Dynamical Systems Perspective Research Articles

Pulsing in the Ahu‘ail‘au pond–spillway system during the 2018 Klauea eruption: a dynamical systems perspective

During the 2018 K $\unicode{x012B}$ lauea lower East Rift Zone eruption, lava from 24 fissures inundated more than 8000 acres of land, destroying more than 700 structures over three months. Eruptive activity eventually focused at a single vent characterized by a continuously fed lava pond that was drained by a narrow spillway into a much wider, slower channelized flow. The spillway exhibited intervals of ‘pulsing’ behaviour in which the lava depth and velocity were observed to oscillate on time scales of several minutes. At the time, this was attributed to variations in vesiculation originating at depth. Here, we construct a toy fluid dynamical model of the pond–spillway system, and present an alternative hypothesis in which pulsing is generated at the surface, within this system. We posit that the appearance of pulsing is due to a supercritical Hopf bifurcation driven by an increase in the Reynolds number. Asymptotics for the limit cycle near the bifurcation point are derived with averaging methods and compare favourably with the cycle periodicity. Because oscillations in the pond were not observable directly due to the elevation of the cone rim and an obscuring volcanic plume, we model the observations using a spatially averaged Saint-Venant model of the spillway forced by the pond oscillator. The predicted spillway cycle periodicity and waveforms compare favourably with observations made during the eruption. The unusually well-documented nature of this eruption enables estimation of the viscosity of the erupting lava.

Reifying actions into artifacts: process–object duality from an embodied perspective on mathematics learning

AbstractGrasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process–object transitions and externalist approaches that stress mathematical artifacts—such as physical manipulatives and formulas—as constituting mathematical objects. We search for a view on process–object duality that overcomes the dualism of mind and body. One such approach is commognition that describes mathematical objects as discursive entities. This paper expands the nature of mathematical objects beyond discourse and highlights the role of learners’ interaction with the environment by adopting ecological onto-epistemology. We develop a functional dynamic systems perspective on process–object duality in mathematics learning emphasizing embodied actions and the re-invention of artifacts’ affordances. As a main result, we reconsider process–object duality as a reification of repetitive actions into a cultural artifact that consists of two steps: (1) forming a new sensory-motor coordination that brings new perception to the fore and (2) crystallizing a new artifact in a mathematical environment that captures this new perception. An empirical example from research on embodied action-based design for trigonometry illustrates our theoretical ideas. Graphical abstract

Qualitative and asymptotic analysis of repulsion systems with diverse applications

Repulsion systems under the discrete dynamical systems perspective are considered. The goal is to study the nonlinear dynamics of an arrangement of individual or cells under various types of repulsion laws. Repulsion is a natural mechanism, so we consider the following laws: global, open, cyclic and bounded repulsion. A first scenario under consideration is when particles are freely located on an Euclidean space and the second scenario where they are restricted to remain in a bounded set. We consider those two scenarios under the previous types of repulsion. The analysis culminates in a unification of two mechanisms: attraction and repulsion via a specific social behaviour problem.

Optimizing with Attractor: A Tutorial

This tutorial presents a novel search system—the Attractor-Based Search System (ABSS)—that can solve the Traveling Salesman Problem very efficiently with optimality guarantee. From the perspective of dynamical systems, a heuristic local search algorithm for an NP-complete combinatorial problem is a discrete dynamical system. In a local search system, an attractor drives the search trajectories into the vicinity of a globally optimal point in the solution space, and the convergence of local search trajectories makes the search system become a global and deterministic system. The attractor contains a small set of the most promising solutions to the problem. The attractor can reduce the problem size exponentially, and thus make the exhaustive search feasible. Therefore, this new search paradigm is called optimizing with attractor. The ABSS consists of two search phases: local search phase and exhaustive search phase. The local search process is used to quickly construct the attractor in the solution space, and the exhaustive search process is used to completely search the attractor to identify the optimal solution. Therefore, the exact optimal solution can be found quickly by combining local search and exhaustive search. This tutorial introduces the concept of an attractor in a local search system, and describes the process of optimizing with the attractor, using the Traveling Salesman Problem as the study platform.

