Dynamical Systems Theory Research Articles

A Developmental Psychobiologist's Commentary on the Future of Affective Science.

A developmental psychobiologist commented on the role of dynamic systems theory in the future of affective science and proposed an alternative paradigm.

A novel investigation of the influence of vaccination on pneumonia disease

In the entire world, pneumonia is one of the leading causes of death, which is particularly dangerous for young children (those under five years old) and the elderly (those over 65). A deterministic susceptible, vaccinated, exposed, infected, and recovered (SVEIR) model is used in this work to mathematically study the dynamics of pneumonia disease and examine stability analysis, basic reproduction numbers, and equilibrium points of dynamical systems theory models. Spatial equilibria are studied to model disease-free equilibria that are locally asymptotic stable. Numerical simulations of the model have been carried out using MATLAB21. The SVEIR flow and its variables for different parameter sets have been studied through numerical simulations. The solution to the issue is provided through the use of illustrated and explicated results. According to research findings, if vaccination rates rise over the necessary vaccination ratio, the sickness will finally vanish from the community.

Psychometric network analysis in rehabilitation research: A methodological demonstration in depression symptoms of veterans and service members at 1 and 2 years after traumatic brain injury.

Psychometric network analysis (PNA) is an application of dynamic systems theory that can inform measurement of complex rehabilitation phenomena such as depressive symptom patterns in veterans and service members (V/SMs) after traumatic brain injury (TBI). This study applied PNA to the Patient Health Questionnaire-9 (PHQ-9), a common measure of depressive symptoms, in a sample of V/SMs with TBI at Years 1 and 2 (Y1-2) postinjury. A sample of 808 V/SMs with TBI participated, 594 contributing PHQ-9 data at Y1 and 585 at Y2. Participants were recruited while or after receiving inpatient postacute rehabilitation from one of five Veterans Affairs Polytrauma Rehabilitation Centers. The networks were stable and invariant over time. At both times, network structure revealed the cardinal depressive symptom "feeling down, depressed, or hopeless," as evidenced by its strength centrality. In the Y1 network, the suicidal ideation node was connected exclusively to the network through the guilt node, and in the Y2 network, the suicidal ideation node formed a second connection through the low mood node. The guilt node was the second most influential node at Y1 but was replaced by anhedonia node at Y2. This study demonstrated the potential of PNA in rehabilitation research and identified the primacy of feeling down, depressed, and hopeless after TBI at both Y1 and Y2, with guilt being the second most influential symptom at Y1, but replaced by anhedonia at Y2, providing supportive evidence that the relationships among depressive symptoms after TBI are dynamic over time. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Unveiling the complexity of L2 learners’ emotions and emotion regulation: A retrodictive qualitative modeling study

AbstractSecond/foreign language (L2) learners often go through a gamut of conflicting experiences embedded within their emotional system, which can be managed through emotion regulation (ER). Motivated by complex dynamic systems theory (CDST), this qualitative study aimed to explore the signature dynamics of positive and negative emotions among six Iranian adolescent English language learners. Additionally, the study investigated how learners manage their emotions in the L2 classroom. Using the CDST as its main theoretical framework and retrodictive qualitative modeling as its main methodology, the current investigation revealed that the learners’ emotional systems were mediated by internal and external contexts, as well as dynamicity. It was further discovered that ER strategies were driven by multiple simultaneous factors influencing emotional systems and the adaptability of those systems to new situations. In the end, we discuss how policymakers, teachers, and researchers can adopt the findings to provide some well‐needed support for classroom‐based L2 learners.

An analysis of the interaction between surface and basal crevasses in ice shelves

Abstract. The prescription of a simple and robust parameterization for calving is one of the most significant open problems in ice sheet modelling. One common approach to the modelling of crevasse propagation in calving in ice shelves is to view crevasse growth as an example of linear elastic fracture mechanics. Prior work has, however, focused on highly idealized crack geometries, with a single fracture incised into a parallel-sided slab of ice. In this paper, we study how fractures growing from opposite sides of such an ice slab interact with each other, focusing on different simple crack arrangements: we consider either perfectly aligned cracks or periodic arrays of laterally offset cracks. We visualize the dynamics of crack growth using simple tools from dynamical systems theory and find that aligned cracks tend to impede each other's growth due to the torques generated by normal stresses on the crack faces, while periodically offset cracks facilitate simultaneous growth of bottom and top cracks. For periodic cracks, the presence of multiple cracks on one side of the ice slab, however, also generates torques that slow crack growth, with widely spaced cracks favouring calving at lower extensional stresses than closely spaced cracks.

