Exploring Limit Cycles of Differential Equations through Information Geometry Unveils the Solution to Hilbert's 16th Problem.

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of |R| to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n≥2 have the maximum number of 2(n-1)(4(n-1)-2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments.

Reminiscences of Half a Century of Life in the World of Theoretical Physics.

Selma Lagerlöf said that culture is what remains when one has forgotten everything we had learned. Without any warranty, through ongoing research tasks, that I will ever attain this high level of wisdom, I simply share here reminiscences that have played, during my life, an important role in my incursions in science, mainly in theoretical physics. I end by presenting some perspectives for future developments.

Metastasis Models: Thermodynamics and Complexity.

The thermodynamic formalism of nonequilibrium systems together with the theory of complex systems and systems biology offer an appropriate theoretical framework to explain the complexity observed at the macroscopic level in physiological phenomena. In turn, they allow the establishment of an appropriate conceptual and operational framework to address the study of phenomena such as the emergence and evolution of cancer.This chapter is organized as follows: In Subheading 1, an integrated vision of these disciplines is offered for the characterization of the emergence and evolution of cancer, seen as a nonlinear dynamic system, temporally and spatially self-organized out of thermodynamic equilibrium. The development of the various mathematical models and different techniques and approaches used in the characterization of cancer metastasis is presented in Subheading 2. Subheading 3 is devoted to the time course of cancer metastasis, with particular emphasis on the epithelial-mesenchymal transition (EMT henceforth) as well as chronotherapeutic treatments. In Subheading 4, models of the spatial evolution of cancer metastasis are presented. Finally, in Subheading 5, some conclusions and remarks are presented.

Interpersonal coordination of opposing player dyads during attacks performed in official football matches.

The purpose of this study was to characterise the interpersonal coordination between opponent players during offensive sequences in official matches and to verify if offensive sequences ended in shots to goal present different coordination patterns when compared than those that ended in defensive tackles. A total of 580 offensive sequences occurred during matches resulting in shots to goal (n = 172) or defensive tackles (n = 408) were analysed. The bidimensional coordinates and technical actions of male professional football players (n = 1160) were obtained using a video-based tracking system. Dyads were defined using a network analysis and composed of the nearest opponent. Interpersonal coordination of the dyads was analysed using the vector coding and the frequency for each coordination pattern was computed. In-phase was predominant for all displacement directions and offensive sequences outcomes, and antiphase was the least frequent. For lateral displacements, offensive sequences ending in shot to goal presented lower frequency for in-phase and higher frequency for offensive player phase than ended in defensive tackle. This information about the relationship of opponent players dyads during decisive moments of the matches provides fundamentals for future research and assists coaches to understand the different behaviours in successful and unsuccessful attacks.

A partially averaged system to model neuron responses to interferential current stimulation.

The interferential current (IFC) therapy is a noninvasive electrical neurostimulation technique intended to activate deep neurons using surface electrodes. In IFC, two independent kilohertz-frequency currents purportedly intersect where an interference field is generated. However, the effects of IFC on neurons within and outside the interference field are not completely understood, and it is unclear whether this technique can reliable activate deep target neurons without side effects. In recent years, realistic computational models of IFC have been introduced to quantify the effects of IFC on brain cells, but they are often complex and computationally costly. Here, we introduce a simplified model of IFC based on the FitzHugh-Nagumo (FHN) model of a neuron. By considering a modified averaging method, we obtain a non-autonomous approximated system, with explicit representation of relevant IFC parameters. For this approximated system we determine conditions under which it reliably approximates the complete FHN system under IFC stimulation, and we mathematically prove its ability to predict nonspiking states. In addition, we perform numerical simulations that show that the interference effect is observed only for a narrow set of IFC parameters and, in particular, for a beat frequency no higher than about 100 [Hz]. Our novel model tailored to the IFC technique contributes to the understanding of neurostimulation modalities using this type of signals, and can have implications in the design of noninvasive electrical stimulation therapies.

Multi-scale mechanobiological model for skeletal muscle hypertrophy.

