Contents. Introduction & recap. The General Linear Model, Part II. F-test and added variance Good & bad models Improvedmodel

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1 DISCOS SPM curse, CRC, Liège, 2009 Cntents The General Linear Mdel, Part II C. Phillips, Centre de Recherches du Cycltrn, ULg, Belgium Based n slides frm: T. Nichls, R. Hensn, S. Kiebel, JB. Pline Intrductin & recap The General Linear Mdel T test F-test and added variance Gd & bad mdels Imprvedmdel mdel HRF and ER fmri «Take hme» message image data parameter estimates Statistical Parametric Map crrected p-values parameter estimates Statistical Parametric Map realignment & mtin mtin crrectin General Linear Linear Mdel Mdel mdel mdel fitting fitting statistic statistic image image crrectin fr fr multiple cmparisns General Linear Linear Mdel Mdel mdel mdel fitting fitting statistic statistic image image nrmalisatin smthing Randm effect effectanalysis anatmical reference kernel design matri Dynamic causal causal mdelling, Functinal & effective cnnectivity, PPI, PPI, Realigned, nrmalised, smthed image data design matri

2 General Linear Mdel What des it mean? General Linear Mdel The mdel can be The mdel uses A set f equatins used t answer a simple linear are used t wide variety f relatinships describe the data. questins. between the Questins abut variables. the data can then be stated as mathematical epressins. GLM is is the basic mdel r general framewrk underlying the analysis f variance and multiple regressin. Functinal neurimaging signal metablic change cerebral bld flw (CBF) physilgical effects bld ygenatin physical effects MR prperties glucse and ygen metablism Bld ygen-level dependent (BOLD) signal - H 15 2 O PET neurnal activity FDG PET cerebral bld vlume (CBV) magnetic field unifrmity (micrscpic) Decay time (T2*) - pening/clsing f inic channels generatin f micrscpic current surces cmbinatin in space and time f the current surces - Measurable electrmagnetic fields n/utside the head fmri (T2*-weighted image) EEG/MEG Surce: Dug Nll s primer Simple fmri eample dataset: can we d better? Vel by vel statistics One One sessin, ne ne subject Passive wrd listening versus rest rest 7 cycles f f rest restand listening Time Time series seriesf f BOLD BOLD respnses in in ne ne vel vel Time Time mdel specificatin parameter estimatin hypthesis test statistic Each epch 6 scans with with7 sec sec TR TR Questin: Is Isthere therea change in in the thebold respnse between listening and and rest? rest? Stimulus functin single vel time series Intensity statistic image r SPM

3 YY N 1 General Linear Mdel p Y = Xβ ε = X β p N N Mdel is isspecified by by N: N: number f f scans scans Design matri X p: p: number f f regressrs Assumptins abut ε 1 ε 1 This Thisisisfr fra SINGLE vel!! Design matri X is isthe thesame fr fr ALL ALL vels!! General Linear Mdel fmri time series: Y 1,,Y s,,y N acquired at times t 1,,t s s,,t N Mdel: Linear cmbinatin f basis functins Mdel: Linear cmbinatin f basis functins Y ) β l l ) s = β 1 f 1 (t s ) β l f l (t s ) β L f L (t s )) ε s f l l (.): basis functins reference wavefrms dummy variables parameters (fied effects) amplitudes f basis functins (regressin slpes) β l l : parameters (fied effects) ε N(0,σ s : residual errrs: ε s ~ 2 ) identically distributed independent, r serially crrelated (Generalised Linear Mdel GLM) Mass univariate apprach Why mdelling? K p K Why? Make inferences abut effects f f interest N Y = Xβ ε Y = N N X p εˆ K βˆ Hw? Mdel? Decmpse data dataint inteffects and and errr Frm statistic using estimates f f effects and and errr data data Use Useany anyavailable knwledge Stimulus functin mdel mdel effects effects estimate errr errr estimate Cntrast: e.g. e.g. [1 [1-1 -1]] statistic

4 Y Parameter estimatin Y = Xβ ε ˆ β 1 = ˆ β2 X Estimate parameters N such that that t= 1 ˆε 2 t εˆ minimal ˆ ε = Y X ˆ β residuals Assume iid iiderrr ˆ β = ( X X ) T 1 X T Least squares parameter estimate Y Mdel cntrasts SPM{t} Cntrast : specifies linear cmbinatin f parameter vectr: c β c = 1 0 b-car amplitude > 0? = ^ ^ β 1 > 0? (β 1 : estimatin f β 1 ) = ^ ^ 1β 1 0β 2 > 0? = ^ test H 0 : c β > 0? T = cntrast f estimated parameters variance estimate T = c β^ s 2 c (X X) X) c SPM{t} t distributin Hw is this cmputed? (t-test) test) Depends n 1 set f degrees f freedm! ν % Estimatin [Y, X] [b, s] Y = X β ε b = (X X) X) X Y (b = e = Y - Xb ε ~ σ 2 N(0,I) (Y : at ne psitin) b = estimatin f β) beta??? images Xb (e e = estimatin f ε) s 2 = (e e/(n( e/(n - p)) (s s = estimatin f σ, n: scans,, p: parametres) 1 image ResMS Test [b, s 2, c] [c b, t] Var(c b) = s 2 c (X X) X) c cmpute fr each cntrast cmpute fr each cntrast c Inf t = c b / sqrt(s 2 c (X X) X) c) c) c b images cn??? cmpute the t images images spm_t??? under the null hypthesis under the null hypthesis H 0 : t ~ Student( df ) df = n-pn

