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Edited by: Paul Bogdan, University of Southern California, United States

Reviewed by: Andras Eke, Semmelweis University, Hungary; Yuankun Xue, University of Southern California, United States; Damian Kelty-Stephen, Grinnell College, United States

This article was submitted to Fractal Physiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The quantification of brain dynamics is essential to its understanding. However, the brain is a system operating on multiple time scales, and characterization of dynamics across time scales remains a challenge. One framework to study such dynamics is that of fractal geometry; and currently there exist several methods for the study of brain dynamics using fractal geometry. We aim to highlight some of the practical challenges of applying fractal geometry to brain dynamics—and as a putative feature for machine learning applications, and propose solutions to enable its wider use in neuroscience. Using intracranially recorded electroencephalogram (EEG) and simulated data, we compared monofractal and multifractal methods with regards to their sensitivity to signal variance. We found that both monofractal and multifractal properties correlate closely with signal variance, thus not being a useful feature of the signal. However, after applying an epoch-wise standardization procedure to the signal, we found that multifractal measures could offer non-redundant information compared to signal variance, power (in different frequency bands) and other established EEG signal measures. We also compared different multifractal estimation methods to each other in terms of reliability, and we found that the Chhabra-Jensen algorithm performed best. Finally, we investigated the impact of sampling frequency and epoch length on the estimation of multifractal properties. Using epileptic seizures as an example event in the EEG, we show that there may be an optimal time scale (i.e., combination of sampling frequency and epoch length) for detecting temporal changes in multifractal properties around seizures. The practical issues we highlighted and our suggested solutions should help in developing robust methods for the application of fractal geometry in EEG signals. Our analyses and observations also aid the theoretical understanding of the multifractal properties of the brain and might provide grounds for new discoveries in the study of brain signals. These could be crucial for the understanding of neurological function and for the developments of new treatments.

Brain dynamics are non-linear and are often considered as one of the most complex natural phenomena, involving several different and interacting temporal scales. For example, fast electric activity, slower chemical reactions, and even slower diffusive processes have been observed in the brain. Interestingly, brain dynamics have also been characterized as “scale-free” (Stam and de Bruin,

Fractals have two specific properties: they consist of parts that are similar to the whole—termed self-similarity, and they have a fractional Hausdorff-Besicovitch dimension, also called fractal dimension (FD) (Mandelbrot,

Objects adequately characterized by a single fractal dimension are referred to as monofractals. However, the fractal formalism has to be extended to capture certain phenomena that cannot be described by a single fractal dimension; these are called multifractals (Stanley et al.,

There is also considerable evidence that brain dynamics are multifractal (Suckling et al.,

To measure the multifractal spectrum in brain dynamics, Multifractal Detrended Fluctuation Analysis (MF-DFA) (Kantelhardt et al.,

The biggest gap in the literature so far, however, is how multifractal properties relate to existing time series signal measures of brain dynamics (e.g., variance of the signal, band power, etc.). A major concern is that complex methods of analysis may not offer a significant advance over simpler, already established methods—this is crucial to a putative feature in machine learning applications, e.g., seizure prediction or detection (Mormann et al.,

To summarize, there is a knowledge gap in three critical areas: (1) which (multi)fractal characterization methodology is best suited for brain signals? (2) what are the optimal estimation parameters (e.g., in terms of recording epoch length) of potentially time varying multifractal properties? (3) what is the relationship between (multi)fractal properties and more traditional and established time series signal measures? To address these questions, we chose to analyse intracranially recorded human electroencephalography (icEEG) data, due to its high temporal resolution and high signal to noise ratio.

To address the questions above, we will first outline four experiments. We will then provide details on monofractal and and multifractal estimation methods, and also show how time series data with known mono- and multifractal properties can be generated to test the performance of the estimation methods. To test the multifractal measures on real-life brain signals, we then applied our analysis on human intracranial EEG. Thus, finally, we will summarize the EEG datasets used in this work.

The original scripts used in this work are available in

Estimation of monofractal properties has been applied to EEG signals in the past with varying and often contrasting results (Esteller et al.,

The fBm was simulated with a Hurst exponent

In order to evaluate the stability of multifractal properties in time, we generated a time series exhibiting stable multifractal properties over time using the p-Model. The time series was then evaluated using an epoch-based approach with the three estimators: MF-DFA, MF-DMA, and Chhabra-Jensen. The stability of the estimator can then simply be assessed as the temporal variability of its output.

