In conscious states, the electrodynamics of the cortex are reported to work near a critical point or phase transition of chaotic dynamics, known as the edge-of-chaos, representing a boundary between stability and chaos. Transitions away from this boundary disrupt cortical information processing and induce a loss of consciousness. The entropy of the electroencephalogram (EEG) is known to decrease as the level of anesthesia deepens. However, whether the chaotic dynamics of electroencephalographic activity shift from this boundary to the side of stability or the side of chaotic enhancement during anesthesia-induced loss of consciousness remains poorly understood. We investigated the chaotic properties of EEGs at two different depths of clinical anesthesia using the maximum Lyapunov exponent, which is mathematically regarded as a formal measure of chaotic nature, using the Rosenstein algorithm. In 14 adult patients, 12 s of electroencephalographic signals were selected during two depths of clinical anesthesia (sevoflurane concentration 2% as relatively deep anesthesia, sevoflurane concentration 0.6% as relatively shallow anesthesia). Lyapunov exponents, correlation dimensions and approximate entropy were calculated from these electroencephalographic signals. As a result, maximum Lyapunov exponent was generally positive during sevoflurane anesthesia, and both maximum Lyapunov exponents and correlation dimensions were significantly greater during deep anesthesia than during shallow anesthesia despite reductions in approximate entropy. The chaotic nature of the EEG might be increased at clinically deeper inhalational anesthesia, despite the decrease in randomness as reflected in the decreased entropy, suggesting a shift to the side of chaotic enhancement under anesthesia.

Fractional-order (FO) chaotic systems exhibit random sequences of significantly greater complexity when compared to integer-order systems. This feature makes FO chaotic systems more secure against various attacks in image cryptosystems. In this study, the dynamical characteristics of the FO Sprott K chaotic system are thoroughly investigated by phase planes, bifurcation diagrams, and Lyapunov exponential spectrums to be utilized in biometric iris image encryption. It is proven with the numerical studies the Sprott K system demonstrates chaotic behaviour when the order of the system is selected as 0.9. Afterward, the introduced FO Sprott K chaotic system-based biometric iris image encryption design is carried out in the study. According to the results of the statistical and attack analyses of the encryption design, the secure transmission of biometric iris images is successful using the proposed encryption design. Thus, the FO Sprott K chaotic system can be employed effectively in chaos-based encryption applications.

This study investigated how data series length and gaps in human kinematic data impact the accuracy of Lyapunov exponents (LyE) calculations with and without cubic spline interpolation. Kinematic time series were manipulated to create various data series lengths (28% and 100% of original) and gap durations (0.05-0.20 s). Longer gaps generally resulted in significantly higher LyE% error values in each plane in noninterpolated data. During cubic spline interpolation, only the 0.20-second gap in frontal plane data resulted in a significantly higher LyE% error. Data series length did not significantly affect LyE% error in noninterpolated data. During cubic spline interpolation, sagittal plane LyE% errors were significantly higher at shorter versus longer data series lengths. These findings suggest that not interpolating gaps in data could lead to erroneously high LyE values and mischaracterization of movement variability. When applying cubic spline, a long gap length (0.20 s) in the frontal plane or a short sagittal plane data series length (1000 data points) could also lead to erroneously high LyE values and mischaracterization of movement variability. These insights emphasize the necessity of detailed reporting on gap durations, data series lengths, and interpolation techniques when characterizing human movement variability using LyE values.

