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, sigma2. 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 sigma2 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 beta for general linear modelling (GLM) of a physiological response to experimental stimulation and we demonstrate nominal type I error control in multiple testing of beta, divided by its standard error, in simulated and biological data under the null hypothesis beta = 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.

Adaptive reconfiguration of fractal small-world human brain functional networks

“…Fractal properties, i.e., self-similarity over several scales of measurement, have been described in many types of neurobiological data including electroencephalographic and functional MRI time series (28)(29)(30)(31), structural MRI measurements of the cortical surface and gray-white matter boundary (32,33), and microscopic data on neuronal dendritic and cerebrovascular arborizations (34). It is also notable that small-world networks have been generated computationally by a fractal growth process (35) and that adaptive rewiring of initially random networks by neurogenesis may allow development and maintenance of smallworld connectivity in the brain (36)(37)(38)(39)(40).…”

Altered temporal correlations in parietal alpha and prefrontal theta oscillations in early-stage Alzheimer disease

“…Indeed, there is considerable evidence pointing to deficits in the functional connectivity (7-9, 37-39), which may be related both to structural atrophy (28) and deficits in cholinergic or other neurotransmitter systems (18,36,40,41). Failures in any of the components of a large-scale circuit are expected also to affect the temporal dynamics of local activity.…”

“…2 A and B). The following filter settings were used (cut-off frequencies, filter order): delta (2-3 Hz, 251), theta (4 -5 Hz, 63), alpha (6 -13 Hz, 28), beta (15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)28), and gamma (30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)13). The mean oscillation amplitude was computed as the time-averaged amplitude envelope.…”

Spontaneous brain activity observed with functional magnetic resonance imaging as a potential biomarker in neuropsychiatric disorders

“…From the power spectral factor f -a , the fractal dimension D f (D f = (3-a)/2) and the Hurst exponent (H) (H = 2-D f ) of a given signal can be estimated (Bullmore et al 2004). The Hurst exponent of the resting-state fMRI has been found to be sensitive to an acute pharmacological challenge (Wink et al 2006) and to the pathological changes of AD (Maxim et al 2005). Because spontaneous activity may have too complex a behavior to be adequately described by a single scaling exponent, multifractal formalism, in which the local scaling behavior in the neighborhood of a singularity is characterized by the Hölder exponent, has also been used to measure endogenous spontaneous fluctuations (Wink et al 2008).…”