A simple method for detecting chaos in nature

Chaotic recurrent neural networks for brain modelling: A review

Even in the absence of external stimuli, the brain is spontaneously active. Indeed, most cortical activity is internally generated by recurrence. Both theoretical and experimental studies suggest that chaotic dynamics characterize this spontaneous activity. While the precise function of brain chaotic activity is still puzzling, we know that chaos confers many advantages. From a computational perspective, chaos enhances the complexity of network dynamics. From a behavioural point of view, chaotic activity could generate the variability required for exploration. Furthermore, information storage and transfer are maximized at the critical border between order and chaos. Despite these benefits, many computational brain models avoid incorporating spontaneous chaotic activity due to the challenges it poses for learning algorithms. In recent years, however, multiple approaches have been proposed to overcome this limitation. As a result, many different algorithms have been developed, initially within the reservoir computing paradigm. Over time, the field has evolved to increase the biological plausibility and performance of the algorithms, sometimes going beyond the reservoir computing framework. In this review article, we examine the computational benefits of chaos and the unique properties of chaotic recurrent neural networks, with a particular focus on those typically utilized in reservoir computing. We also provide a detailed analysis of the algorithms designed to train chaotic RNNs, tracing their historical evolution and highlighting key milestones in their development. Finally, we explore the applications and limitations of chaotic RNNs for brain modelling, consider their potential broader impacts beyond neuroscience, and outline promising directions for future research.

Assessing the impact of seawater blockade on coastal lake degradation using Chironomidae larvae

Coastal ecosystems, such as lakes and lagoons, are unique and valuable water bodies whose proper functioning depends on hydrological connectivity with the sea or ocean. Human interventions, such as the construction of storm surge barriers, that block the periodic and free influx of seawater into lakes induce a permanent freshwater state. This study presents such disturbances, considered as environmental stressors, initiating changes in the assemblages of Chironomidae larvae inhabiting the bottom of Lake Jamno (southern coast of the Baltic Sea). Changes in the structure of this assemblages were assessed during a long-term study (2010-20), which considered two periods: a time of free seawater intrusion (FF) and seven years of blocked influx (BF). The findings indicate that, following the activation of storm surge barriers, the α-diversity of larvae consistently decreased throughout the lake. Concurrently, the density of Chironomidae larvae decreased by over 20%, although their biomass increased. In the last year of the study with functioning gates, the diversity of the studied insects was drastically reduced and was limited to only two genus: Chironomus sp. and Procladius sp., which serve as indicators of disturbances in aquatic ecosystems undergoing changes in line with deterministic chaos theory. The information provided indicates that periodic increases in salinity significantly affect the structure of Chironomidae larvae, though it should be considered as a component of several other parameters (EC, temperature, or nutrients).

Focal Infrared Neural Stimulation Propagates Dynamical Transformations in Auditory Cortex

Significance

Infrared neural stimulation (INS) has emerged as a potent neuromodulation technology, offering safe and focal stimulation with superior spatial recruitment profiles compared to conventional electrical methods. However, the neural dynamics induced by INS stimulation remain poorly understood. Elucidating these dynamics will help develop new INS stimulation paradigms and advance its clinical application.

Aim

In this study, we assessed the local network dynamics of INS entrainment in the auditory thalamocortical circuit using the chronically implanted rat model; our approach focused on measuring INS energy-based local field potential (LFP) recruitment induced by focal thalamocortical stimulation. We further characterized linear and nonlinear oscillatory LFP activity in response to single-pulse and periodic INS and performed spectral decomposition to uncover specific LFP band entrainment to INS. Finally, we examined spike-field transformations across the thalamocortical synapse using spike-LFP coherence coupling.

Results

We found that INS significantly increases LFP amplitude as a log-linear function of INS energy per pulse, primarily entraining to LFP β and γ bands with synchrony extending to 200 Hz in some cases. A subset of neurons demonstrated nonlinear, chaotic oscillations linked to information transfer across cortical circuits. Finally, we utilized spike-field coherences to correlate spike coupling to LFP frequency band activity and suggest an energy-dependent model of network activation resulting from INS stimulation.

Conclusions

We show that INS reliably drives robust network activity and can potently modulate cortical field potentials across a wide range of frequencies in a stimulus parameter-dependent manner. Based on these results, we propose design principles for developing full coverage, all-optical thalamocortical auditory neuroprostheses.

