Characterizing the functional MRI response using Tikhonov regularization

Abstract

The problem of evaluating an averaged functional magnetic resonance imaging (fMRI) response for repeated block design experiments was considered within a semiparametric regression model with autocorrelated residuals. We applied functional data analysis (FDA) techniques that use a least-squares fitting of B-spline expansions with Tikhonov regularization. To deal with the noise autocorrelation, we proposed a regularization parameter selection method based on the idea of combining temporal smoothing with residual whitening. A criterion based on a generalized chi(2)-test of the residuals for white noise was compared with a generalized cross-validation scheme. We evaluated and compared the performance of the two criteria, based on their effect on the quality of the fMRI response. We found that the regularization parameter can be tuned to improve the noise autocorrelation structure, but the whitening criterion provides too much smoothing when compared with the cross-validation criterion. The ultimate goal of the proposed smoothing techniques is to facilitate the extraction of temporal features in the hemodynamic response for further analysis. In particular, these FDA methods allow us to compute derivatives and integrals of the fMRI signal so that fMRI data may be correlated with behavioral and physiological models. For example, positive and negative hemodynamic responses may be easily and robustly identified on the basis of the first derivative at an early time point in the response. Ultimately, these methods allow us to verify previously reported correlations between the hemodynamic response and the behavioral measures of accuracy and reaction time, showing the potential to recover new information from fMRI data.

Changes in Functional Magnetic Resonance Imaging Cortical Activation with Cross Education to an Immobilized Limb

“…After the images were converted to Talairach coordinates, the region of interest (ROI) was mapped for each subject individually before the signal parameters were extracted. Given the noisy nature of the fMRI response, the BOLD hemodynamic response ftinction of the activated voxels in an ROI is first smoothed using a generalization of the BOLDfold approach (43). A time course is extracted for each participant that represents an average for the design blocks, which was then averaged across the participants in each group (e.g., CAST-TRAIN vs CAST) to provide information about the characteristics of the hemodynamic response function over time.…”

The relationship between naming reaction time and functional MRI parameters in Broca's area

“…In order to obtain a smoothed fMRI signal for a block, we incorporated spline interpolation in our analysis method. With the spline functions incorporated into a GLM approach, we effectively computed the hemodynamic response averaged over all the blocks and used that as a reference function for computing the activation maps [33]. The method is a generalization of the BOLDfold approach, which is suitable for repeated block design experiments in which the comparison function, normally used in a regression style GLM analysis, is defined empirically through periodic data folding.…”

“…For optimal sensitivity, the experiment used a blocked design, as described above, and was analyzed using a previously developed and validated regularized BOLDfold approach [15,18,23,27,29,[31][32][33]. The BOLDfold method of analysis requires that sufficient time elapse between task conditions for the hemodynamic response function (HRF or BOLD function) to fully return to baseline levels.…”

Nonparametric inference of the hemodynamic response using multi-subject fMRI data

“…Bai et al (2009) and Wang et al (2011) constructed nonparametric estimates of the HRF in the frequency domain. Nonparametric methods in the time domain mainly fall into two types: representing the HRF with a linear combination of functional bases (Aguirre et al, 1998;Vakorin et al, 2007;Woolrich et al, 2004;Zarahn, 2002), or treating the HRF at every unit time point as a free parameter (Dale, 1999;Lange et al, 1999). In this paper we adopt the latter approach in the time domain to develop nonparametric estimation and inferences for HRFs.…”

“…Another example is given in Marrelec et al (2001Marrelec et al ( , 2003, where the HRF is represented by orthogonal functional bases and a smoothness constraint is imposed through regularizing the norm of its second order derivative. Similarly, representing the HRF by spline bases, Vakorin et al (2007) and Zhang et al (2007) used Tikhonov regularization (Tikhonov and Arsenin, 1977). The estimator proposed by Casanova et al (2008Casanova et al ( , 2009) combines Tikhonov regularization and generalized cross validation (Wahba, 1990) (referred to Tik-GCV hereafter), greatly reducing the computational burden involved in parameter selection.…”