Lyra cosmologies with the dynamical system perspective

In this study, we revisit the homogeneous and isotropic, spatially curved model in the Lyra geometry framework. We apply the dynamical system approach to study the Universe evolution in model. The present cosmic dynamical system formulation is independent of the model parameter which yields the Universe expansion phases dominated by the stiff fluid, radiation, dark matter and dark energy. These expansion phases in the Lyra model are consistent with the General relativity model. In addition, the displacement vector field of the Lyra geometry model will give rise to the stiff matter dominated phase during early times of the Universe evolution. The cosmographic parameters, EoS parameter and statefinder diagnostic have been utilized in the dynamical system to illustrate the cosmic evolution of the Universe. The numerical solution technique is used to illustrate graphical behaviors of the cosmological parameters. On the basis of dynamical system and numerical analysis, we exhibit that the model explains accelerating universe expansion with transition from the decelerating phase and it will lead to the Λ cold dark matter (ΛCDM) model in the near future.

Sources of richness and ineffability for phenomenally conscious states.

Conscious states-state that there is something it is like to be in-seem both rich or full of detail and ineffable or hard to fully describe or recall. The problem of ineffability, in particular, is a longstanding issue in philosophy that partly motivates the explanatory gap: the belief that consciousness cannot be reduced to underlying physical processes. Here, we provide an information theoretic dynamical systems perspective on the richness and ineffability of consciousness. In our framework, the richness of conscious experience corresponds to the amount of information in a conscious state and ineffability corresponds to the amount of information lost at different stages of processing. We describe how attractor dynamics in working memory would induce impoverished recollections of our original experiences, how the discrete symbolic nature of language is insufficient for describing the rich and high-dimensional structure of experiences, and how similarity in the cognitive function of two individuals relates to improved communicability of their experiences to each other. While our model may not settle all questions relating to the explanatory gap, it makes progress toward a fully physicalist explanation of the richness and ineffability of conscious experience-two important aspects that seem to be part of what makes qualitative character so puzzling.

The interacting vacuum energy models with spatial curvature: A dynamical system perspective with observational constraints

In this paper, we study the cosmic dynamics of varying vacuum models where the dark matter interacts with the vacuum energy. We consider the homogeneous and isotropic spacetime with spatial curvature and apply the dynamical system technique to the varying vacuum models by specifying the form of energy exchange rate [Formula: see text] between the dark energy and dark matter. Further, we utilize the cosmographic parameters and statefinder parameters in the terms of dynamical variables of the cosmological dynamical system to explore the cosmic dynamics of the universe. The Milne universe solution exists as a consequence of spatially curved geometry in the model. The models also yield radiation- matter- and dark energy-dominated phases in order and thus explain the late-time accelerated expansion of the universe. The strength of interaction terms will affect the existence of cosmological phases in the model but there will always be an attractor corresponding to the accelerating universe. We numerically solve the system to illustrate the evolution of cosmological quantities and dynamical variables in the models. The role of curvature is visible during the transition from the decelerated phase into the accelerating phase. The numerical solutions affirm that the cosmological parameters in the models are consistent with their corresponding observational values.

Glycolytic oscillations under periodic drivings.

In many living organisms displaying circadian rhythms, the intake of energy often occurs in a periodic manner. Glycolysis is a prototypical biochemical reaction that exhibits a self-sustained oscillation under continuous injection of glucose. Here we study the effect of periodic injection of glucose on the glycolytic oscillation from a dynamical systems perspective. In particular, we employ Goldbeter's allosteric model of phosphofructokinase as a model system for glycolytic oscillations, and explore the effect of periodic substrate influx of varying frequencies and amplitudes by building the phase diagrams of Lyapunov exponents and oscillatory periods. When the frequency of driving is tuned around the harmonic and sub/super-harmonic conditions of the natural frequency, the system is entrained to a frequency-locked state, forming an entrainment band that broadens with an increasing amplitude of driving. On the other hand, if the amplitude is substantial, the system may transition, albeit infrequent, to a chaotic state which defies prediction of dynamical behaviour. Our study offers in-depth understandings into the controllability of glycolytic oscillation as well as explaining physical underpinnings that enable the synchronous oscillations among a dense population of cells.

