Theory predicts that indirect interactions in ecological networks sustain species diversity through oscillatory dynamics. However, a framework linking interaction structure to the presence, type, and complexity of these cycles is lacking. Here, we develop an analytical toolbox combining invasion graphs with a mathematical decomposition of interaction matrices into symmetric and antisymmetric components. We find that invasion cycles-closed loops of species invasions-are suppressed when symmetric interactions dominate, reflecting strong self-limitation. Conversely, antisymmetric dominance, indicating competitive asymmetries, leads to the well-known cycles of single-species invasion such as rock-paper-scissors as well as novel multispecies invasion patterns, in which several species simultaneously invade each transition of the cycle. As asymmetries increase, more complex cycles involving both sequential and simultaneous invasions emerge. Yet this potential for cycles is suppressed as variability in intrinsic growth rates increases. Our work clarifies when interactions drive cycles and introduces a simple ratio that assesses symmetric versus antisymmetric contributions in the interaction matrix, constraining cycle emergence and the number of species they can sustain.

AbstractCooperation, the mutual benefit that individuals of different species obtain when interacting, is ubiquitous in nature. Despite their prevalence, we know little about the mechanisms particular to cooperation that maintain biodiversity. To address this gap, we introduce cooperation into structural stability, a general framework for understanding how species interactions determine the long-term persistence of species within communities. This approach allows the identification of three distinct processes. First, cooperation increases the opportunities for species to coexist more than interspecific competition does, improving species persistence across a wider range of environmental conditions. Second, cooperation creates intertwined biodiversity, where the existence of some species begets the presence of others. Third, cooperation promotes alternative structures of community assembly due to variations in species' performance. These structures diversify the pathways to species coexistence. In conclusion, our work suggests that cooperation, relative to interspecific competition, maximizes the maintenance of biodiversity.

Understanding consciousness requires bridging theoretical models and clinically measurable brain dynamics. This review integrates three complementary frameworks that converge on a dynamical view of conscious processing: continuous formulations of Integrated Information Theory (IIT), attractor-landscape modeling of brain-state transitions, and perturbational complexity metrics from transcranial magnetic stimulation combined with electroencephalography (TMS-EEG). Continuous-time IIT formalizes how integrated information evolves across temporal hierarchies, while dynamical-systems approaches show that consciousness emerges near criticality, where metastable attractors enable flexible transitions between partially synchronized states. Perturbational-complexity indices capture these properties empirically, quantifying the brain’s capacity for integration and differentiation even without behavioral responsiveness. Across anesthesia, disorders of consciousness, epilepsy, and neurodegeneration, TMS-EEG biomarkers reveal reduced complexity and altered synchronization consistent with structural and functional disconnection. Integrating multimodal data—diffusion MRI, fMRI, EEG, and causal perturbations—is consistent with individualized modeling of consciousness-related dynamics. Standardized protocols, mechanistically interpretable machine learning, and longitudinal validation are essential for clinical translation. By uniting information-theoretic, dynamical, and empirical perspectives, this framework offers a reproducible foundation for consciousness biomarkers that mechanistically link brain dynamics to subjective experience, paving the way for precision applications in neurology, psychiatry, and anesthesia.

This paper introduces a formal definition of intrinsic information as an inherent property of dynamical systems, characterized by their global topological and geometric structures. Unlike standard approaches that treat information as an epistemic or communication-theoretic tool, we propose that information possesses a specific shape defined by attractors, Morse decompositions, and Lyapunov landscapes. We argue that for any real-world phenomenon, the regularity of measurements on its observables allows for the derivation of a dynamical system whose informational content is objective and measurable. To substantiate this link, we provide concrete applications in ecology and neuroscience, offering specific metrics that quantify organizational complexity. Finally, our framework introduces the idea that information, by expressing itself through formal and geometric shape, acts as a structural constraint with explanatory power. This characterization of intrinsic information enables a rigorous scientific approach to its properties, including measurement, perturbation analysis, and the study of transient and asymptotic dynamics. We position this work as not merely technical, but as a robust research program that offers a new, in-depth perspective on the reality that science addresses across multiple domains.

