event-related fmri christian ruff with thanks to: rik henson

Event-related Event-related fMRIfMRIEvent-related Event-related fMRIfMRI

Christian RuffChristian Ruff

With thanks to: Rik Henson

RealignmentRealignment SmoothingSmoothing

NormalisationNormalisation

General linear modelGeneral linear model

Statistical parametric map (SPM)Statistical parametric map (SPM)Image time-seriesImage time-series

Parameter estimatesParameter estimates

Design matrixDesign matrix

TemplateTemplate

KernelKernel

Gaussian Gaussian field theoryfield theory

p <0.05p <0.05

StatisticalStatisticalinferenceinference

OverviewOverviewOverviewOverview

1. Block/epoch vs. event-related fMRI1. Block/epoch vs. event-related fMRI

2. (Dis)Advantages of efMRI2. (Dis)Advantages of efMRI

3. GLM: Convolution3. GLM: Convolution

4. BOLD impulse response4. BOLD impulse response

5. Temporal Basis Functions5. Temporal Basis Functions

6. Timing Issues6. Timing Issues

7. Design Optimisation 7. Design Optimisation – “Efficiency”– “Efficiency”








Designs: Block/epoch- vs event-Designs: Block/epoch- vs event-relatedrelated

Designs: Block/epoch- vs event-Designs: Block/epoch- vs event-relatedrelated

U1 P1 U3U2 P2

Data

Model

P = PleasantU = Unpleasant

Block/epoch designs examine responses to series of similar stimuli

U1 U2 U3 P1 P2 P3

Event-related designs account for response to each single stimulus

~4s

1. Randomised1. Randomised trialtrial order orderc.f. confounds of blocked designs (Johnson et al 1997)c.f. confounds of blocked designs (Johnson et al 1997)

1. Randomised1. Randomised trialtrial order orderc.f. confounds of blocked designs (Johnson et al 1997)c.f. confounds of blocked designs (Johnson et al 1997)

Advantages of event-related fMRIAdvantages of event-related fMRIAdvantages of event-related fMRIAdvantages of event-related fMRI

Blocked designs may trigger expectations and cognitive setsBlocked designs may trigger expectations and cognitive sets

……

Pleasant (P)Unpleasant (U)

Intermixed designs can minimise this by stimulus randomisationIntermixed designs can minimise this by stimulus randomisation

…… …… ………………

eFMRI: Stimulus randomisationeFMRI: Stimulus randomisationeFMRI: Stimulus randomisationeFMRI: Stimulus randomisation

Unpleasant (U) Unpleasant (U) Unpleasant (U)Pleasant (P) Pleasant (P)

1. Randomised trial order 1. Randomised trial order c.f. confounds of blocked designs (Johnson et al 1997)c.f. confounds of blocked designs (Johnson et al 1997)

2. Post hoc / subjective classification of trials2. Post hoc / subjective classification of trialse.g, according to subsequent memory (Gonsalves & Paller 2000)e.g, according to subsequent memory (Gonsalves & Paller 2000)




eFMRI: post-hoc classification of eFMRI: post-hoc classification of trialstrials

eFMRI: post-hoc classification of eFMRI: post-hoc classification of trialstrials

Gonsalves, P & Paller, K.A. (2000). Gonsalves, P & Paller, K.A. (2000). Nature Neuroscience, 3Nature Neuroscience, 3 (12):1316-21 (12):1316-21

Items with wrongItems with wrong memory of picture („hat“) were associated with more memory of picture („hat“) were associated with more occipital activity occipital activity at encodingat encoding than items with correct rejection („brain“) than items with correct rejection („brain“)

„„was shown as was shown as picture“picture“

„„was was notnot shown shown as picture“as picture“

Participant response:Participant response:



3. Some events can only be indicated by subject (in time)3. Some events can only be indicated by subject (in time)e.g, spontaneous perceptual changes (Kleinschmidt et al 1998)e.g, spontaneous perceptual changes (Kleinschmidt et al 1998)





eFMRI: “on-line” event-definitioneFMRI: “on-line” event-definitioneFMRI: “on-line” event-definitioneFMRI: “on-line” event-definition




4. Some trials cannot be blocked due to stimulus context or interactions4. Some trials cannot be blocked due to stimulus context or interactionse.g, “oddball” designs (Clark et al., 2000)e.g, “oddball” designs (Clark et al., 2000)






time

OddballOddball

…

eFMRI: Stimulus contexteFMRI: Stimulus contexteFMRI: Stimulus contexteFMRI: Stimulus context