Skeletal muscle adaptation is correlated to training exercise by triggering different signaling pathways that target many functions; in particular, the IGF1-AKT pathway controls protein synthesis and degradation. These two functions regulate the adaptation in size and strength of muscles. Computational models for muscle adaptation have focused on: the biochemical description of signaling pathways or the mechanical description of muscle function at organ scale; however, an interrelation between these two models should be considered to understand how an adaptation in muscle size affects the protein synthesis rate. In this research, a dynamical model for the IGF1-AKT signaling pathway is linked to a continuum-mechanical model describing the active and passive mechanical response of a muscle; this model is used to study the impact of the adaptive muscle geometry on the protein synthesis at the fiber scale. This new computational model links the signaling pathway to the mechanical response by introducing a growth tensor, and links the mechanical response to the signaling pathway through the evolution of the protein synthesis rate. The predicted increase in cross sectional area (CSA) due to an 8 weeks training protocol excellently agreed with experimental data. Further, our results show that muscle growth rate decreases, if the correlation between protein synthesis and CSA is negative. The outcome of this study suggests that multi-scale models coupling continuum mechanical properties and molecular functions may improve muscular therapies and training protocols.

Accurate characterization of dynamic microbial gene expression and growth rate profiles.

Genetic circuits are subject to variability due to cellular and compositional contexts. Cells face changing internal states and environments, the cellular context, to which they sense and respond by changing their gene expression and growth rates. Furthermore, each gene in a genetic circuit operates in a compositional context of genes which may interact with each other and the host cell in complex ways. The context of genetic circuits can, therefore, change gene expression and growth rates, and measuring their dynamics is essential to understanding natural and synthetic regulatory networks that give rise to functional phenotypes. However, reconstruction of microbial gene expression and growth rate profiles from typical noisy measurements of cell populations is difficult due to the effects of noise at low cell densities among other factors. We present here a method for the estimation of dynamic microbial gene expression rates and growth rates from noisy measurement data. Compared to the current state-of-the-art, our method significantly reduced the mean squared error of reconstructions from simulated data of growth and gene expression rates, improving the estimation of timing and magnitude of relevant shapes of profiles. We applied our method to characterize a triple-reporter plasmid library combining multiple transcription units in different compositional and cellular contexts in Escherichia coli. Our analysis reveals cellular and compositional context effects on microbial growth and gene expression rate dynamics and suggests a method for the dynamic ratiometric characterization of constitutive promoters relative to an in vivo reference.

Resultados en psicoterapia individual de adultos desde la perspectiva de los Sistemas Dinámicos: una revisión sistemática / Adult individual psychotherapy results from the Dynamic Systems perspective: a systematic review

Resumen: Existe un variado número de investigaciones que emplea nociones de la perspectiva de sistemas dinámicos (SD) para describir procesos de cambio en psicoterapia, conceptualizándolo como un sistema no lineal autoorganizado que presenta procesos emergentes y variaciones estructurales. Se realizó una revisión sistemática de la investigación en psicoterapia individual con pacientes adultos abordada desde esta perspectiva. La revisión se sustentó en la metodología PRISMA rastreando los principales conceptos de la perspectiva SD aplicados a la psicoterapia individual de adultos (entre 1997 y 2019), en los idiomas inglés y español, utilizando las bases de datos electrónicas PsycINFO y ProQuest. La selección final incluyó 34 estudios, tanto estudios de caso como estudios naturalistas, que abordaron diferentes variables de proceso y resultado de la psicoterapia. Los resultados resaltan la forma en que dichos conceptos ayudan a comprender el cambio de los pacientes como un proceso no lineal, destacando sus características de autoorganización, transiciones desde estados que generan sufrimiento psicológico a estados más saludables, y la formación de patrones emergentes en diferentes etapas de la psicoterapia. Se discuten algunos aspectos derivados (p.e. rol de la alianza, y de las intervenciones clínicas) que pueden ser abordados en el trabajo terapéutico.

Abstract: There is a diverse body of research that utilizes notions of the dynamical systems (DS) perspective to describe change processes in psychotherapy, understanding it as a non-linear self-organized system that presents emergent processes and structural variations. A systematic review of research in individual psychotherapy with adult patients addressed from this perspective has been carried out. The review was carried out supported by the PRISMA methodology tracking the main concepts of the DS perspective applied to individual psychotherapy of adults (between 1997 and 2019), in English and Spanish, using the electronic databases PsycINFO and ProQuest. The final selection included 34 studies, both case studies and naturalistic studies, covering different process and outcome variables of psychotherapy. The results highlight how such concepts help to understand patients' change as a nonlinear process, emphasizing its self-organizing characteristics, transitions from states that generate psychological distress to healthier states, and the formation of emergent patterns at different stages of psychotherapy. Some related aspects (e.g. role of the alliance, and of clinical interventions) that can be considered in the therapeutic work are discussed.

