Characterizing the functional MRI response using Tikhonov regularization

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.

[1]  Karl J. Friston,et al.  Analysis of fMRI Time-Series Revisited—Again , 1995, NeuroImage.

[2]  Li Hai Tan,et al.  Correlation between temporal response of fMRI and fast reaction time in a language task. , 2004, Magnetic resonance imaging.

[3]  Juan Manuel Peña,et al.  Totally positive bases for shape preserving curve design and optimality of B-splines , 1994, Comput. Aided Geom. Des..

[4]  S. Sourbron,et al.  Choice of the regularization parameter for perfusion quantification with MRI. , 2004, Physics in medicine and biology.

[5]  Xianhong Xie,et al.  Optimal spline smoothing of fMRI time series by generalized cross-validation , 2003, NeuroImage.

[6]  N. Davies,et al.  Significance levels of the Box-Pierce portmanteau statistic in finite samples , 1977 .

[7]  Henry W. Altland,et al.  Applied Functional Data Analysis , 2003, Technometrics.

[8]  R. Buxton The Elusive Initial Dip , 2001, NeuroImage.

[9]  R. Buxton,et al.  Dynamics of blood flow and oxygenation changes during brain activation: The balloon model , 1998, Magnetic resonance in medicine.

[10]  A Adler,et al.  Objective selection of hyperparameter for EIT , 2006, Physiological measurement.

[11]  G. Wahba Smoothing noisy data with spline functions , 1975 .

[12]  Karl J. Friston,et al.  To Smooth or Not to Smooth? Bias and Efficiency in fMRI Time-Series Analysis , 2000, NeuroImage.

[13]  G. Wahba Spline models for observational data , 1990 .

[14]  Karl J. Friston,et al.  Analysis of fMRI Time-Series Revisited , 1995, NeuroImage.

[15]  Ravi S. Menon,et al.  Mental chronometry using latency-resolved functional MRI. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[16]  G. Box,et al.  On a measure of lack of fit in time series models , 1978 .

[17]  Chung-Ming Chen,et al.  Cross-reference weighted least square estimates for positron emission tomography , 1998, IEEE Transactions on Medical Imaging.

[18]  William D. O'Brien,et al.  A regularized inverse approach to ultrasonic pulse-echo imaging , 2006, IEEE Transactions on Medical Imaging.

[19]  Karl J. Friston,et al.  Human Brain Function , 1997 .

[20]  A. Shmuel,et al.  Imaging brain function in humans at 7 Tesla , 2001, Magnetic resonance in medicine.

[21]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[22]  Ron Borowsky,et al.  The Role of the Left Hemisphere in Motor Control of Touch: A Functional Magnetic Resonance Imaging Analysis , 2002, Brain and Cognition.

[23]  Dae-Shik Kim,et al.  Origin of Negative Blood Oxygenation Level—Dependent fMRI Signals , 2002, Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism.

[24]  R. Turner,et al.  Event-Related fMRI: Characterizing Differential Responses , 1998, NeuroImage.

[25]  Ravi S. Menon,et al.  On the characteristics of functional magnetic resonance imaging of the brain. , 1998, Annual review of biophysics and biomolecular structure.

[26]  E. DeYoe,et al.  Functional magnetic resonance imaging (FMRI) of the human brain , 1994, Journal of Neuroscience Methods.

[27]  Karl J. Friston,et al.  Statistical parametric maps in functional imaging: A general linear approach , 1994 .

[28]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[29]  N. Logothetis,et al.  Negative functional MRI response correlates with decreases in neuronal activity in monkey visual area V1 , 2006, Nature Neuroscience.

[30]  Richard Bowtell,et al.  Detecting activations in event‐related fMRI using analysis of variance , 1999, Magnetic resonance in medicine.

[31]  M. D’Esposito,et al.  Empirical Analyses of BOLD fMRI Statistics , 1997, NeuroImage.

[32]  E. Bullmore,et al.  Statistical methods of estimation and inference for functional MR image analysis , 1996, Magnetic resonance in medicine.

[33]  Mark D'Esposito,et al.  Variation of BOLD hemodynamic responses across subjects and brain regions and their effects on statistical analyses , 2004, NeuroImage.

[34]  Ron Borowsky,et al.  Functional MRI activation maps from empirically defined curve fitting , 2005 .

[35]  B. Chalmond Modeling and inverse problems in image analysis , 2003 .

[36]  Chong Gu Smoothing Spline Anova Models , 2002 .

[37]  Lars Kai Hansen,et al.  Modeling the hemodynamic response in fMRI using smooth FIR filters , 2000, IEEE Transactions on Medical Imaging.