Industrial development is commonly described as a sequence of technological stages, from automation to artificial intelligence. This study examines whether successive industrial paradigms—from Industry 3.0 to the emerging Industry 6.0—can be more adequately understood as transformations in technological rationality rather than merely technological upgrades. The analysis adopts a conceptual–philosophical methodology informed by targeted review of peer-reviewed literature indexed in Scopus and Web of Science, integrating Kuhn’s notion of paradigms with Peircean inferential logic. Through systematic comparison of technological configurations, problem-framing practices, and epistemic assumptions, the study maps each paradigm onto a dominant mode of inference. The findings indicate that Industry 3.0 privileges deductive rule-based control, Industry 4.0 relies on inductive data-driven optimization, Industry 5.0 foregrounds hermeneutic interpretation and normative judgment, and prospective Industry 6.0 can be coherently interpreted as oriented toward abductive hypothesis generation within human–AI systems. Industrial change thus emerges as a reconfiguration of epistemic limits rather than a linear trajectory of technical improvement. The analysis concludes that expanding machine intelligence does not eliminate human authority but intensifies epistemic responsibility, understood as the obligation to determine relevance, value, and legitimacy in socio-technical systems.

The intrinsic information of a given physical system refers to the structure, state, and dynamics inherent to the system. Our aim is to show that this type of information possesses structure and can be accurately described geometrically, as it generates genuine information fields that effectively explain the observed relationships and the dynamics of natural phenomena. Since intrinsic information is universal, its description as a mathematical information field also is. Moreover, thischaracterization of intrinsic information allows for a scientificapproach of its properties, measurements, analysisof perturbations, transient and asymptotic dynamicsin order to explain phenomena in different areas. We consider the value of this approach as a step forwards in foundation of new perspectives in science. Thus, the value of this work is not mainly technical, but of a new in-depth look at the reality that science deals with every day in multiple areas.

In this work we present a web version of Marlo diagrams, which are an innovative way to visualize basic principles of logical reasoning, returning to the tradition of the Quantification of the Predicate. The page contains concise instructions, multiple examples of classical reasoning, and more than one hundred proposed exercises. Thus, it is a teaching resource that can be executed on computers, mobile phones, and tablets as an ICT tool for the development of critical thinking. We present the essential definitions of Marlo diagrams and their operations and provide JavaScript code examples to illustrate their implementation. We also present the line of research that we are currently working on to expand the scope of Marlo diagrams by adding disjunctions to their regions.

In this paper, we study in detail the structure of the global attractor for the Lotka–Volterra system with a Volterra–Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in Hofbauer and Schreiber (J Math Biol 85:54, 2022) and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.

Alternative perspectives on the maintenance of biodiversity and the assembly of ecological communities suggest that both processes cannot be investigated simultaneously. In this concept and synthesis, we challenge this view by presenting major theoretical advances in structural stability and permanence theory. These advances, which provide complementary views, allow studying the short- and long-term dynamics of ecological communities as changes in species richness, composition, and abundance. Here, the global attractor, technically named informational structure (IS), is the central element to construct from information of species' intrinsic growth rates and their strength and sign of interactions. The global attractor has four main properties: (1) It contains all the limits of what is feasible and unfeasible of the dynamical behavior of an ecological system, therefore, (2) it provides a thorough characterization of all combinations of species' richness and composition in which species can coexist (i.e., feasible and stable equilibrium), (3) as well as all connections (paths) of assembly between coexisting communities. Importantly, (4) such topology of coexisting communities and their connections changes when environmental (abiotic and biotic) variation affects the ability of species to grow and interact with others. Overall, these four properties allow switching from a traditional evaluation of species coexistence at equilibrium to a much more realistic nonequilibrium perspective where changes in the structure of the global attractor underlie the transient ecological dynamics. Several fields in ecology can benefit from the study of an IS. For instance, it can serve to evaluate community responses after the end of a perturbation, to design restoration trajectories, to study the consequences of biological invasions on the persistence of native species within communities, or to assess ecosystem health status. We illustrate this latter possibility with empirical observations of 7 years in Mediterranean annual grasslands. We document that extremely wet or dry years generate ISs supporting few coexisting communities and few assembly paths. The remaining communities distinguish winners from losers of ongoing climate change and indicate the limits to future community assembly opportunities. A fully tractable operational framework is readily available to understand and predict the assembly and dynamics of ecological communities in an ever-changing world.