5. More accurate models even for blocked designs?5. More accurate models even for blocked designs?e.g., “state-item” interactions (Chawla et al, 1999)e.g., “state-item” interactions (Chawla et al, 1999)





5. More accurate models even for blocked designs?5. More accurate models even for blocked designs?e.g., “state-item” interactions (Chawla et al, 1999)e.g., “state-item” interactions (Chawla et al, 1999)


P1 P2 P3

“Event” model may capture state-item interactions (with longer SOAs)

U1 U2 U3

Blocked Design

“Epoch” model assumes constant neural processes throughout block

Data

Model

U1 U2 U3 P1 P2 P3

eFMRI: “Event” model of block-eFMRI: “Event” model of block-designs designs

eFMRI: “Event” model of block-eFMRI: “Event” model of block-designs designs

Data

Model

Convolved with HRF

=>

Series of eventsDelta

functions

• DesignsDesigns can be can be blockedblocked or or intermixed, intermixed, BUTBUT models models for blocked designs can be for blocked designs can be epochepoch- or - or eventevent-related-related

• EpochsEpochs are periods of sustained stimulation are periods of sustained stimulation (e.g, box-car functions)(e.g, box-car functions)

• EventsEvents are impulses (delta-functions) are impulses (delta-functions)

• Near-identical regressors can be created by Near-identical regressors can be created by 1) sustained epochs, 2) rapid series of 1) sustained epochs, 2) rapid series of events (SOAs<~3s)events (SOAs<~3s)

• In SPM8, all conditions are specified in In SPM8, all conditions are specified in terms of their 1) terms of their 1) onsetsonsets and 2) and 2) durationsdurations

… … epochs: variable or constant durationepochs: variable or constant duration … … events: zero durationevents: zero duration

• DesignsDesigns can be can be blockedblocked or or intermixed, intermixed, BUTBUT models models for blocked designs can be for blocked designs can be epochepoch- or - or eventevent-related-related

• EpochsEpochs are periods of sustained stimulation are periods of sustained stimulation (e.g, box-car functions)(e.g, box-car functions)

• EventsEvents are impulses (delta-functions) are impulses (delta-functions)

• Near-identical regressors can be created by Near-identical regressors can be created by 1) sustained epochs, 2) rapid series of 1) sustained epochs, 2) rapid series of events (SOAs<~3s)events (SOAs<~3s)

• In SPM8, all conditions are specified in In SPM8, all conditions are specified in terms of their 1) terms of their 1) onsetsonsets and 2) and 2) durationsdurations

… … epochs: variable or constant durationepochs: variable or constant duration … … events: zero durationevents: zero duration

“Classic” Boxcar function

Sustained epoch

Modeling block designs: epochs vs Modeling block designs: epochs vs eventsevents

Modeling block designs: epochs vs Modeling block designs: epochs vs eventsevents

• Blocks of trials can be modelled as boxcars Blocks of trials can be modelled as boxcars or runs of eventsor runs of events

• BUT: interpretation of the parameter BUT: interpretation of the parameter estimates may differestimates may differ

• Consider an experiment presenting words Consider an experiment presenting words at different rates in different blocks:at different rates in different blocks:

• An “epoch” model will estimate An “epoch” model will estimate parameter that increases with rate, parameter that increases with rate, because the parameter reflects because the parameter reflects response per blockresponse per block

• An “event” model may estimate An “event” model may estimate parameter that parameter that decreasesdecreases with rate, with rate, because the parameter reflects because the parameter reflects response per wordresponse per word

• Blocks of trials can be modelled as boxcars Blocks of trials can be modelled as boxcars or runs of eventsor runs of events

• BUT: interpretation of the parameter BUT: interpretation of the parameter estimates may differestimates may differ

• Consider an experiment presenting words Consider an experiment presenting words at different rates in different blocks:at different rates in different blocks:

• An “epoch” model will estimate An “epoch” model will estimate parameter that increases with rate, parameter that increases with rate, because the parameter reflects because the parameter reflects response per blockresponse per block

• An “event” model may estimate An “event” model may estimate parameter that parameter that decreasesdecreases with rate, with rate, because the parameter reflects because the parameter reflects response per wordresponse per word

