A multiscale symbolic approach to decoding delta and ripple oscillation bands as biomarkers for epileptiform discharges.

We use a multiscale symbolic approach to study the complex dynamics of temporal lobe refractory epilepsy employing high-resolution intracranial electroencephalogram (iEEG). We consider the basal and preictal phases and meticulously analyze the dynamics across frequency bands, focusing on high-frequency oscillations up to 240 Hz. Our results reveal significant periodicities and critical time scales within neural dynamics across frequency bands. By bandpass filtering neural signals into delta, theta, alpha, beta, gamma, and ripple high-frequency bands (HFO), each associated with specific neural processes, we examine the distinct nonlinear dynamics. Our method introduces a reliable approach to pinpoint intrinsic time lag scales τ within frequency bands of the basal and preictal signals, which are crucial for the study of refractory epilepsy. Using metrics such as permutation entropy (H), Fisher information (F), and complexity (C), we explore nonlinear patterns within iEEG signals. We reveal the intrinsic τmax that maximize complexity within each frequency band, unveiling the nonlinear subtle patterns of the temporal structures within the basal and preictal signal. Examining the H×F and C×F values allows us to identify differences in the delta band and a band between 200 and 220 Hz (HFO 6) when comparing basal and preictal signals. Differences in Fisher information in the delta and HFO 6 bands before seizures highlight their role in capturing important system dynamics. This offers new perspectives on the intricate relationship between delta oscillations and HFO waves in patients with focal epilepsy, highlighting the importance of these patterns and their potential as biomarkers.

Chaotic dynamics of the Hénon map and neuronal input-output: A comparison with neurophysiological data.

In this study, the Hénon map was analyzed using quantifiers from information theory in order to compare its dynamics to experimental data from brain regions known to exhibit chaotic behavior. The goal was to investigate the potential of the Hénon map as a model for replicating chaotic brain dynamics in the treatment of Parkinson's and epilepsy patients. The dynamic properties of the Hénon map were compared with data from the subthalamic nucleus, the medial frontal cortex, and a q-DG model of neuronal input-output with easy numerical implementation to simulate the local behavior of a population. Using information theory tools, Shannon entropy, statistical complexity, and Fisher's information were analyzed, taking into account the causality of the time series. For this purpose, different windows over the time series were considered. The findings revealed that neither the Hénon map nor the q-DG model could perfectly replicate the dynamics of the brain regions studied. However, with careful consideration of the parameters, scales, and sampling used, they were able to model some characteristics of neural activity. According to these results, normal neural dynamics in the subthalamic nucleus region may present a more complex spectrum within the complexity-entropy causality plane that cannot be represented by chaotic models alone. The dynamic behavior observed in these systems using these tools is highly dependent on the studied temporal scale. As the size of the sample studied increases, the dynamics of the Hénon map become increasingly different from those of biological and artificial neural systems.

Hydrological changes caused by the construction of dams and reservoirs: The CECP analysis.

We investigated the influence of the construction of cascade dams and reservoirs on the predictability and complexity of the streamflow of the São Francisco River, Brazil, by using complexity entropy causality plane (CECP) in its standard and weighted form. We analyzed daily streamflow time series recorded in three fluviometric stations: São Francisco (upstream of cascade dams), Juazeiro (downstream of Sobradinho dam), and Pão de Açúcar station (downstream of Sobradinho and Xingó dams). By comparing the values of CECP information quantifiers (permutation entropy and statistical complexity) for the periods before and after the construction of Sobradinho (1979) and Xingó (1994) dams, we found that the reservoirs' operations changed the temporal variability of streamflow series toward the less predictable regime as indicated by higher entropy (lower complexity) values. Weighted CECP provides some finer details in the predictability of streamflow due to the inclusion of amplitude information in the probability distribution of ordinal patterns. The time evolution of streamflow predictability was analyzed by applying CECP in 2 year sliding windows that revealed the influence of the Paulo Alfonso complex (located between Sobradinho and Xingó dams), construction of which started in the 1950s and was identified through the increased streamflow entropy in the downstream Pão de Açúcar station. The other streamflow alteration unrelated to the construction of the two largest dams was identified in the upstream unimpacted São Francisco station, as an increase in the entropy around 1960s, indicating that some natural factors could also play a role in the decreased predictability of streamflow dynamics.

High-frequency oscillations in the ripple bands and amplitude information coding: Toward a biomarker of maximum entropy in the preictal signals.