Creating Refined Datasets for Better Chaos Detection

In recent years, the analysis of signal properties (especially biomedical signals) has become an important research direction. One interesting feature of signals is their potential to be chaotic. This article concerns the issues of classification of real signals or synthetic ones in the context of detecting chaotic properties. In previous works, datasets of synthetic signals were created based on well-known chaotic and non-chaotic dynamical systems. They were published and used to train classifiers. This paper extends the previous studies and proposes a method for obtaining/extracting signals to force classifiers to learn to detect chaos. The proposed method allows the generation of groups of signals with similar initial conditions. The property of chaotic dynamical systems was used here, which consists of the strong dependence of the signal courses on a small change in the initial conditions. This method is based on reconstructing multidimensional phase space and data clustering. An additional goal of the work is to create referential datasets with so-called refined signals using the described method and to make them publicly available. The usefulness of the new datasets was confirmed during a simple experiment with the usage of the LSTM neural network.

Spatio-temporal pattern formation of living organisms at the edge of chaos

Understanding spatio-temporal dynamics is essential for predicting how populations fluctuate over time and space. Theoretical models have highlighted the ecological complexity of spatio-temporal dynamics, which can lead to the emergence of complex patterns, including nonlinear dynamics and chaotic behavior, important mechanisms for maintaining of biodiversity. However, these dynamics are difficult to observe experimentally due to a lack of temporal and spatial resolution. Here, we show that even a single-species system exhibits complex spatio-temporal patterns without external forcing where order and chaos coexist (edge of chaos). Automated analyses of experimental dynamics of cells of a ciliate on a microfluidic chip environment with 50 interconnected patches documented pattern formation, including chaos-like dynamics, using several analytical methods. Different initial conditions caused changes in patterns, revealing the complexity and principal unpredictability of self-organized pattern formation. A model containing the stochastic fluctuations of the experiment verified the deterministic nature of patterns. The results show the intrinsic complexity of ecological systems, challenging predictions in nature conservation. Our results bridge the gap between theoretical models and experimental observations, offering new insights into the fundamental nature of living systems and their spatio-temporal organization.

Modeling Cortical Versus Hippocampal Network Dysfunction in a Human Brain Assembloid Model of Epilepsy and Intellectual Disability

Neurodevelopmental disorders often impair multiple cognitive domains. For instance, a genetic epilepsy syndrome might cause seizures due to cortical hyperexcitability and present with memory impairments arising from hippocampal dysfunction. This study examines how a single disorder differentially affects distinct brain regions by using human patient iPSC-derived cortical- and hippocampal-ganglionic eminence assembloids to model Developmental and Epileptic Encephalopathy 13 (DEE-13), a condition arising from gain-of-function mutations in the SCN8A gene. While cortical assembloids showed network hyperexcitability akin to epileptogenic tissue, hippocampal assembloids did not, and instead displayed network dysregulation patterns similar to in vivo hippocampal recordings from epilepsy patients. Predictive computational modeling, immunohistochemistry, and single-nucleus RNA sequencing revealed changes in excitatory and inhibitory neuron organization that were specific to hippocampal assembloids. These findings highlight the unique impacts of a single pathogenic variant across brain regions and establish hippocampal assembloids as a platform for studying neurodevelopmental disorders.

Modeling free tumor growth: Discrete, continuum, and hybrid approaches to interpreting cancer development

Tumor growth dynamics serve as a critical aspect of understanding cancer progression and treatment response to mitigate one of the most pressing challenges in healthcare. The in silico approach to understanding tumor behavior computationally provides an efficient, cost-effective alternative to wet-lab examinations and are adaptable to different environmental conditions, time scales, and unique patient parameters. As a result, this paper explored modeling of free tumor growth in cancer, surveying contemporary literature on continuum, discrete, and hybrid approaches. Factors like predictive power and high-resolution simulation competed against drawbacks like simulation load and parameter feasibility in these models. Understanding tumor behavior in different scenarios and contexts became the first step in advancing cancer research and revolutionizing clinical outcomes.

Selectively advantageous instability in biotic and pre-biotic systems and implications for evolution and aging

Rules of biology typically involve conservation of resources. For example, common patterns such as hexagons and logarithmic spirals require minimal materials, and scaling laws involve conservation of energy. Here a relationship with the opposite theme is discussed, which is the selectively advantageous instability (SAI) of one or more components of a replicating system, such as the cell. By increasing the complexity of the system, SAI can have benefits in addition to the generation of energy or the mobilization of building blocks. SAI involves a potential cost to the replicating system for the materials and/or energy required to create the unstable component, and in some cases, the energy required for its active degradation. SAI is well-studied in cells. Short-lived transcription and signaling factors enable a rapid response to a changing environment, and turnover is critical for replacement of damaged macromolecules. The minimal gene set for a viable cell includes proteases and a nuclease, suggesting SAI is essential for life. SAI promotes genetic diversity in several ways. Toxin/antitoxin systems promote maintenance of genes, and SAI of mitochondria facilitates uniparental transmission. By creating two distinct states, subject to different selective pressures, SAI can maintain genetic diversity. SAI of components of synthetic replicators favors replicator cycling, promoting emergence of replicators with increased complexity. Both classical and recent computer modeling of replicators reveals SAI. SAI may be involved at additional levels of biological organization. In summary, SAI promotes replicator genetic diversity and reproductive fitness, and may promote aging through loss of resources and maintenance of deleterious alleles.