=3 =5

=9=11

Rate = 1/4s Rate = 1/2s

Epochs vs eventsEpochs vs eventsEpochs vs eventsEpochs vs events

1. 1. Less efficient for detecting effects than are blocked designs Less efficient for detecting effects than are blocked designs (see later…) (see later…)

2. Some psychological processes have to/may be better blocked 2. Some psychological processes have to/may be better blocked (e.g., if difficult to switch between states, or to reduce surprise effects)(e.g., if difficult to switch between states, or to reduce surprise effects)

1. 1. Less efficient for detecting effects than are blocked designs Less efficient for detecting effects than are blocked designs (see later…) (see later…)

2. Some psychological processes have to/may be better blocked 2. Some psychological processes have to/may be better blocked (e.g., if difficult to switch between states, or to reduce surprise effects)(e.g., if difficult to switch between states, or to reduce surprise effects)

Disadvantages of intermixed designsDisadvantages of intermixed designsDisadvantages of intermixed designsDisadvantages of intermixed designs



2. (Dis)advantages of efMRI2. (Dis)advantages of efMRI





• Function of blood oxygenation, flow, Function of blood oxygenation, flow, volume (Buxton et al, 1998)volume (Buxton et al, 1998)

• Peak (max. oxygenation) 4-6s Peak (max. oxygenation) 4-6s poststimulus; baseline after 20-30spoststimulus; baseline after 20-30s

• Initial undershoot can be observed Initial undershoot can be observed (Malonek & Grinvald, 1996)(Malonek & Grinvald, 1996)

• Similar across V1, A1, S1…Similar across V1, A1, S1…

• … … but possible differences across:but possible differences across: other regions (Schacter et al 1997) other regions (Schacter et al 1997)

individuals (Aguirre et al, 1998)individuals (Aguirre et al, 1998)

• Function of blood oxygenation, flow, Function of blood oxygenation, flow, volume (Buxton et al, 1998)volume (Buxton et al, 1998)

• Peak (max. oxygenation) 4-6s Peak (max. oxygenation) 4-6s poststimulus; baseline after 20-30spoststimulus; baseline after 20-30s

• Initial undershoot can be observed Initial undershoot can be observed (Malonek & Grinvald, 1996)(Malonek & Grinvald, 1996)

• Similar across V1, A1, S1…Similar across V1, A1, S1…

• … … but possible differences across:but possible differences across: other regions (Schacter et al 1997) other regions (Schacter et al 1997)

individuals (Aguirre et al, 1998)individuals (Aguirre et al, 1998)

BriefStimulus

Undershoot

InitialUndershoot

Peak

BOLD impulse responseBOLD impulse responseBOLD impulse responseBOLD impulse response

• Early event-related fMRI studies Early event-related fMRI studies used a long Stimulus Onset used a long Stimulus Onset Asynchrony (SOA) to allow BOLD Asynchrony (SOA) to allow BOLD response to return to baselineresponse to return to baseline

• However, overlap between However, overlap between successive responses at short SOAs successive responses at short SOAs can be accommodated if the BOLD can be accommodated if the BOLD response is explicitly modeled, response is explicitly modeled, particularly if responses are assumed particularly if responses are assumed to superpose linearlyto superpose linearly

• Short SOAs are more sensitive; see Short SOAs are more sensitive; see laterlater

• Early event-related fMRI studies Early event-related fMRI studies used a long Stimulus Onset used a long Stimulus Onset Asynchrony (SOA) to allow BOLD Asynchrony (SOA) to allow BOLD response to return to baselineresponse to return to baseline

• However, overlap between However, overlap between successive responses at short SOAs successive responses at short SOAs can be accommodated if the BOLD can be accommodated if the BOLD response is explicitly modeled, response is explicitly modeled, particularly if responses are assumed particularly if responses are assumed to superpose linearlyto superpose linearly

• Short SOAs are more sensitive; see Short SOAs are more sensitive; see laterlater

BriefStimulus

Undershoot

InitialUndershoot

Peak

BOLD impulse responseBOLD impulse responseBOLD impulse responseBOLD impulse response