Intracranial electroencephalography (iEEG) can directly record local field potentials (LFPs) from a large set of neurons in the vicinity of the electrode. To search for possible epileptic biomarkers and to determine the epileptogenic zone that gives rise to seizures, we investigated the dynamics of basal and preictal signals. For this purpose, we explored the dynamics of the recorded time series for different frequency bands considering high-frequency oscillations (HFO) up to 240 Hz. We apply a Hilbert transform to study the amplitude and phase of the signals. The dynamics of the different frequency bands in the time causal entropy-complexity plane, H × C, is characterized by comparing the dynamical evolution of the basal and preictal time series. As the preictal states evolve closer to the time in which the epileptic seizure starts, the, H × C, dynamics changes for the higher frequency bands. The complexity evolves to very low values and the entropy becomes nearer to its maximal value. These quasi-stable states converge to equiprobable states when the entropy is maximal, and the complexity is zero. We could, therefore, speculate that in this case, it corresponds to the minimization of Gibbs free energy. In this case, the maximum entropy is equivalent to the principle of minimum consumption of resources in the system. We can interpret this as the nature of the system evolving temporally in the preictal state in such a way that the consumption of resources by the system is minimal for the amplitude in frequencies between 220-230 and 230-240 Hz.

Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis.

The main motivation of this paper is to introduce the permutation Jensen-Shannon distance, a symbolic tool able to quantify the degree of similarity between two arbitrary time series. This quantifier results from the fusion of two concepts, the Jensen-Shannon divergence and the encoding scheme based on the sequential ordering of the elements in the data series. The versatility and robustness of this ordinal symbolic distance for characterizing and discriminating different dynamics are illustrated through several numerical and experimental applications. Results obtained allow us to be optimistic about its usefulness in the field of complex time-series analysis. Moreover, thanks to its simplicity, low computational cost, wide applicability, and less susceptibility to outliers and artifacts, this ordinal measure can efficiently handle large amounts of data and help to tackle the current big data challenges.

Abnormal EEG signal energy in the elderly: A wavelet analysis of event-related potentials during a stroop task.

BACKGROUND: Previous work showed that elderly with excess in theta activity in their resting state electroencephalogram (EEG) are at higher risk of cognitive decline than those with a normal EEG. By using event-related potentials (ERP) during a counting Stroop task, our prior work showed that elderly with theta excess have a large P300 component compared with normal EEG group. This increased activity could be related to a higher EEG signal energy used during this task. NEW METHOD: By wavelet analysis applied to ERP obtained during a counting Stroop task we quantified the energy in the different frequency bands of a group of elderly with altered EEG. RESULTS: In theta and alpha bands, the total energy was higher in elderly subjects with theta excess, specifically in the stimulus categorization window (258-516 ms). Both groups solved the task with similar efficiency. COMPARISON WITH EXISTING METHODS: The traditional ERP analysis in elderly compares voltage among conditions and groups for a given time window, while the frequency composition is not usually examined. We complemented our previous ERP analysis using a wavelet methodology. Furthermore, we showed the advantages of wavelet analysis over Short Time Fourier Transform when exploring EEG signal during this task. CONCLUSIONS: The higher EEG signal energy in ERP might reflect undergoing neurobiological mechanisms that allow the elderly with theta excess to cope with the cognitive task with similar behavioral results as the normal EEG group. This increased energy could promote a metabolic and cellular dysregulation causing a greater decline in cognitive function.

Statistical complexity is maximized close to criticality in cortical dynamics.

Complex systems are typically characterized as an intermediate situation between a complete regular structure and a random system. Brain signals can be studied as a striking example of such systems: cortical states can range from highly synchronous and ordered neuronal activity (with higher spiking variability) to desynchronized and disordered regimes (with lower spiking variability). It has been recently shown, by testing independent signatures of criticality, that a phase transition occurs in a cortical state of intermediate spiking variability. Here we use a symbolic information approach to show that, despite the monotonical increase of the Shannon entropy between ordered and disordered regimes, we can determine an intermediate state of maximum complexity based on the Jensen disequilibrium measure. More specifically, we show that statistical complexity is maximized close to criticality for cortical spiking data of urethane-anesthetized rats, as well as for a network model of excitable elements that presents a critical point of a nonequilibrium phase transition.

Information Theoretic Measures and Their Applications.

The concept of entropy, an ever-growing physical magnitude that measured the degree of decay of order in a physical system, was introduced by Rudolf Clausius in 1865 through an elegant formulation of the second law of thermodynamics [...].

Estimating ecoacoustic activity in the Amazon rainforest through Information Theory quantifiers.

Automatic monitoring of biodiversity by acoustic sensors has become an indispensable tool to assess environmental stress at an early stage. Due to the difficulty in recognizing the Amazon's high acoustic diversity and the large amounts of raw audio data recorded by the sensors, the labeling and manual inspection of this data is not feasible. Therefore, we propose an ecoacoustic index that allows us to quantify the complexity of an audio segment and correlate this measure with the biodiversity of the soundscape. The approach uses unsupervised methods to avoid the problem of labeling each species individually. The proposed index, named the Ecoacoustic Global Complexity Index (EGCI), makes use of Entropy, Divergence and Statistical Complexity. A distinguishing feature of this index is the mapping of each audio segment, including those of varied lengths, as a single point in a 2D-plane, supporting us in understanding the ecoacoustic dynamics of the rainforest. The main results show a regularity in the ecoacoustic richness of a floodplain, considering different temporal granularities, be it between hours of the day or between consecutive days of the monitoring program. We observed that this regularity does a good job of characterizing the soundscape of the environmental protection area of Mamirauá, in the Amazon, differentiating between species richness and environmental phenomena.