Epilepsy surgery is effective for patients with medication-resistant seizures, however 20-40% of them are not seizure free after surgery. Aim of this study is to evaluate the role of linear and non-linear EEG features to predict post-surgical outcome. We included 123 paediatric patients who underwent epilepsy surgery at Bambino Gesù Children Hospital (January 2009-April 2020). All patients had long term video-EEG monitoring. We analysed 1-min scalp interictal EEG (wakefulness and sleep) and extracted 13 linear and non-linear EEG features (power spectral density (PSD), Hjorth, approximate entropy, permutation entropy, Lyapunov and Hurst value). We used a logistic regression (LR) as feature selection process. To quantify the correlation between EEG features and surgical outcome we used an artificial neural network (ANN) model with 18 architectures. LR revealed a significant correlation between PSD of alpha band (sleep), Mobility index (sleep) and the Hurst value (sleep and awake) with outcome. The fifty-four ANN models gave a range of accuracy (46-65%) in predicting outcome. Within the fifty-four ANN models, we found a higher accuracy (64.8% ± 7.6%) in seizure outcome prediction, using features selected by LR. The combination of PSD of alpha band, mobility and the Hurst value positively correlate with good surgical outcome.

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.
UNASSIGNED: The online version contains supplementary material available at 10.1007/s11071-023-08285-1.

To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When do the signs of the invasion growth rates determine coexistence? (ii) When signs are sufficient, which invasion growth rates need to be positive? We focus on deterministic models and equate coexistence with permanence, i.e., a global attractor bounded away from extinction. For models satisfying certain technical assumptions, we introduce invasion graphs where vertices correspond to proper subsets of species (communities) supporting an ergodic measure and directed edges correspond to potential transitions between communities due to invasions by missing species. These directed edges are determined by the signs of invasion growth rates. When the invasion graph is acyclic (i.e. there is no sequence of invasions starting and ending at the same community), we show that permanence is determined by the signs of the invasion growth rates. In this case, permanence is characterized by the invasibility of all [Formula: see text] communities, i.e., communities without species i where all other missing species have negative invasion growth rates. To illustrate the applicability of the results, we show that dissipative Lotka-Volterra models generically satisfy our technical assumptions and computing their invasion graphs reduces to solving systems of linear equations. We also apply our results to models of competing species with pulsed resources or sharing a predator that exhibits switching behavior. Open problems for both deterministic and stochastic models are discussed. Our results highlight the importance of using concepts about community assembly to study coexistence.

OBJECTIVE: Noninvasive screening of hypo- and hyperkalemia can prevent fatal arrhythmia in end-stage renal disease (ESRD) patients, but current methods for monitoring of serum potassium (K+) have important limitations. We investigated changes in nonlinear dynamics and morphology of the T wave in the electrocardiogram (ECG) of ESRD patients during hemodialysis (HD), assessing their relationship with K+ and designing a K+ estimator.
METHODS: ECG recordings from twenty-nine ESRD patients undergoing HD were processed. T waves in 2-min windows were extracted at each hour during an HD session as well as at 48 h after HD start. T wave nonlinear dynamics were characterized by two indices related to the maximum Lyapunov exponent (λt, λwt) and a divergence-related index (η). Morphological variability in the T wave was evaluated by three time warping-based indices (dw, reflecting morphological variability in the time domain, and da and daNL, in the amplitude domain). K+was measured from blood samples extracted during and after HD. Stage-specific and patient-specific K+ estimators were built based on the quantified indices and leave-one-out cross-validation was performed separately for each of the estimators.
RESULTS: The analyzed indices showed high inter-individual variability in their relationship with K+. Nevertheless, all of them had higher values at the HD start and 48 h after it, corresponding to the highest K+. The indices η and dw were the most strongly correlated with K+ (median Pearson correlation coefficient of 0.78 and 0.83, respectively) and were used in univariable and multivariable linear K+ estimators. Agreement between actual and estimated K+ was confirmed, with averaged errors over patients and time points being 0.000 ± 0.875 mM and 0.046 ± 0.690 mM for stage-specific and patient-specific multivariable K+ estimators, respectively.
CONCLUSIONS: ECG descriptors of T wave nonlinear dynamics and morphological variability allow noninvasive monitoring of K+ in ESRD patients.
CONCLUSIONS: ECG markers have the potential to be used for hypo- and hyperkalemia screening in ESRD patients.