5 Cntents Mdel cntrasts SPM{F} Tests multiple linear hyptheses : Des X 1 mdel anything? Intrductin & recap F-test and added variance Gd & bad mdels Imprvedmdel mdel HRF and ER fmri «Take hme» message X 0 X 1 S 2 X 0 H 0 : True mdel is X 0 additinal variance accunted fr by tested effects S 2 0 > S 2 F = errr variance estimate This mdel? Or this ne? F ~ ( S S 2 ) / S 2 X 0 F test (SPM{F}) : a reduced mdel r multi-dimensinal cntrasts? Tests multiple linear hyptheses. X 1 (β 3-4 ) X 0 This mdel? Or this ne? H 0 : True mdel is X 0 ^ H 0 : β 3-4 = (0 0) ^ test H 0 : c β = 0? c = Hw is this cmputed? (F-test) Estimatin [Y, X] [b, s] Y = X β ε Y = X 0 β 0 ε 0 Estimatin [Y, X 0 ] [b 0, s 0 ] (nt really like that ) b 0 = (X 0 X 0 ) X 0 Y e 0 = Y - X 0 b 0 (e 0 = estimatin f s 2 0 = (e 0 e 0 /(n - p 0 )) (s 0 = estimatin f Test [b, s, c] [ess, F] F = (e 0 e 0 - e e)/(pe)/(p - p 0 ) / s 2 image estimatin f ε 0 ) under the null hypthesis : F ~ F(df1 df1,df2) p - p n-p 0 ε ~ N(0, σ 2 I) ε 0 ~ N(0, σ 2 0 I) X 0 : X Reduced estimatin f σ 0, n: : # time bins, p 0 : # parameters) image (e 0 e 0 - e e)/(pe)/(p - p 0 ) : spm_ess??? image f F : spm_f???

6 F distributin T and F test: take hme... Depends n 2 sets f degrees f freedm! T tests are simple cmbinatins f the betas; they are either psitive r negative (b1 b2 is different frm b2 b1) F tests can be viewed as testing fr the additinal variance eplained by a larger mdel wrt.. a simpler mdel, r F test the sum f the squares f ne r several cmbinatins f the betas in testing single cntrast with an F test, fr e. b1 b2, the result will be the same as testing b2 b1. It will be eactly the square f the t-test, t test, testing fr bth psitive and negative effects, and the p-value p will be twice as big. «Additinal variance» : Again «Additinal variance» : Again Testing fr the green Independent cntrasts crrelated regressrs, fr eampl green: subject age yellw: subject scre

7 «Additinal variance» : Again Testing fr the red «Additinal variance» : Again Testing fr the green crrelated cntrasts Entirely crrelated cntrasts? Nn estimable! «Additinal variance» : Again Cntents Testing fr the green and yellw If significant? Culd be G r Y! Intrductin & recap F-test and added variance Gd & bad mdels Imprvedmdel mdel HRF and ER fmri «Take hme» message

8 Abadmdel... True signal (---) and bserved signal Abadmdel... b 1 = 0.22 b 2 = 0.11 Mdel (green, peak at 6sec) and TRUE signal (blue, peak at 3sec) Fitting : b1 = 0.2, mean =.11 = P( b1 = 0 ) = 0.1 (t-test b1>0) P( b1 = 0 ) = 0.2 (F-test b1 0) Y X β ε Nise Residual Variance = 0.3 (still cntains sme signal) Test fr the green regressr nt significant A «better» mdel... A better mdel... True signal bserved signal b 1 = 0.22 b 2 = 2.15 b 3 = 0.11 Mdel (green and red) and true signal (blue ---) Red regressr : tempral derivative f the green regressr = Residual Var = 0.2 P( b1 = 0 ) = 0.07 (t test b1>0) Glbal fit (blue) and partial fit (green & red) Adjusted and fitted signal Y X β ε P( [b1 b2] = [0 0] ) = (F test [b1 b2] [0 0]) Nise (a smaller variance) Test f the green regressr almst significant Test F very significant Test f the red regressr very significant

9 Summary... Crrelatin between regressrs The residuals shuld be lked at...(nn randm structure?) We rather test fleible mdels if there is little a priri infrmatin, and precise nes with a lt a priri infrmatin In general, use the F-tests t lk fr an verall effect, then lk at the betas r the adjusted signal t characterise the rigin f the signal True signal Mdel (green and red) Fitting (blue : glbal fit) Interpreting the test n a single parameter (ne functin) can be very cnfusing: cf the delay r magnitude situatin Nise Crrelatin between regressrs Crrelatin between regressrs - 2 Y b 1 = 0.79 b 2 = 0.85 b3 = 0.06 = X β ε Residual var. = 0.2 P( b1 = 0 ) = 0.08 (t test b1>0) P( b2 = 0 ) = 0.07 (t test b2>0) P( [b1 b2] = 0 ) = (F test [b1 b2] 0 ) true signal Mdel : red regressr rthgnalised with respect t the green ne = remve every thing that can crrelate with the green regressr Fit Nise

10 Crrelatin between regressrs -2 Y b 1 = b 2 = b3 = = X β ε Residual var. = 0.2 P( b1 = 0 ) = (t test b1>0) P( b2 = 0 ) = 0.07 (t test b2>0) P( [b1 b2] = 0 ) = (F test [b1 b2] 0) Design rthgnality : «eplre design» Black = cmpletely crrelated White = cmpletely rthgnal 1 2 Crr(1,1) Crr(1,2) Beware : when there is mre than 2 regressrs (C1,C2,C3...), yu may think that there is little crrelatin (light grey) between them, but C1 C2 C3 may be crrelated with C4 C5 cmpletely crrelated... Summary... Y = Xb e X = Cnd 1 Cnd 2 Mean C2 Mean = C1C2 C1 We are implicitly testing additinal effect nly, s we may miss the signal if there is sme crrelatin in the mdel using t tests Orthgnalisatin is nt generally needed - parameters and test n the changed regressr dn t t change It is always simpler (when pssible!) t have rthgnal (uncrrelated) regressrs Parameters are nt unique in general! Sme cntrasts have n meaning: NON ESTIMABLE Eample here : c = [1 0 0] is nt estimable ( = n specific infrmatin in the first regressr); c = [1-1 0] is estimable. In case f crrelatin, use F-tests F t see the verall significance. There is generally n way t decide where the «cmmn» part shared by tw regressrs shuld be attributed t In case f crrelatin and yu need t rthgnlise a part f the design matri, there is n need t re-fit a new mdel : the cntrast nly shuld change.

11 Cntents Hemdynamic Respnse Functin Intrductin & Recap F-test and added variance Gd & bad mdels Imprved mdel Haemdynamic respnse functin High pass filter Serial crrelatin Glbal effect HRF and ER fmri «Takehme»message message % signal change = (pint baseline)/baseline usually 0.5-3% initial dip -mre fcal and ptentially a better measure -smewhat elusive s far, nt everyne can find it time t rise signal begins t rise sn after stimulus start time t peak signal peaks 4-6 sec after stimulus begins pst stimulus undersht signal suppressed after stimulatin ends Haemdynamic respnse functin Imprved mdel Functin f bld ygenatin, flw, vlume (Butn et al, 1998) Peak Cnvlve stimulus functin with mdel f BOLD respnse Peak (ma. ygenatin) 4-6s pststimulus; baseline after 20-30s Initial undersht can be bserved (Malnek & Grinvald, 1996) Brief Stimulus Undersht Haemdynamic respnse functin fitted data Similar acrss V1, A1, S1 but differences acrss: ther regins (Schacter et al 1997) individuals (Aguirre et al, 1998) Initial Undersht

12 Lw frequency nuisance effects Physilgical nise Drifts physical physilgical Aliased high frequency effects cardiac (~1 Hz) respiratry (~0.25 Hz) Pwer in the lw frequencies Pwer spectrum High pass filter high pass filter implemented by by residuals f f DCT DCT set set Y = Xβ ε raw fmri time series GLM fitted adjusted fr glbal & lw Hz effects t t 1 f = r ( t ) cs rπ t N t 1 fitted b-car scaled fr glbal changes discrete csine transfrm set set fitted high-pass filter residuals

13 High pass filter Y = Xβ ε Serial crrelatin (fmri) General Linear Mdel Generalised Linear Mdel data and and three different mdels fmri time series are aut-crrelated: adaptatin f general linear mdel necessary fr valid test estimatin f autcrrelatin Optimal lw-pass filter Serial crrelatin Errr cvariance matri Y 2 = 1 ε ε ~ N (0, σ ) t ay t t with t Y = Xβ ε Cv(ε ) i.i.d. autregressive prcess f rder 1 (AR(1)) N N autcvariance functin Cv(Y ) AR(1) sampled errr cvariance matrices (10 3 vels) N N Serial crrelatin

14 Serial crrelatins Y = Xβ ε ε ~ N(0,σ 2 V) intrinsic autcrrelatin V Restricted Maimum Likelihd Y = Xβ ε Cv(ε )? bserved Prblem: Estimate σ 2 V at each vel and make inference abut c T β Mdel: Mdel V as linear cmbinatin f m variance cmpnents Q 1 V V= = λ 1 Q 1 λ 2 Q 2 λ m Q m Assumptins: V is the same at each vel σ 2 is different at each vel Eample: Fr ne fmri sessin, use 2 variance cmpnents. Chice f Q 1 and Q 2 mtivated by autregressive mdel f rder 1 plus white nise (AR(1)wn) Q 1 Q 2 Q 2 ReML estimated crrelatin matri Serial crrelatins estimatin estimatin ^ β = (X T X) X T Y unbiased, rdinary least squares estimate Cmpute sample cvariance matri f data at all activated vels: C Y = Σ k Y k Y T k /K Imprtant: Data Y kk must be high-pass filtered. Mdel C Y as C Y = X β β T X T Σ λ i Q i ii and estimate hyperparameters λ ii using Restricted Maimum Likelihd (ReML) ^ ^ Estimate V by V = N Σ λ i Q i i i /trace(σ λ i Q i i i ) Estimate σ 2 at each vel in the usual way by σ^ 2 = (RY) T (RY) / trace(r V) unbiased where R = I X(X T X) X C Y C ^ Y y θˆ j = Xθ ε j, OLS j = X y Ordinary least-squares j j Cˆ ε = Cv ˆ ( ε) = ReML( ˆ θ T 1 1 j, ML = ( X V X ) 2 2 passes (first pass fr frselectin f f vels) mre accurate estimate f f V T c θ t = T SE ( c θ ) Estimatin in SPM j vel j X T y y, X, Q) T V j ReML ReML (pled estimate) 1 y Maimum Likelihd Assume, at at vel j: j: ε, j = σ C jv T 2 T 2 2 SE ( c θ ) = σˆ c ( V 1/ X ) ( V 1/ X ) j T c

15 Serial crrelatins inference inference Inference: T test null hypthesis c T β = 0, cmpute t-value by dividing size f effect by its standard deviatin: t = c T β ^ / std[c T β] ^ where std[c T ^ β] β] = sqrt(σ 22 c' c' (X T X) X T V X (X T X) c ) but std[c T ^ β] β] is nt a χ 2 variable because f V Apprimating χ 2 distributin using Satterthwaite apprimatin: Var[σ 2 ]= 2σ 4 trace(r V R V ) / trace(r V ) 2 ν = 2E[σ 2 ] 2 /Var[σ 2 ] = trace(rv) 2 / trace(rvrv) effective degrees f f freedm Use t-distributin with ν degrees f freedm t cmpute p-value fr t Abslute value f BOLD signal is meaningless fmri signal f an individual vel acrss scans and sessins Scale the scans by the sessin glbal mean fmri Glbal scaling artefact PET Glbal effects: AnCva Scale each scan by its wn glbal mean? N scaling. Vel 1, n effect Vel 2, ps. effect cnditin 2 rcbf rcbf Stimulus Glbal Vel 1 Vel 2 Vel 1, neg. effect With scaling: vel/glbal Vel 2, "n" effect cnditin 1 single subject activatin AnCva classic way way t t include a nuisance cvariate int int a cmparisn assumptins linear linear / parallel parallel Cnstant Cnstant acrss acrss cnditins cnditins rcbf αk 2 αk 1 g.. gcbf ζ k 1 gcbf

16 Single subject activatin (AnCva) Prprtinal scaling by gcbf Y qj =α qj =α q q μ ζ (g (g qj - qj -g..) ) ε jq jq ε jq ~ jq N(0,σ 2 2 )) Null hypthesis (at (at this this vel) vel) H 0 :α 0 :α 1 = 1 α 2 2 parameter vectr β = (α (α 1, 1, α 2, 2, α 3, 3, μ, μ, ζ )) T T cntrast weights c activatin: H 1 :α 1 :α 2 > 2 α 1 1 (( ) 0) deactivatin: H 1 :α 1 :α 1 > 1 α 2 2 ( ( ) 0) F --test H 0 : 0 : α 1 = 1 α 2 = 2 α 3 = 3 0 rcbf μ αk 2 μ αk 3 μ αk 1 ζ k 1 g.. gcbf scale gcbf t t50ml/min/dl Y qj = Y qj Y qj / qj /(g (g qj / qj / )) statistics n n adjusted data data scales variance e.g. e.g. single single subject activatin Y qj = Y qj α q q ε qj qj ε qj ~ qj N(0,σ 2 2 )) Y qj = qj α q (g q (g qj /50) qj /50) ε qj ε qj ε qj ~ ε qj N(0, N(0, σ 2 2 (g (g qj /50) qj /50) 2 2 )) rcbf (adj) rcbf rcbf (adj) Y Y 0 0 rcbf gcbf 50 gcbf Cnfunded cvariates Cntents E.g. glbal effects frequently crrect fr glbal changes nuisance effect? glbal mean affected by respnse? Mtin effects in fmri artefactual deactivatins rcbf rcbf g.. gcbf Intrductin & Recap F-test and added variance Gd & bad mdels Imprvedmdel mdel HRF and ER fmri «Take hme» message g.. gcbf

17 => BOLD Impulse Respnse Epch vs Event-related fmri Functin f bld ygenatin, flw, vlume (Butn et al, 1998) Peak (ma. ygenatin) 4-6s pststimulus; baseline after 20-30s Initial undersht can be bserved (Malnek & Grinvald, 1996) Similar acrss V1, A1, S1 but differences acrss: ther regins (Schacter et al 1997) individuals (Aguirre et al, 1998) Brief Stimulus Peak Undersht PET Blcked cnceptin (scans assigned t cnditins) A B A B Cnditin A Scans 1-10 Scans Scans Design Matri Cnditin B fmri Epch cnceptin (scans treated as timeseries) Bcar functin fmri Event-related cnceptin Delta functins Cnvlved with HRF Overview Advantages f Event-related fmri 1. Advantages f efmri 2. BOLD impulse respnse 3. General Linear Mdel 4. Tempral Basis Functins 5. Timing Issues 6. Design Optimisatin 1. Randmised trial rder c.f. cnfunds f blcked designs

18 Blcked Data Mdel O = Old Wrds N = New Wrds Advantages f Event-related fmri 1. Randmised trial rder c.f. cnfunds f blcked designs 2. Pst hc / subjective classificatin f trials e.g, accrding t subsequent memry O1 O2 O3 N1 N2 N3 Randmised O1 N1 O2 O3 N2 Event-Related R = Wrds Later Remembered F = Wrds Later Frgtten ~4s Advantages f Event-related fmri 1. Randmised trial rder c.f. cnfunds f blcked designs 2. Pst hc / subjective classificatin f trials e.g, accrding t subsequent memry 3. Sme events can nly be indicated by subject (in (in time) e.g,, spntaneus perceptual changes R R F R F Data Mdel

19 Advantages f Event-related fmri 1. Randmised trial rder c.f. cnfunds f blcked designs 2. Pst hc / subjective classificatin f trials e.g, accrding t subsequent memry Number f Perceptual Reversals Inter Reversal Time (s) Inter Reversal Time (s) 3. Sme events can nly be indicated by subject (in (in time) e.g,, spntaneus perceptual changes 4. Sme trials cannt be blcked e.g, ddball designs Advantages f Event-related fmri Oddball 1. Randmised trial rder c.f. cnfunds f blcked designs 2. Pst hc / subjective classificatin f trials e.g, accrding t subsequent memry 3. Sme events can nly be indicated by subject (in (in time) e.g,, spntaneus perceptual changes 4. Sme trials cannt be blcked e.g, ddball designs Time 5. Mre accurate mdels even fr blcked designs? e.g, state-item interactins

20 Epch mdel Blcked Design Data Mdel Disadvantage f Randmised Designs O1 O2 O3 N1 N2 N3 Event mdel 1. Less efficient fr detecting effects than are blcked designs (see later ) 2. Sme psychlgical prcesses may be better blcked (eg task-switching, switching, attentinal instructins) O1 O2 O3 N1 N2 N3 Overview BOLD Impulse Respnse 1. Advantages f efmri 2. BOLD impulse respnse 3. General Linear Mdel 4. Tempral Basis Functins 5. Timing Issues 6. Design Optimisatin Functin f f bld ygenatin, flw, vlume (Butn et et al, 1998) Peak (ma. ygenatin) 4-6s 4 pststimulus; ; baseline after 20-30s Initial undersht can be be bserved (Malnek( & Grinvald, 1996) Similar acrss V1, A1, S1 Brief Stimulus Peak Undersht but differences acrss: ther regins (Schacter( et et al al 1997) individuals (Aguirre et et al, 1998) Initial Undersht

