Synchronization is the phenomenon according which coupled nonlinear oscillators exhibit an unison rhythm and behave as a single oscillator without the need of any external control device, it is an emergent property of the system largely studied with many relevant applications. We present some results about global synchronization for identical coupled oscillators via networks or simplicial complexes. Then we will consider topological signals, i.e., dynamical variables defined on nodes, links, triangles, etc. of higher-order networks. The latter are attracting increasing attention from scholars but the investigation of their collective phenomena is only at its infancy. We combine topology and nonlinear dynamics to determine the conditions for global synchronization of topological signals on simplicial complexes [T. Carletti, L. Giambagli and G. Bianconi, Phys. Rev. Letters, 130, 187401 (2023)]. We show that topological obstruction impedes odd dimensional signals to globally synchronize. We provide results where the coupling can be realized via the Laplace matrix or the Dirac one [T. Carletti, L. Giambagli, R. Muolo and G. Bianconi, J. Phys. Complex. 6 025009 (2025)], allowing thus to couple signals of different dimension.

Acknowledgment. The presented work is the result of several projects realized with several colleagues, among which Prof. Ginestra Bianconi, Lorenzo Giambagli and Riccardo Muolo.

A wealth of evidence shows that real-world networks are endowed with the small-world property, i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultrasmall-world organization, whereby the graph’s diameter is independent of the network size over several orders of magnitude, is still unknown. I will show that the “six degrees of separation” is the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. Moreover, the emergence of such a regularity is compatible with all other features, such as clustering and scale-freeness, that normally 

characterize the structure of social networks. Thus, simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.

Modeling real systems by simplicial complexes of different architecture enables investigating geometry-embedded interactions of various orders and their impact on the emergent collective behavior, as a key feature of complexity[1]. In this context, the three-body interactions embedded in exact triangles or triangle faces of simplexes play an essential role beyond the leading pairwise couplings. However, their role critically depends on the studied dynamics. We present some results of the field-driven spin systems with the competing pairwise and triangle-embedded interactions [2], demonstrating that both types of interactions promote collective (self-organized) dynamics on the hysteresis loop in the assembly of triangles [1,2]. In contrast, studies of the phase synchronization processes revealed that pairwise coupling is necessary to provide the transition to a synchronized state. Meanwhile, the increasing triangle-embedded interactions strength tends to reduce the order and induces the hysteresis behavior even in the high-dimensional simplicial complexes with the spectral dimension ds>4; Fig.1. We show how the hysteresis properties, cluster formation and partial synchronization occur depending on the architecture of the assembly characterized by its spectral dimension and the dimension of shared faces, and the distribution of internal frequencies [3,4,5]. These synchronization features are demonstrated in studies of model nanoassembles and human connectome networks.

Figure 1. The segment of a structure of 5-cliques self-assembled under geometric compatibility rules.

REFERENCES

[1] Tadic, B., and Melnik, R., Dynamics 1(2), 181 (2021); EPJB 97, 68 (2024)

[2] Tadic, B., and Gupte, N., Europhys. Lett. 132, 60008 (2021)

[3] Chutani, M., Tadic, B., and Gupte, N., Phys. Rev. E 104(3), 034206 (2021).

[4] Sahoo, S., Tadic, B., Chutani, M., and Gupte, N., Phys. Rev. E 108(3), 034309 (2023)

[5] Sahoo, S. and Gupte, G., Entropy 27(3),233 (2025).

Handling regime shifts and non-stationary time series in deep learning systems presents a significant challenge. In online learning, when new information is introduced, it can disrupt previously stored data and alter the model's overall paradigm, especially with non-stationary data sources. Therefore, it is crucial for neural systems to quickly adapt to new paradigms while preserving essential past knowledge relevant to the overall problem.