Cortical timescales and the modular organization of structural and functional brain networks

Recent years have seen growing interest in characterizing the properties of regional brain dynamics and their relationship to other features of brain structure and function. In particular, multiple studies have observed regional differences in the "timescale" over which activity fluctuates during periods of quiet rest. In the cerebral cortex, these timescales have been associated with both local circuit properties as well as patterns of inter-regional connectivity, including the extent to which each region exhibits widespread connectivity to other brain areas. In the current study, we build on prior observations of an association between connectivity and dynamics in the cerebral cortex by investigating the relationship between BOLD fMRI timescales and the modular organization of structural and functional brain networks. We characterize network community structure across multiple scales and find that longer timescales are associated with greater within-community functional connectivity and diverse structural connectivity. We also replicate prior observations of a positive correlation between timescales and structural connectivity degree. Finally, we find evidence for preferential functional connectivity between cortical areas with similar timescales. We replicate these findings in an independent dataset. These results contribute to our understanding of functional brain organization and structure-function relationships in the human brain, and support the notion that regional differences in cortical dynamics may in part reflect the topological role of each region within macroscale brain networks.

Enhancement of solid-liquid mixing state quality in a mechanical stirred reactor with serial-chaotic rotation generated by basic speed method

In this work, a novel controllable chaotic stirring strategy that basic speed with chaotic mappings is proposed to enhance the solid-liquid mixing state quality. Specially, the modern statistical image analysis technique is introduced to explore the intensification mechanism of the mixing process. Results show that the best experimental conditions are obtained by studying the influence of factors such as the type of chaotic mapping, the speed change time, and the basic speed on the mixing state quality. Moreover, the case in which the basic speed is set to 150 r/min generated by the cascaded Logistic-Cubic chaotic mapping is the best while the speed change time is set to 5 s and the fluctuation threshold is 30. The mixing time of this case is 50 s and the shortest, energy consumption is 1.64 × 104 W/m3 and appropriate, the solid particle suspension quality is 83 and the best.

Criticality supports cross-frequency cortical-thalamic information transfer during conscious states

Consciousness is thought to be regulated by bidirectional information transfer between the cortex and thalamus, but the nature of this bidirectional communication - and its possible disruption in unconsciousness - remains poorly understood. Here, we present two main findings elucidating mechanisms of corticothalamic information transfer during conscious states. First, we identify a highly preserved spectral channel of cortical-thalamic communication that is present during conscious states, but which is diminished during the loss of consciousness and enhanced during psychedelic states. Specifically, we show that in humans, mice, and rats, information sent from either the cortex or thalamus via δ/θ/α waves (∼1-13 Hz) is consistently encoded by the other brain region by high γ waves (52-104 Hz); moreover, unconsciousness induced by propofol anesthesia or generalized spike-and-wave seizures diminishes this cross-frequency communication, whereas the psychedelic 5-methoxy-N,N-dimethyltryptamine (5-MeO-DMT) enhances this low-to-high frequency interregional communication. Second, we leverage numerical simulations and neural electrophysiology recordings from the thalamus and cortex of human patients, rats, and mice to show that these changes in cross-frequency cortical-thalamic information transfer may be mediated by excursions of low-frequency thalamocortical electrodynamics toward/away from edge-of-chaos criticality, or the phase transition from stability to chaos. Overall, our findings link thalamic-cortical communication to consciousness, and further offer a novel, mathematically well-defined framework to explain the disruption to thalamic-cortical information transfer during unconscious states.

Chaos in a bacterial stress response

Cellular responses to environmental changes are often highly heterogeneous and exhibit seemingly random dynamics. The astonishing insight of chaos theory is that such unpredictable patterns can, in principle, arise without the need for any random processes, i.e., purely deterministically without noise. However, while chaos is well understood in mathematics and physics, its role in cell biology remains unclear because the complexity and noisiness of biological systems make testing difficult. Here, we show that chaos explains the heterogeneous response of Escherichia coli cells to oxidative stress. We developed a theoretical model of the gene expression dynamics and demonstrate that chaotic behavior arises from rapid molecular feedbacks that are coupled with cell growth dynamics and cell-cell interactions. Based on theoretical predictions, we then designed single-cell experiments to show we can shift gene expression from periodic oscillations to chaos on demand. Our work suggests that chaotic gene regulation can be employed by cell populations to generate strong and variable responses to changing environments.