GLM for a single voxel:

y(t) = u(t) h() + (t)

u(t) = neural causes (stimulus train)

u(t) = (t - nT)

h() = hemodynamic (BOLD) response

h() = ßi fi ()

fi() = temporal basis functions

y(t) = ßi fi (t - nT) + (t)

y = X ß + ε

GLM for a single voxel:

y(t) = u(t) h() + (t)

u(t) = neural causes (stimulus train)

u(t) = (t - nT)

h() = hemodynamic (BOLD) response

h() = ßi fi ()

fi() = temporal basis functions

y(t) = ßi fi (t - nT) + (t)

y = X ß + ε

Design Matrix

convolution

T 2T 3T ...

u(t) h()= ßi fi ()

sampled each scan

General General Linear Linear (Convolution) (Convolution) ModelModelGeneral General Linear Linear (Convolution) (Convolution) ModelModel

Auditory words

every 20s

SPM{F}SPM{F}

0 time {secs} 300 time {secs} 30

Sampled every TR = 1.7s

Design matrix, Design matrix, XX

[x(t)[x(t)ƒƒ11(() | x(t)) | x(t)ƒƒ22(() |...]) |...]…

Gamma functions ƒGamma functions ƒii(() of ) of

peristimulus time peristimulus time (Orthogonalised)(Orthogonalised)

General General Linear Model in SPMLinear Model in SPMGeneral General Linear Model in SPMLinear Model in SPM

Temporal basis functionsTemporal basis functionsTemporal basis functionsTemporal basis functions

• Fourier SetFourier SetWindowed sines & cosinesWindowed sines & cosinesAny shape (up to frequency limit)Any shape (up to frequency limit)Inference via F-testInference via F-test

• Fourier SetFourier SetWindowed sines & cosinesWindowed sines & cosinesAny shape (up to frequency limit)Any shape (up to frequency limit)Inference via F-testInference via F-test


• Finite Impulse ResponseFinite Impulse ResponseMini “timebins” (selective averaging)Mini “timebins” (selective averaging)AAny shapeny shape (up to bin-width (up to bin-width))Inference via F-testInference via F-test

• Finite Impulse ResponseFinite Impulse ResponseMini “timebins” (selective averaging)Mini “timebins” (selective averaging)AAny shapeny shape (up to bin-width (up to bin-width))Inference via F-testInference via F-test


• Fourier Set / FIRFourier Set / FIRAny shape (up to frequency limit / bin Any shape (up to frequency limit / bin

width)width)Inference via F-testInference via F-test

• Gamma FunctionsGamma FunctionsBounded, asymmetrical (like BOLD)Bounded, asymmetrical (like BOLD)Set of different lagsSet of different lagsInference via F-testInference via F-test

• Fourier Set / FIRFourier Set / FIRAny shape (up to frequency limit / bin Any shape (up to frequency limit / bin

width)width)Inference via F-testInference via F-test



• Fourier Set / FIRFourier Set / FIRAny shape (up to frequency limit / bin width)Any shape (up to frequency limit / bin width)Inference via F-testInference via F-test


• ““Informed” Basis SetInformed” Basis SetBest guess of canonical BOLD responseBest guess of canonical BOLD responseVariability captured by Taylor expansion Variability captured by Taylor expansion “Magnitude” inferences via t-test“Magnitude” inferences via t-test…?…?

• Fourier Set / FIRFourier Set / FIRAny shape (up to frequency limit / bin width)Any shape (up to frequency limit / bin width)Inference via F-testInference via F-test


• ““Informed” Basis SetInformed” Basis SetBest guess of canonical BOLD responseBest guess of canonical BOLD responseVariability captured by Taylor expansion Variability captured by Taylor expansion “Magnitude” inferences via t-test“Magnitude” inferences via t-test…?…?


““Informed” Basis SetInformed” Basis Set (Friston et al. 1998)(Friston et al. 1998)

• Canonical HRF (2 gamma functions)Canonical HRF (2 gamma functions)



Canonical




plusplus Multivariate Taylor expansion in: Multivariate Taylor expansion in:

time (time (Temporal DerivativeTemporal Derivative))





CanonicalTemporal






width (width (Dispersion DerivativeDispersion Derivative))






CanonicalTemporalDispersion







• ““Magnitude” inferences via t-test on Magnitude” inferences via t-test on canonical parameterscanonical parameters (providing (providing canonical is a reasonable fit)canonical is a reasonable fit)















• ““Latency” inferences via testLatency” inferences via testss on on ratioratio of of derivativederivative : : canonical parameterscanonical parameters







