Detecting fMRI activation allowing for unknown latency of the hemodynamic response
Jonathan E. Taylor1, Keith J. Worsley2
1Department of Statistics, Stanford University
2Department of Mathematics and Statistics, McGill University

Summary

Several authors have suggested allowing for unknown latency of the hemodynamic response by incorporation of hemodynamic derivative terms into the linear model for the statistical analysis of fMRI data [1,2]. We use random field theory to provide a P-value for local maxima of two test statistics that have been recently proposed for detecting activation based on this analysis [3,4].

Methods

Let Y(t) be the pre-whitened fMRI data at time t, with pre-whitened stimulus s(t), pre-whitened drift and other covariates z(t), and hrf h(t) with unknown delay or latency shift δ, modeled as

Y(t) = (s*h)(t-δ)β+z(t)γ+ε(t) ~ x1(t1+x2 (t2+z(t)γ+ε(t)

where x1=(s*h), x2=-(s*h'), * is convolution, ' is derivative, β1=β, β2=βδ and ε(t)~N(0,σ2) independently [1,2].

To detect activation allowing for unknown delay, the T statistic for β1 is the best if there is no delay shift, but if there is delay shift then we will lose sensitivity. The F statistic for both β1 and β2 is sensitive to any linear combination of the hrf and its derivative, but it will waste sensitivity on unrealistic linear combinations such as negative BOLD responses to the stimulus.

To overcome this, two alternatives have been proposed: a "one-sided" F statistic that equals the usual F statistic but which takes the value zero if the estimated β1 is negative [3], and the cone T statistic for testing the regressor x1(t) + x2(t)δ for fixed δ, maximized over all values of |δ|<Δ [4]. The approximate P-values of local maxima of these two test statitics, based on random field thoery, are given below [5].

Results

The resulting sensitivities, with Δ=2s, plotted as a function of the shift of the hrf δ, are shown in Figure 1.

For shifts less than ~1s, the simple T statistic is the most sensitive, but for larger shifts the cone T statistic is the most sensitive. For very large shifts the one-sided F statistic is the most sensitive, but the simple F statistic is always the least sensitive. The same holds for the block design but the differences are less pronounced. Our overall recommendation is to use the simple T statistic if the shift in the hrf is less than 1s, and the cone T statistic if not.

Application

We applied the above test statistics to detecting a hot stimulus in a pain perception experiment with blocks of 9s hot and 9s warm stimuli, interspersed with 9s rest [6]. For a whole brain search of 1184cm3 with an estimated 8.78mm FWHM average smoothing, the P=0.05 thresholds for the test statistics are given in Table 1. For a Δ=2s hrf shift range, the cone angle Θ21 was 60.9 +/- 1.7o. Figure 3 shows the four thresholded images (labels in Table 1). The volume of detected activation remains roughly the same (see Table 1) but the cone T statistic detects the most activation (left primary somatosensory area and left and right thalamus).

Table 1. Thresholds and activated volumes at P=0.05.
Statistic Threshold Activated volume (cm3)
(a) T statistic for hrf 5.15 4.0
(b) F statistic for hrf and derivative 5.80 2.9
(c) One-sided F statistic 5.63 3.8
(d) Cone T statistic 5.46 4.3
Threshold is on the same scale of minimum distance of the rejection region from the origin. Activated volume is the volume of the statistics above the threshold.

References

[1] Henson et al. (2002), Neuroimage, 15:83-97.

[2] Liao et al. (2002), Neuroimage, 16:593-606.

[3] Calhoun et al. (2004), Neuroimage, 22:252-257.

[4] Friman et al. (2003), Neuroimage, 19:837-845.

[5] Worsley and Taylor (2006), Neuroimage, in press.

[6] Worsley et al. (2002), Neuroimage, 15:1-15.