21 BOLD Impulse Respnse Overview Early event-related fmri studies used a lng Stimulus Onset Asynchrny (SOA) t t allw BOLD respnse t t return t t baseline Hwever, if if the BOLD respnse is is eplicitly mdelled,, verlap between successive respnses at at shrt SOAs can be be accmmdated particularly if if respnses are assumed t t superpse linearly Shrt SOAs are mre sensitive Brief Stimulus Initial Undersht Peak Undersht 1. Advantages f efmri 2. BOLD impulse respnse 3. General Linear Mdel 4. Tempral Basis Functins 5. Timing Issues 6. Design Optimisatin General Linear (Cnvlutin) Mdel General Linear Mdel (in SPM) GLM fr a single vel: Y(t) = (t) h(t) ε (t) h(t)= ß i f i (t) Auditry wrds every 20s (t) = stimulus train (delta functins) (t) = δ(t - nt) h(t) = hemdynamic (BOLD) respnse h(t) = ß ii f i (t) i f i (t) i = tempral basis functins Y(t) = ß ii f i i (t (t- nt) ε T 2T 3T... cnvlutin sampled each scan Design Matri (Orthgnalised) Gamma functins ƒ i (u) f peristimulus time u Sampled every TR = 1.7s Design matri, X [ƒ 1 (u) (t) ƒ 2 (u) (t)...] SPM{F} 0 time {secs{ secs} } 30

22 Overview Tempral Basis Functins 1. Advantages f efmri 2. BOLD impulse respnse 3. General Linear Mdel 4. Tempral Basis Functins 5. Timing Issues 6. Design Optimisatin Cannical Infrmed Basis Set (Fristn et al. 1998) Cannical HRF (2 (2 gamma functins) Tempral Basis Functins Tempral Basis Functins Cannical Tempral Infrmed Basis Set (Fristn et al. 1998) Cannical HRF (2 (2 gamma functins) plus Multivariate Taylr epansin in: time (Tempral Derivative) Cannical Tempral Dispersin Infrmed Basis Set (Fristn et al. 1998) Cannical HRF (2 (2 gamma functins) plus Multivariate Taylr epansin in: time (Tempral Derivative) width (Dispersin Derivative)

23 Tempral Basis Functins Tempral Basis Functins Cannical Tempral Dispersin Infrmed Basis Set (Fristn et al. 1998) Cannical HRF (2 (2 gamma functins) plus Multivariate Taylr epansin in: time (Tempral Derivative) width (Dispersin Derivative) Cannical Tempral Dispersin Infrmed Basis Set (Fristn et al. 1998) Cannical HRF (2 (2 gamma functins) plus Multivariate Taylr epansin in: time (Tempral Derivative) width (Dispersin Derivative) Magnitude inferences via t-test t test n cannical parameters (prviding cannical is is a gd fit mre later) Magnitude inferences via t-test t test n cannical parameters (prviding cannical is is a gd fit mre later) Latency inferences via tests n n rati f f derivative : cannical parameters (mre later ) Tempral Basis Functins Furier Set Windwed sines & csines Any shape (up t t frequency limit) Inference via F-testF Tempral Basis Functins Finite Impulse Respnse (FIR) Mini timebins (selective averaging) Any shape (up t t bin-width) Inference via F-testF

24 Tempral Basis Functins Furier Set Windwed sines & csines Any shape (up t t frequency limit) Inference via F-testF Gamma Functins Bunded, asymmetrical (like BOLD) Set f f different lags Inference via F-testF Tempral Basis Functins Furier Set Windwed sines & csines Any shape (up t t frequency limit) Inference via F-testF Gamma Functins Bunded, asymmetrical (like BOLD) Set f f different lags Inference via F-testF Infrmed Basis Set Best guess f f cannical BOLD respnse Variability captured by Taylr epansin Magnitude inferences via t-test? test? Tempral Basis Functins Tempral Basis Sets: Which One? In this eample (rapid mtr respnse t faces, Hensn et al, 2001) Cannical Tempral Dispersin FIR cannical tempral dispersin derivatives appear sufficient may nt be fr mre cmple trials (eg stimulus-delay-respnse) but then such trials better mdelled with separate neural cmpnents (ie activity n lnger delta functin) cnstrained HRF (Zarahn, 1999)

25 Overview Timing Issues 1. Advantages f efmri 2. BOLD impulse respnse 3. General Linear Mdel 4. Tempral Basis Functins 5. Timing Issues 6. Design Optimisatin Typical TR fr 48 slice EPI at 3mm spacing is ~ 4s Scans TR=4s Timing Issues Timing Issues Typical TR fr 48 slice EPI at 3mm spacing is is ~ 4s Scans TR=4s Typical TR fr 48 slice EPI at 3mm spacing is is ~ 4s Scans TR=4s Sampling at [0,4,8,12 ] pst- stimulus may miss peak signal Stimulus (synchrnus) SOA=8s Effective sampling rate=4s Sampling at [0,4,8,12 ] ] pst- stimulus may miss peak signal Higher effective sampling by: 1. Asynchrny, eg. SOA=1.5TR Stimulus (asynchrnus) SOA=6s Effective sampling rate=2s