• ““Latency” inferences via testLatency” inferences via testss on on ratioratio of of derivativederivative : : canonical parameterscanonical parameters



• Long Stimulus Onset Asychrony (SOA)Can ignore overlap between responses (Cohen et al 1997)

… but long SOAs are less sensitive• Fully counterbalanced designs

Assume response overlap cancels (Saykin et al 1999)Include fixation trials to “selectively average” response even at short SOA (Dale & Buckner, 1997)

… but often unbalanced, e.g. when events defined by subject• Define HRF from pilot scan on each subject

May capture inter-subject variability (Zarahn et al, 1997)

… but not interregional variability • Numerical fitting of highly parametrised response functions

Separate estimate of magnitude, latency, duration (Kruggel et al 1999)

… but computationally expensive for every voxel

Other approaches (e.g., outside SPM)Other approaches (e.g., outside SPM)Other approaches (e.g., outside SPM)Other approaches (e.g., outside SPM)

+ FIR+ Dispersion+ TemporalCanonical

… canonical + temporal + dispersion derivatives appear sufficient to capture most activity

… may not be true for more complex trials (e.g. stimulus-prolonged delay (>~2 s)-response)

… but then such trials better modelled with separate neural components (i.e., activity no longer delta function) + constrained HRF (Zarahn, 1999)

In this example (rapid motor response to faces, Henson et al, 2001)…

Which temporal basis set?Which temporal basis set?Which temporal basis set?Which temporal basis set?

• Typical TR for 60 slice EPI at 3mm Typical TR for 60 slice EPI at 3mm spacing is ~ 4sspacing is ~ 4s


Scans TR=4s

Timing issues: SamplingTiming issues: SamplingTiming issues: SamplingTiming issues: Sampling


• Sampling at [0,4,8,12…] post- Sampling at [0,4,8,12…] poststimulus may miss peak signalstimulus may miss peak signal


• Sampling at [0,4,8,12…] post- Sampling at [0,4,8,12…] poststimulus may miss peak signalstimulus may miss peak signal Stimulus (synchronous)

Scans TR=4s

SOA=8s

Sampling rate=4s



• Sampling at [0,4,8,1Sampling at [0,4,8,122…] post- …] poststimulus may miss peak signalstimulus may miss peak signal

• Higher effective sampling by: Higher effective sampling by:

1. Asynchrony1. Asynchronye.g.,e.g., SOA=1.5TR SOA=1.5TR




1. Asynchrony1. Asynchronye.g.,e.g., SOA=1.5TR SOA=1.5TR

Stimulus (asynchronous) SOA=6s

Sampling rate=2s


Scans TR=4s




1. Asynchrony1. Asynchronye.g.,e.g., SOA=1.5TR SOA=1.5TR 2. Random Jitter 2. Random Jitter

e,g., e,g., SOA=(2±0.5)TRSOA=(2±0.5)TR





e,g., e,g., SOA=(2±0.5)TRSOA=(2±0.5)TR

Stimulus (random jitter)

Sampling rate=2s


Scans TR=4s





e,g., e,g., SOA=(2±0.5)TRSOA=(2±0.5)TR

• Better response characterisation Better response characterisation (Miezin et al, 2000)(Miezin et al, 2000)





e,g., e,g., SOA=(2±0.5)TRSOA=(2±0.5)TR

• Better response characterisation Better response characterisation (Miezin et al, 2000)(Miezin et al, 2000)

Stimulus (random jitter)

Sampling rate=2s


Scans TR=4s

x2 x3

T=16, TR=2s

Scan0 1

o

T0=9 oT0=16

Timing issues: Slice-TimingTiming issues: Slice-TimingTiming issues: Slice-TimingTiming issues: Slice-Timing

T1 = 0 s

T16 = 2 s

• ““Slice-timing Problem”:Slice-timing Problem”: Slices acquired at different times, yet model Slices acquired at different times, yet model

is the same for all slicesis the same for all slices different results (using canonical HRF) for different results (using canonical HRF) for

different reference slices different reference slices (slightly less problematic if middle slice is (slightly less problematic if middle slice is

selected as reference, and with short TRs)selected as reference, and with short TRs)

• Solutions:Solutions:

1. Temporal interpolation of data1. Temporal interpolation of data… but less good for longer TRs… but less good for longer TRs

