Display Complex Dynamics Research Articles

Effects of Colored Noise in the Dynamic Motions and Conformational Exploration of Enzymes

The intracellular environment displays complex dynamics influenced by factors such as molecular crowding and the low Reynolds number of the cytoplasm. Enzymes exhibiting active matter properties further heighten this complexity which can lead to memory effects. Molecular simulations often neglect these factors, treating the environment as a “thermal bath” using the Langevin equation (LE) with white noise. One way to consider these factors is by using colored noise instead within the generalized Langevin equation (GLE) framework, which allows for the incorporation of memory effects that have been observed in experimental data. We investigated the structural and dynamic differences in Shikimate kinase (SK) using LE and GLE simulations. Our results suggest that GLE simulations, which reveal significant changes, could be utilized for assessing conformational motions’ impact on catalytic reactions.

A transformer model for cause-specific hazard prediction

BackgroudModelling discrete-time cause-specific hazards in the presence of competing events and non-proportional hazards is a challenging task in many domains. Survival analysis in longitudinal cohorts often requires such models; notably when the data is gathered at discrete points in time and the predicted events display complex dynamics. Current models often rely on strong assumptions of proportional hazards, that is rarely verified in practice; or do not handle sequential data in a meaningful way. This study proposes a Transformer architecture for the prediction of cause-specific hazards in discrete-time competing risks. Contrary to Multilayer perceptrons that were already used for this task (DeepHit), the Transformer architecture is especially suited for handling complex relationships in sequential data, having displayed state-of-the-art performance in numerous tasks with few underlying assumptions on the task at hand.ResultsUsing synthetic datasets of 2000–50,000 patients, we showed that our Transformer model surpassed the CoxPH, PyDTS, and DeepHit models for the prediction of cause-specific hazard, especially when the proportional assumption did not hold. The error along simulated time outlined the ability of our model to anticipate the evolution of cause-specific hazards at later time steps where few events are observed. It was also superior to current models for prediction of dementia and other psychiatric conditions in the English longitudinal study of ageing cohort using the integrated brier score and the time-dependent concordance index. We also displayed the explainability of our model’s prediction using the integrated gradients method.ConclusionsOur model provided state-of-the-art prediction of cause-specific hazards, without adopting prior parametric assumptions on the hazard rates. It outperformed other models in non-proportional hazards settings for both the synthetic dataset and the longitudinal cohort study. We also observed that basic models such as CoxPH were more suited to extremely simple settings than deep learning models. Our model is therefore especially suited for survival analysis on longitudinal cohorts with complex dynamics of the covariate-to-outcome relationship, which are common in clinical practice. The integrated gradients provided the importance scores of input variables, which indicated variables guiding the model in its prediction. This model is ready to be utilized for time-to-event prediction in longitudinal cohorts.

Emergence of complex oscillatory dynamics in the neuronal networks with long activity time of inhibitory synapses

The brain displays complex dynamics, including collective oscillations, and extensive research has been conducted to understand their generation. However, our understanding of how biological constraints influence these oscillations is incomplete. This study investigates the essential properties of neuronal networks needed to generate oscillations resembling those in the brain. A simple discrete-time model of interconnected excitable elements is developed, capable of closely resembling the complex oscillations observed in biological neural networks. In the model, synaptic connections remain active for a duration exceeding individual neuron activity. We show that the inhibitory synapses must exhibit longer activity than excitatory synapses to produce a diverse range of the dynamical states, including biologically plausible oscillations. Upon meeting this condition, the transition between different dynamical states can be controlled by external stochastic input to the neurons. The study provides a comprehensive explanation for the emergence of distinct dynamical states in neural networks based on specific parameters.

Permutation entropy-based characterization of speckle patterns generated by semiconductor laser light

Semiconductor lasers with optical feedback are stochastic nonlinear systems that can display complex dynamics and abrupt changes when their operation conditions change. Even very small changes can lead to large variations in the spatial and spectral properties of the laser emission. This makes a semiconductor laser with feedback an ideal system for conducting controlled experiments to test data analysis tools to detect and characterize transitions. Here, we identify feedback-induced transitions by analyzing speckle patterns that are generated after the laser light propagates in an optical fiber. Speckle patterns result from the interference of multiple modes, and their statistical properties are understood, but a direct mathematical model does not exist. Here we show the versatility of the correlation length and the permutation entropy as measures for characterizing speckle patterns. Combining entropy and correlation analysis with speckle contrast analysis, we uncover changes that occur when the laser current increases from below to well above the threshold, which unveils the effects of optical feedback on the coherence of the laser emission.