A detailed characterization of complex networks using Information Theory.

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdos-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

Inhibitory autapse mediates anticipated synchronization between coupled neurons.

Two identical autonomous dynamical systems unidirectionally coupled in a sender-receiver configuration can exhibit anticipated synchronization (AS) if the receiver neuron also receives a delayed negative self-feedback. Recently, AS was shown to occur in a three-neuron motif with standard chemical synapses where the delayed inhibition was provided by an interneuron. Here, we show that a two-neuron model in the presence of an inhibitory autapse, which is a massive self-innervation present in the cortical architecture, may present AS. The GABAergic autapse regulates the internal dynamics of the receiver neuron and acts as the negative delayed self-feedback required by dynamical systems in order to exhibit AS. In this biologically plausible scenario, a smooth transition from the usual delayed synchronization (DS) to AS typically occurs when the inhibitory conductance is increased. The phenomenon is shown to be robust when model parameters are varied within a physiological range. For extremely large values of the inhibitory autapse the system undergoes to a phase-drift regime in which the receiver is faster than the sender. Furthermore, we show that the inhibitory autapse promotes a faster internal dynamics of the free-running Receiver when the two neurons are uncoupled, which could be the mechanism underlying anticipated synchronization and the DS-AS transition.

Bandt-Pompe symbolization dynamics for time series with tied values: A data-driven approach.

In 2002, Bandt and Pompe [Phys. Rev. Lett. 88, 174102 (2002)] introduced a successfully symbolic encoding scheme based on the ordinal relation between the amplitude of neighboring values of a given data sequence, from which the permutation entropy can be evaluated. Equalities in the analyzed sequence, for example, repeated equal values, deserve special attention and treatment as was shown recently by Zunino and co-workers [Phys. Lett. A 381, 1883 (2017)]. A significant number of equal values can give rise to false conclusions regarding the underlying temporal structures in practical contexts. In the present contribution, we review the different existing methodologies for treating time series with tied values by classifying them according to their different strategies. In addition, a novel data-driven imputation is presented that proves to outperform the existing methodologies and avoid the false conclusions pointed by Zunino and co-workers.

An analysis of high-frequency cryptocurrencies prices dynamics using permutation-information-theory quantifiers.

This paper discusses the dynamics of intraday prices of 12 cryptocurrencies during the past months' boom and bust. The importance of this study lies in the extended coverage of the cryptoworld, accounting for more than 90% of the total daily turnover. By using the complexity-entropy causality plane, we could discriminate three different dynamics in the data set. Whereas most of the cryptocurrencies follow a similar pattern, there are two currencies (ETC and ETH) that exhibit a more persistent stochastic dynamics, and two other currencies (DASH and XEM) whose behavior is closer to a random walk. Consequently, similar financial assets, using blockchain technology, are differentiated by market participants.

Analysis of ischaemic crisis using the informational causal entropy-complexity plane.

In the present work, an ischaemic process, mainly focused on the reperfusion stage, is studied using the informational causal entropy-complexity plane. Ischaemic wall behavior under this condition was analyzed through wall thickness and ventricular pressure variations, acquired during an obstructive flow maneuver performed on left coronary arteries of surgically instrumented animals. Basically, the induction of ischaemia depends on the temporary occlusion of left circumflex coronary artery (which supplies blood to the posterior left ventricular wall) that lasts for a few seconds. Normal perfusion of the wall was then reestablished while the anterior ventricular wall remained adequately perfused during the entire maneuver. The obtained results showed that system dynamics could be effectively described by entropy-complexity loops, in both abnormally and well perfused walls. These results could contribute to making an objective indicator of the recovery heart tissues after an ischaemic process, in a way to quantify the restoration of myocardial behavior after the supply of oxygen to the ventricular wall was suppressed for a brief period.

Rhythmic activities of the brain: Quantifying the high complexity of beta and gamma oscillations during visuomotor tasks.