Varying vacuum models with spatial curvature: a dynamical system perspective

Varying vacuum models with spatial curvature: a dynamical system perspective

From cross-lagged effects to feedback effects: Further insights into the estimation and interpretation of bidirectional relations.

Bidirectional relations have long been of interest in psychology and other social behavioral sciences. In recent years, the widespread use of intensive longitudinal data has provided new opportunities to examine dynamic bidirectional relations between variables. However, most previous studies have focused on the effect of one variable on the other (i.e., cross-lagged effects) rather than the overall effect representing the dynamic interplay between two variables (i.e., feedback effects), which we believe may be due to a lack of relevant methodological guidance. To quantify bidirectional relations as a whole, this study attempted to provide guidance for the estimation and interpretation of feedback effects based on dynamic structural equation models. First, we illustrated the estimation procedure for the average and person-specific feedback effects. Then, to facilitate the interpretation of feedback effects, we established an empirical benchmark by quantitatively synthesizing the results of relevant empirical studies. Finally, we used a set of empirical data to demonstrate how feedback effects can help (a) test theories based on bidirectional relations and (b) reveal correlates of individual differences in bidirectional relations. We also discussed the broad application prospects of feedback effects from a dynamic systems perspective. This study provides guidance for applied researchers interested in further examining feedback effects in bidirectional relations, and the shift from focusing on cross-lagged effects only to a comprehensive consideration of feedback effects may provide new insights into the study of bidirectional relations.

Adaptive oscillators support Bayesian prediction in temporal processing.

Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model-both widely used for temporal estimation and prediction-and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors-in a Bayesian sense-that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.

Integrability and dynamics of the Poisson–Boltzmann equation in simple geometries

In this article, we conduct a systematic study of the integrability and dynamics of the PB equation in plane, cylinder and sphere geometries. Specifically, for the PB equation with single species and zero permanent charges, explicit solutions are available in the plane and cylinder geometries, while non-integrability in the sphere geometry is demonstrated by the differential Galoisian approach. Furthermore, we establish a stable manifold result on a noncompact straight-line solution of the PB equation, providing a unified dynamical system perspective on the problem of single charged membrane. Notably, in the case of plane geometry, our results reveal that the concave-upward profile of the differential capacitance curve arises from the interplay between anions and cations present in the electrolyte, as opposed to the saddle structure of the PB equation.

Cosmic dynamics of isotropic models with inhomogeneous EoS: A dynamical system perspective

We examine the homogeneous and isotropic models sourced by the fluid having an inhomogenous equation of state (EoS) using qualitative methods. The flat Friedmann–Lemaitre–Robertson–Walker (FLRW) model explains the accelerating universe expansion of the present era having past of decelerating era dominated by radiation and matter. The open FLRW model incorporates the radiation, matter and dark energy dominated era in order with the spatial curvature playing role just before the accelerated universe evolution. The dominating behaviors of radiation, matter and spatial curvature affect the evolution trajectories of the deceleration and effective EoS parameter, which are obtained by the numerical solutions of autonomous system. The model parameters affect the nature of dark energy and therefore the EoS parameter may also lie in the quintessence or phantom regions. The closed FLRW model having constant positive curvature may yield the bouncing and turnaround universe evolution with dark energy-dominated phase at late-times. The application of statefinder diagnostic in the cosmological dynamical systems highlights that the universe may approach the [Formula: see text] cold dark matter ([Formula: see text]CDM) universe as a particular case in the future.