Karyotype as code of codes: An inheritance platform to shape the pattern and scale of evolution

Organismal evolution displays complex dynamics in phase and scale which seem to trend towards increasing biocomplexity and diversity. For over a century, such amazing dynamics have been cleverly explained by the apparently straightforward mechanism of natural selection: all diversification, including speciation, results from the gradual accumulation of small beneficial or near-neutral alterations over long timescales. However, although this has been widely accepted, natural selection makes a crucial assumption that has not yet been validated. Specifically, the informational relationship between small microevolutionary alterations and large macroevolutionary changes in natural selection is unclear. To address the macroevolution-microevolution relationship, it is crucial to incorporate the concept of organic codes and particularly the “karyotype code” which defines macroevolutionary changes. This concept piece examines the karyotype from the perspective of two-phased evolution and four key components of information management. It offers insight into how the karyotype creates and preserves information that defines the scale and phase of macroevolution and, by extension, microevolution. We briefly describe the relationship between the karyotype code, the genetic code, and other organic codes in the context of generating evolutionary novelties in macroevolution and imposing constraints on them as biological routines in microevolution. Our analyses suggest that karyotype coding preserves many organic codes by providing system-level inheritance, and similar analyses are needed to classify and prioritize a large number of different organic codes based on the phases and scales of evolution. Finally, the importance of natural information self-creation is briefly discussed, leading to a call to integrate information and time into the relationship between matter and energy.

Embedding the Dynamics of Forced Nonlinear Systems in Multistable Memristor Circuits

A well-known feature of memristors is that they makes the circuit dynamics much richer than that generated by classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$RLC$ </tex-math></inline-formula> circuits containing nonlinear resistors. In the case of circuits with ideal memristors, such a multistability property, i.e., the coexistence of many different attractors for a fixed set of parameters, is connected to the fact that the state space is composed of a continuum of invariant manifolds where either convergent or oscillatory and more complex behaviors can be displayed. In this paper we investigate the possibility of designing memristor circuits where known attractors are embedded into the invariant manifolds. We consider a class of forced nonlinear systems containing several systems which are known to display complex dynamics, and we investigate under which conditions the dynamics of any given system of the class can be reproduced by a circuit composed of a two-terminal (one port) element connected to a flux-controlled memristor. It is shown that an input-less circuit is capable to replicate the system attractors generated by varying the constant forcing input, once the parameters of the two-terminal element and the memristor nonlinear characteristic are suitably selected. Indeed, there is a one-one correspondence between the dynamics generated by the nonlinear system for a constant value of the input and that displayed on one of the invariant manifolds of the input-less memristor circuit. Some extensions concerning the case of non-constant forcing terms and the use of charge-controlled memristors are also provided. The results are illustrated via FitzHugh-Nagumo model and Duffing oscillator.

Emergent dynamics in excitable flow systems.

Flow networks can describe many natural and artificial systems. We present a model for a flow system that allows for volume accumulation, includes conduits with a nonlinear relation between current and pressure difference, and can be applied to networks of arbitrary topology. The model displays complex dynamics, including self-sustained oscillations in the absence of any dynamics in the inputs and outputs. In this work we analytically show the origin of self-sustained oscillations for the one-dimensional case. We numerically study the behavior of systems of arbitrary topology under different conditions: we discuss their excitability, the effect of different boundary conditions, and wave propagation when the network has regions of conduits with linear conductance.

