar size” New in SPM2 A1-A2 B1-B2 p(A1=A2)

P 2 p(A1=A2+B1=B2)

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P 2 p(A1=A2+B1=B2)

P 2 p(A1=A2+B1=B2)


27 A taxonomy of design Categorical designs Categorical designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Parametric designs Linear - Cognitive components and dimensions Nonlinear- Polynomial expansions Factorial designs Factorial designs Categorical- Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric- Linear and nonlinear interactions - Psychophysiological Interactions

28 Nonlinear parametric responses Inverted ‘U’ response to increasing word presentation rate in the DLPFC SPM{F} Polynomial expansion: f(x) ~ f(x) ~   x +   x 2 +... …(N-1)th order for N levels Linear Quadratic E.g, F-contrast [0 1 0] on Quadratic Parameter =>

29 A taxonomy of design Categorical designs Categorical designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Parametric designs Linear - Cognitive components and dimensions Nonlinear- Polynomial expansions Factorial designs Factorial designs Categorical- Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric- Linear and nonlinear interactions - Psychophysiological Interactions

30 Interactions and pure insertion Presence of an interaction can show a failure of pure insertion (using earlier example)…Presence of an interaction can show a failure of pure insertion (using earlier example)… A1A2 B2B1 Task (1/2) Viewing Naming Stimuli (A/B) Objects Colours Visual ProcessingV Object Recognition R Phonological RetrievalP Object viewingR,V Colour viewingV Object namingP,R,V,RxP Colour namingP,V Naming-specific object recognition viewing naming viewing naming Object - Colour (Object – Colour) x (Viewing – Naming) [1 -1 0 0] - [0 0 1 -1] = [1 -1]  [1 -1] = [1 -1 -1 1] [ R,V - V ] - [ P,R,V,RxP - P,V ] = R – R,RxP = RxP

31 A taxonomy of design Categorical designs Categorical designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Parametric designs Linear - Cognitive components and dimensions Nonlinear- Polynomial expansions Factorial designs Factorial designs Categorical- Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric- Linear and nonlinear interactions - Psychophysiological Interactions

32 SPM{Z} Attentional modulation of V1 - V5 contribution Attention V1 V5 attention no attention V1 activity V5 activity time V1 activity Psycho-physiological Interaction (PPI) Parametric, factorial design, in which one factor is psychological (eg attention)...and other is physiological (viz. activity extracted from a brain region of interest)

33 Psycho-physiological Interaction (PPI) PPIs tested by a GLM with form:PPIs tested by a GLM with form: y = (V1  A).  1 + V1.  2 + A.  3 +  c = [1 0 0] However, the interaction term of interest, V1  A, is the product of V1 activity and Attention block AFTER convolution with HRFHowever, the interaction term of interest, V1  A, is the product of V1 activity and Attention block AFTER convolution with HRF We are really interested in interaction at neural level, but:We are really interested in interaction at neural level, but: (HRF  V1)  (HRF  A)  HRF  (V1  A) (unless A low frequency, eg, blocked; so problem for event-related PPIs) SPM2 can effect a deconvolution of physiological regressors (V1), before calculating interaction term and reconvolving with the HRFSPM2 can effect a deconvolution of physiological regressors (V1), before calculating interaction term and reconvolving with the HRF Deconvolution is ill-constrained, so regularised using smoothness priors (using ReML)Deconvolution is ill-constrained, so regularised using smoothness priors (using ReML) New in SPM2

34 OverviewOverview 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity 1. General Linear Model Design Matrix Global normalisation 2. fMRI timeseries Highpass filtering HRF convolution Temporal autocorrelation 3. Statistical Inference Gaussian Field Theory 4. Random Effects 5. Experimental Designs 6. Effective Connectivity

35 Effective vs. functional connectivity No connection between B and C, yet B and C correlated because of common input from A, eg: A = V1 fMRI time-series B = 0.5 * A + e1 C = 0.3 * A + e2 Correlations: ABC 1 0.491 0.300.121 A B C 0.49 0.31 -0.02  2 =0.5, ns. Functional connectivity Effective connectivity

36 Dynamic Causal Modelling PPIs allow a simple (restricted) test of effective connectivityPPIs allow a simple (restricted) test of effective connectivity Structural Equation Modelling is more powerful (Buchel & Friston, 1997)Structural Equation Modelling is more powerful (Buchel & Friston, 1997) However in SPM2, Dynamic Causal Modelling (DCM) is preferredHowever in SPM2, Dynamic Causal Modelling (DCM) is preferred DCMs are dynamic models specified at the neural levelDCMs are dynamic models specified at the neural level The neural dynamics are transformed into predicted BOLD signals using a realistic biological haemodynamic forward model (HDM)The neural dynamics are transformed into predicted BOLD signals using a realistic biological haemodynamic forward model (HDM) The neural dynamics comprise a deterministic state-space model and a bilinear approximation to model interactions between variablesThe neural dynamics comprise a deterministic state-space model and a bilinear approximation to model interactions between variables New in SPM2

37 Dynamic Causal Modelling The variables consist of:The variables consist of: connections between regions self-connections direct inputs (eg, visual stimulations) contextual inputs (eg, attention) Connections can be bidirectionalConnections can be bidirectional Variables estimated using EM algorithmVariables estimated using EM algorithm Priors are:Priors are: empirical (for haemodynamic model) principled (dynamics to be convergent) shrinkage (zero-mean, for connections) Inference using posterior probabilitiesInference using posterior probabilities Methods for Bayesian model comparisonMethods for Bayesian model comparison New in SPM2 direct inputs - u 1 (e.g. visual stimuli) z 2 V5 z 1 V1 y1y1 z 3 SPC contextual inputs - u 2 (e.g. attention) y2y2 y3y3 z = f(z,u,  z )  Az + uBz + Cu y = h(z,  h ) +  z = state vector u = inputs  = parameters (connection/haemodynamic ).

