Wavelet statistics of functional MRI data and the general linear model

Wavelets and statistical analysis of functional magnetic resonance images of the human brain

Wavelets provide an orthonormal basis for multiresolution analysis and decorrelation or ‘whitening’ of nonstationary time series and spatial processes. Wavelets are particularly well suited to analysis of biological signals and images, such as human brain imaging data, which often have fractal or scale-invariant properties. We briefly define some key properties of the discrete wavelet transform (DWT) and review its applications to statistical analysis of functional magnetic resonance imaging (fMRI) data. We focus on time series resampling by ‘wavestrapping’ of wavelet coefficients, methods for efficient linear model estimation in the wavelet domain, and wavelet-based methods for multiple hypothesis testing, all of which are somewhat simplified by the decorrelating property of the DWT.

Model-based clustering of meta-analytic functional imaging data

We present a method for the analysis of meta-analytic functional imaging data. It is based on Activation Likelihood Estimation (ALE) and subsequent model-based clustering using Gaussian mixture models, expectation-maximization (EM) for model fitting, and the Bayesian Information Criterion (BIC) for model selection. Our method facilitates the clustering of activation maxima from previously performed imaging experiments in a hierarchical fashion. Regions with a high concentration of activation coordinates are first identified using ALE. Activation coordinates within these regions are then subjected to model-based clustering for a more detailed cluster analysis. We demonstrate the usefulness of the method in a meta-analysis of 26 fMRI studies investigating the well-known Stroop paradigm.

WSPM: Wavelet-based statistical parametric mapping

Recently, we have introduced an integrated framework that combines wavelet-based processing with statistical testing in the spatial domain. In this paper, we propose two important enhancements of the framework. First, we revisit the underlying paradigm; i.e., that the effect of the wavelet processing can be considered as an adaptive denoising step to "improve" the parameter map, followed by a statistical detection procedure that takes into account the non-linear processing of the data. With an appropriate modification of the framework, we show that it is possible to reduce the spatial bias of the method with respect to the best linear estimate, providing conservative results that are closer to the original data. Second, we propose an extension of our earlier technique that compensates for the lack of shift-invariance of the wavelet transform. We demonstrate experimentally that both enhancements have a positive effect on performance. In particular, we present a reproducibility study for multi-session data that compares WSPM against SPM with different amounts of smoothing. The full approach is available as a toolbox, named WSPM, for the SPM2 software; it takes advantage of multiple options and features of SPM such as the general linear model.

HBM functional imaging analysis contest data analysis in wavelet space

An analysis of the Functional Imaging Analysis Contest (FIAC) data is presented using spatial wavelet processing. This technique allows the image to be filtered adaptively according to the data itself, rather than relying on a predetermined filter. This adaptive filtering leads to better estimation of the parameters and contrasts in terms of mean squared error. It will be shown that by introducing a slight bias into the estimation, a large reduction in the variance can be achieved, leading to better overall mean squared error estimates. As no single filter needs to be preselected, results containing many scales of information can be found. In the FIAC data, it is shown that both small-scale and large-scale (smoother, more dispersed) effects occur. The combination of small- and large-scale effects detected in the FIAC data would be easy to miss using conventional single filter analysis.

The parcellation of cortical areas using replicator dynamics in fMRI

In this paper, we show that replicator dynamics can be used as an exploratory analysis tool to detect subregions of cortical areas on the basis of the similarity between fMRI time series. As similarity measure, we propose to use canonical correlation, a multivariate extension to the typically employed Pearson's correlation coefficient. We applied the replicator process to data obtained from two different experimental paradigms in the search for subregions within the left lateral frontal cortex (LFC). In both cases, the replicator process resulted in a parcellation that corresponds to a recently suggested subdivision of the LFC in anterior-posterior direction. Most notably, these results were very consistent when compared across different measurements of a single subject and across a group of subjects.

Emotional task-dependent low-frequency fluctuations and methylphenidate: Wavelet scaling analysis of 1/f-type fluctuations in fMRI of the cerebellar vermis

Ion channel currents, neural firing patterns, and brain BOLD signals display 1/f-type fluctuations or fractal properties in time. By design, fMRI methods attempt to minimize the contribution of variance from low-frequency physiological 1/f-noise. New fMRI methods are described to visualize and measure 1/f-type BOLD fluctuations in volunteers recalling affectively neutral or emotional memories or meditating (i.e., attending to breathing) then retrospectively rating emotional content. A wavelet scaling exponent (alpha) was used to characterize signals from 0.015625 to 0.5Hz in cerebellar lobules VIII to X of the vermis (posterior inferior vermis; PIV), a region coordinating balance, eye tracking, locomotion, and vascular tone, and a possible site of pathology in attention deficit hyperactivity disorder (ADHD).

RESULTS

Changes in alpha and emotional measures were correlated in PIV voxels (r = 0.622, d.f .= 14, P < 0.0005), but not other regions examined. In contrast, conventional means and standard deviations of PIV voxels were unchanged. Methylphenidate, shown to decrease slow oscillations in rodent basal ganglia [Ruskin DN, Bergstrom DA, Shenker A, Freeman LE, Baek D, Walters JR. Drugs used in the treatment of attention-deficit/hyperactivity disorder affect postsynaptic firing rate and oscillation without preferential dopamine autoreceptor action. Biol Psychiatry 2001;49:340-50.], abolished task-dependent alpha changes in the PIV of an adult with ADHD. Wavelet analysis of long BOLD time series appears well suited to fractal physiology and studies of pharmacologically modulated cerebellar-thalamic-cortical function in ADHD or other psychiatric disorders.

