Fractal Time Series: Background, Estimation Methods, and Performances.

Over the past 40 years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.

Fractal Dimension as a discriminative feature for high accuracy classification in motor imagery EEG-based brain-computer interface.

BACKGROUND AND OBJECTIVE: The brain-computer interface (BCI) technology acquires human brain electrical signals, which can be effectively and successfully used to control external devices, potentially supporting subjects suffering from motor impairments in the interaction with the environment. To this aim, BCI systems must correctly decode and interpret neurophysiological signals reflecting the intention of the subjects to move. Therefore, the accurate classification of single events in motor tasks represents a fundamental challenge in ensuring efficient communication and control between users and BCIs. Movement-associated changes in electroencephalographic (EEG) sensorimotor rhythms, such as event-related desynchronization (ERD), are well-known features of discriminating motor tasks. Fractal dimension (FD) can be used to evaluate the complexity and self-similarity in brain signals, potentially providing complementary information to frequency-based signal features. METHODS: In the present work, we introduce FD as a novel feature for subject-independent event classification, and we test several machine learning (ML) models in behavioural tasks of motor imagery (MI) and motor execution (ME). RESULTS: Our results show that FD improves the classification accuracy of ML compared to ERD. Furthermore, unilateral hand movements have higher classification accuracy than bilateral movements in both MI and ME tasks. CONCLUSIONS: These results provide further insights into subject-independent event classification in BCI systems and demonstrate the potential of FD as a discriminative feature for EEG signals.

Brain Dynamics Complexity as a Signature of Cognitive Decline in Parkinson's Disease.

BACKGROUND: Higuchi's fractal dimension (FD) captures brain dynamics complexity and may be a promising method to analyze resting-state functional magnetic resonance imaging (fMRI) data and detect the neuronal interaction complexity underlying Parkinson's disease (PD) cognitive decline. OBJECTIVES: The aim was to compare FD with a more established index of spontaneous neural activity, the fractional amplitude of low-frequency fluctuations (fALFF), and identify through machine learning (ML) models which method could best distinguish across PD-cognitive states, ranging from normal cognition (PD-NC), mild cognitive impairment (PD-MCI) to dementia (PDD). Finally, the aim was to explore correlations between fALFF and FD with clinical and cognitive PD features. METHODS: Among 118 PD patients age-, sex-, and education matched with 35 healthy controls, 52 were classified with PD-NC, 46 with PD-MCI, and 20 with PDD based on an extensive cognitive and clinical evaluation. fALFF and FD metrics were computed on rs-fMRI data and used to train ML models. RESULTS: FD outperformed fALFF metrics in differentiating between PD-cognitive states, reaching an overall accuracy of 78% (vs. 62%). PD showed increased neuronal dynamics complexity within the sensorimotor network, central executive network (CEN), and default mode network (DMN), paralleled by a reduction in spontaneous neuronal activity within the CEN and DMN, whose increased complexity was strongly linked to the presence of dementia. Further, we found that some DMN critical hubs correlated with worse cognitive performance and disease severity. CONCLUSIONS: Our study indicates that PD-cognitive decline is characterized by an altered spontaneous neuronal activity and increased temporal complexity, involving the CEN and DMN, possibly reflecting an increased segregation of these networks. Therefore, we propose FD as a prognostic biomarker of PD-cognitive decline. © 2023 The Authors. Movement Disorders published by Wiley Periodicals LLC on behalf of International Parkinson and Movement Disorder Society.