Driven dust-charge fluctuation and chaotic ion dynamics in the plasma sheath and presheath regions

Possible existence of chaotic oscillations in ion dynamics in the sheath and presheath regions of a dusty plasma, induced by externally driven dust-charge fluctuation, is presented in this work. In a complex plasma, dust charge fluctuation occurs continuously with time due to the variation of electron and ion currents flowing into the dust particles. In most of the works related to dust-charge fluctuation, theoretically it is assumed that the average dust-charge fluctuation follows the plasma perturbation, while in reality, the dust-charge fluctuation is a semirandom phenomenon, fluctuating about some average value. The very cause of dust-charge fluctuation in a dusty plasma also points to the fact that these fluctuations can be driven externally by changing electron and ion currents to the dust particles. With the help of a hybrid-particle in cell-Monte Carlo collision (h-PIC-MCC) code in this work, we use the plasma sheath as a candidate for driving the dust-charge fluctuation by periodically exposing the sheath-side wall to UV radiation, causing photoemission of electrons, which in turn drive the dust-charge fluctuation. We show that this driven dust-charge fluctuation can induce a chaotic response in the ion dynamics in the sheath and the presheath regions.

Cortical timescales and the modular organization of structural and functional brain networks

Recent years have seen growing interest in characterizing the properties of regional brain dynamics and their relationship to other features of brain structure and function. In particular, multiple studies have observed regional differences in the "timescale" over which activity fluctuates during periods of quiet rest. In the cerebral cortex, these timescales have been associated with both local circuit properties as well as patterns of inter-regional connectivity, including the extent to which each region exhibits widespread connectivity to other brain areas. In the current study, we build on prior observations of an association between connectivity and dynamics in the cerebral cortex by investigating the relationship between BOLD fMRI timescales and the modular organization of structural and functional brain networks. We characterize network community structure across multiple scales and find that longer timescales are associated with greater within-community functional connectivity and diverse structural connectivity. We also replicate prior observations of a positive correlation between timescales and structural connectivity degree. Finally, we find evidence for preferential functional connectivity between cortical areas with similar timescales. We replicate these findings in an independent dataset. These results contribute to our understanding of functional brain organization and structure-function relationships in the human brain, and support the notion that regional differences in cortical dynamics may in part reflect the topological role of each region within macroscale brain networks.

Natural Exponential and Three-Dimensional Chaotic System

Existing chaotic system exhibits unpredictability and nonrepeatability in a deterministic nonlinear architecture, presented as a combination of definiteness and stochasticity. However, traditional two-dimensional chaotic systems cannot provide sufficient information in the dynamic motion and usually feature low sensitivity to initial system input, which makes them computationally prohibitive in accurate time series prediction and weak periodic component detection. Here, a natural exponential and three-dimensional chaotic system with higher sensitivity to initial system input conditions showing astonishing extensibility in time series prediction and image processing is proposed. The chaotic performance evaluated theoretically and experimentally by Poincare mapping, bifurcation diagram, phase space reconstruction, Lyapunov exponent, and correlation dimension provides a new perspective of nonlinear physical modeling and validation. The complexity, robustness, and consistency are studied by recursive and entropy analysis and comparison. The method improves the efficiency of time series prediction, nonlinear dynamics-related problem solving and expands the potential scope of multi-dimensional chaotic systems.

Estimating divergent forest carbon stocks and sinks via a knife set approach

Forest carbon stocks and sinks (CSSs) have been widely estimated using climate classification tables and linear regression (LR) models with common independent variables (IVs) such as the average diameter at breast height (DBH) of stems and root shoot ratio. However, this approach is relatively ineffective when the explanatory power of IVs is lower than that of unobservable variables. Various environmental and anthropogenic factors affect target variables that cause the correlation between them to be chaotic. Here, we designed a knife set (KS) approach combining LR models and the wandering through random forests (WTF) algorithm and applied it in a specific case of Phyllostachys edulis (Carrière) J. Houz. (P. edulis) forests, which have an irregular relationship between their belowground carbon (BGC) stocks and average DBH. We then validated the KS approach performed by cluster computing to estimate the aboveground carbon (AGC) and BGC stocks and the total net primary production (TNPP). The estimated CSSs were compared to the benchmark of the methodology that applied Tier 1 in the Intergovernmental Panel on Climate Change (IPCC) Guidelines for National Greenhouse Gas Inventories via 10-fold cross validation, and the KS approach significantly increased precision and accuracy of estimations. Our approach provides general insights to accurately estimate forest CSSs relying on evidence-based field data, even if some target variables are divergent in specific forest types. We also pointed out the reason why current fancy models containing machine learning (ML) or deep learning algorithms are not effective in predicting the target variables of certain chaotic systems is perhaps that the total explanatory power of observable variables is less than that of the total unobservable variables. Quantifying unobservable variables into observable variables is a linchpin of future works related to chaotic system estimation.