To assess whether the chosen multifractal metrics contribute any non-redundant features about the signal in addition to more established signal metrics, we analyzed human EEG signals recorded intracranially. Again, we used an epoch-based approach, and we compared the multifractal metrics to a number of conventional signal metrics (mean, standard deviation, line length, bandpower) on each epoch. The similarity between signal features was evaluated using Pearson correlation and Mutual Information (Guyon and Elisseeff,

Finally, we also evaluated the impact of the multifractal estimation parameters in the characterization of a seizure. We used intracranial EEG signals recorded from four patients undergoing pre-surgical planning, the signals were originally sampled at 5,000 Hz. For this analysis, down-sampled versions were evaluated with epochs of different sizes. To assess the effect of sampling frequency, we down-sampled the signal to 4,000, 3,000, 2,500, 2,000, 1,000, 800, 750, 600, 500, 400, 300, and 250 Hz. For each sampling frequency, we evaluated different epoch sizes (1,024, 2,048, 4,096, 8,192, and 16,384 points).

We defined a difference in multifractal spectrum width (Δα^{†}) during the seizure compared to the background as the effect size (Cohen's D) between the ictal and interictal periods:

where < Δα^{†} > represents the mean and

To estimate the monofractal properties from a time series, we used two established estimation approaches: Higuchi method (Higuchi,

Mandelbrot (

However, we also note that more generally, the fractal dimension

The Higuchi method measures the fractal dimension FD of a time series. It consists of constructing series with elements of an original time series and measuring their lengths (Higuchi,

If the average curve length < _{m}(_{m} over _{Hig}.

The routine used in the estimation of Higuchi fractal dimension FD is available at

The Detrended Fluctuation Analysis (DFA) method is an alternative method (Peng et al.,

The method consists of the following steps: Initially the time series with

Where _{l} windows of length

The local trend (_{l}(_{l} represents the total number of windows. In the following step, Equation (6) is obtained for several window lengths (

The code used here is available in the Physionet repository (

In this section, we describe three multifractal spectrum estimators: Multifractal Detrended Moving Average (Gu and Zhou,

Multifractal properties are represented as spectra (Figure

Essentially, _{F}) of the subset (_{s}) of the time series _{s}) that has a the Hölder exponent α (van den Berg,

Multifractal singularity spectrum with a characteristic parabolic shape. The spectrum width (Δα) and height (Δ

To characterize the function, or singularity spectrum

Multifractal Detrended Moving Average (MF-DMA) is one of the most commonly used methods for the estimation of multifractal measures. The method of calculation consists of the following steps (Gu and Zhou,

We then calculate the moving average over time windows of length

A detrended version of the signal is obtained by the subtraction:

The resulting series is then divided in _{l} disjoint sets of points of size

A generalized

and

It is possible to find a power-law relationship between ^{q}(

The multifractal “mass exponent” (Biswas and Cresswell,

where _{f} is the fractal dimension of the support measure. For a single-channel time series, _{f} = 1. The spectrum,

It is important to note that the Legendre transform is known to cause problems in multifractal spectra derivations if some heterogeneities are present in the signal, as has been reported elsewhere (Chhabra and Jensen,

The Multifractal Detrended Fluctuation Analysis (MF-DFA) method is essentially a generalization of the DFA approach (Kantelhardt et al.,

It is then divided into

where _{ν} represents the fitting in the epoch ν obtained via linear regression. The overall q-th order fluctuation functions can be obtained as:

A log-log plot of _{q}(

Multifractal spectra can be obtained in a more direct way, without the need for the Legendre transform using the Chhabra-Jensen (CJ) method (Chhabra and Jensen,

First, we define:

where _{i}(

and

A numerical approximation to the equations above is provided by the measures

α and

The algorithmic summary of the Chhabra-Jensen method consists of the following steps:

The algorithm has as input the time series, a range of

The time series is divided into non-overlapping epochs of length

The measures

α and

A rejection criterion is also used, where all ^{2} < 0.9 in the linear regression are not considered.

The code used in this study to calculate the multifractal spectrum is available at:

To fully test methods of estimating the monofractal dimension from time series, we computationally produced time series that are known to be fractal (used for Experiment 1). We generated fractional Brownian motion (fBm) (Mandelbrot and Van Ness,

Impact of the signal standard deviation on monofractal scaling exponent estimation.

