Complexity synchronization in living matter: a mini review.

Fractal time series have been argued to be ubiquitous in human physiology and some of the implications of that ubiquity are quite remarkable. One consequence of the omnipresent fractality is complexity synchronization (CS) observed in the interactions among simultaneously recorded physiologic time series discussed herein. This new kind of synchronization has been revealed in the interaction triad of organ-networks (ONs) consisting of the mutually interacting time series generated by the brain (electroencephalograms, EEGs), heart (electrocardiograms, ECGs), and lungs (Respiration). The scaled time series from each member of the triad look nothing like one another and yet they bear a deeply recorded synchronization invisible to the naked eye. The theory of scaling statistics is used to explain the source of the CS observed in the information exchange among these multifractal time series. The multifractal dimension (MFD) of each time series is a measure of the time-dependent complexity of that time series, and it is the matching of the MFD time series that provides the synchronization referred to as CS. The CS is one manifestation of the hypothesis given by a "Law of Multifractal Dimension Synchronization" (LMFDS) which is supported by data. Therefore, the review aspects of this paper are chosen to make the extended range of the LMFDS hypothesis sufficiently reasonable to warrant further empirical testing.

Complexity synchronization in emergent intelligence.

In this work, we use a simple multi-agent-based-model (MABM) of a social network, implementing selfish algorithm (SA) agents, to create an adaptive environment and show, using a modified diffusion entropy analysis (DEA), that the mutual-adaptive interaction between the parts of such a network manifests complexity synchronization (CS). CS has been shown to exist by processing simultaneously measured time series from among organ-networks (ONs) of the brain (neurophysiology), lungs (respiration), and heart (cardiovascular reactivity) and to be explained theoretically as a synchronization of the multifractal dimension (MFD) scaling parameters characterizing each time series. Herein, we find the same kind of CS in the emergent intelligence of groups formed in a self-organized social interaction without macroscopic control but with biased self-interest between two groups of agents playing an anti-coordination game. This computational result strongly suggests the existence of the same CS in real-world social phenomena and in human-machine interactions as that found empirically in ONs.

Complexity Synchronization of Organ Networks.

The transdisciplinary nature of science as a whole became evident as the necessity for the complex nature of phenomena to explain social and life science, along with the physical sciences, blossomed into complexity theory and most recently into complexitysynchronization. This science motif is based on the scaling arising from the 1/f-variability in complex dynamic networks and the need for a network of networks to exchange information internally during intra-network dynamics and externally during inter-network dynamics. The measure of complexity adopted herein is the multifractal dimension of the crucial event time series generated by an organ network, and the difference in the multifractal dimensions of two organ networks quantifies the relative complexity between interacting complex networks. Information flows from dynamic networks at a higher level of complexity to those at lower levels of complexity, as summarized in the 'complexity matching effect', and the flow is maximally efficient when the complexities are equal. Herein, we use the scaling of empirical datasets from the brain, cardiovascular and respiratory networks to support the hypothesis that complexity synchronization occurs between scaling indices or equivalently with the matching of the time dependencies of the networks' multifractal dimensions.

Complexity synchronization: a measure of interaction between the brain, heart and lungs.

Herein we address the measurable consequences of the network effect (NE) on time series generated by different parts of the brain, heart, and lung organ-networks (ONs), which are directly related to their inter-network and intra-network interactions. Moreover, these same physiologic ONs have been shown to generate crucial event (CE) time series, and herein are shown, using modified diffusion entropy analysis (MDEA) to have scaling indices with quasiperiodic changes in complexity, as measured by scaling indices, over time. Such time series are generated by different parts of the brain, heart, and lung ONs, and the results do not depend on the underlying coherence properties of the associated time series but demonstrate a generalized synchronization of complexity. This high-order synchrony among the scaling indices of EEG (brain), ECG (heart), and respiratory time series is governed by the quantitative interdependence of the multifractal behavior of the various physiological ONs' dynamics. This consequence of the NE opens the door for an entirely general characterization of the dynamics of complex networks in terms of complexity synchronization (CS) independently of the scientific, engineering, or technological context. CS is truly a transdisciplinary effect.

EEG phase synchronization during absence seizures.