Emergence and algorithmic information dynamics of systems and observers.

One of the challenges of defining emergence is that one observer's prior knowledge may cause a phenomenon to present itself as emergent that to another observer appears reducible. By formalizing the act of observing as mutual perturbations between dynamical systems, we demonstrate that the emergence of algorithmic information does depend on the observer's formal knowledge, while being robust vis-a-vis other subjective factors, particularly: the choice of programming language and method of measurement; errors or distortions during the observation; and the informational cost of processing. This is called observer-dependent emergence (ODE). In addition, we demonstrate that the unbounded and rapid increase of emergent algorithmic information implies asymptotically observer-independent emergence (AOIE). Unlike ODE, AOIE is a type of emergence for which emergent phenomena will be considered emergent no matter what formal theory an observer might bring to bear. We demonstrate the existence of an evolutionary model that displays the diachronic variant of AOIE and a network model that displays the holistic variant of AOIE. Our results show that, restricted to the context of finite discrete deterministic dynamical systems, computable systems and irreducible information content measures, AOIE is the strongest form of emergence that formal theories can attain. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

LOICA: Integrating Models with Data for Genetic Network Design Automation.

Genetic design automation tools are necessary to expand the scale and complexity of possible synthetic genetic networks. These tools are enabled by abstraction of a hierarchy of standardized components and devices. Abstracted elements must be parametrized from data derived from relevant experiments, and these experiments must be related to the part composition of the abstract components. Here we present Logical Operators for Integrated Cell Algorithms (LOICA), a Python package for designing, modeling, and characterizing genetic networks based on a simple object-oriented design abstraction. LOICA uses classes to represent different biological and experimental components, which generate models through their interactions. These models can be parametrized by direct connection to data contained in Flapjack so that abstracted components of designs can characterize themselves. Models can be simulated using continuous or stochastic methods and the data published and managed using Flapjack. LOICA also outputs SBOL3 descriptions and generates graph representations of genetic network designs.

An integrative model of cancer cell differentiation with immunotherapy.

In order to improve cancer treatments, cancer cell differentiation and immunotherapy are the subjects of several studies in different branches of interdisciplinary sciences. In this work, we develop a new population model that integrates other complementary ones, thus emphasizing the relationship between cancer cells at different differentiation stages and the main immune system cells. For this new system, specific ranges were found where transdifferentiation of differentiated cancer cells can occur. In addition, a specific therapy against cancer stem cells was analysed by simulating cytotoxic cell vaccines. In reference to the latter, the different combinations of parameters that optimize it were studied.

External Stimuli on Neural Networks: Analytical and Numerical Approaches.

Based on the behavior of living beings, which react mostly to external stimuli, we introduce a neural-network model that uses external patterns as a fundamental tool for the process of recognition. In this proposal, external stimuli appear as an additional field, and basins of attraction, representing memories, arise in accordance with this new field. This is in contrast to the more-common attractor neural networks, where memories are attractors inside well-defined basins of attraction. We show that this procedure considerably increases the storage capabilities of the neural network; this property is illustrated by the standard Hopfield model, which reveals that the recognition capacity of our model may be enlarged, typically, by a factor 102. The primary challenge here consists in calibrating the influence of the external stimulus, in order to attenuate the noise generated by memories that are not correlated with the external pattern. The system is analyzed primarily through numerical simulations. However, since there is the possibility of performing analytical calculations for the Hopfield model, the agreement between these two approaches can be tested-matching results are indicated in some cases. We also show that the present proposal exhibits a crucial attribute of living beings, which concerns their ability to react promptly to changes in the external environment. Additionally, we illustrate that this new approach may significantly enlarge the recognition capacity of neural networks in various situations; with correlated and non-correlated memories, as well as diluted, symmetric, or asymmetric interactions (synapses). This demonstrates that it can be implemented easily on a wide diversity of models.

Synthetic Birdsongs as a Tool to Induce, and Iisten to, Replay Activity in Sleeping Birds.

