Temporal Basis FunctionsTemporal Basis Functions

Melanie BolyMelanie Boly

Methods for Dummies 27 Jan 2010Methods for Dummies 27 Jan 2010

Used to model our fMRI signalUsed to model our fMRI signal

A basis function is the combining of a number of functions to describe a A basis function is the combining of a number of functions to describe a more complex function.more complex function.

What’s a basis function then…?What’s a basis function then…?

Fourier analysis

The complex wave at the top can be decomposed into the sum of the three simpler waves shown below.

f(t)=h1(t)+h2(t)+h3(t)

f(t)

h1(t)

h2(t)

h3(t)

Temporal Basis Functions for fMRITemporal Basis Functions for fMRI

In fMRI we need to describe a function of % signal change over In fMRI we need to describe a function of % signal change over time.time.

There are various different basis sets that we could use to There are various different basis sets that we could use to approximate the signal.approximate the signal.

Finite Impulse Response (FIR)

Fourier

HRFHRF

BriefStimulus

Undershoot

InitialUndershoot

Peak

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

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

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

Similar across V1, A1, S1…… but differences across: other regions (Schacter et

al 1997) individuals (Aguirre et al, 1998)

Temporal Basis Functions for fMRITemporal Basis Functions for fMRI

Better though to use functions that make use Better though to use functions that make use of our knowledge of the shape of the HRF.of our knowledge of the shape of the HRF.

One gamma function alone provides a One gamma function alone provides a reasonably good fit to the HRF. They are reasonably good fit to the HRF. They are asymmetrical and can be set at different lags.asymmetrical and can be set at different lags. However they lack an undershoot.However they lack an undershoot.

If we add two of them together we get the If we add two of them together we get the canonical HRF.canonical HRF.

General (convoluted) Linear ModelGeneral (convoluted) Linear Model

Ex: Auditory words

every 20s

Sampled every TR = 1.7s Design matrix, Design matrix, XX …

HRF ƒHRF ƒii(() of ) of

peristimulus time peristimulus time

Limits of HRFLimits of HRF

General shape of the BOLD impulse response similar across General shape of the BOLD impulse response similar across early sensory regions, such as V1 and S1. early sensory regions, such as V1 and S1.

Variability across higher cortical regions.Variability across higher cortical regions.

Considerable variability across people. Considerable variability across people.

These types of variability can be These types of variability can be accommodated by expanding the HRF in terms accommodated by expanding the HRF in terms of temporal basis functions.of temporal basis functions.

Canonical HRF (2 gamma Canonical HRF (2 gamma functions)functions) plusplus Multivariate Taylor Multivariate Taylor expansion in:expansion in:

time (time (Temporal DerivativeTemporal Derivative))width (width (Dispersion DerivativeDispersion Derivative))

The temporal derivative can The temporal derivative can model (small) differences in the model (small) differences in the latency of the peak response.latency of the peak response.

The dispersion derivative can The dispersion derivative can model (small) differences in the model (small) differences in the duration of the peak response.duration of the peak response.

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

General (convoluted) Linear ModelGeneral (convoluted) Linear Model

Ex: 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

General (convoluted) Linear ModelGeneral (convoluted) Linear Model

Ex: 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

REVIEW DESIGN

These plots show the haemodynamic response at a single voxel. The left plot shows the HRF as estimated using the simple model. Lack of fit is corrected, on the right using a more flexible model with basis functions.

F-tests allow for any “canonical-like” responsesF-tests allow for any “canonical-like” responses

T-tests on canonical HRF alone (at 1st level) can be improved by derivatives T-tests on canonical HRF alone (at 1st level) can be improved by derivatives reducing residual error, and can be interpreted as “amplitude” differences, reducing residual error, and can be interpreted as “amplitude” differences, assumingassuming canonical HRF canonical HRF is good fit… is good fit…

Comparison of the fitted Comparison of the fitted responseresponse

Which temporal basis functions…?Which temporal basis functions…?

Which temporal basis functions…?Which temporal basis functions…?

+ FIR+ Dispersion+ TemporalCanonical

…canonical + temporal + dispersion derivatives appear sufficient…may not be for more complex trials (eg stimulus-delay-response)…but then such trials better modelled with separate neural components

(ie activity no longer delta function) + constrained HRF (Zarahn, 1999)

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

Putting them into your design matrixPutting them into your design matrix

Left Right Mean 1 0 0 -1 0 0 0

Non-linear effectsNon-linear effects

Underadditivity at short SOAsUnderadditivity at short SOAsLinearPrediction

VolterraPrediction

Implicationsfor Efficiency

Putting them into your design matrixPutting them into your design matrix

Thanks to…Thanks to… Rik Henson’s slides: Rik Henson’s slides:

www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt

Previous years’ presenters’ slidesPrevious years’ presenters’ slides

Guillaume Flandin, Antoinette Nicolle Guillaume Flandin, Antoinette Nicolle