Electroencephalography (EEG) signals depict the electrical activity that takes place at the surface of the brain and provide an important tool for understanding a variety of cognitive processes. The EEG is the product of synchronized activity of the brain, and variations in EEG oscillations patterns reflect the underlying changes in neuronal synchrony. Our aim is to characterize the complexity of the EEG rhythmic oscillations bands when the subjects perform a visuomotor or imagined cognitive tasks (imagined movement), providing a causal mapping of the dynamical rhythmic activities of the brain as a measure of attentional investment. We estimate the intrinsic correlational structure of the signals within the causality entropy-complexity plane H×C, where the enhanced complexity in the gamma 1, gamma 2, and beta 1 bands allows us to distinguish motor-visual memory tasks from control conditions. We identify the dynamics of the gamma 1, gamma 2, and beta 1 rhythmic oscillations within the zone of a chaotic dissipative behavior, whereas in contrast the beta 2 band shows a much higher level of entropy and a significant low level of complexity that correspond to a non-invertible cubic map. Our findings enhance the importance of the gamma band during attention in perceptual feature binding during the visuomotor/imagery tasks.

Nonlinear dynamics of river runoff elucidated by horizontal visibility graphs.

Horizontal Visibility Graphs (HVGs) are a recently developed method to construct networks from time series. The values of the time series are considered as the nodes of the network and are linked to each other if there is no larger value between them, such as they can "see" each other. The network properties reflect the nonlinear dynamics of the time series. For some classes of stochastic processes and for periodic time series, analytical results can be obtained for network-derived quantities such as the degree distribution, the local clustering coefficient distribution, the mean path length, and others. HVGs have the potential to discern between deterministic-chaotic and correlated-stochastic time series. Here, we investigate the sensitivity of the HVG methodology to properties and pre-processing of real-world data, i.e., time series length, the presence of ties, and deseasonalization, using a set of around 150 runoff time series from managed rivers at daily resolution from Brazil with an average length of 65 years. We show that an application of HVGs on real-world time series requires a careful consideration of data pre-processing steps and analysis methodology before robust results and interpretations can be obtained. For example, one recent analysis of the degree distribution of runoff records reported pronounced sub-exponential "long-tailed" behavior of North American rivers, whereas another study of South American rivers showed hyper-exponential "short-tailed" behavior resembling correlated noise. We demonstrate, using the dataset of Brazilian rivers, that these apparently contradictory results can be reconciled by minor differences in data-preprocessing (here: small differences in subtracting the seasonal cycle). Hence, data-preprocessing that is conventional in hydrology ("deseasonalization") changes long-term correlations and the overall runoff dynamics substantially, and we present empirical consequences and extensive simulations to investigate these issues from a HVG methodological perspective. After carefully accounting for these methodological aspects, the HVG analysis reveals that the river runoff dataset shows indeed complex behavior that appears to stem from a superposition of short-term correlated noise and "long-tailed behaviour," i.e., highly connected nodes. Moreover, the construction of a dam along a river tends to increase short-term correlations in runoff series. In summary, the present study illustrates the (often substantial) effects of methodological and data-preprocessing choices for the interpretation of river runoff dynamics in the HVG framework and its general applicability for real-world time series.

Causal Shannon-Fisher Characterization of Motor/Imagery Movements in EEG.

The electroencephalogram (EEG) is an electrophysiological monitoring method that allows us to glimpse the electrical activity of the brain. Neural oscillations patterns are perhaps the best salient feature of EEG as they are rhythmic activities of the brain that can be generated by interactions across neurons. Large-scale oscillations can be measured by EEG as the different oscillation patterns reflected within the different frequency bands, and can provide us with new insights into brain functions. In order to understand how information about the rhythmic activity of the brain during visuomotor/imagined cognitive tasks is encoded in the brain we precisely quantify the different features of the oscillatory patterns considering the Shannon-Fisher plane H × F . This allows us to distinguish the dynamics of rhythmic activities of the brain showing that the Beta band facilitate information transmission during visuomotor/imagined tasks.

Multiscale permutation entropy analysis of laser beam wandering in isotropic turbulence.

We have experimentally quantified the temporal structural diversity from the coordinate fluctuations of a laser beam propagating through isotropic optical turbulence. The main focus here is on the characterization of the long-range correlations in the wandering of a thin Gaussian laser beam over a screen after propagating through a turbulent medium. To fulfill this goal, a laboratory-controlled experiment was conducted in which coordinate fluctuations of the laser beam were recorded at a sufficiently high sampling rate for a wide range of turbulent conditions. Horizontal and vertical displacements of the laser beam centroid were subsequently analyzed by implementing the symbolic technique based on ordinal patterns to estimate the well-known permutation entropy. We show that the permutation entropy estimations at multiple time scales evidence an interplay between different dynamical behaviors. More specifically, a crossover between two different scaling regimes is observed. We confirm a transition from an integrated stochastic process contaminated with electronic noise to a fractional Brownian motion with a Hurst exponent H=5/6 as the sampling time increases. Besides, we are able to quantify, from the estimated entropy, the amount of electronic noise as a function of the turbulence strength. We have also demonstrated that these experimental observations are in very good agreement with numerical simulations of noisy fractional Brownian motions with a well-defined crossover between two different scaling regimes.