Birdsong is a complex vocal behavior, which emerges out of the interaction between a nervous system and a highly nonlinear vocal device, the syrinx. In this work we discuss how low dimensional dynamical systems, interpretable in terms of the biomechanics involved, are capable of synthesizing realistic songs. We review the experimental and conceptual steps that lead to the formulation of low dimensional dynamical systems for the song system and describe the tests that quantify their success. In particular, we show how to evaluate computational models by comparing the responses of highly selective neurons to the bird's own song and to synthetic copies generated mathematically. Beyond testing the hypothesis behind the model's construction, these low dimensional models allow designing precise stimuli in order to explore the sensorimotor integration of acoustic signals.

Aplicación de una ley matemática exponencial a la dinámica cardíaca en 16 horas: estudio realizado con 250 pacientes / Application of an exponential mathematical law to cardiac dynamics in 16 hours: a study of 250 patients

Resumen Introducción: Los sistemas dinámicos y la geometría fractal han sido el sustrato para el advenimiento de una ley matemática aplicada al diagnóstico de la dinámica cardíaca en 21 horas. Objetivo: Confirmar la aplicabilidad clínica de la ley matemática exponencial en 16 horas a partir de un estudio de concordancia diagnóstica frente a la norma de referencia. Materiales y método: Se realizó un estudio con 250 registros electrocardiográficos continuos y ambulatorios; 50 pertenecían a pacientes normales y 200 a pacientes con diversas enfermedades cardíacas. Se simuló la secuencia de frecuencias cardíacas y se construyeron los atractores correspondientes. Se calculó la dimensión fractal y la ocupación del atractor en el espacio generalizado de box-counting. Por último, se estableció el diagnóstico fisicomatemático en 16 y 21 horas y se efectuó la validación estadística. Resultados: Los espacios de ocupación para normalidad en la rejilla pequeña se encontraron entre 205 y 372, y entre 56 y 201 para dinámicas patológicas, lo cual permitió evidenciar la capacidad del método para diferenciar normalidad de enfermedad a través de la ocupación espacial de los atractores con base en la ley matemática en 16 horas. Se hallaron valores de sensibilidad y especificidad del 100% y un coeficiente kappa del orden de 1, luego de comparar el diagnóstico fisicomatemático frente a la norma de referencia. Conclusión: La ley matemática exponencial en 16 horas demostró su utilidad como herramienta de ayuda diagnóstica y predictiva, lo cual permitió diferenciar normalidad y estados evolutivos hacia enfermedad y agudización.

Abstract Introduction: Dynamic systems and fractal geometry have been the substrate for the rising of a mathematical law applied to the diagnosis of cardiac dynamics in 21 hours. Objective: To confirm the clinical applicability of the exponential mathematical law in 16 hours, with a study of diagnostic agreement against the Gold Standard. Materials and method: It was made a study with 250 ambulatory and continuous electrocardiographic recordings, 50 belonged to normal patients and 200 to patients with various cardiac pathologies. The sequence of heart rates was simulated, and attractors were constructed. It was calculated the fractal dimension of the attractor and its occupation in the generalized Box-Counting space. Finally, it was determined the physical-mathematical diagnostic in 16 and 21 hours, and statistical validation was performed. Results: The occupation spaces in the small grid were between 205 and 372 for normality, and between 56 and 201 for pathologic dynamics, which demonstrated the ability of the method to differentiate normal condition from sickness, through spatial occupation of attractors according to mathematical law in 16 hours. There were obtained values of sensitivity and specificity of 100% and Kappa coefficient was 1, after comparing the physic-mathematical analysis against the Gold Standard. Conclusion: The exponential mathematical law in 16 hours proved its utility as diagnostic and predictive tool support, allowing to differentiate normal, developmental stages to disease and exacerbation.

Model-based estimation of transmissibility and reinfection of SARS-CoV-2 P.1 variant.

Background: The SARS-CoV-2 variant of concern (VOC) P.1 (Gamma variant) emerged in the Amazonas State, Brazil, in November 2020. The epidemiological consequences of its mutations have not been widely studied, despite detection of P.1 in 36 countries, with local transmission in at least 5 countries. A range of mutations are seen in P.1, ten of them in the spike protein. It shares mutations with VOCs previously detected in the United Kingdom (B.1.1.7, Alpha variant) and South Africa (B.1.351, Beta variant). Methods: We estimated the transmissibility and reinfection of P.1 using a model-based approach, fitting data from the national health surveillance of hospitalized individuals and frequency of the P.1 variant in Manaus from December-2020 to February-2021. Results: Here we estimate that the new variant is about 2.6 times more transmissible (95% Confidence Interval: 2.4-2.8) than previous circulating variant(s). Manaus already had a high prevalence of individuals previously affected by the SARS-CoV-2 virus and our fitted model attributed 28% of Manaus cases in the period to reinfections by P.1, confirming the importance of reinfection by this variant. This value is in line with estimates from blood donors samples in Manaus city. Conclusions: Our estimates rank P.1 as one of the most transmissible among the SARS-CoV-2 VOCs currently identified, and potentially as transmissible as the posteriorly detected VOC B.1.617.2 (Delta variant), posing a serious threat and requiring measures to control its global spread.