Note that there are alternative methods to generate monofractal time series (Davies and Harte,

Similarly to the fBm, we also used a computational procedure to generate time series that are known to be multifractal (for Experiment 2) based on the _{1} as its only input and is often mentioned in literature (Meneveau and Sreenivasan, _{L} = _{1}, it is possible to establish a second fraction in which a second parameter will be given by _{2} = 1−_{1}. The heights of each interval will thus be given by _{1}ϵ_{L}, and _{2}ϵ_{L}, respectively. This procedure is repeated for each remaining segment, selecting left or right for _{1} randomly (Meneveau and Sreenivasan,

We employed the

Intracranial EEG data segments extracted from recordings in patients undergoing evaluation for epilepsy surgery were used for Experiments 3 and 4. In order to evaluate the effect of EEG signal variance change on multifractal properties (Experiment 3), we specifically looked for one recording, where the signal variance changes dramatically over time. One such recording was found in one patient (male, 28 years old, temporal lobe epilepsy, recorded at the National Hospital for Neurology and Neurosurgery (NHNN) (UCLH NHS Foundation Trust, Queen Square, London, UK), patient ID: “NHNN1”) near one seizure event. We used a 60-min recording segment around the epileptic seizure for our analysis. The seizure onset and offset were marked by expert clinicians, independent of this research project. Note that we used this segment specifically due to the dramatic change in signal variance, which actually occurred before the seizure and evolves over about 15 min. We do not make conclusions about the seizure event itself at this stage, but rather use this recording as an example to illustrate a technical point about multifractal property estimation from EEG.

To analyse the possible changes in multifractal properties during seizures (Experiment 4), we used a different dataset: Intracranial EEG from four subjects were retrieved from the ieeg.org repository (

The anonymized data analyzed in this study were recorded in patients undergoing evaluation for epilepsy surgery. iEEG.org portal provided EEG data and ethical approval for analyzing the data was provided by Mayo Clinic IRB (Brinkmann et al.,

For NHNN data, the subject gave informed written consent, and the study was approved by the Joint Research Ethics Committee of the NHNN (UCLH NHS Foundation Trust) and UCL Queen Square Institute of Neurology, Queen Square, London, UK.

Unless stated otherwise, we have applied the same pre-processing and analysis parameters to the computationally generated time series and the human EEG recordings and performed the fractal and multifractal estimations on 1,024-sample epochs. In Experiments 1 and 2, we were specifically interested in the effect of signal variance on the (multi) fractal estimation, and for comparison we also subjected the EEG signal to a standardization procedure, as follows:

where <

The Chhabra-Jensen method requires as input a distribution function over the domain of positive real numbers, which is incompatible with EEG data which contain positive and negative values. Hence, we propose the use of a sigmoid-transformation here (Equation 27) to map the time series onto positive values, in order to apply the Chhabra-Jensen method. Example sigmoid functions and correspondingly transformed EEG signal are shown in Appendix Figure

The parameter

We evaluated the relationship between monofractal measures and signal variance using a simulated time series based on fractional Brownian motion (fBm), where its signal variance is modulated by a modified ramp function. The modulation function is shown in Figure

We estimated the monofractal dimension of this simulated signal using two standard methods: Higuchi and DFA. We observe that both methods appear to be affected by the changing signal variance (Figures

In conclusion, monofractal properties derived for each epoch from DFA and Higuchi methods (with, or without signal standardization) correlate highly with the signal standard deviation of the epoch. Therefore, in epoch-based approaches (e.g., for application such as detecting or predicting epileptic seizures), the monofractal properties cannot be regarded as a new useful EEG feature of an epoch that is not redundant to standard deviation of the epoch. Thus we turn our attention to multifractal properties of the signal next.

In the following, we will denote the epoch-wise estimates of multifractal width Δα and height Δ^{†} and Δ^{†} for the measure of the epoch-based standardized time series).

This experiment was designed to assess the reliability of the different multifractal estimation methods over time. In other words, if the multifractal properties of the time series remain constant over different epochs, then we expect the multifractal estimation method to show the same output over these different epochs. Note that the accuracy of these methods (i.e., the method outputting the expected multifractal measures of a predefined multifractal object with known multifractal properties) has been demonstrated elsewhere (Chhabra and Jensen,

Figure ^{†} and Δ^{†} were clearly different from zero. The (Δα^{†}, Δ^{†}) output variances over time for the MF-DFA, MF-DMA, and Chhabra-Jensen estimation methods were: (0.018, 0.18), (4.17e-4, 0.0028), and (2.3e-30, 6.5e-30), respectively. In addition, the MF-DFA output violated the theoretical topological limit of Δ^{†} = 1, again indicating problems in the MF-DFA method, potentially due to the inversion of multifractal spectrum (Mukli et al.,

Comparison of three multifractal spectrum estimation methods (MF-DFA, MF-DMA, and Chhabra-Jensen) for p-Model simulated time series. ^{†} and ^{†}.