38 Dynamic Causal Modelling New in SPM2 Z2Z2 stimuli u 1 context u 2 Z1Z1 + + - -  + u1u1 u2u2 z2z2 z1z1 - - 

39 Dynamic Causal Modelling New in SPM2 V1IFG V5 SPC Motion Photic Attention.82 (100%).42 (100%).37 (90%).69 (100%).47 (100%).65 (100%).52 (98%).56 (99%) Friston et al. (2003) Büchel & Friston (1997) Effects Photic – dots vs fixation Motion – moving vs static Attenton – detect changes Attention modulates the backward- connections IFG→SPC and SPC→V5Attention modulates the backward- connections IFG→SPC and SPC→V5 The intrinsic connection V1→V5 is insignificant in the absence of motionThe intrinsic connection V1→V5 is insignificant in the absence of motion

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41 Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995) Statistical parametric maps in functional imaging: A general linear approach” Human Brain Mapping 2:189-210 Worsley KJ & Friston KJ (1995) Analysis of fMRI time series revisited — again” NeuroImage 2:173-181 Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (2000) “To smooth or not to smooth” NeuroImage Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5:179-197 Holmes AP, Friston KJ (1998) “Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754 Worsley KJ, Marrett S, Neelin P, Evans AC (1992) “A three-dimensional statistical analysis for CBF activation studies in human brain”Journal of Cerebral Blood Flow and Metabolism 12:900-918 Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC (1995) “A unified statistical approach for determining significant signals in images of cerebral activation” Human Brain Mapping 4:58-73 Friston KJ, Worsley KJ, Frackowiak RSJ, Mazziotta JC, Evans AC (1994) Assessing the Significance of Focal Activations Using their Spatial Extent” Human Brain Mapping 1:214-220 Cao J (1999) The size of the connected components of excursion sets of  2, t and F fields” Advances in Applied Probability (in press) Worsley KJ, Marrett S, Neelin P, Evans AC (1995) Searching scale space for activation in PET images” Human Brain Mapping 4:74-90 Worsley KJ, Poline J-B, Vandal AC, Friston KJ (1995) Tests for distributed, non-focal brain activations” NeuroImage 2:183-194 Friston KJ, Holmes AP, Poline J-B, Price CJ, Frith CD (1996) Detecting Activations in PET and fMRI: Levels of Inference and Power” Neuroimage 4:223-235 Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995) Statistical parametric maps in functional imaging: A general linear approach” Human Brain Mapping 2:189-210 Worsley KJ & Friston KJ (1995) Analysis of fMRI time series revisited — again” NeuroImage 2:173-181 Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (2000) “To smooth or not to smooth” NeuroImage Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5:179-197 Holmes AP, Friston KJ (1998) “Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754 Worsley KJ, Marrett S, Neelin P, Evans AC (1992) “A three-dimensional statistical analysis for CBF activation studies in human brain”Journal of Cerebral Blood Flow and Metabolism 12:900-918 Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC (1995) “A unified statistical approach for determining significant signals in images of cerebral activation” Human Brain Mapping 4:58-73 Friston KJ, Worsley KJ, Frackowiak RSJ, Mazziotta JC, Evans AC (1994) Assessing the Significance of Focal Activations Using their Spatial Extent” Human Brain Mapping 1:214-220 Cao J (1999) The size of the connected components of excursion sets of  2, t and F fields” Advances in Applied Probability (in press) Worsley KJ, Marrett S, Neelin P, Evans AC (1995) Searching scale space for activation in PET images” Human Brain Mapping 4:74-90 Worsley KJ, Poline J-B, Vandal AC, Friston KJ (1995) Tests for distributed, non-focal brain activations” NeuroImage 2:183-194 Friston KJ, Holmes AP, Poline J-B, Price CJ, Frith CD (1996) Detecting Activations in PET and fMRI: Levels of Inference and Power” Neuroimage 4:223-235 Some References

42 PCA/SVD and Eigenimages A time-series of 1D images 128 scans of 32 “voxels” Expression of 1st 3 “eigenimages” Eigenvalues and spatial “modes” The time-series ‘reconstituted’

43 PCA/SVD and Eigenimages Y (DATA) time voxels Y = USV T = s 1 U 1 V 1 T + s 2 U 2 V 2 T +... +... APPROX. OF Y U1U1U1U1 = s1s1s1s1 V1V1V1V1APPROX. + s 2 U2U2U2U2 V2V2V2V2 + s 3 APPROX. OF Y U3U3U3U3 V3V3V3V3

44 Time x Condition interaction Time x condition interactions (i.e. adaptation) assessed with the SPM{T}

45 Minimise the difference between the observed (S) and implied (  ) covariances by adjusting the path coefficients (B) The implied covariance structure: x= x.B + z x= z.(I - B) -1 x : matrix of time-series of Regions 1-3 B: matrix of unidirectional path coefficients Variance-covariance structure: x T. x =  = (I-B) -T. C.(I-B) -1 where C= z T z x T.x is the implied variance covariance structure  C contains the residual variances (u,v,w) and covariances The free parameters are estimated by minimising a [maximum likelihood] function of S and  Structural Equation Modelling (SEM) 1 3 2zz z

46 Attention - No attention Attention No attention 0.76 0.47 0.75 0.43 Changes in “effective connectivity”

47 PP = Second-order Interactions V5 V1 V1xPP V5  2 =11, p<0.01 0.14 Modulatory influence of parietal cortex on V1 to V5


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