Complexity in psychological self-ratings: implications for research and practice

BackgroundPsychopathology research is changing focus from group-based “disease models” to a personalized approach inspired by complex systems theories. This approach, which has already produced novel and valuable insights into the complex nature of psychopathology, often relies on repeated self-ratings of individual patients. So far, it has been unknown whether such self-ratings, the presumed observables of the individual patient as a complex system, actually display complex dynamics. We examine this basic assumption of a complex systems approach to psychopathology by testing repeated self-ratings for three markers of complexity: memory, the presence of (time-varying) short- and long-range temporal correlations; regime shifts, transitions between different dynamic regimes; and sensitive dependence on initial conditions, also known as the “butterfly effect,” the divergence of initially similar trajectories.MethodsWe analyzed repeated self-ratings (1476 time points) from a single patient for the three markers of complexity using Bartels rank test, (partial) autocorrelation functions, time-varying autoregression, a non-stationarity test, change point analysis, and the Sugihara-May algorithm.ResultsSelf-ratings concerning psychological states (e.g., the item “I feel down”) exhibited all complexity markers: time-varying short- and long-term memory, multiple regime shifts, and sensitive dependence on initial conditions. Unexpectedly, self-ratings concerning physical sensations (e.g., the item “I am hungry”) exhibited less complex dynamics and their behavior was more similar to random variables.ConclusionsPsychological self-ratings display complex dynamics. The presence of complexity in repeated self-ratings means that we have to acknowledge that (1) repeated self-ratings yield a complex pattern of data and not a set of (nearly) independent data points, (2) humans are “moving targets” whose self-ratings display non-stationary change processes including regime shifts, and (3) long-term prediction of individual trajectories may be fundamentally impossible. These findings point to a limitation of popular statistical time series models whose assumptions are violated by the presence of these complexity markers. We conclude that a complex systems approach to mental health should appreciate complexity as a fundamental aspect of psychopathology research by adopting the models and methods of complexity science. Promising first steps in this direction, such as research on real-time process monitoring, short-term prediction, and just-in-time interventions, are discussed.

Experimental Validation of Multiple Attractors in a Simple Chaotic Circuit

In this paper, a simple jerk circuit that allows studying the dynamical behavior of a three-dimensional autonomous chaotic system with only one nonlinear term is further investigated by numerical simulations and experimental validation. Depending on a single tuning parameter, the chaotic system is theoretically studied using standard techniques such as equilibrium analysis, bifurcation diagram, and Lyapunov exponents. Subsequently, the circuit that models the chaotic system is implemented to validate theoretical prediction experimentally. Despite the simple structure of the jerk circuit, experimental study of Fourier spectra has shown that the jerk circuit displays complex dynamics characterized by periodic limit cycles and aperiodic strange attractors. In addition, the jerk circuit has exhibited a wide tuning range and experimental results have shown good agreement with theoretical prediction except for few cases where numerical simulation has failed to accurately match experimental results due to sensitivity to initial conditions which is a signature of chaotic nonlinear systems.

Complex dynamics of a novel 3D autonomous system without linear terms having line of equilibria: coexisting bifurcations and circuit design

In recent years, an increasing interest has been devoted to the discovery of new chaotic systems with special properties. In this paper, a novel and singular 3D autonomous system without linear terms is introduced. The singularity of the model is that it is dissipative, possesses rotation symmetry and line of equilibria thus displays complex dynamics. The nonlinear behaviour of the introduced model is studied in terms of bifurcation diagrams, Lyapunov exponent plots, time series, frequency spectra two-parameter diagrams, Lyapunov stability diagrams as well as basins of attraction. Some interesting phenomena are found including, for instance, periodic oscillations, chaotic oscillations, periodic windows, symmetry restoring crises scenario, the coexistence of multiple bifurcations and offset-boosting property while monitoring the system parameters. Coexistence of attractors discovered in this work includes up to six competing attractors with one point attractor. Compared to some few cases previously reported (system without linear terms) (Xu and Wang in Opt Int J Light Electron Opt 125:2526–2530, 2014. https://doi.org/10.1007/s11071-016-3170-x; Kengne in Commun Nonlinear Sci Numer Simul 52:62–76, 2017. https://doi.org/10.1016/j.cnsns.2017.04.017; Mobayen et al. in Int J Syst Sci 49:1–15, 2018. https://doi.org/10.1080/00207721.2017.1410251; Zhang et al. in Int J Non-linear Mech 106:1–12, 2018. https://doi.org/10.1016/j.ijnonlinmec.2018.08.012; Pham et al. in Chaos Solitons Fractals 120:213–221, 2019. https://doi.org/10.1016/j.chaos.2019.02.003), the model considered in this work is the only one in which such type of complex dynamics has already been found. A suitable analog simulator (electronics circuit) is designed and used to support the theoretical analysis.

Wavelet Analysis of the Effect of Injection Strategies on Cycle to Cycle Variation GDI Optical Engine under Clean and Fouled Injector

High fluctuation in cyclic variations influences engine combustion negatively, leading to higher fuel consumption, lower performance, and drivability problems. This paper examines the impacts of injection strategies (injection pressure, injection timing and injection duration) on the cyclic variation of gasoline spark ignition (SI) optical engine under clean and fouled injectors. The principal oscillatory modes of the cycle to cycle variation have been identified, and the engine cycles over which these modes may persist are described. Through the wavelet power spectrum, the presence of short, intermediate and long-term periodicities in the pressure signal have been detected. It was noticed that depending on the clean and fouled injector, the long and intermediate-term periodicities may span many cycles, whereas the short-period oscillations tend to appear intermittently. Information of these periodicities could be helpful to promote efficient control strategies for better combustion. The outcomes report that the cyclic variation of the IMEP happens on multiple timescales, and thus, display complex dynamics. As the fouled injector used, mainly intermittent high-frequency fluctuations are recognized.

Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization

Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.

How seasonal forcing influences the complexity of a predator-prey system

Almost all population communities are strongly influenced by their seasonally varying living environments. We investigate the influence of seasons on populations via a periodically forced predator-prey system with a nonmonotonic functional response. We study four seasonality mechanisms via a continuation technique. When the natural death rate is periodically varied, we get six different bifurcation diagrams corresponding to different bifurcation cases of the unforced system. If the carrying capacity is periodic, two different bifurcation diagrams are obtained. Here we cannot get a 'universal diagram' like the one in the periodically forced system with monotonic Holling type Ⅱ functional response; that is, the two elementary seasonality mechanisms have different effects on the population. When both the natural death rate and the carrying capacity are forced with two different seasonality mechanisms, the phenomena that arise are to some extent different. The bifurcation results also show that each seasonality mechanism can display complex dynamics such as multiple attractors including stable cycles of different periods, quasi-periodic solutions, chaos, switching between these attractors and catastrophic transitions. In addition, we give some orbits in phase space and corresponding Poincaré sections to illustrate different attractors.

Spiking and bursting patterns of fractional-order Izhikevich model

Bursting and spiking oscillations play major roles in processing and transmitting information in the brain through cortical neurons that respond differently to the same signal. These oscillations display complex dynamics that might be produced by using neuronal models and varying many model parameters. Recent studies have shown that models with fractional order can produce several types of history-dependent neuronal activities without the adjustment of several parameters. We studied the fractional-order Izhikevich model and analyzed different kinds of oscillations that emerge from the fractional dynamics. The model produces a wide range of neuronal spike responses, including regular spiking, fast spiking, intrinsic bursting, mixed mode oscillations, regular bursting and chattering, by adjusting only the fractional order. Both the active and silent phase of the burst increase when the fractional-order model further deviates from the classical model. For smaller fractional order, the model produces memory dependent spiking activity after the pulse signal turned off. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. On the network level, the response of the neuronal network shifts from random to scale-free spiking. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents.

Stable and Dynamic Coding for Working Memory in Primate Prefrontal Cortex.

Working memory (WM) provides the stability necessary for high-level cognition. Influential theories typically assume that WM depends on the persistence of stable neural representations, yet increasing evidence suggests that neural states are highly dynamic. Here we apply multivariate pattern analysis to explore the population dynamics in primate lateral prefrontal cortex (PFC) during three variants of the classic memory-guided saccade task (recorded in four animals). We observed the hallmark of dynamic population coding across key phases of a working memory task: sensory processing, memory encoding, and response execution. Throughout both these dynamic epochs and the memory delay period, however, the neural representational geometry remained stable. We identified two characteristics that jointly explain these dynamics: (1) time-varying changes in the subpopulation of neurons coding for task variables (i.e., dynamic subpopulations); and (2) time-varying selectivity within neurons (i.e., dynamic selectivity). These results indicate that even in a very simple memory-guided saccade task, PFC neurons display complex dynamics to support stable representations for WM.SIGNIFICANCE STATEMENT Flexible, intelligent behavior requires the maintenance and manipulation of incoming information over various time spans. For short time spans, this faculty is labeled “working memory” (WM). Dominant models propose that WM is maintained by stable, persistent patterns of neural activity in prefrontal cortex (PFC). However, recent evidence suggests that neural activity in PFC is dynamic, even while the contents of WM remain stably represented. Here, we explored the neural dynamics in PFC during a memory-guided saccade task. We found evidence for dynamic population coding in various task epochs, despite striking stability in the neural representational geometry of WM. Furthermore, we identified two distinct cellular mechanisms that contribute to dynamic population coding.