Next, we evaluated the relationship between multifractal signal properties and other widely used conventional EEG measures (such as signal variance). Figure ^{†}). Finally, signal line length also shows a very different temporal profile from Δα^{†}. A similar figure showing the variation of Δ^{†} metrics is available in Appendix Figure

Temporal dynamics of multifractal spectrum width compared with conventional measures for human intracranial EEG. ^{†}).

Figure ^{†}). We also note that Δα is highly correlated with DFA and DFA^{†} estimates (ρ = 0.74 and ρ = 0.71, respectively) while it is markedly reduced for Δα^{†} (|ρ| < 0.3). The analysis based on the mutual information (Ince et al.,

Correlation between multifractal spectrum and conventional EEG measures for human icEEG data (from Figure

The relationships of the multifractal properties and specific EEG frequency band power are shown in Figure ^{†}, Δ^{†}, and signal power in the classical EEG bands are low (|ρ| < 0.3). A supplementary analysis of EEG time series data containing different sleep stages (which are known to be dominated by specific frequencies) shows similar results (see Appendix ^{†} (using epoch-wise standardization of the time series) in the subsequent analysis.

Comparison of multifractal measures with classical spectral band power. Scatter plot matrix comparing both standardized multifractal spectrum width and height (Δα^{†} and Δ^{†}) with the δ, θ, α, β, and γ average band power in each epoch. Each scatter point is derived from a single epoch of the time series. The diagonal of the matrix features the histograms for each measure. The lower triangle contains the scatter plots for each pair of measures. The upper triangle shows the Pearson correlation for each pair of measure, where the size of the font additionally corresponds to the correlation coefficient to provide an additional visual cue. The icEEG data underlying this figure is shown in Figure

The variation of the multifractal spectrum width Δα^{†} for different combinations of epoch sizes and sampling frequencies is shown in Figure ^{†} during the ictal period (marked by the red lines). To quantify this effect, Figure ^{†} distributions plotted against epoch duration (in seconds). In this plot, we included 15 different sampling frequencies, and also data from three different EEG channels (all in the seizure onset zone). A peak in ^{†} during a seizure can be best captured when using 1 s epochs (regardless of sampling frequency). This effect was not found for the sampling frequency or epoch length separately. Similar results for additional patients are shown on

Influence of EEG sampling frequency and epoch length on multifractal spectrum width around and during an epileptic seizure. ^{†}) in a 15-min intracranial EEG segment containing one seizure (onset and offset marked by the red lines). The signal was initially sampled at 5,000 Hz. Each column shows Δα^{†} for 5,000, 2,500, 500, and 100 Hz sampling rates. Different epoch sizes were used ranging from 1,024 to 16,384 samples (in each row). ^{†}) and epoch duration in seconds (obtained by dividing the number of sampling points by the sampling rate of the signal). Channel 1 is the data shown in

In this study, we have explored the monofractal and multifractal properties of human EEG recordings and used simulated data to test the performance of fractal property estimation methods. Although mono- and multi-fractal approaches have been widely employed in the study of physiological signals in humans (Ivanov et al.,

One of our key observations is that monofractal estimators are tightly correlated with signal variance—even following epoch-wise standardization, whereas multifractal properties following epoch-wise standardization are no longer tightly correlated with signal variance. This may appear to be a curious and non-intuitive observation that, to our knowledge, has not been reported before.

To interpret this observation, it is worth noting the relationship between monofractal and multifractal analyses. Essentially, in multifractal analysis, at the point for which

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, _{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,

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. interictal epochs, i.e., a hard separation that may not represent continuous phenomena accurately. Future work should take into account that multifractal properties may be continuously changing over time (a striking example is shown in Appendix Figure

Previous studies reported that the brain is characterized by critical dynamics (Eguíluz et al.,

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.,

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,

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,

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

Furthermore, the slow temporal changes in multifractal dynamics need to be characterized in a systematic way. Using epileptic seizures as an example, Appendix Figure

Finally, it is well-recognized that epileptic seizures are spatio-temporal processes (see e.g., Wang et al.,

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.

The datasets analyzed for this study can be found in

LF, LL, MW, and YW: conceptualization, project administration, resources, and writing of the original draft. LF: data curation, funding acquisition, investigation, and visualization. LF and YW: formal analysis and validation. JM, LF, LL, MW, and YW: methodology. JM, LF, and YW: software. LL, MW, and YW: supervision. JM, LF, LL, MW, ML, NS, and YW: writing of the review, and editing.

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.

The authors acknowledge the use of the UCL Legion High Performance Computing Facility (Legion@UCL), and associated support services, in the completion of this work.

The authors would like to thank Benjamin H. Brinkmann, Joost Wagenaar, and Hoameng Ung from

The Supplementary Material for this article can be found online at: