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

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Temporal Basis Functions Melanie Boly Methods for Dummies 27 Jan 2010

Used to model our fMRI signal Used to model our fMRI signal A basis function is the combining of a number of functions to describe a more complex function. A basis function is the combining of a number of functions to describe a more complex function. 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 fMRI In fMRI we need to describe a function of % signal change over time. 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. There are various different basis sets that we could use to approximate the signal. Finite Impulse Response (FIR) Fourier

HRF Brief Stimulus Undershoot Initial Undershoot 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 fMRI Better though to use functions that make use of our knowledge of the shape of the HRF. 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. 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. However they lack an undershoot. If we add two of them together we get the canonical HRF. If we add two of them together we get the canonical HRF.

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

Fits of a boxcar epoch model with (red) and without (black) convolution by a canonical HRF, together with the data (blue). HRF versus boxcar

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

Canonical HRF (2 gamma functions) plus Multivariate Taylor expansion in: 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. “Informed” Basis Set (Friston et al. 1998)

General (convoluted) Linear Model Ex: Auditory words every 20s SPM{F} 0 time {secs} 30 Sampled every TR = 1.7s Design matrix, X [x(t)  ƒ 1 (  ) | x(t)  ƒ 2 (  ) |...] … Gamma functions ƒ i (  ) of peristimulus time 

General (convoluted) Linear Model Ex: Auditory words every 20s SPM{F} 0 time {secs} 30 Sampled every TR = 1.7s Design matrix, X [x(t)  ƒ 1 (  ) | x(t)  ƒ 2 (  ) |...] … Gamma functions ƒ i (  ) of 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” 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… Comparison of the fitted response

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 matrix Left Right Mean 1 0 0 -1 0 0 0

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

Putting them into your design matrix

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

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