Quantile estimation to derive optimized test thresholds for random field statistics

We present a numerical method to estimate the true threshold values in random fields needed to determine the significance of apparent signals observed in noisy images. To accomplish this, a quantile estimation algorithm is applied to derive the threshold with a predefined confidence interval from a large number of simulated random fields. Also, a computationally efficient method for generating a random field simulation is presented using resampling techniques. Applying these techniques, thresholds have been determined for a large variety of parameter settings (smoothness, voxel size, brain shape, type of statistics). By means of interpolation techniques, thresholds for additional arbitrary settings can be quickly derived without the need to run individual simulations. Compared to the parametric approach of Worsley et al. (1996) (Worsley, K.J., Marrett, S., Neelin P., Vandal, A.C., Friston, K.J., Evans, A.C., 1996. A unified statistical approach for determining significant signals in images of cerebral activation. Hum. Brain Mapp. 4, 58-73) and Friston et al. (1991) (Friston, K.J., Frith, C.D., Liddle, P.F., Frackowiak, R.S. 1991. Comparing functional (PET) images: the assessment of significant change. J. Cereb. Blood Flow Metab. 11(4), 690-699), and to the Bonferroni approach, these optimized thresholds lead to higher levels of significance (i.e., lower p values) with a specific amount of activation especially with fields of moderate smoothness (i.e., with a relative full width half maximum between 2 and 6). Alternatively, the threshold for a specified level of significance can be lowered. This improved statistical sensitivity is illustrated by the analysis of an actual event related functional magnetic resonance data set, and its limitations are tested by determining the false positive rate with experimental MR noise data. The grid of estimated threshold values as well as the interpolation algorithm to derive thresholds for arbitrary parameter settings are made available over the internet (http://neuro2.med.uni-magdeburg.de/quantile_estimation).

Wavelets and functional magnetic resonance imaging of the human brain

The discrete wavelet transform (DWT) is widely used for multiresolution analysis and decorrelation or "whitening" of nonstationary time series and spatial processes. Wavelets are naturally appropriate for analysis of biological data, such as functional magnetic resonance images of the human brain, which often demonstrate scale invariant or fractal properties. We provide a brief formal introduction to key properties of the DWT and review the growing literature on its application to fMRI. We focus on three applications in particular: (i) wavelet coefficient resampling or "wavestrapping" of 1-D time series, 2- to 3-D spatial maps and 4-D spatiotemporal processes; (ii) wavelet-based estimators for signal and noise parameters of time series regression models assuming the errors are fractional Gaussian noise (fGn); and (iii) wavelet shrinkage in frequentist and Bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps. We conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fMRI data.

Wavelet variance components in image space for spatiotemporal neuroimaging data

Neuroimaging studies place great emphasis on not only the estimation but also the standard error estimates of underlying parameters derived from a temporal model. This allows inferences to be made about the signal estimates and resulting conclusions to be drawn about the underlying data. It can often be advantageous to interrogate temporal models after spatial transformation of the data into the wavelet domain. Wavelet bases provide a multiresolution decomposition of the spatial data dimension and an ensuing reduction in spatial correlation. However, widespread acceptance of these wavelet techniques has been hampered by the limited ability to reconstruct both parametric and error estimates into the image domain after analysis of temporal models in the wavelet domain. This paper introduces a derivation and a fast implementation of a method for the calculation of the variance of the parametric images obtained from wavelet filters. The technique is proposed for a class of estimators that have been shown to be useful in neuroimaging studies. The techniques are demonstrated for both functional magnetic resonance imaging (fMRI) and positron emission tomography (PET) data sets.

Integrated wavelet processing and spatial statistical testing of fMRI data

We introduce an integrated framework for detecting brain activity from fMRI data, which is based on a spatial discrete wavelet transform. Unlike the standard wavelet-based approach for fMRI analysis, we apply the suitable statistical test procedure in the spatial domain. For a desired significance level, this scheme has one remaining degree of freedom, characterizing the wavelet processing, which is optimized according to the principle of minimal approximation error. This allows us to determine the threshold values in a way that does not depend on data. While developing our framework, we make only conservative assumptions. Consequently, the detection of activation is based on strong evidence. We have implemented this framework as a toolbox (WSPM) for the SPM2 software, taking advantage of multiple options and functions of SPM such as the setup of the linear model and the use of the hemodynamic response function. We show by experimental results that our method is able to detect activation patterns; the results are comparable to those obtained by SPM even though statistical assumptions are more conservative.

Wavelet-based multifractal analysis of fMRI time series

Functional magnetic resonance imaging (fMRI) time series are investigated with a multifractal method based on the Wavelet Modulus Maxima (WTMM) method to extract local singularity ("fractal") exponents. The spectrum of singularity exponents of each fMRI time series is quantified by spectral characteristics including its maximum and the corresponding dimension. We found that the range of Hölder exponents in voxels with activation is close to 1, whereas exponents are close to 0.5 in white matter voxels without activation. The maximum dimension decreases going from white matter to gray matter, and is lower still for activated time series. The full-width-at-half-maximum of the spectra is higher in activated areas. The proposed method becomes particularly effective when combining these spectral characteristics into a single parameter. Using these multifractal parameters, it is possible to identify activated areas in the human brain in both hybrid and in vivo fMRI data sets without knowledge of the stimulation paradigm applied.