Synchronization transitions caused by time-varying coupling functions.

Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase- synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Search Strategies in the Perceptual-Motor Workspace and the Acquisition of Coordination, Control, and Skill.

In this paper we re-visit and elaborate-on the theoretical framework of learning as searching within the perceptual-motor workspace for a solution to the task. The central focus is the nature of search strategies to locate and create stable equilibrium regions in the perceptual-motor workspace and how these strategies relate to the emergent movement forms in the acquisition of coordination, control, and skill. In the ecological theory of perception and action, the enhanced stability of performance occurs through the attunement of the perceptual systems to the task dynamics together with modifications of action as task and intrinsic dynamics cooperate and/or compete. Thus, through practice in this search process, individuals adapt to the pick-up of task relevant perceptual variables and change their movement form according to the stability of the performed action and its outcome in relation to the task demands. Contemporary experimental findings have revealed features of the search process given the interaction of individual intrinsic dynamics in the context of task requirements and principles that drive the change - e.g., exploitation of more tolerant task-space solutions and emergence of compensatory mechanisms. Finally, we outline how the search strategy framework relates to traditional learning-related phenomena: including the dynamical pathways of learning, learning curves, factors of learning, individuality, motor development, and sport and rehabilitation interventions.

Influence of technological progress and renewability on the sustainability of ecosystem engineers populations.

Overpopulation and environmental degradation due to inadequate resource-use are outcomes of human's ecosystem engineering that has profoundly modified the world's landscape. Despite the age-old concern that unchecked population and economic growth may be unsustainable, the prospect of societal collapse remains contentious today. Contrasting with the usual approach to modeling human-nature interactions, which are based on the Lotka-Volterra predator-prey model with humans as the predators and nature as the prey, here we address this issue using a discrete-time population dynamics model of ecosystem engineers. The growth of the population of engineers is modeled by the Beverton-Holt equation with a density-dependent carrying capacity that is proportional to the number of usable habitats. These habitats (e.g., farms) are the products of the work of the individuals on the virgin habitats (e.g., native forests), hence the denomination engineers of ecosystems to those agents. The human-made habitats decay into degraded habitats, which eventually regenerate into virgin habitats. For slow regeneration resources, we find that the dynamics is dominated by rounds of prosperity and collapse, in which the population reaches vanishing small densities. However, increase of the efficiency of the engineers to explore the resources eliminates the dangerous oscillatory patterns of feast and famine and leads to a stable equilibrium that balances population growth and resource availability. This finding supports the viewpoint of growth optimists that technological progress may avoid collapse.

Modeling the Epigenetic Landscape in Plant Development.

Computational mechanistic models enable a systems-level understanding of plant development by integrating available molecular experimental data and simulating their collective dynamical behavior. Boolean gene regulatory network dynamical models have been extensively used as a qualitative modeling framework for such purpose. More recently, network modeling protocols have been extended to model the epigenetic landscape associated with gene regulatory networks. In addition to understanding the concerted action of interconnected genes, epigenetic landscape models aim to uncover the patterns of cell state transition events that emerge under diverse genetic and environmental background conditions. In this chapter we present simple protocols that naturally extend gene regulatory network modeling and demonstrate their use in modeling plant developmental processes under the epigenetic landscape framework. We focus on conceptual clarity and practical implementation, providing directions to the corresponding technical literature. The protocols presented here can be applied to any well-characterized gene regulatory network in plants, animals, or human disease.

Probabilistic Inference for Dynamical Systems.

A general framework for inference in dynamical systems is described, based on the language of Bayesian probability theory and making use of the maximum entropy principle. Taking the concept of a path as fundamental, the continuity equation and Cauchy's equation for fluid dynamics arise naturally, while the specific information about the system can be included using the maximum caliber (or maximum path entropy) principle.