go to site The mono- and multifractal properties we are investigating here are essentially describing different properties of the multifractal spectrum. We further observed that the Chhabra-Jensen method is the most reliable out of the three multifractal estimation methods. As was pointed out in the original publication Chhabra and Jensen, , this is most likely due to the fact that the Chhabra-Jensen method avoids a Legendre transform that the other methods require.
The Legendre transformation requires smoothing of the D q curve and can lead to errors. For further advantages of the Chhabra-Jensen method, the reader is referred to the original publication Chhabra and Jensen, A recent development, FMF method Mukli et al. Finally, our analysis highlighted the importance of choosing an adequate epoch size given a sampling frequency, in order to study events such as epileptic seizures. However, our study was based on the analysis of ictal vs.
Future work should take into account that multifractal properties may be continuously changing over time a striking example is shown in Appendix Figure E7 in Supplementary Material , and an explicitly time based approach may be needed. Along similar lines, our finding of a optimal time scale may be due to the non-stationary nature of the multifractal properties. Further theoretical work may have to develop a temporally resolved multifractal estimator, in order to fully understand this aspect.
It has been further suggested that more than one scaling exponent would be necessary to properly characterize the brain's critical dynamics Suckling et al. Hence, it has been proposed that using additional, higher-order statistical moments can better characterize such data Fraiman and Chialvo, In this work, we contribute a complementary observation: while monofractal measures of EEG appeared to essentially follow the slow changes of signal variance, multifractal characterization is capable of revealing new information.
In terms of generative processes that can produce monofractal properties, it has been suggested that a property called Self-Organized Criticality SOC Bak et al. SOC describes the capacity of a system to evolve naturally into a critical state a state in which a minimum perturbation could lead to events of all sizes. Such phenomena display power-law distributions and fractal properties as signatures Bak and Paczuski, SOC behavior has been linked to physiological control mechanisms, such as in human heart rate variability Goldberger et al.
The analysis and understanding of the non-classical SOC is, however, still under development.
In this context, our multifractal spectral analyses of human EEG data suggest that cerebral phenomena should not be modeled by a single avalanche model classical SOC , in agreement with findings in a previous study Fraiman and Chialvo, Moreover, it is hypothesized that brain dynamics are non-ergodic Bianco et al. Thus, multifractal analyses could provide a new paradigm for studying brain function and structure, as previously suggested in other studies of normal Suckling et al.
Furthermore, generative processes displaying multifractal properties could help understanding the observed multifractal changes on a mechanistic level.
The University of Melbourne Library. Bengtsson, H. Burrough P. Generic aspects of complexity in brain imaging data and other biological systems. Goodchild M. The supports of all four patterns are fractals.
We want to emphasize that the conclusions from our work are drawn on the basis that slow changes in signal fractal features can be captured by using an epoch-wise feature extraction procedure. It is also from a feature redundancy perspective that we argue for the need of multifractal approaches over monofractal measures. We do not dispute the usefulness of monofractal measures in other general applications. In our work, we essentially performed a feature selection procedure using correlation and mutual information Guyon and Elisseeff, We evaluated how different signal feature compare on an epoch-wise basis.
Feature selection is crucial to obtain faster and cost effective models, and avoids overfitting of the available data. It might also help achieving a deeper insight into the nature of the studied phenomena Blum and Langley, ; Liu et al. A fundamental observation in our work is that an optimal time scales may exist for specific physiological processes such as epileptic seizures in terms of their multifractal dynamics Figure 7 and Appendix B in Supplementary Material.
This is further supported by similar findings in monofractal analysis Eke et al. The implications of this observation are that certain scaling exponents will only exist in specific time scales and the diversity of scaling exponents will depend on the duration of the epoch. If this is indeed the case, a temporally resolved not epoch-based multifractal method should be developed in future to adequately characterize brain dynamics.
Furthermore, the slow temporal changes in multifractal dynamics need to be characterized in a systematic way. Using epileptic seizures as an example, Appendix Figure E7 in Supplementary Material shows that dramatic changes in multifractal properties can sometimes be seen before an epileptic seizure. This observation requires further investigation to address questions such as: are all epileptic seizures characterized by pre-ictal changes in multifractal properties?
Do other physiological processes, such as sleep, influence this finding? To answer these questions, we will most likely also need well-characterized experimental conditions, where seizures can be triggered in a controlled manner. Finally, it is well-recognized that epileptic seizures are spatio-temporal processes see e. Data-driven unsupervised approaches, such as dimensionality reduction, may help summarize spatial aspects. Additionally, the challenge will be to develop a spatio-temporal multifractal analysis approach that can also deal with the challenges of low spatial sampling resolution in EEG recordings.
Our work has highlighted several challenges that need to be considered when analysing multifractal properties of EEG signals; namely choice of the appropriate estimation method, estimation parameters, and the influence of the time series variance on signal features. We have suggested some solutions to these problems, such as the used of the Chhabra-Jensen approach combined with an epoch-wise standardization approach, which has shown potential capabilities as a signal feature for machine learning applications.
We have also highlighted possible process-specific challenges. In terms of epileptic seizures, future work is required to analyse a larger number of patients in order to draw firmer conclusions on the potential clinical relevance of multifractal analyses. Furthermore, the study of mechanistic generative models of EEG may shed light on why those multifractal changes occur. For example, a generative process of potential interest could feature a modified version of Bak—Tang—Wiesenfeld model Bak et al. In this paper, we have analyzed the monofractal and multifractal properties of human EEG recordings.
We have shown that monofractal estimates are influenced by the standard deviation of the time series, thus not capturing features beyond signal variance. For multifractal estimation, we have shown that the Chhabra-Jensen approach is the most stable, and we have developed a method of signal pre-processing to remove the influence caused by the variance of the signal. Using the suggested approach, the multifractal estimates do not correlate with traditional EEG measures, thus yielding additional information about the signal and being a relevant signal feature.
Finally, our results also indicate a preferential time scale to identify differences in multifractal properties between ictal and interictal state recordings in patients with epilepsy. LF: data curation, funding acquisition, investigation, and visualization. LF and YW: formal analysis and validation. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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