This paper proposes a novel neural network training algorithm called Lyapunov Learning. This approach leverages the properties of nonlinear chaotic dynamical systems to prepare the model for potential regime shifts. Drawing inspiration from Stuart Kauffman's Adjacent Possible theory, we leverage local unexplored regions of the solution space to enable flexible adaptation. The neural network is designed to operate at the edge of chaos, where the maximum Lyapunov exponent, indicative of a system's sensitivity to small perturbations, evolves around zero over time.

Our approach demonstrates effective and significant improvements in experiments involving regime shifts in non-stationary systems. In particular, we train a neural network to deal with an abrupt change in Lorenz's chaotic system parameters. The neural network equipped with Lyapunov learning significantly outperforms the regular training, increasing the loss ratio by about 96%.

The Earth system is a very complex and dynamical one basing on various feedbacks. This makes predictions and risk analysis even of very strong (sometime extreme) events as floods, landslides, heatwaves, and earthquakes etc. a challenging task. A recently developed approach via complex networks will be presented. This leads to an inverse problem: Is there a backbone-like structure underlying the climate system? To treat this problem, we have proposed some methods to reconstruct and analyze a complex network from spatio-temporal data based on a nonlinear causality analysis. This approach enables us to uncover relations in the climate system even between far away tipping points as Amazonia and the Tibetan Plateau or Arctic and Southwest China, but also to follow-up rather short phenomena as tropical cyclones.

Then extreme events are studied from this perspective by means of the nonlinear event synchronization analysis. This way we discover relations to global and regional circulation patterns in oceans and atmosphere, which lead to construct substantially better predictions, in particular for the onset of the Indian Summer Monsoon, the Indian Ocean Dipole, or heat waves in Europe.

The behavioral and signaling patterns observed in social animal groups—ranging from honeybees to fish schools—have inspired extensive research into collective decision-making, aiming to uncover the mechanisms behind their remarkable ability to reach consensus without centralized control. In this talk, we examine how key features such as quorum-dependent recruitment dynamics, inhibitory signaling, and individual variability give rise to robust group-level decisions. Drawing from empirical findings in animal behavior and theoretical models, we explore how non-linear social interactions and stochastic processes can drive flexible yet reliable consensus formation. We introduce a modeling framework that captures these principles and demonstrate how they affect the group's ability to balance speed, accuracy, and cohesion in decision-making. This work not only enhances our understanding of biological collectives, but also informs the design of decentralized algorithms for applications in robotics and distributed systems.

Optoelectronic oscillators with time-delayed feedback are of interest for generating precise frequencies, waveforms with chaotic dynamics and for exploring the dynamics of networks of coupled nonlinear oscillators.  We will describe experiments with an optoelectronic oscillator where single photons are detected, and one can watch the progression from shot noise to the birth of a chaotic attractor.  The dynamics of coupled nonlinear optoelectronic oscillators and synchronization will be described.  Can heterogeneity of large numbers of nonlinear oscillators help with synchronization of their dynamics?  Since this is the year of the quantum (IYQ 25) that recognizes 100 years since the initial development of quantum mechanics, we conclude by considering a basic question: how much information can be conveyed through detection of a single photon?

Tipping represents a phenomenon that makes a large transition from one stable state to another due to a small perturbation. It has been reported in nature, climate, ocean systems and ecology and engineering systems. Past studies revealed interesting tipping behaviours in non-autonomous systems such as bifurcation tipping, rate-induced tipping and noise induced tipping. Each of them shows specific characteristic features and their ultimate effect on the tipping element in several model systems. I will discuss here, in brief, what is the meaning of tipping and how it appears in some natural systems. Finally, I will share our experience in ecological models how they respond to environmental variability leading to tipping.

In one study, we considered the 1-dimension Spruce budworm model, that shows coexisting states of an undesirable outbreak of large population and a desirable low population of budworms. We observed bifurcation tipping between the alternate states when the carrying capacity of the system is varied with a linear rate. The tipping shows a delayed transition to an alternate state as usually reported in many other systems. However, we reported an interesting tipping behaviour in the system, when it is perturbed with two successive pulse-like external forcing mimicking the exemplary natural events such as the appearance of two successive earthquake events. We noticed that even if a large pulse of external pulse fails to induce a tipping, a relatively weaker pulse followed by the large one is able to induce tipping. While such tipping due to successive external perturbations is destructive in the case of earthquake, it is beneficial for budworm population when a transition occurs from an outbreak state to a desirable low population saving a vegetation. 

