Temporal Basis Functions

1. Temporal Basis Functions Methods for Dummies 27 Jan 2010 Melanie Boly

2. What’s a basis function then…? • Used to model our fMRI signal • A basis function is the combining of a number of functions to describe a more complex function. f(t) h1(t) h2(t) h3(t) 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)

3. Temporal Basis Functions for fMRI • In fMRI we need to describe a function of % signal change over time. • There are various different basis sets that we could use to approximate the signal. Finite Impulse Response (FIR) Fourier

4. Peak Brief Stimulus Undershoot Initial Undershoot HRF 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)

5. Temporal Basis Functions for fMRI • Better though to use functions that make use of our knowledge of the shape of the HRF. • One gamma function alone provides a reasonably good fit to the HRF. They are asymmetrical and can be set at different lags. • However they lack an undershoot. • If we add two of them together we get the canonical HRF.

6. Ex: Auditory words every 20s HRF ƒi() of peristimulus time  Sampled every TR = 1.7s Design matrix, X … General (convoluted) Linear Model

7. Limits of HRF • General shape of the BOLD impulse response similar across early sensory regions, such as V1 and S1. • Variability across higher cortical regions. • Considerable variability across people. • These types of variability can be accommodated by expanding the HRF in terms of temporal basis functions.

8. “Informed” Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor expansion in: time (Temporal Derivative) width (Dispersion Derivative) The temporal derivative can model (small) differences in the latency of the peak response. The dispersion derivative can model (small) differences in the duration of the peak response.

9. Ex: Auditory words every 20s Gamma functions ƒi() of peristimulus time  SPM{F} Sampled every TR = 1.7s Design matrix, X [x(t)ƒ1() | x(t)ƒ2() |...] … 0 time {secs} 30 General (convoluted) Linear Model

10. Ex: Auditory words every 20s Gamma functions ƒi() of peristimulus time  SPM{F} Sampled every TR = 1.7s Design matrix, X [x(t)ƒ1() | x(t)ƒ2() |...] … 0 time {secs} 30 General (convoluted) Linear Model REVIEW DESIGN

11. Comparison of the fitted response 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” responses T-tests on canonical HRF alone (at 1st level) can be improved by derivatives reducing residual error, and can be interpreted as “amplitude” differences, assuming canonical HRF is good fit…

12. Which temporal basis functions…?

13. Which temporal basis functions…? In this example (rapid motor response to faces, Henson et al, 2001)… Canonical + Temporal + Dispersion + FIR …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)

14. Left Right Mean 1 0 0 -1 0 0 0 Putting them into your design matrix

15. Underadditivity at short SOAs Linear Prediction Implications for Efficiency Volterra Prediction Non-linear effects

16. Putting them into your design matrix

17. Thanks to… • Rik Henson’s slides: www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt • Previous years’ presenters’ slides • Guillaume Flandin, Antoinette Nicolle