Stability of Cross-Feeding Polymorphisms in Microbial Communities

Cross-feeding, a relationship wherein one organism consumes metabolites excreted by another, is a ubiquitous feature of natural and clinically-relevant microbial communities and could be a key factor promoting diversity in extreme and/or nutrient-poor environments. However, it remains unclear how readily cross-feeding interactions form, and therefore our ability to predict their emergence is limited. In this paper we developed a mathematical model parameterized using data from the biochemistry and ecology of an E. coli cross-feeding laboratory system. The model accurately captures short-term dynamics of the two competitors that have been observed empirically and we use it to systematically explore the stability of cross-feeding interactions for a range of environmental conditions. We find that our simple system can display complex dynamics including multi-stable behavior separated by a critical point. Therefore whether cross-feeding interactions form depends on the complex interplay between density and frequency of the competitors as well as on the concentration of resources in the environment. Moreover, we find that subtly different environmental conditions can lead to dramatically different results regarding the establishment of cross-feeding, which could explain the apparently unpredictable between-population differences in experimental outcomes. We argue that mathematical models are essential tools for disentangling the complexities of cross-feeding interactions.

The transmembrane protein Crumbs displays complex dynamics during follicular morphogenesis and is regulated competitively by Moesin and aPKC

The transmembrane protein Crumbs (Crb) functions in apical polarity and epithelial integrity. To better understand its role in epithelial morphogenesis, we examined Crb localization and dynamics in the late follicular epithelium of Drosophila. Crb was unexpectedly dynamic during middle-to-late stages of egg chamber development, being lost from the marginal zone (MZ) in stage 9 before abruptly returning at the end of stage 10b, then undergoing a pulse of endocytosis in stage 12. The reappearance of MZ Crb is necessary to maintain an intact adherens junction and MZ. Although Crb has been proposed to interact through its juxtamembrane domain with Moesin (Moe), a FERM domain protein that regulates the cortical actin cytoskeleton, the functional significance of this interaction is poorly understood. We found that whereas the Crb juxtamembrane domain was not required for adherens junction integrity, it was necessary for MZ localization of Moe, aPKC and F-actin. Furthermore, Moe and aPKC functioned antagonistically, suggesting that Moe limits Crb levels by reducing its interactions with the apical Par network. Additionally, Moe mutant cells lost Crb from the apical membrane and accumulated excess Crb at the MZ, suggesting that Moe regulates Crb distribution at the membrane. Together, these studies reveal reciprocal interactions between Crb, Moe and aPKC during cellular morphogenesis.

The transmembrane protein Crumbs displays complex dynamics during follicular morphogenesis and is regulated competitively by Moesin and aPKC.

The transmembrane protein Crumbs (Crb) functions in apical polarity and epithelial integrity. To better understand its role in epithelial morphogenesis, we examined Crb localization and dynamics in the late follicular epithelium of Drosophila. Crb was unexpectedly dynamic during middle-to-late stages of egg chamber development, being lost from the marginal zone (MZ) in stage 9 before abruptly returning at the end of stage 10b, then undergoing a pulse of endocytosis in stage 12. The reappearance of MZ Crb is necessary to maintain an intact adherens junction and MZ. Although Crb has been proposed to interact through its juxtamembrane domain with Moesin (Moe), a FERM domain protein that regulates the cortical actin cytoskeleton, the functional significance of this interaction is poorly understood. We found that whereas the Crb juxtamembrane domain was not required for adherens junction integrity, it was necessary for MZ localization of Moe, aPKC and F-actin. Furthermore, Moe and aPKC functioned antagonistically, suggesting that Moe limits Crb levels by reducing its interactions with the apical Par network. Additionally, Moe mutant cells lost Crb from the apical membrane and accumulated excess Crb at the MZ, suggesting that Moe regulates Crb distribution at the membrane. Together, these studies reveal reciprocal interactions between Crb, Moe and aPKC during cellular morphogenesis.

Nonlinear Dynamics of Hysteresis-Based Load Controls

Direct load control can be achieved by varying the hysteresis band of switchable loads, thereby changing their on/off durations. Such hysteresis-based control methods may, however, display complex dynamics that must be thoroughly understood in order to design safe control mechanisms. This paper explores the dynamical behavior of a group of hysteresis-based PEV chargers. Of interest is the change in the total power demand of the group as the hysteresis- band limits are varied. A detailed state-space model is used to capture the dynamics of the load aggregation. This model suggests that for certain control inputs, e.g. periodic ramp signals, the system may display rich dynamical behavior. It is observed that structural stability of the system may be disrupted as certain characteristics of the input signal are varied. The paper explores these phenomena through a bifurcation analysis of the load population dynamics. The results identify performance limitations that govern the responsiveness of fast-acting demand control.

A Toolbox of Fluorescence Microscopic Approaches Reveals Dynamics and Assembly of a Membrane-Associated Protein

A Toolbox of Fluorescence Microscopic Approaches Reveals Dynamics and Assembly of a Membrane-Associated Protein