Absence seizures-generalized rhythmic spike-and-wave discharges (SWDs) are the defining property of childhood (CAE) and juvenile (JAE) absence epilepsies. Such seizures are the most compelling examples of pathological neuronal hypersynchrony. All the absence detection algorithms proposed so far have been derived from the properties of individual SWDs. In this work, we investigate EEG phase synchronization in patients with CAE/JAE and healthy subjects to explore the possibility of using the wavelet phase synchronization index to detect seizures and quantify their disorganization (fragmentation). The overlap of the ictal and interictal probability density functions was high enough to preclude effective seizure detection based solely on changes in EEG synchronization. We used a machine learning classifier with the phase synchronization index (calculated for 1 s data segments with 0.5 s overlap) and the normalized amplitude as features to detect generalized SWDs. Using 19 channels (10-20 setup), we identified 99.2% of absences. However, the overlap of the segments classified as ictal with seizures was only 83%. The analysis showed that seizures were disorganized in approximately half of the 65 subjects. On average, generalized SWDs lasted about 80% of the duration of abnormal EEG activity. The disruption of the ictal rhythm can manifest itself as the disappearance of epileptic spikes (with high-amplitude delta waves persisting), transient cessation of epileptic discharges, or loss of global synchronization. The detector can analyze a real-time data stream. Its performance is good for a six-channel setup (Fp1, Fp2, F7, F8, O1, O2), which can be implemented as an unobtrusive EEG headband. False detections are rare for controls and young adults (0.03% and 0.02%, respectively). In patients, they are more frequent (0.5%), but in approximately 82% cases, classification errors are caused by short epileptiform discharges. Most importantly, the proposed detector can be applied to parts of EEG with abnormal EEG activity to quantitatively determine seizure fragmentation. This property is important because a previous study reported that the probability of disorganized discharges is eight times higher in JAE than in CAE. Future research must establish whether seizure properties (frequency, length, fragmentation, etc.) and clinical characteristics can help distinguish CAE and JAE.

Temporal complexity measure of reaction time series: Operational versus event time.

INTRODUCTION: Detrended fluctuation analysis (DFA) is a well-established method to evaluate scaling indices of time series, which categorize the dynamics of complex systems. In the literature, DFA has been used to study the fluctuations of reaction time Y(n) time series, where n is the trial number. METHODS: Herein we propose treating each reaction time as a duration time that changes the representation from operational (trial number) time n to event (temporal) time t, or X(t). The DFA algorithm was then applied to the X(t) time series to evaluate scaling indices. The dataset analyzed is based on a Go-NoGo shooting task that was performed by 30 participants under low and high time-stress conditions in each of six repeated sessions over a 3-week period. RESULTS: This new perspective leads to quantitatively better results in (1) differentiating scaling indices between low versus high time-stress conditions and (2) predicting task performance outcomes. CONCLUSION: We show that by changing from operational time to event time, the DFA allows discrimination of time-stress conditions and predicts performance outcomes.

The Fractal Tapestry of Life: II Entailment of Fractional Oncology by Physiology Networks.

This is an essay advocating the efficacy of using the (noninteger) fractional calculus (FC) for the modeling of complex dynamical systems, specifically those pertaining to biomedical phenomena in general and oncological phenomena in particular. Herein we describe how the integer calculus (IC) is often incapable of describing what were historically thought to be simple linear phenomena such as Newton's law of cooling and Brownian motion. We demonstrate that even linear dynamical systems may be more accurately described by fractional rate equations (FREs) when the experimental datasets are inconsistent with models based on the IC. The Network Effect is introduced to explain how the collective dynamics of a complex network can transform a many-body noninear dynamical system modeled using the IC into a set of independent single-body fractional stochastic rate equations (FSREs). Note that this is not a mathematics paper, but rather a discussion focusing on the kinds of phenomena that have historically been approximately and improperly modeled using the IC and how a FC replacement of the model better explains the experimental results. This may be due to hidden effects that were not anticapated in the IC model, or to an effect that was acknowledged as possibly significant, but beyond the mathematical skills of the investigator to Incorporate into the original model. Whatever the reason we introduce the FRE used to describe mathematical oncology (MO) and review the quality of fit of such models to tumor growth data. The analytic results entailed in MO using ordinary diffusion as well as fractional diffusion are also briefly discussed. A connection is made between a time-dependent fractional-order derivative, technically called a distributed-order parameter, and the multifractality of time series, such that an observed multifractal time series can be modeled using a FRE with a distributed fractional-order derivative. This equivalence between multifractality and distributed fractional derivatives has not received the recognition in the applications literature we believe it warrants.