A carbon price hybrid forecasting model based on data multi-scale decomposition and machine learning

Accurate carbon price forecasting is of great significance to the operation of carbon financial markets. However, limited by the non-linearity and non-stationarity of the carbon price, the accurate and reliable predictions are difficult. To address the issue of applicability and accuracy, a novel carbon price hybrid model based on decomposition, entropy, and machine learning methods is proposed, named as CEEMDAN-PE-LSTM-RVM. Adopting the advanced structure (i.e., the prediction under classification), the proposed model owns reliable performance in face of the cases with different complexity. Furthermore, the relationship between the data feature and prediction accuracy is discussed to provide a benchmark for judging the reliability of the prediction, in which the chaos degree is introduced as a feature to characterize carbon price quantitatively. The performance of the proposed model is evaluated through historical data of four representative carbon prices. The results show that the average MAPE and RMSE of the proposed model achieve 1.7027 and 0.7993, respectively, which is significantly greater than others; the proposed model owns great robustness, which is less affected by the complexity of predicted objects. Thus, the proposed model provides a reliable tool for carbon financial markets.

Detecting the dynamical instability of complex time series via partitioned entropy

A method is proposed to detect the dynamical instability of complex time series. We focus on how the partitioned entropy of an initially localized region of the attractor evolves in time and show that its growth rate corresponds to the first Lyapunov exponent. To avoid spurious detection of the dynamical instability, a criterion is further introduced to distinguish chaos from limit cycles or tori. Numerical experiments using prototypical models of chaotic systems demonstrate that the growth rate of the partitioned entropy indeed provides a good estimate of the first Lyapunov exponent. The method is also shown to be robust against observational noise and dynamical noise. Analysis of experimental data measured from a physical model of the vocal folds highlights the practical applicability of the present method to real-world data. Advantages of the present method over conventional methods are also discussed.

Eight Weeks of Vitamin C Supplementation Restores the Lost Correlation between Serum Leptin and C-reactive Protein (CRP) in Patients with Type 2 Diabetes; A Randomized, Double-blind, Parallel-group, Placebo-controlled Clinical Trial

OBJECTIVE

Inflammation is a well-described factor in the pathophysiology of type 2 diabetes mellitus (DM), which has been a suspect in the alteration of correlations between CRP and leptin in patients with type 2 DM.

AIM

This study aimed to show the effect of vitamin C as an antioxidant on the correlation of the serum levels of C-reactive protein (CRP) and leptin in patients with type 2 DM.

METHODS

We recruited 70 patients with longstanding T2DM and randomly assigned them into two groups; one received 500 mg/day of vitamin C, and the other received a placebo for eight weeks. Both groups were matched regarding baseline characteristics such as age, gender, weight, and diabetic medications.

RESULTS

Out of 70 individuals, 57 participants were left in the study. After eight weeks of follow-up, leptin level was significantly increased in the Vitamin C group (MD = 3.48 change = 24%, p-value = 0.001) but did not change in the placebo group. Other markers such as Fasting plasma glucose, HbA1c, Creatinine, uric acid, Urea, cholesterol, HDL, LDL, TG, AST, ALT, insulin, and CRP did not significantly change in both groups (p value > 0.05). The significant changes in the leptin level among the vitamin C group also remained after controlling for age, BMI, Blood pressure (BP), Triglyceride (TG), and cholesterol. Also, the correlation between serum CRP and leptin became significant in the vitamin C group after eight weeks of follow-up but not in the placebo group. (rs = 0.730, p < 0.001 vs. rs = 0.286, p-value = 0.266 in placebo group).

CONCLUSION

This study shows vitamin C can restore CRP-leptin correlation in patients with type 2 diabetes and increase serum leptin levels. More studies are needed to clarify the mechanism of this restoration.

CLINICAL TRIAL REGISTRATION NUMBER

IRCT20160811029306N1.

Rethinking the Prevalence and Relevance of Chaos in Ecology

Chaos was proposed in the 1970s as an alternative explanation for apparently noisy fluctuations in population size. Although readily demonstrated in models, the search for chaos in nature proved challenging and led many to conclude that chaos is either rare or nigh impossible to detect. However, in the intervening half-century, it has become clear that ecosystems are replete with the enabling conditions for chaos. Chaos has been repeatedly demonstrated under laboratory conditions and has been found in field data using updated detection methods. Together, these developments indicate that the apparent rarity of chaos was an artifact of data limitations and overreliance on low-dimensional population models. We invite readers to reevaluate the relevance of chaos in ecology, and we suggest that chaos is not as rare or undetectable as previously believed.