Sufficient conditions on the continuous spectrum for ergodic Schrödinger operators

We investigate the spectral types of families of discrete one-dimensional Schrödinger operators {Hω}ω∈Ω , where each H ω has a potential function V ω (n) = f(T n ω) for n∈Z , T is an ergodic homeomorphism on a compact space Ω and f:Ω→R is a continuous function. The main analytical tool of the paper is Gordon’s lemma, of which we provide a complete proof. We demonstrate that a generic operator Hω∈{Hω}ω∈Ω has purely continuous spectrum if {Tnα}n≥0 is dense in Ω for a certain α ∈ Ω. Furthermore, we extend this result to the case where {Ω, T} satisfies topological repetition property, a concept introduced by Boshernitzan and Damanik (2008, []). Theorems presented in this paper weaken the hypotheses of previous research and enable us to reach the same conclusion as those authors. Our paper presents a comprehensive study of the topological and metric repetition properties, highlighting their binding role between the theory of dynamical systems and spectral theory. We also discuss the implications of these properties on the qualitative description of specific physical systems.

FateNet: an integration of dynamical systems and deep learning for cell fate prediction.

Understanding cellular decision-making, particularly its timing and impact on the biological system such as tissue health and function, is a fundamental challenge in biology and medicine. Existing methods for inferring fate decisions and cellular state dynamics from single-cell RNA sequencing data lack precision regarding decision points and broader tissue implications. Addressing this gap, we present FateNet, a computational approach integrating dynamical systems theory and deep learning to probe the cell decision-making process using scRNA-seq data. By leveraging information about normal forms and scaling behavior near bifurcations common to many dynamical systems, FateNet predicts cell decision occurrence with higher accuracy than conventional methods and offers qualitative insights into the new state of the biological system. Also, through in-silico perturbation experiments, FateNet identifies key genes and pathways governing the differentiation process in hematopoiesis. Validated using different scRNA-seq data, FateNet emerges as a user-friendly and valuable tool for predicting critical points in biological processes, providing insights into complex trajectories. github.com/ThomasMBury/fatenet.

The dynamics of changes in linguistic complexity and writing scores in timed argumentative writing among beginning-level EFL learners

Abstract This study investigates longitudinal changes in linguistic complexity and holistic scores in EFL timed argumentative writing among beginning-level learners using Complex Dynamic Systems Theory. It also explores relationships among linguistic complexity over time and with writing scores. Over nine months, 42 Japanese EFL learners completed six timed essays. Linguistic complexity was assessed through lexical, phraseological, and syntactic features. Findings indicate linear growth in writing scores, lexical sophistication, and noun phrase (NP) complexity, alongside stable phrasal complexity and non-linear subordination. Interrelated growth patterns included connections between lexical sophistication and phrasal complexity, and phraseological complexity and subordination. Competitive dynamics were observed between lexical diversity and NP complexity, and phraseological complexity and phrasal sophistication. Linear associations emerged between writing scores and lexical sophistication or syntactic complexity, while a non-linear relationship occurred with phraseological sophistication. This study illuminates the intricate dynamics of linguistic complexity and L2 writing scores among beginning-level learners in argumentative writing.

Automating the discovery of partial differential equations in dynamical systems

Abstract Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm’s performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.

The critical dynamics of hippocampal seizures

Epilepsy is defined by the abrupt emergence of harmful seizures, but the nature of these regime shifts remains enigmatic. From the perspective of dynamical systems theory, such critical transitions occur upon inconspicuous perturbations in highly interconnected systems and can be modeled as mathematical bifurcations between alternative regimes. The predictability of critical transitions represents a major challenge, but the theory predicts the appearance of subtle dynamical signatures on the verge of instability. Whether such dynamical signatures can be measured before impending seizures remains uncertain. Here, we verified that predictions on bifurcations applied to the onset of hippocampal seizures, providing concordant results from in silico modeling, optogenetics experiments in male mice and intracranial EEG recordings in human patients with epilepsy. Leveraging pharmacological control over neural excitability, we showed that the boundary between physiological excitability and seizures can be inferred from dynamical signatures passively recorded or actively probed in hippocampal circuits. Of importance for the design of future neurotechnologies, active probing surpassed passive recording to decode underlying levels of neural excitability, notably when assessed from a network of propagating neural responses. Our findings provide a promising approach for predicting and preventing seizures, based on a sound understanding of their dynamics.

Dynamic behavior and driving region of spray combustion instability in a backward-facing step combustor.

We numerically study the dynamic behavior and driving region of spray combustion instability in a backward-facing step combustor using analytical methodologies based on dynamical systems theory, symbolic dynamics, complex networks, and machine learning. The global dynamic behavior of a heat release rate field represents low-dimensional chaotic oscillations with deterministically aperiodic intercycle dynamics. Spray combustion instability is driven in the formation and separation region of a large-scale organized vortex induced by the hydrodynamic shear layer instability at the edge of the backstep. This region corresponds fairly to that of the hub in an acoustic-energy-flux-based spatial network. The feature importance in a random forest is valid for clarifying the feedback coupling of spray combustion instability.