EDITORIAL article Front. Comput. Neurosci., 01 November 2023 Volume 17 - 2023 | https://doi.org/10.3389/fncom.2023.1310392

This paper presents an introduction to dynamic epistemic logic through some formal systems that allow to represent knowledge and beliefs of one or more agents, as well as epistemic actions that can modify them. Specifically, we present the public announcements logic, action models and plausibility models.

Este trabajo presenta una introducción a la lógica epistémica dinámica a través de diversos sistemas formales que permiten representar conocimientos y creencias de uno o varios agentes, así como acciones epistémicas que pueden modificarlos. Concretamente, presentamos la lógica de anuncios públicos, modelos de acción y modelos de plausibilidad.

This book explores a different pragmatic approach to algorithmic complexity rooted or motivated by the theoretical foundations of algorithmic probability

In this chapter, we review a series of topics relevant to psychological science in which the Algorithmic Complexity of Short Strings (ACSS), as estimated using the methods described in the first part of this book, proved useful. These topics are remarkably diverse, including fields such as development, working memory, reasoning, aesthetic preferences, visual cognition, randomness perception and production, language evolution [1], and even belief in conspiracy theories [2].

The self-organising global dynamics underlying brain states emerge from complex recursive nonlinear interactions between interconnected brain regions. Until now, most efforts of capturing the causal mechanistic generating principles have supposed underlying stationarity, being unable to describe the non-stationarity of brain dynamics, i.e. time-dependent changes. Here, we present a novel framework able to characterise brain states with high specificity, precisely by modelling the time-dependent dynamics. Through describing a topological structure associated to the brain state at each moment in time (its attractor or 'information structure'), we are able to classify different brain states by using the statistics across time of these structures hitherto hidden in the neuroimaging dynamics. Proving the strong potential of this framework, we were able to classify resting-state BOLD fMRI signals from two classes of post-comatose patients (minimally conscious state and unresponsive wakefulness syndrome) compared with healthy controls with very high precision.

The model transform fits exactly the parameters of a suitable model to empirical or simulated data in each point in time and/or space. We describe several examples of concrete model transforms and their applications. The model transform allows simple theoretical models to be applied to complex empirical systems in each short interval of time or/and in each local neighborhood. The model can be chosen to identify, for instance, the temporal evolution of the attractor landscape for empirical systems which depict a complex dynamics over time.

Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this concept, defined as Global Structural Stability. This approach would fit with the proper mathematical concept of structural stability, in which we find a full description of the complex dynamics on the phase space due to nonlinear dynamics. This fact can be analyzed as an informational field grounded in a global attractor whose structure can be completely characterized. These attractors are stable under perturbation and suppose the minimal structurally stable sets. We also study in detail, mathematically and computationally, the zones characterizing different levels of biodiversity in bipartite graphs describing mutualistic antagonistic systems of population dynamics. In particular, we investigate the dependence of the region of maximal biodiversity of a system on its connectivity matrix. On the other hand, as the network topology does not completely determine the robustness of the dynamics of a complex network, we study the correlation between structural stability and several graph measures. A systematic study on synthetic and biological graphs is presented, including 10 mutualistic networks of plants and seed-dispersal and 1000 random synthetic networks. We compare the role of centrality measures and modularity, concluding the importance of just cooperation strength among nodes when describing areas of maximal biodiversity. Indeed, we show that cooperation parameters are the central role for biodiversity while other measures act as secondary supporting functions.

The algorithmic complexity (hence also called program-size complexity) of a bit string is defined as the length of the shortest binary computer program that prints out the string (see ( 1.8 ) in Sect. 1.3 ). However, no general, finite and deterministic procedure exists to calculate algorithmic complexity. For a given string there are infinite many programs producing it.