Persistence and anti-persistence in treadmill walking.

BACKGROUND: Strong, long-range persistent correlations in stride time (ST) and length (SL) are the fundamental traits of treadmill gait. Our recent work showed that the ST and SL time series' statistical properties originated from the superposition of large-scale trends and small-scale fluctuations (residuals). Trends served as the control manifolds about which ST and SL fluctuated. RESEARCH QUESTION: Do random changes in treadmill belt speed affect the trend properties and ST/SL scaling exponents? METHODS: We used Multivariate Adaptive Regression Splines (MARS) to determine gait trends during a walk on a treadmill whose belt speed was perturbed by a strong random noise (coefficient of variation was equal to 0.075, 0.1, and 0.13 for treadmill speed 0.8 m/s, 1.2 m/s, and 1.6 m/s, respectively). Then, we calculated the ST/SL scaling exponents of the experimental time series and the corresponding MARS residuals with the madogram estimator. RESULTS: Except for the ST at the lowest treadmill speed, the normalized trend duration was at least two times greater than that for the unperturbed walk. The Cauchy distribution scale parameter, which served as a measure of the width of SL and ST trend slope distributions, was at v=1.2m/s, almost 50% and 25% smaller than the unperturbed values. The differences were even greater at v=1.6 m/s: 73% and 83%. Apart from ST at v=0.8m/s, the ST/SL scaling indices were close to 0.5. For all speeds, the ST and SL MARS residuals were strongly anti-persistent. At v=1.2m/s, the corresponding scaling exponents were equal to 0.37±0.10 and 0.25±0.09. SIGNIFICANCE: At normal and moderate treadmill speeds, in the presence of random belt speed perturbations, strongly anti-persistent fluctuations about gentle, persistent trends can lead to weak persistence/antipersistence of ST/SL time series.

Fractional Calculus and the Future of Science.

The invitation to contribute to this anthology of articles on the fractional calculus (FC) encouraged submissions in which the authors look behind the mathematics and examine what must be true about the phenomenon to justify the replacement of an integer-order derivative with a non-integer-order (fractional) derivative (FD) before discussing ways to solve the new equations [...].

Hypothetical Control of Fatal Quarrel Variability.

Wars, terrorist attacks, as well as natural catastrophes typically result in a large number of casualties, whose distributions have been shown to belong to the class of Pareto's inverse power laws (IPLs). The number of deaths resulting from terrorist attacks are herein fit by a double Pareto probability density function (PDF). We use the fractional probability calculus to frame our arguments and to parameterize a hypothetical control process to temper a Lévy process through a collective-induced potential. Thus, the PDF is shown to be a consequence of the complexity of the underlying social network. The analytic steady-state solution to the fractional Fokker-Planck equation (FFPE) is fit to a forty-year fatal quarrel (FQ) dataset.

Absence Seizure Detection Algorithm for Portable EEG Devices.

Absence seizures are generalized nonmotor epileptic seizures with abrupt onset and termination. Transient impairment of consciousness and spike-slow wave discharges (SWDs) in EEG are their characteristic manifestations. This type of seizure is severe in two common pediatric syndromes: childhood (CAE) and juvenile (JAE) absence epilepsy. The appearance of low-cost, portable EEG devices has paved the way for long-term, remote monitoring of CAE and JAE patients. The potential benefits of this kind of monitoring include facilitating diagnosis, personalized drug titration, and determining the duration of pharmacotherapy. Herein, we present a novel absence detection algorithm based on the properties of the complex Morlet continuous wavelet transform of SWDs. We used a dataset containing EEGs from 64 patients (37 h of recordings with almost 400 seizures) and 30 age and sex-matched controls (9 h of recordings) for development and testing. For seizures lasting longer than 2 s, the detector, which analyzed two bipolar EEG channels (Fp1-T3 and Fp2-T4), achieved a sensitivity of 97.6% with 0.7/h detection rate. In the patients, all false detections were associated with epileptiform discharges, which did not yield clinical manifestations. When the duration threshold was raised to 3 s, the false detection rate fell to 0.5/h. The overlap of automatically detected seizures with the actual seizures was equal to ~96%. For EEG recordings sampled at 250 Hz, the one-channel processing speed for midrange smartphones running Android 10 (about 0.2 s per 1 min of EEG) was high enough for real-time seizure detection.