2. 2. More general basis set (e.g., withMore general basis set (e.g., withtemporal derivatives)temporal derivatives)

… but inferences via F-test… but inferences via F-test

• ““Slice-timing Problem”:Slice-timing Problem”: Slices acquired at different times, yet model Slices acquired at different times, yet model

is the same for all slicesis the same for all slices different results (using canonical HRF) for different results (using canonical HRF) for

different reference slices different reference slices (slightly less problematic if middle slice is (slightly less problematic if middle slice is

selected as reference, and with short TRs)selected as reference, and with short TRs)

• Solutions:Solutions:

1. Temporal interpolation of data1. Temporal interpolation of data… but less good for longer TRs… but less good for longer TRs

2. 2. More general basis set (e.g., withMore general basis set (e.g., withtemporal derivatives)temporal derivatives)

… but inferences via F-test… but inferences via F-test

Timing issues: Slice-timingTiming issues: Slice-timingTiming issues: Slice-timingTiming issues: Slice-timing

Bottom SliceTop Slice

SPM{t} SPM{t}

TR=3s

Interpolated

SPM{t}

Derivative

SPM{F}








7. Design Optimisation – “Efficiency”7. Design Optimisation – “Efficiency”







7. Design Optimisation – “Efficiency”7. Design Optimisation – “Efficiency”

• HRF can be viewed as a filter HRF can be viewed as a filter (Josephs & Henson, 1999)(Josephs & Henson, 1999)

• We want to maximise the signal We want to maximise the signal passed by this filterpassed by this filter

• Dominant frequency of canonical Dominant frequency of canonical HRF is ~0.04 HzHRF is ~0.04 Hz

The most efficient design is a The most efficient design is a sinusoidal modulation of neural sinusoidal modulation of neural activity with period ~24sactivity with period ~24s

• (e.g., boxcar with 12s on/ 12s off)(e.g., boxcar with 12s on/ 12s off)

• HRF can be viewed as a filter HRF can be viewed as a filter (Josephs & Henson, 1999)(Josephs & Henson, 1999)

• We want to maximise the signal We want to maximise the signal passed by this filterpassed by this filter

• Dominant frequency of canonical Dominant frequency of canonical HRF is ~0.04 HzHRF is ~0.04 Hz

The most efficient design is a The most efficient design is a sinusoidal modulation of neural sinusoidal modulation of neural activity with period ~24sactivity with period ~24s

• (e.g., boxcar with 12s on/ 12s off)(e.g., boxcar with 12s on/ 12s off)

Design EfficiencyDesign EfficiencyDesign EfficiencyDesign Efficiency

=

=

A very “efficient” design!

Stimulus (“Neural”) HRF Predicted Data

Sinusoidal modulation, f = 1/33sSinusoidal modulation, f = 1/33s

=

=

Blocked-epoch (with small SOA) quite “efficient”

HRF Predicted DataStimulus (“Neural”)

Blocked, epoch = 20sBlocked, epoch = 20sBlocked, epoch = 20sBlocked, epoch = 20s

=

“Effective HRF” (after highpass filtering)(Josephs & Henson, 1999)

Very ineffective: Don’t have long (>60s) blocks!

=


Blocked (80s), SOAmin=4s, highpass filter = 1/120s

Blocked (80s), SOAmin=4s, highpass filter = 1/120s

=

=

Randomised design spreads power over frequencies


Randomised, SOAmin=4s, highpass filter = 1/120sRandomised, SOAmin=4s, highpass filter = 1/120s

• T-statistic for a given contrast: T-statistic for a given contrast: T = T = cTb / var(cTb)

• For maximum T, we want minimum standard error of For maximum T, we want minimum standard error of contrast estimates ( contrast estimates (var(cTb)) maximum precision maximum precision

• Var(cTb) = sqrt(2cT(XTX)-1c) (i.i.d)

• If we assume that noise variance (If we assume that noise variance (2) is unaffected by changes in is unaffected by changes in X, t X, then our precision for given parameters is proportional to the hen our precision for given parameters is proportional to the design efficiency: design efficiency: e(c,X) = e(c,X) = { { ccT T ((XXTTXX))-1 -1 cc } }-1-1

We can We can influence influence ee (a priori) by the spacing and sequencing of (a priori) by the spacing and sequencing of epochs/events in our design matrix epochs/events in our design matrix

e e is specific for a given contrast!is specific for a given contrast!