Dynamical Systems Characterisation and Reduced Order Modelling of Thermoacoustics in a Lean Direct Injection (LDI) Hydrogen Combustor

Abstract In this study, the thermoacoustic instabilities of partially premixed hydrogen flames in a lean direct injection (LDI) multi-cluster combustor are characterised using dynamical systems theory. The combustor is operated at a range of bulk velocities (30 - 90 m/s) and equivalence ratios(0.2-0.6), and time-resolved pressure oscillations and integrated OH* chemiluminescence measurements were taken. The thermoacoustic system reveals a variety of dynamical states in pressure such as period-1 Limit Cycle Oscillation (LCO) with a single characteristic frequency, period-2 LCO with two characteristic frequencies, intermittent, quasi-periodic and chaotic states as either bulk velocity or equivalence ratio is varied. At a bulk velocity of 30 m/s, as the equivalence ratio is gradually decreased from 0.6 to 0.2, the dynamical behaviour follows a sequence from an intermittent state to a period-1 LCO, then to a quasi-periodic state, and eventually reaches a chaotic state. As the equivalence ratio is decreased for a bulk velocity of 60 m/s, the pressure oscillations evolve from a period-2 LCO to quasi-periodic state before flame blows off. The emergence of period-2 and quasi-periodic states indicates the presence of strong non-linear interactions among the cavity acoustic modes. The emergence of period-2 and quasi-periodic states indicates strong non-linear interactions among the cavity acoustic modes. These modes and their spatial behaviour are investigated using a Helmholtz solver, showing that hydrogen flames can excite a wide range of cavity acoustic modes

Analyzing the Kuralay-II equation: bifurcation, chaos, and sensitivity insights through conformable derivative and Jacobi elliptic function expansion

This study explores the intricate dynamics of the Kuralay-II equation by employing the conformable derivative. Using the Galilean transformation, we can establish a dynamical system related to the equation. We investigate bifurcation methods in this derived system using planar dynamical systems theory. By introducing a perturbed term, we thoroughly investigate the possibility of chaotic behaviors in the Kuralay-II equation using comprehensive two-phase portraiture. Through careful analysis, we have determined that even small changes in the initial conditions have little impact on the stability of the solution, which has been confirmed by employing the Runge–Kutta method. In addition, we obtain exact solutions for the Kuralay-II equation by the Jacobi elliptic function expansion method. Graphical results of some solutions are showcased, offering a comprehensive evaluation using MATLAB across various dimensions. This study has yielded significant findings, such as the discovery of bifurcation points, the determination of conditions for chaos, and the analysis of stability under perturbations. These results have enhanced our understanding of the behavior of the Kuralay-II equation.

Enhanced cellular longevity arising from environmental fluctuations

Cellular longevity is regulated by both genetic and environmental factors. However, the interactions of these factors in the context of aging remain largely unclear. Here, we formulate a mathematical model for dynamic glucose modulation of a core gene circuit in yeast aging, which not only guided the design of pro-longevity interventions but also revealed the theoretical principles underlying these interventions. We introduce the dynamical systems theory to capture two general means for promoting longevity-the creation of a stable fixed point in the "healthy" state of the cell and the "dynamic stabilization" of the system around this healthy state through environmental oscillations. Guided by the model, we investigate how both of these can be experimentally realized by dynamically modulating environmental glucose levels. The results establish a paradigm for theoretically analyzing the trajectories and perturbations of aging that can be generalized to aging processes in diverse cell types and organisms.

Comparing parameter-reduction methods on a biophysical model of an auditory hair cell

Biophysical models describing complex cellular phenomena typically include systems of nonlinear differential equations with many free parameters. While experimental measurements can fix some parameters, those describing internal cellular processes frequently remain inaccessible. Hence, a proliferation of free parameters risks overfitting the data, limiting the model's predictive power. In this study, we develop systematic methods, applying statistical analysis and dynamical-systems theory, to reduce parameter count in a biophysical model. We demonstrate our techniques on a five-variable computational model designed to describe active, mechanical motility of auditory hair cells. Specifically, we use two statistical measures, the total-effect and PAWN indices, to rank each free parameter by its influence on selected, core properties of the model. With the resulting ranking, we fix most of the less influential parameters, yielding a five-parameter model with refined predictive power. We validate the theoretical model with experimental recordings of active hair-bundle motility, specifically by using Akaike and Bayesian information criteria after obtaining maximum-likelihood fits. As a result, we determine the system's most influential parameters, which illuminate the key biophysical elements of the cell's overall features. Even though we demonstrate with a concrete example, our techniques provide a general framework, applicable to other biophysical systems. Published by the American Physical Society 2024

A network control theory pipeline for studying the dynamics of the structural connectome.

Network control theory (NCT) is a simple and powerful tool for studying how network topology informs and constrains the dynamics of a system. Compared to other structure-function coupling approaches, the strength of NCT lies in its capacity to predict the patterns of external control signals that may alter the dynamics of a system in a desired way. An interesting development for NCT in the neuroscience field is its application to study behavior and mental health symptoms. To date, NCT has been validated to study different aspects of the human structural connectome. NCT outputs can be monitored throughout developmental stages to study the effects of connectome topology on neural dynamics and, separately, to test the coherence of empirical datasets with brain function and stimulation. Here, we provide a comprehensive pipeline for applying NCT to structural connectomes by following two procedures. The main procedure focuses on computing the control energy associated with the transitions between specific neural activity states. The second procedure focuses on computing average controllability, which indexes nodes' general capacity to control the dynamics of the system. We provide recommendations for comparing NCT outputs against null network models, and we further support this approach with a Python-based software package called 'network control theory for python'. The procedures in this protocol are appropriate for users with a background in network neuroscience and experience in dynamical systems theory.

Dynamical analysis and the soliton solutions of (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative

This paper investigates the soliton solutions and dynamical analysis of (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative, which is transformed into the ordinary differential equation. By using the second-order complete discriminant system, the soliton solutions are presented. By utilizing the theory of planar dynamical system, the phase portraits of the dynamical system and its disturbance system are drawn. Moreover, three-dimensional, two-dimensional, and contour plots of soliton solutions for (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative have also been plotted.

Multi-condensate state as a functional strategy to optimize the cell signaling output

The existence of multiple biomolecular condensates inside living cells is a peculiar phenomenon not compatible with the predictions of equilibrium statistical mechanics. In this work, we address the problem of multiple condensates state (MCS) from a functional perspective. We combine Langevin dynamics, reaction-diffusion simulation, and dynamical systems theory to demonstrate that MCS can indeed be a function optimization strategy. Using Arp2/3 mediated actin nucleation pathway as an example, we show that actin polymerization is maximum at an optimal number of condensates. For a fixed amount of Arp2/3, MCS produces a greater response compared to its single condensate counterpart. Our analysis reveals the functional significance of the condensate size distribution which can be mapped to the recent experimental findings. Given the spatial heterogeneity within condensates and non-linear nature of intracellular networks, we envision MCS to be a generic functional solution, so that structures of network motifs may have evolved to accommodate such configurations.

Waddington landscape for prototype learning in generalized Hopfield networks

Networks in machine learning offer examples of complex high-dimensional dynamical systems inspired by and reminiscent of biological systems. Here, we study the learning dynamics of generalized Hopfield networks, which permit visualization of internal memories. These networks have been shown to proceed through a “feature-to-prototype” transition, as the strength of network nonlinearity is increased, wherein the learned, or terminal, states of internal memories transition from mixed to pure states. Focusing on the prototype learning dynamics of the internal memories, we observe stereotypical dynamics of memories wherein similar subgroups of memories sequentially split at well-defined saddles. The splitting order is interpretable and reproducible from one simulation to the other. The dynamics prior to splits are robust to variations in many features of the system. To develop a more rigorous understanding of these global dynamics, we study smaller subsystems that exhibit similar properties to the full system. Within these smaller systems, we combine analytical calculations with numerical simulations to study the dynamics of the feature-to-prototype transition, and the emergence of saddle points in the learning landscape. We exhibit regimes where saddles appear and disappear through saddle-node bifurcations, qualitatively changing the distribution of learned memories as the strength of the nonlinearity is varied—allowing us to systematically investigate the mechanisms that underlie the emergence of the learning dynamics. Several features of the learning dynamics are reminiscent of the Waddington's caricature of cellular differentiation, and we attempt to make this analogy more precise. Memories can thus differentiate in a predictive and controlled way, revealing bridges between experimental biology, dynamical systems theory, and machine learning. Published by the American Physical Society 2024

Stabilization of aperiodic sampled-data switched affine systems to hybrid limit cycles

This paper deals with the stabilization of aperiodic sampled-data switched affine systems to a predetermined hybrid limit cycle using a hybrid dynamical system representation and a control Lyapunov function approach. Some preliminaries on the hybrid dynamical system formalism provide the framework for modeling switched affine systems followed by definitions on hybrid limit cycles and related notions. The main result, based on simple Linear Matrix Inequalities (LMI), guarantees that the solutions to the closed-loop system converge to a hybrid limit cycle defined by the states, functioning modes with their corresponding dwell times. The theoretical results are evaluated on academic examples and demonstrate the potential and the originality of the method over the recent literature.