Application of 0-1 test for chaos on forward converter to study the nonlinear dynamics

DC–DC converters has significant role in the applied power electronic systems, distributed power systems, computers, home appliances and communication equipment. A converter must remain within the specified range of operation. The main goal of this paper is to discuss the nonlinear behavior of forward converter and highlighted the application of the 0-1 test by applying it on the forward converter. As forward converter may contains electronic components, which cause instability in the system. So, it is necessary to understand its behavior when specifications of components are changed. To study chaotic behavior, 0-1 test will be applied on the forward converter, which is a novel technique outperform in unearthing the subtle chaotic behavior in deterministic dynamical systems. The forward converter goes from period-1, period-2, period-4 and finally become chaotic when the load resistance is varied. This variation in the behavior of the forward converter are analysis through 0-1 test for chaos. Moreover, time series plot, phase portrait and Bifurcation diagram for forward converter is also drawn for the validation of results obtained from 0-1 test. Test algorithm is applied via MATLAB and simulation of forward converter via MultiSim by varying its load resistance.

Chaos is not rare in natural ecosystems

Chaotic dynamics are thought to be rare in natural populations but this may be due to methodological and data limitations, rather than the inherent stability of ecosystems. Following extensive simulation testing, we applied multiple chaos detection methods to a global database of 172 population time series and found evidence for chaos in >30%. In contrast, fitting traditional one-dimensional models identified <10% as chaotic. Chaos was most prevalent among plankton and insects and least among birds and mammals. Lyapunov exponents declined with generation time and scaled as the -1/6 power of body mass among chaotic populations. These results demonstrate that chaos is not rare in natural populations, indicating that there may be intrinsic limits to ecological forecasting and cautioning against the use of steady-state approaches to conservation and management.

An Alternative Approach for Identifying Nonlinear Dynamics of the Cascade Logistic-Cubic System

The 0-1 test for chaos, which is a simple binary method, has been widely used to detect the nonlinear behaviors of the non-cascade chaotic dynamics. In this paper, the validity checks of the 0-1 test for chaos to the popular cascade Logistic-Cubic (L-C) system is conducted through exploring the effects of sensitivity parameters. Results show that the periodic, weak-chaotic, and strong-chaotic states of the cascade L-C system can be effectively identified by the introduced simple method for detecting chaos. Nevertheless, the two sensitivity parameters, including the frequency ω and the amplitude α, are critical for the chaos indicator (i.e., the median of asymptotic growth rate, Km) when the cascade dynamic is detected by the method. It is found that the effect of α is more sensitive than that of ω on Km regarding the three dynamical states of the cascade L-C system. Meanwhile, it is recommended that the three states are identified according to the change of K with α from zero to ten since the periodic and weak-chaotic states cannot be identified when the α is greater than a certain constant. In addition, the modified mean square displacement Dc*(n) fails to distinguish its periodic and weak-chaotic states, whereas it can obviously distinguish the above two and strong-chaotic states. This work is therefore invaluable to gaining insight into the understanding of the complex nonlinearity of other different cascade dynamical systems with indicator comparison.

Complex bio rhythms

Complex biorhythms are characteristic of ubiquitous phenomena appearing in many disciplines of human knowledge. This Special Issue collects articles devoted to different complex biorhythms phenomena such as cardiac dynamics, Covid-19 dynamics, dynamics of neural networks, cell dynamics, and a few articles devoted to general methods. It furnishes a rich overview of the field and can stimulate and inspire further researches.

Forecasting for COVID-19 has failed

Epidemic forecasting has a dubious track-record, and its failures became more prominent with COVID-19. Poor data input, wrong modeling assumptions, high sensitivity of estimates, lack of incorporation of epidemiological features, poor past evidence on effects of available interventions, lack of transparency, errors, lack of determinacy, consideration of only one or a few dimensions of the problem at hand, lack of expertise in crucial disciplines, groupthink and bandwagon effects, and selective reporting are some of the causes of these failures. Nevertheless, epidemic forecasting is unlikely to be abandoned. Some (but not all) of these problems can be fixed. Careful modeling of predictive distributions rather than focusing on point estimates, considering multiple dimensions of impact, and continuously reappraising models based on their validated performance may help. If extreme values are considered, extremes should be considered for the consequences of multiple dimensions of impact so as to continuously calibrate predictive insights and decision-making. When major decisions (e.g. draconian lockdowns) are based on forecasts, the harms (in terms of health, economy, and society at large) and the asymmetry of risks need to be approached in a holistic fashion, considering the totality of the evidence.