The Fractal Tapestry of Life: A Review of Fractal Physiology.

Since its inception the science of physiology, like many other non-physical disciplines, has been guided in its development by the mechanical models of physics. That strategy has proven to be extraordinarily successful, even surviving the introduction of fractals into the modeling strategy. That is until quite recently. The true complexity of physiologic networks has been revealed with the development and implementation of ever more sensitive sensors and mathematically sophisticated data processing techniques. These developments have led to a divergence of the modeling strategies appropriate for the physical sciences from those for the life sciences. Therefore, we review how far the fractal concept has taken us into the non-mechanical interpretation of physiology. What emerges in this brief review of fractal physiology is the increasing importance of criticality, the cooperative nature of networks in physiologic behavior, and the importance of the fractional calculus in characterizing the dynamics of living systems. We draw some further inferences from the review and speculate as to what research directions might be most productive for continuing future success.

Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: "is there a more optimal way to optimize?". Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.

Caputo Fractional Derivative and Quantum-Like Coherence.

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Sir Isaac Newton Stranger in a Strange Land.

The theme of this essay is that the time of dominance of Newton's world view in science is drawing to a close. The harbinger of its demise was the work of Poincaré on the three-body problem and its culmination into what is now called chaos theory. The signature of chaos is the sensitive dependence on initial conditions resulting in the unpredictability of single particle trajectories. Classical determinism has become increasingly rare with the advent of chaos, being replaced by erratic stochastic processes. However, even the probability calculus could not withstand the non-Newtonian assault from the social and life sciences. The ordinary partial differential equations that traditionally determined the evolution of probability density functions (PDFs) in phase space are replaced with their fractional counterparts. Allometry relation is proven to result from a system's complexity using exact solutions for the PDF of the Fractional Kinetic Theory (FKT). Complexity theory is shown to be incompatible with Newton's unquestioning reliance on an absolute space and time upon which he built his discrete calculus.

Significance of trends in gait dynamics.

Trends in time series generated by physiological control systems are ubiquitous. Determining whether trends arise from intrinsic system dynamics or originate outside of the system is a fundamental problem of fractal series analysis. In the latter case, it is necessary to filter out the trends before attempting to quantify correlations in the noise (residuals). For over two decades, detrended fluctuation analysis (DFA) has been used to calculate scaling exponents of stride time (ST), stride length (SL), and stride speed (SS) of human gait. Herein, rather than relying on the very specific form of detrending characteristic of DFA, we adopt Multivariate Adaptive Regression Splines (MARS) to explicitly determine trends in spatio-temporal gait parameters during treadmill walking. Then, we use the madogram estimator to calculate the scaling exponent of the corresponding MARS residuals. The durations of ST and SL trends are determined to be independent of treadmill speed and have distributions with exponential tails. At all speeds considered, the trends of ST and SL are strongly correlated and are statistically independent of their corresponding residuals. The averages of scaling exponents of ST and SL MARS residuals are slightly smaller than 0.5. Thus, contrary to the interpretation prevalent in the literature, the statistical properties of ST and SL time series originate from the superposition of large scale trends and small scale fluctuations. We show that trends serve as the control manifolds about which ST and SL fluctuate. Moreover, the trend speed, defined as the ratio of instantaneous values of SL and ST trends, is tightly controlled about the treadmill speed. The strong coupling between the ST and SL trends ensures that the concomitant changes of their values correspond to movement along the constant speed goal equivalent manifold as postulated by Dingwell et al. 10.1371/journal.pcbi.1000856.

Complex Periodicity and Synchronization.

A recent experiment proves the therapeutic effect of arm-in-arm walking, showing that if an aged participant walks in close synchrony with a young companion, the complexity matching effect results in the restoration of complexity in the former. A clear manifestation of complexity restoration is a perfect synchronization. The authors of this interesting experiment leave open two important problems. The first is the measure of complexity that is interpreted as a degree of multifractality. The second problem is the lack of a theoretical derivation of synchronization, which is experimentally observed with no theoretical derivation. The main goal of this paper is to establish a physiological foundation of these important results based on the recent advances on the dynamics of the brain, interpreted as a system at criticality. Criticality is a phenomenon requiring the cooperative interaction of units, the neurons of the brain, and is hypothesized as the main source of cognition. Using the criticality-induced intelligence, we define complexity as a property of crucial events, a form of temporal complexity, and we prove that the perfect synchronization is due to the interaction between the two systems, with the more complex system restoring the temporal complexity of the less complex system. The phenomenon of temporal complexity is characterized by ergodicity breaking that has made it difficult in the past to derive the perfect synchronization generated by complexity matching. For this reason, we supplement the main result of this paper with a comparison between complexity matching and complexity management.

Recent Advances in Systems and Network Medicine: Meeting Report from the First International Conference in Systems and Network Medicine.

The First International Conference in Systems and Network Medicine gathered together 200 global thought leaders, scientists, clinicians, academicians, industry and government experts, medical and graduate students, postdoctoral scholars and policymakers. Held at Georgetown University Conference Center in Washington D.C. on September 11-13, 2019, the event featured a day of pre-conference lectures and hands-on bioinformatic computational workshops followed by two days of deep and diverse scientific talks, panel discussions with eminent thought leaders, and scientific poster presentations. Topics ranged from: Systems and Network Medicine in Clinical Practice; the role of -omics technologies in Health Care; the role of Education and Ethics in Clinical Practice, Systems Thinking, and Rare Diseases; and the role of Artificial Intelligence in Medicine. The conference served as a unique nexus for interdisciplinary discovery and dialogue and fostered formation of new insights and possibilities for health care systems advances.

Changes in Interictal Pretreatment and Posttreatment EEG in Childhood Absence Epilepsy.

Spike and wave discharges (SWDs) are a characteristic manifestation of childhood absence epilepsy (CAE). It has long been believed that they unpredictably emerge from otherwise almost normal interictal EEG. Herein, we demonstrate that pretreatment closed-eyes theta and beta EEG wavelet powers of CAE patients (20 girls and 10 boys, mean age 7.4 ± 1.9 years) are much higher than those of age-matched healthy controls at multiple sites of the 10-20 system. For example, at the C4 site, we observed a 100 and 63% increase in power of theta and beta rhythms, respectively. We were able to compare the baseline and posttreatment wavelet power in 16 patients. Pharmacotherapy brought about a statistically significant decrease in delta and theta wavelet power in all the channels, e.g., for C4 the reduction was equal to 45% (delta) and 63% (theta). The less pronounced attenuation of posttreatment beta waves was observed in 13 channels (36% at C4 site). The beta and theta wavelet power were positively correlated with the percentage of time in seizure (defined as the ratio of the duration of all absences which patients experienced to the duration of recording) for majority of channels. We hypothesize that the increased theta and beta powers result from cortical hyperexcitability and propensity for epileptic spike generation, respectively. We argue that the distinct features of CAE wavelet power spectrum may be used to define an EEG biomarker which could be used for diagnosis and monitoring of patients.

Diffusion Entropy vs. Multiscale and Rényi Entropy to Detect Progression of Autonomic Neuropathy.

We review the literature to argue the importance of the occurrence of crucial events in the dynamics of physiological processes. Crucial events are interpreted as short time intervals of turbulence, and the time distance between two consecutive crucial events is a waiting time distribution density with an inverse power law (IPL) index µ, with µ < 3 generating non-stationary behavior. The non-stationary condition is characterized by two regimes of the IPL index: (a) perennial non-stationarity, with 1 < µ < 2 and (b) slow evolution toward the stationary regime, with 2 < µ < 3. Human heartbeats and brain dynamics belong to the latter regime, with healthy physiological processes tending to be closer to the border with the perennial non-stationary regime with µ = 2. The complexity of cognitive tasks is associated with the mental effort required to address a difficult task, which leads to an increase of µ with increasing task difficulty. On this basis we explore the conjecture that disease evolution leads the IPL index µ moving from the healthy condition µ = 2 toward the border with Gaussian statistics with µ = 3, as the disease progresses. Examining heart rate time series of patients affected by diabetes-induced autonomic neuropathy of varying severity, we find that the progression of cardiac autonomic neuropathy (CAN) indeed shifts µ from the border with perennial variability, µ = 2, to the border with Gaussian statistics, µ = 3 and provides a novel, sensitive index for assessing disease progression. We find that at the Gaussian border, the dynamical complexity of crucial events is replaced by Gaussian fluctuation with long-time memory.