Exploring the dynamics of intentional sensorimotor desynchronization using phasing performance in music.

Humans tend to synchronize spontaneously to rhythmic stimuli or with other humans, but they can also desynchronize intentionally in certain situations. In this study, we investigate the dynamics of intentional sensorimotor desynchronization using phasing performance in music as an experimental paradigm. Phasing is a compositional technique in modern music that requires musicians to desynchronize from each other in a controlled manner. A previous case study found systematic nonlinear trajectories in the phasing performance between two expert musicians, which were explained by coordination dynamics arising from the interaction between the intrinsic tendency of synchronization and the intention of desynchronization. A recent exploratory study further examined the dynamics of phasing performance using a simplified task of phasing against a metronome. Here we present a further analysis and modeling of the data from the exploratory study, focusing on the various types of phasing behavior found in non-expert participants. Participants were instructed to perform one phasing lap, and individual trials were classified as successful (1 lap), unsuccessful (> 1 laps), or incomplete (0 lap) based on the number of laps made. It was found that successful phasing required a gradual increment of relative phase and that different types of failure (unsuccessful vs. incomplete) were prevalent at slow vs. fast metronome tempi. The results are explained from a dynamical systems perspective, and a dynamical model of phasing performance is proposed which captures the interaction of intrinsic dynamics and intentional control in an adaptive-frequency oscillator coupled to a periodic external stimulus. It is shown that the model can replicate the multiple types of phasing behavior as well as the effect of tempo observed in the human experiment. This study provides further evidence that phasing performance is governed by the nonlinear dynamics of rhythmic coordination. It also demonstrates that the musical technique of phasing provides a unique experimental paradigm for investigating human rhythmic behavior.

Initial conditions combine with sensory evidence to induce decision-related dynamics in premotor cortex

We used a dynamical systems perspective to understand decision-related neural activity, a fundamentally unresolved problem. This perspective posits that time-varying neural activity is described by a state equation with an initial condition and evolves in time by combining at each time step, recurrent activity and inputs. We hypothesized various dynamical mechanisms of decisions, simulated them in models to derive predictions, and evaluated these predictions by examining firing rates of neurons in the dorsal premotor cortex (PMd) of monkeys performing a perceptual decision-making task. Prestimulus neural activity (i.e., the initial condition) predicted poststimulus neural trajectories, covaried with RT and the outcome of the previous trial, but not with choice. Poststimulus dynamics depended on both the sensory evidence and initial condition, with easier stimuli and fast initial conditions leading to the fastest choice-related dynamics. Together, these results suggest that initial conditions combine with sensory evidence to induce decision-related dynamics in PMd.

Heterogeneity of dynamic synergetic configurations of salience network in children with autism spectrum disorder.

Atypical functional connectivity (FC) patterns have been identified in autism spectrum disorders (ASD), especially within salience network (SN) and between SN and default mode network (DMN) and central executive network (CEN). But whether the dynamic configuration of intra-SN and inter-SN (SN with DMN and CEN) FC in ASD is also heterogeneous remains unknown. Based on the resting-state functional magnetic resonance imaging data from 105 ASD and 102 typically-developing controls (TC), we calculated the time-varying FC of intra-SN and inter-SN (SN with DMN and CEN). Then, the joint recurrence features for the time-varying FC were calculated to assess how the SN dynamically recruits different configurations of network segregation and integration in ASD, that is, synergies, from the dynamical systems perspective. We analyzed the differences in synergetic patterns between ASD subtypes obtained by k-means clustering algorithm based on the synergy of SN and TC, and investigated the relationships between synergy of SN and severity of clinical symptoms of ASD for ASD subtypes. Two ASD subtypes were revealed, where the synergy of SN in ASD subtype 1 has lower stability and periodicity compared to the TC, and ASD subtype 2 exhibits the opposite alteration. Synergy of SN for ASD subtype 1 and 2 was found to predict the severity of communication impairments and restricted and repetitive behaviors in ASD, respectively. These results suggest the existence of subtypes with distinct patterns of the synergy of SN in ASD, and provide insight into the complex pathophysiological mechanism of clinical manifestations of ASD.

Exploring models of running vacuum energy with viscous dark matter from a dynamical system perspective

Running vacuum models and viscous dark matter scenarios beyond perfect fluid idealization are two appealing theoretical strategies that have been separately studied as alternatives to solve some problems rooted in the ΛCDM cosmological model. In this paper, we combine these two notions in a single cosmological setting and investigate their cosmological implications, paying particular attention in the interplay between these two constituents in different cosmological periods. Specifically, we consider a well-studied running vacuum model inspired by renormalization group, and a recently proposed general parameterization for the bulk viscosity ξ. By employing dynamical system analysis, we explore the physical aspects of the new phase space that emerges from the combined models and derive stability conditions that ensure complete cosmological dynamics. We identify four distinct classes of models and find that the critical points of the phase space are non-trivially renewed compared to the single scenarios. We then proceed, in a joint and complementary way to the dynamical system analysis, with a detailed numerical exploration to quantify the impact of both the running parameter and the bulk viscosity coefficient on the cosmological evolution. Thus, for some values of the model parameters, numerical solutions show qualitative differences from the ΛCDM model, which is phenomenologically appealing in light of cosmological observations.

A Multivariate Method for Dynamic System Analysis: Multivariate Detrended Fluctuation Analysis Using Generalized Variance.

Fractal fluctuations are a core concept for inquiries into human behavior and cognition from a dynamic systems perspective. Here, we present a generalized variance method for multivariate detrended fluctuation analysis (mvDFA). The advantage of this extension is that it can be applied to multivariate time series and considers intercorrelation between these time series when estimating fractal properties. First, we briefly describe how fractal fluctuations have advanced a dynamic system understanding of cognition. Then, we describe mvDFA in detail and highlight some of the advantages of the approach for simulated data. Furthermore, we show how mvDFA can be used to investigate empirical multivariate data using electroencephalographic recordings during a time-estimation task. We discuss this methodological development within the framework of interaction-dominant dynamics. Moreover, we outline how the availability of multivariate analyses can inform theoretical developments in the area of dynamic systems in human behavior.

Emergence of belief-like representations through reinforcement learning.

To behave adaptively, animals must learn to predict future reward, or value. To do this, animals are thought to learn reward predictions using reinforcement learning. However, in contrast to classical models, animals must learn to estimate value using only incomplete state information. Previous work suggests that animals estimate value in partially observable tasks by first forming "beliefs"-optimal Bayesian estimates of the hidden states in the task. Although this is one way to solve the problem of partial observability, it is not the only way, nor is it the most computationally scalable solution in complex, real-world environments. Here we show that a recurrent neural network (RNN) can learn to estimate value directly from observations, generating reward prediction errors that resemble those observed experimentally, without any explicit objective of estimating beliefs. We integrate statistical, functional, and dynamical systems perspectives on beliefs to show that the RNN's learned representation encodes belief information, but only when the RNN's capacity is sufficiently large. These results illustrate how animals can estimate value in tasks without explicitly estimating beliefs, yielding a representation useful for systems with limited capacity.

Processing misconceptions: dynamic systems perspectives on thinking and learning

The study of complex dynamic systems is central to biology. In this essay, I argue that thinking and learning can also be understood as phenomena that emerge from the continuous interactivity of dynamic systems. I first present and apply core concepts of dynamic systems theories to both biological and cognitive systems. I then use these ideas to explain how a dynamic systems perspective can recast the way we think about misconceptions, a central concept in the field of education research. Rather than model misconceptions as object-like entities that students either have or do not have, misconceptions can be modeled as patterns that emerge from continuous cognitive processes. I end by discussing how adopting a dynamic systems perspective suggests a need for research that uses methods designed to study processes in time and can inspire educators to embrace and value variation and fluctuation in students’ thinking and learning.