Consciousness is supported by near-critical slow cortical electrodynamics

Mounting evidence suggests that during conscious states, the electrodynamics of the cortex are poised near a critical point or phase transition and that this near-critical behavior supports the vast flow of information through cortical networks during conscious states. Here, we empirically identify a mathematically specific critical point near which waking cortical oscillatory dynamics operate, which is known as the edge-of-chaos critical point, or the boundary between stability and chaos. We do so by applying the recently developed modified 0-1 chaos test to electrocorticography (ECoG) and magnetoencephalography (MEG) recordings from the cortices of humans and macaques across normal waking, generalized seizure, anesthesia, and psychedelic states. Our evidence suggests that cortical information processing is disrupted during unconscious states because of a transition of low-frequency cortical electric oscillations away from this critical point; conversely, we show that psychedelics may increase the information richness of cortical activity by tuning low-frequency cortical oscillations closer to this critical point. Finally, we analyze clinical electroencephalography (EEG) recordings from patients with disorders of consciousness (DOC) and show that assessing the proximity of slow cortical oscillatory electrodynamics to the edge-of-chaos critical point may be useful as an index of consciousness in the clinical setting.

Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning

Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use in control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), and by encoding the control signal and initial position into the losses of a physics-informed neural network (PINN), the nonlinear system is forced to exhibit the desired trajectory given the control signal. PIDOC receives signals as physics commands and learns from the chaotic data output from the nonlinear van der Pol system, where the output of the PINN is the control. Applied to a benchmark problem, PIDOC successfully implements control with a higher stochasticity for higher-order terms. PIDOC has also been proven to be capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet do not change the overall control quality. For highly nonlinear systems, PIDOC is not able to execute control with a high accuracy compared with the benchmark problem. The depth and width of the neural network structure do not greatly change the convergence of PIDOC based on case studies of van der Pol systems with low and high nonlinearities. Surprisingly, enlarging the control signal does not help to improve the control quality. The proposed framework can potentially be applied to many nonlinear systems for nonlinear controls.

Abrupt shifts in 21st-century plankton communities

Marine microbial communities sustain ocean food webs and mediate global elemental cycles. These communities will change with climate; these changes can be gradual or foreseeable but likely have much more substantial consequences when sudden and unpredictable. In a complex virtual marine microbial ecosystem, we find that climate change–driven shifts over the 21st century are often abrupt, large in amplitude and extent, and unpredictable using standard early warning signals. Phytoplankton with unique resource needs, especially fast-growing species such as diatoms, are more prone to abrupt shifts. Abrupt shifts in biomass, productivity, and community structure are concentrated in Atlantic and Pacific subtropics. Abrupt changes in environmental variables such as temperature and nutrients rarely precede these ecosystem shifts, indicating that rapid community restructuring can occur in response to gradual environmental changes, particularly in nutrient supply rate ratios.

Chaotic Dynamics by Some Quadratic Jerk Systems

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

p27Kip1, an Intrinsically Unstructured Protein with Scaffold Properties

The Cyclin-dependent kinase (CDK) regulator p27^Kip1 is a gatekeeper of G1/S transition. It also regulates G2/M progression and cytokinesis completion, via CDK-dependent or -independent mechanisms. Recently, other important p27^Kip1 functions have been described, including the regulation of cell motility and migration, the control of cell differentiation program and the activation of apoptosis/autophagy. Several factors modulate p27^Kip1 activities, including its level, cellular localization and post-translational modifications. As a matter of fact, the protein is phosphorylated, ubiquitinated, SUMOylated, O-linked N-acetylglicosylated and acetylated on different residues. p27^Kip1 belongs to the family of the intrinsically unstructured proteins and thus it is endowed with a large flexibility and numerous interactors, only partially identified. In this review, we look at p27^Kip1 properties and ascribe part of its heterogeneous functions to the ability to act as an anchor or scaffold capable to participate in the construction of different platforms for modulating cell response to extracellular signals and allowing adaptation to environmental changes.

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks

Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, [Formula: see text], that determines the strength of the correlation of the noise. To predict [Formula: see text] the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the [Formula: see text] value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same [Formula: see text] parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.

A review of dynamical systems approaches for the detection of chaotic attractors in cancer networks

Cancers are complex dynamical systems. They remain the leading cause of disease-related pediatric mortality in North America. To overcome this burden, we must decipher the state-space attractor dynamics of gene expression patterns and protein oscillations orchestrated by cancer stemness networks. The review provides an overview of dynamical systems theory to steer cancer research in pattern science. While most of our current tools in network medicine rely on statistical correlation methods, causality inference remains primitively developed. As such, a survey of attractor reconstruction methods and machine algorithms for the detection of causal structures applicable in experimentally derived time series cancer datasets is presented. A toolbox of complex systems approaches are discussed for reconstructing the signaling state space of cancer networks, interpreting causal relationships in their time series gene expression patterns, and assisting clinical decision making in computational oncology. As a proof of concept, the applicability of some algorithms are demonstrated on pediatric brain cancer datasets and the requirement of their time series analysis is highlighted.

Nonlinear delay differential equations and their application to modeling biological network motifs

Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.

Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.