26 Timing Issues Timing Issues Typical TR fr 48 slice EPI at 3mm spacing is is ~ 4s Scans TR=4s Typical TR fr 48 slice EPI at 3mm spacing is is ~ 4s Scans TR=4s Sampling at [0,4,8,12 ] ] pst- stimulus may miss peak signal Higher effective sampling by: Asynchrny, eg.. SOA=1.5TR 2. Randm Jitter, eg. SOA=(2±0.5)TR Stimulus (randm jitter) Effective sampling rate < 2s Sampling at [0,4,8,12 ] ] pst- stimulus may miss peak signal Higher effective sampling by: Asynchrny, eg.. SOA=1.5TR Randm Jitter, eg. SOA=(2±0.5)TR Better respnse characte- risatin (Miezin et al, 2000) Stimulus (randm jitter) Effective sampling rate < 2s Timing Issues Timing Issues but Slice-timing Prblem (Hensn et al, 1999) Slices acquired at different times, yet mdel is is the same fr all slices but Slice-timing Prblem (Hensn et et al, 1999) Slices acquired at at different times, yet mdel is is the same fr all all slices => different results (using cannical HRF) fr different reference slices Tp Slice SPM{t} Bttm Slice SPM{t} TR=3s

27 Timing Issues Timing Issues but Slice-timing Prblem (Hensn et et al, 1999) Slices acquired at at different times, yet mdel is is the same fr all all slices => different results (using cannical HRF) fr different reference slices Slutins: 1. Tempral interplatin f f data but less gd fr lnger TRs Tp Slice Bttm Slice TR=3s SPM{t} SPM{t} Interplated SPM{t} but Slice-timing Prblem (Hensn et et al, 1999) Slices acquired at at different times, yet mdel is is the same fr all all slices => different results (using cannical HRF) fr different reference slices Slutins: Tempral interplatin f f data but less gd fr lnger TRs 2. Mre general basis set (e.g., with tempral derivatives) but inferences via F-testF Tp Slice Bttm Slice TR=3s SPM{t} SPM{t} Interplated SPM{t} Derivative SPM{F} BOLD Respnse Latency (Linear) Assume the real respnse, r(t), is a scaled (by α) versin f the cannical, f(t), but delayed by a small amunt dt: r(t) = α f(tdt) ~ α f(t) α f (t) dt 1 st -rder Taylr If the fitted respnse, R(t), is mdelled by the cannical tempral derivative: R(t) = ß 1 f(t) ß 2 f (t) GLM fit Then cannical and derivative parameter estimates, ß 1 and ß 2, are such that: BOLD Respnse Latency: eample Psitive Negative cnstant derivative HRF α = ß 1 dt = ß 2 / ß 1 ie, Latency can be apprimated by the rati f derivativet-cannical parameter estimates (within limits f first-rder apprimatin, /- 1s) Designmatri events, events, SOA SOA ~18s, ~18s, TR TR 3s 3s

28 BOLD Respnse Latency (Linear) Neural Respnse Latency? Delayed Respnses (green/ yellw) Cannical Basis Functins Cannical Derivative Neural A. Decreased BOLD A. Smaller Peak Parameter Estimates ß 1 ß 2 ß 1 ß 2 ß 1 ß 2 B. Advanced C. Shrtened (same integrated) B. Earlier Onset C. Earlier Peak Actual latency, dt, vs. ß 2 / ß 1 ß 2 /ß 1 Face repetitin reduces latency as well as magnitude f fusifrm respnse D. Shrtened (same maimum) D. Smaller Peak and earlier Peak BOLD Respnse Latency (Iterative) BOLD Respnse Latency (Iterative) Peak Delay Onset Delay Height Numerical fitting f eplicitly parameterised cannical HRF (Hensn et al, 2001) Distinguishes between Onset and Peak latency unlike tempral derivative and which may be imprtant fr interpreting neural changes (see previus slide) Distributin f parameters tested nnparametrically (Wilcn s T ver subjects) Neural D. Shrtened (same maimum) 240ms Peak Delay wt(11)=14, p<.05 N difference in Onset Delay, wt(11)= % Height Change wt(11)=5, p<.001 BOLD D. Smaller Peak and earlier Peak Mst parsimnius accunt is that repetitin reduces duratin f neural activity

29 Overview Fied SOA = 16s Stimulus ( Neural ) HRF Predicted Data 1. Advantages f efmri 2. BOLD impulse respnse 3. General Linear Mdel 4. Tempral Basis Functins 5. Timing Issues 6. Design Optimisatin = Nt particularly efficient Fied SOA = 4s Stimulus ( Neural ) HRF Predicted Data Randmised, SOA min = 4s Stimulus ( Neural ) HRF Predicted Data = = Very Inefficient Mre Efficient