• T-statistic for a given contrast: T-statistic for a given contrast: T = T = cTb / var(cTb)

• For maximum T, we want minimum standard error of For maximum T, we want minimum standard error of contrast estimates ( contrast estimates (var(cTb)) maximum precision maximum precision

• Var(cTb) = sqrt(2cT(XTX)-1c) (i.i.d)

• If we assume that noise variance (If we assume that noise variance (2) is unaffected by changes in is unaffected by changes in X, t X, then our precision for given parameters is proportional to the hen our precision for given parameters is proportional to the design efficiency: design efficiency: e(c,X) = e(c,X) = { { ccT T ((XXTTXX))-1 -1 cc } }-1-1

We can We can influence influence ee (a priori) by the spacing and sequencing of (a priori) by the spacing and sequencing of epochs/events in our design matrix epochs/events in our design matrix

e e is specific for a given contrast!is specific for a given contrast!

Design efficiencyDesign efficiency

• Design parametrised by:Design parametrised by:

SOASOAminmin Minimum SOA Minimum SOA

p(t)p(t) Probability of event Probability of event

at each at each SOASOAminmin





Design efficiency: Trial spacingDesign efficiency: Trial spacing





• DeterministicDeterministicp(t)=1 iff t=nSOAminp(t)=1 iff t=nSOAmin












• Stationary stochastic Stationary stochastic p(t)=constant<1p(t)=constant<1






• Stationary stochastic Stationary stochastic p(t)=constant<1p(t)=constant<1







• Stationary stochastic Stationary stochastic p(t)=constantp(t)=constant

• Dynamic stochasticDynamic stochasticp(t) varies (e.g., p(t) varies (e.g.,

blocked)blocked)






• Stationary stochastic Stationary stochastic p(t)=constantp(t)=constant

• Dynamic stochasticDynamic stochasticp(t) varies (e.g., p(t) varies (e.g.,

blocked)blocked)

Blocked designs most efficient! (with small SOAmin)


• However, block designs are often However, block designs are often not advisable due to interpretative not advisable due to interpretative difficulties (see before)difficulties (see before)

• Event trains may then be Event trains may then be constructed by modulating the constructed by modulating the event probabilities in a dynamic event probabilities in a dynamic stochastic fashionstochastic fashion

• This can result in intermediate This can result in intermediate levels of efficiencylevels of efficiency

• However, block designs are often However, block designs are often not advisable due to interpretative not advisable due to interpretative difficulties (see before)difficulties (see before)

• Event trains may then be Event trains may then be constructed by modulating the constructed by modulating the event probabilities in a dynamic event probabilities in a dynamic stochastic fashionstochastic fashion

• This can result in intermediate This can result in intermediate levels of efficiencylevels of efficiency


3 sessions with 128 scansFaces, scrambled facesSOA always 2.97 sCycle length 24 s

e

4s smoothing; 1/60s highpass filtering4s smoothing; 1/60s highpass filtering4s smoothing; 1/60s highpass filtering4s smoothing; 1/60s highpass filtering

• Design parametrised by:Design parametrised by:SOASOAminmin Minimum SOA Minimum SOA

ppii((hh)) Probability of event-type Probability of event-type i i

given history given history hh of last of last mm events events

• With With nn event-types event-types ppii((hh)) is a is a n x nn x n

Transition MatrixTransition Matrix

• Example: Randomised ABExample: Randomised AB

AA BBAA 0.50.5

0.5 0.5

BB 0.50.5 0.50.5

=> => ABBBABAABABAAA...ABBBABAABABAAA...

• Design parametrised by:Design parametrised by:SOASOAminmin Minimum SOA Minimum SOA

ppii((hh)) Probability of event-type Probability of event-type i i

given history given history hh of last of last mm events events

• With With nn event-types event-types ppii((hh)) is a is a n x nn x n

Transition MatrixTransition Matrix

• Example: Randomised ABExample: Randomised AB

AA BBAA 0.50.5

0.5 0.5

BB 0.50.5 0.50.5

=> => ABBBABAABABAAA...ABBBABAABABAAA...

Differential Effect (A-B)

Common Effect (A+B)

Design efficiency: Trial sequencingDesign efficiency: Trial sequencing


• Example: Alternating ABExample: Alternating AB AA BB

AA 001 1 BB 11 00

=> => ABABABABABAB...ABABABABABAB...

