Temporal Basis Functions Methods for Dummies 27 Jan

Temporal Basis Functions Methods for Dummies 27 Jan 2010 Melanie Boly

What’s a basis function then…? l Used to model our f. MRI signal l A basis function is the combining of a number of functions to describe a more complex function. f(t) Fourier analysis h 1(t) h 2(t) The complex wave at the top can be decomposed into the sum of the three simpler waves shown below. f(t)=h 1(t)+h 2(t)+h 3(t)

Temporal Basis Functions for f. MRI l l In f. MRI 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

HRF Function of blood oxygenation, flow, volume (Buxton et al, 1998) Peak (max. oxygenation) 4 -6 s poststimulus; baseline after 20 -30 s Initial undershoot can be observed (Malonek & Grinvald, 1996) Similar across V 1, A 1, S 1… … but differences across: other regions (Schacter et al 1997) individuals (Aguirre et al, 1998) Brief Stimulus Undershoot Initial Undershoot

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

General (convoluted) Linear Model Ex: Auditory words every 20 s HRF ƒi( ) of peristimulus time Sampled every TR = 1. 7 s Design matrix, X …

Limits of HRF l General shape of the BOLD impulse response similar across early sensory regions, such as V 1 and S 1. l Variability across higher cortical regions. l Considerable variability across people. l These types of variability can be accommodated by expanding the HRF in terms of temporal basis functions.

“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.

General (convoluted) Linear Model Ex: Auditory words every 20 s Gamma functions ƒi( ) of peristimulus time Sampled every TR = 1. 7 s Design matrix, X [x(t) ƒ 1( ) | x(t) ƒ 2( ) |. . . ] SPM{F} … 0 time {secs} 30

General (convoluted) Linear Model Ex: Auditory words every 20 s Gamma functions ƒi( ) of peristimulus time Sampled every TR = 1. 7 s Design matrix, X [x(t) ƒ 1( ) | x(t) ƒ 2( ) |. . . ] SPM{F} … 0 time {secs} REVIEW DESIGN 30

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 1 st level) can be improved by derivatives reducing residual error, and can be interpreted as “amplitude” differences, assuming canonical HRF is good fit…

Which temporal basis functions…?

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)

Putting them into your design matrix Left Right 10 Mean 0 -1 0 00

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

Putting them into your design matrix

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