30 Blcked, SOA min = 4s Stimulus ( Neural ) HRF Predicted Data = Blcked, epch = 20s Stimulus ( Neural ) HRF Predicted Data = = Even mre Efficient Blcked-epch (with small SOA) and Time-Freq equivalences Sinusidal mdulatin, f = 1/33s Stimulus ( Neural ) HRF Predicted Data Blcked (80s), SOA min =4s, highpass filter = 1/120s Stimulus ( Neural ) HRF Predicted Data = = Effective HRF (after highpass filtering) (Jsephs & Hensn, 1999) = = The mst efficient design f all! Dn t have lng (>60s) blcks!

31 Randmised, SOA min =4s, highpass filter = 1/120s Stimulus ( Neural ) HRF Predicted Data = T = c T β // std(c T β) β) Design Efficiency std(c T β) β) = sqrt(σ 2 c T (X T X) -1-1 c) c) (i.i.d) Fr ma. T, T, want min. cntrast variability (Fristn et et al, 1999) Events (A-B) If If assume that nise variance (σ( (σ 2 ))is unaffected by changes in in X X then want maimal efficiency, e: e: = e(c,x) ) = { { c T (X T X) -1-1 c } -1-1 = maimal bandpassed signal energy (Jsephs & Hensn, 1999) (Randmised design spreads pwer ver frequencies) Efficiency - Multiple Event-types types Efficiency - Multiple Event-types types Design parametrised by: SOA min min Minimum SOA p i (h) i Prbability f f event- type iigiven histry h f f last m events Differential Effect (A-B) Eample: Alternating AB A B A 0 1 B 1 0 Permuted (A-B) With n event-types types p i (h) i is is a n m n Transitin Matri Eample: Randmised AB A B A B => => ABBBABAABABAAA... Cmmn Effect (AB) 4s smthing; 1/60s highpass filtering => ABABABABABAB... Eample: Permuted AB A B AA 0 1 AB BA BB 1 0 => ABBAABABABBA... Alternating (A-B) 4s smthing; 1/60s highpass filtering

32 Efficiency - Multiple Event-types types Efficiency - Cnclusins Eample: Null events A B A B => AB-BAA--B---ABB... BAA--B---ABB... Efficient fr differential and main effects at shrt SOA Equivalent t stchastic SOA (Null Event like third unmdelled event-type) type) Selective averaging f data (Dale & Buckner 1997) Null Events (A-B) Null Events (AB) 4s smthing; 1/60s highpass filtering Optimal design fr ne cntrast may nt be ptimal fr anther Blcked designs generally mst efficient with shrt SOAs (but earlier restrictins and prblems f interpretatin ) With randmised designs, ptimal SOA fr differential effect (A-B) is minimal SOA (assuming n saturatin), whereas ptimal SOA fr main effect (AB) is 16-20s Inclusin f null events imprves efficiency fr main effect at shrt SOAs (at cst f efficiency fr differential effects) If If rder cnstrained, intermediate SOAs (5-20s) can be ptimal; if if SOA cnstrained, pseud-randmised designs can be ptimal (but may intrduce cntetsensitivity) Cntents Way t prceed Prepare yur questins. ALL the questins! Intrductin & Recap Gd & bad mdels Imprvedmdel mdel HRF and ER fmri «Take hme» message Find a mdel which allws cntrasts that translates these questins. takes int accunt ALL the effects (interactin, sessins,etc) Devise task & stimulus presentatin. Acquire the data & analyse. Nt the ther way rund!!!

33 Three Stages f an Eperiment 1. Sledgehammer Apprach brute frce eperiment : pwerful stimulus & dn t try t cntrl fr everything lk at was dne befre r by thers run a cuple f subjects -- see if it lks prmising if it desn t lk great, tweak the stimulus r task try t be a subject yurself s yu can ntice any prblems with stimuli r subject strategies Three Stages f an Eperiment 1. Sledgehammer Apprach 2. Real Eperiment at sme pint, yu have t stp changing things and cllect enugh subjects run with the same cnditins t publish it hw many subjects d yu need sme psychphysical studies test tw r three subjects, many studies test 6-10 subjects, randm effects analysis requires at least 15 subjects,... sme subjects WILL be rejected, s acquire mre than the minimum! can run all subjects in ne r tw days pr: minimize setup and variability cn: bad magnet day means a lt f wasted time make sure all the data are treated the same way. (script) Three Stages f an Eperiment 1. Sledgehammer Apprach 2. Real Eperiment 3. Whipped Cream eperiment after the real eperiment wrks, then think abut a whipped cream versin ging straight t whipped cream is a huge endeavr, especially if yu re new t imaging and it gives yu a secnd paper!