• Example: Alternating ABExample: Alternating AB AA BB

AA 001 1 BB 11 00

=> => ABABABABABAB...ABABABABABAB... Alternating (A-B)

Permuted (A-B)

• Example: Permuted ABExample: Permuted AB

AA BB

AAAA 0 0 11

ABAB 0.50.5 0.5 0.5

BABA 0.50.5 0.50.5

BBBB 1 1 00

=> => ABBAABABABBA...ABBAABABABBA...



• Example: Null eventsExample: Null events

AA BB

AA 0.330.330.330.33

BB 0.330.33 0.330.33

=> => AB-BAA--B---ABB...AB-BAA--B---ABB...

• Efficient for differential Efficient for differential andand main main effects at short SOAeffects at short SOA

• Equivalent to stochastic SOA (Null Equivalent to stochastic SOA (Null Event like third unmodelled event-Event like third unmodelled event-type) type)

• Example: Null eventsExample: Null events

AA BB

AA 0.330.330.330.33

BB 0.330.33 0.330.33

=> => AB-BAA--B---ABB...AB-BAA--B---ABB...

• Efficient for differential Efficient for differential andand main main effects at short SOAeffects at short SOA

• Equivalent to stochastic SOA (Null Equivalent to stochastic SOA (Null Event like third unmodelled event-Event like third unmodelled event-type) type)

Null Events (A+B)

Null Events (A-B)


• Optimal design for one contrast may not be optimal for another Optimal design for one contrast may not be optimal for another

• Blocked designs generally most efficient (with short SOAs, given optimal Blocked designs generally most efficient (with short SOAs, given optimal block length is not exceeded)block length is not exceeded)

• However, However, psychological efficiencypsychological efficiency often dictates intermixed designs, and often dictates intermixed designs, and often also sets limits on SOAsoften also sets limits on SOAs

• With randomised designs, optimal SOA for differential effect (A-B) is With randomised designs, optimal SOA for differential effect (A-B) is minimal SOA (>2 seconds, and assuming no saturation), whereas optimal minimal SOA (>2 seconds, and assuming no saturation), whereas optimal SOA for main effect (A+B) is 16-20sSOA for main effect (A+B) is 16-20s

• Inclusion of null events improves efficiency for main effect at short SOAs (at Inclusion of null events improves efficiency for main effect at short SOAs (at cost of efficiency for differential effects)cost of efficiency for differential effects)

• If order constrained, intermediate SOAs (5-20s) can be optimal If order constrained, intermediate SOAs (5-20s) can be optimal

• If SOA constrained, pseudorandomised designs can be optimal (but may If SOA constrained, pseudorandomised designs can be optimal (but may introduce context-sensitivity)introduce context-sensitivity)

• Optimal design for one contrast may not be optimal for another Optimal design for one contrast may not be optimal for another

• Blocked designs generally most efficient (with short SOAs, given optimal Blocked designs generally most efficient (with short SOAs, given optimal block length is not exceeded)block length is not exceeded)

• However, However, psychological efficiencypsychological efficiency often dictates intermixed designs, and often dictates intermixed designs, and often also sets limits on SOAsoften also sets limits on SOAs

• With randomised designs, optimal SOA for differential effect (A-B) is With randomised designs, optimal SOA for differential effect (A-B) is minimal SOA (>2 seconds, and assuming no saturation), whereas optimal minimal SOA (>2 seconds, and assuming no saturation), whereas optimal SOA for main effect (A+B) is 16-20sSOA for main effect (A+B) is 16-20s

• Inclusion of null events improves efficiency for main effect at short SOAs (at Inclusion of null events improves efficiency for main effect at short SOAs (at cost of efficiency for differential effects)cost of efficiency for differential effects)

• If order constrained, intermediate SOAs (5-20s) can be optimal If order constrained, intermediate SOAs (5-20s) can be optimal

• If SOA constrained, pseudorandomised designs can be optimal (but may If SOA constrained, pseudorandomised designs can be optimal (but may introduce context-sensitivity)introduce context-sensitivity)

Design efficiency: ConclusionsDesign efficiency: Conclusions

End: OverviewEnd: OverviewEnd: OverviewEnd: Overview















event-related fmri christian ruff with thanks to: rik henson

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