Invasive dynamics for a predator-prey system with Allee effect in both populations and a special emphasis on predator mortality

We considered a non-linear predator-prey model with an Allee effect on both populations on a two spatial dimension reaction-diffusion setup. Special importance to predator mortality was given as it may be often controlled through human-made harvesting processes. The local dynamics of the model was studied through boundedness, equilibrium, and stability analysis. An extensive numerical stability analysis was performed and found that bi-stability is not possible for the non-spatial model. By analyzing the spatial model, we found the condition for successful invasion and the persistence region of the species based on the predator Allee effect and its mortality parameter. Four different dynamics in this region of the parameter space are mainly explored. First, the Allee effect on both populations leads to various new types of species spread. Second, for a high value of per-capita growth rate, two completely new spreads (e.g., sun surface, colonial) have been found depending on the Allee effect parameter. Third, the Allee coefficient on the predator population leads to spatiotemporal chaos via a patchy spread for both linear and quadratic mortality rates. Finally, a more rigorous analysis is performed to study the chaotic nature of the system within the whole persistence domain. We have studied the possibility of chaos through temporal variation in different invasion regions. Furthermore, the chaotic fluctuation is studied through the sensitivity of initial conditions and by investigating the dominant Lyapunov exponent value.

Cancer: A turbulence problem

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Application of nonlinear dynamics to cancer ecosystems.

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Chemical turbulence and strange attractor models in tumor growth, invasion and pattern formation are investigated.

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Computational algorithms for detecting such structures are proposed.

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Complex systems applications to cancer dynamics.

Cancers are complex, adaptive ecosystems. They remain the leading cause of disease-related death among children in North America. As we approach computational oncology and Deep Learning Healthcare, our mathematical models of cancer dynamics must be revised. Recent findings support the perspective that cancer-microenvironment interactions may consist of chaotic gene expressions and turbulent protein flows during pattern formation. As such, cancer pattern formation, protein-folding and metastatic invasion are discussed herein as processes driven by chemical turbulence within the framework of complex systems theory. To conclude, cancer stem cells are presented as strange attractors of the Waddington landscape.

Approaching Adversarial Example Classification with Chaos Theory

Adversarial examples are one of the most intriguing topics in modern deep learning. Imperceptible perturbations to the input can fool robust models. In relation to this problem, attack and defense methods are being developed almost on a daily basis. In parallel, efforts are being made to simply pointing out when an input image is an adversarial example. This can help prevent potential issues, as the failure cases are easily recognizable by humans. The proposal in this work is to study how chaos theory methods can help distinguish adversarial examples from regular images. Our work is based on the assumption that deep networks behave as chaotic systems, and adversarial examples are the main manifestation of it (in the sense that a slight input variation produces a totally different output). In our experiments, we show that the Lyapunov exponents (an established measure of chaoticity), which have been recently proposed for classification of adversarial examples, are not robust to image processing transformations that alter image entropy. Furthermore, we show that entropy can complement Lyapunov exponents in such a way that the discriminating power is significantly enhanced. The proposed method achieves 65% to 100% accuracy detecting adversarials with a wide range of attacks (for example: CW, PGD, Spatial, HopSkip) for the MNIST dataset, with similar results when entropy-changing image processing methods (such as Equalization, Speckle and Gaussian noise) are applied. This is also corroborated with two other datasets, Fashion-MNIST and CIFAR 19. These results indicate that classifiers can enhance their robustness against the adversarial phenomenon, being applied in a wide variety of conditions that potentially matches real world cases and also other threatening scenarios.

A Review of the Serrated-Flow Phenomenon and Its Role in the Deformation Behavior of High-Entropy Alloys

High-entropy alloys (HEAs) are a novel class of alloys that have many desirable properties. The serrated flow that occurs in high-entropy alloys during mechanical deformation is an important phenomenon since it can lead to significant changes in the microstructure of the alloy. In this article, we review the recent findings on the serration behavior in a variety of high-entropy alloys. Relationships among the serrated flow behavior, composition, microstructure, and testing condition are explored. Importantly, the mechanical-testing type (compression/tension), testing temperature, applied strain rate, and serration type for certain high-entropy alloys are summarized. The literature reveals that the serrated flow can be affected by experimental conditions such as the strain rate and test temperature. Furthermore, this type of phenomenon has been successfully modeled and analyzed, using several different types of analytical methods, including the mean-field theory formalism and the complexity-analysis technique. Importantly, the results of the analyses show that the serrated flow in HEAs consists of complex dynamical behavior. It is anticipated that this review will provide some useful and clarifying information regarding the serrated-flow mechanisms in this material system. Finally, suggestions for future research directions in this field are proposed, such as the effects of irradiation, additives (such as C and Al), the presence of nanoparticles, and twinning on the serrated flow behavior in HEAs.