Fractional Gaussian noise, functional MRI and Alzheimer's disease

Abstract

Fractional Gaussian noise (fGn) provides a parsimonious model for stationary increments of a self-similar process parameterised by the Hurst exponent, H, and variance, σ². Fractional Gaussian noise with H < 0.5 demonstrates negatively autocorrelated or antipersistent behaviour; fGn with H > 0.5 demonstrates 1/f, long memory or persistent behaviour; and the special case of fGn with H = 0.5 corresponds to classical Gaussian white noise. We comparatively evaluate four possible estimators of fGn parameters, one method implemented in the time domain and three in the wavelet domain. We show that a wavelet-based maximum likelihood (ML) estimator yields the most efficient estimates of H and σ² in simulated fGn with 0 < H < 1. Applying this estimator to fMRI data acquired in the “resting” state from healthy young and older volunteers, we show empirically that fGn provides an accommodating model for diverse species of fMRI noise, assuming adequate preprocessing to correct effects of head movement, and that voxels with H > 0.5 tend to be concentrated in cortex whereas voxels with H < 0.5 are more frequently located in ventricles and sulcal CSF. The wavelet-ML estimator can be generalised to estimate the parameter vector β for general linear modelling (GLM) of a physiological response to experimental stimulation and we demonstrate nominal type I error control in multiple testing of β, divided by its standard error, in simulated and biological data under the null hypothesis β = 0. We illustrate these methods principally by showing that there are significant differences between patients with early Alzheimer's disease (AD) and age-matched comparison subjects in the persistence of fGn in the medial and lateral temporal lobes, insula, dorsal cingulate/medial premotor cortex, and left pre- and postcentral gyrus: patients with AD had greater persistence of resting fMRI noise (larger H) in these regions. Comparable abnormalities in the AD patients were also identified by a permutation test of local differences in the first-order autoregression AR(1) coefficient, which was significantly more positive in patients. However, we found that the Hurst exponent provided a more sensitive metric than the AR(1) coefficient to detect these differences, perhaps because neurophysiological changes in early AD are naturally better described in terms of abnormal salience of long memory dynamics than a change in the strength of association between immediately consecutive time points. We conclude that parsimonious mapping of fMRI noise properties in terms of fGn parameters efficiently estimated in the wavelet domain is feasible and can enhance insight into the pathophysiology of Alzheimer's disease.

Introduction

It is well-known that functional magnetic resonance imaging (fMRI) time series typically demonstrate complex and locally variable autocorrelation structure, even when the data have been acquired with the subject “at rest”. There is preliminary evidence that fMRI noise often has long memory in time, or 1/f spectral properties, meaning it is positively autocorrelated and there is disproportionate power in the spectrum at low frequencies (Zarahn et al., 1997); for review, see Bullmore et al. (2004) and references therein. However, there may also be high-frequency events like spikes or transients, or more sustained bursts of negative autocorrelation, in fMRI noise.

Not only is fMRI noise diverse, its presumably multiple sources are incompletely known, and are likely to vary in impact from one data set to the next. Head movement, for example, is an individually variable but common source of long memory noise caused by slow rotation or translation of the subject's head through an imperfectly homogeneous magnetic or RF field during scanning (Bullmore et al., 1999a). Cardiac and/or respiratory cycle-related pulsations may also contribute noise with properties depending in part on the sampling rate of data acquisition (repetition time, TR) and the proportion of cerebrospinal fluid represented in a voxel (Cordes et al., 2001, Purdon and Weisskoff, 1999). There are inevitably also instrumental and thermal sources of noise.

To date, one of the most successful modelling strategies for fMRI noise G = (G₁,G₂,…,G_n), n being the number of time points, has been the adoption of autoregressive, linear time invariant models (Bullmore et al., 1996, Bullmore et al., 2001, Dale, 1999, Locascio et al., 1997, Purdon and Weisskoff, 1999, Worsley et al., 2002) of the form

$$G_t=\sum _{i=1}^p{\eta }_i{G}_{t-i}+{\epsilon }_t,{\epsilon }_t\stackrel{iid}{\sim }N(0,{\sigma }^2)\text{,}$$

where p is the order of the autoregressive AR(p) process and t = 1,2,…,n. However, AR models will require many parameters to account for long-range autocorrelated processes. The variability of autocorrelation between voxels (Bullmore et al., 1996, Worsley et al., 2002) suggests that it might be appropriate to adapt the order of AR process to each individual time series, which can be automated using model selection criteria such as the Bayesian information criterion (Fadili and Bullmore, 2002), but this is not always done in practice.

In this paper, we consider an alternative class of models, called fractional Gaussian noise or fGn, as a new approach to statistical modelling of fMRI noise. More formally, we are interested in the model that fMRI noise is distributed as a fractional Gaussian noise specified completely by only two parameters, its Hurst exponent H and its variance σ², i.e., G ∼ N(0,Σ), with covariance matrix Σ ≔ Σ(H,σ²). The main potential advantage of fGn compared to AR(p) models is its simplicity and parsimony: Only two parameters need to be estimated, with no complications concerning optimal model order specification.

In the rest of this paper, we rehearse some key mathematical properties of fractional Gaussian noise; we comparatively evaluate four possible estimators of fGn parameters; and we apply the optimal, wavelet-based maximum likelihood estimator to analysis of fMRI noise properties in resting data acquired from healthy volunteers and patients with early Alzheimer's disease, motivated by the hypothesis that there might be disease-related changes in the Hurst exponent of fMRI noise (Jeong, 2004).