In our second study, we investigate partial tipping in three ecological models against periodic environmental variation: 1-dimensional equilibrium Spruce budworm model, 2-prey-predator model with coexisting steady state and a limit cycle and a 3-species food-chain model with coexisting chaotic and limit cycle states. In the equilibrium and non-equilibrium systems, we find a partial tipping when the tipping occurs only for a fraction of initial points selected from the basin of attraction of the bistable systems. Most importantly, the partial tipping shows a frequency response, with a highest probability of tipping at a critical frequency of the periodic forcing, may be identified as a resonance frequency. Another crucial aspect of such partial tipping is the choice of initial points that are to be selected close to the basin boundary; only exception is the 3-dimensional chaotic system where the partial tipping occurs for arbitrary choice of initial conditions not necessarily to be selected from locations close to the basin boundary.

References

1. When do multiple pulses of environmental variation trigger tipping in an ecological system? A. Basak, S. K. Dana, N. Bairagi, U. Feudel, Chaos: An Interdisciplinary Journal of

Nonlinear Science 34, 093105 (2024).

2. Partial tipping in bistable ecological systems under periodic environmental variability. A. Basak, S. K. Dana, N. Bairagi, Chaos: An Interdisciplinary Journal of Nonlinear Science 34, 083130 (2024).

The broad field of vibrational dynamics of mechanical and electromechanical oscillators associated with energy harvesting mechanisms has grown considerably in recent years. Ambient vibrations, wind and sound are sources of mechanical energy that are highly available in nature and therefore renewable. The general objective of this research is to analyse new approaches and tools to improve the capture of electrical energy from ambient mechanical vibrations. In particular, we studied the impact of synchronisation, resonance and chaotic behaviour on the process of energy harvesting from mechanical oscillators. Using Lagrangian formalism and Euler-Bernoulli beam theory combined with Kirchoff's electrical laws, the equations of the electromechanical systems are established and their numerical resolution is performed via Runge Kutta's algorithm. The results show that the parameters of the systems used have a significant influence on synchronisation and on the energy harvested. More specifically, it shows that phase synchronisation can increase the energy efficiency of multiple oscillator systems. Furthermore, in fluid-structure interaction, the power recovered increases with the amplitude of wind speed fluctuations and with tip mass, hence the advantage of using the cantilever beam with tip mass.

We live in a cyclic environment, and most, if not all, our bodily processes have adjusted to this cycle of 24 hours. An internal clock inside our body enable us to adjust these processes to the external 24-h cycle of day and night. Disturbance of this 24-h rhythm is shown to be associated to disease, like cardiovascular disease, Alzheimer’s disease, metabolic diseases, and many others. Studies in a wide variety of species, from bacteria to mammals, have shown that there are three physiological features conserved across species in this ‘circadian’ (about a day) clock. The first feature is that this clock is an endogenous clock that keeps its 24-h oscillation even under constant conditions, without external stimuli. The second feature is the temperature compensation of this clock. Ordinarily, chemical reactions accelerate at higher temperatures and decelerate at lower temperatures. However, this clock keeps its 24-h period over the physiological range of temperatures. The final feature is that this clock is able to adjust its phase to changes in the external cycle, i.e., it can be entrained. These interesting features are brought about at different levels of organization, and uses different time scales to keep the circadian rhythms of our body attuned to the external cycle. In this talk I will describe some of the mechanisms that underlie this mysterious clockwork, which keeps us in sync with the external environment, by keeping a constant and robust rhythm, while also being able to adapt to changes in the environment (e.g., jet lag).

Abstraction is the process of extracting the essential features from raw data while ignoring irrelevant details. This is similar to the process of focusing on large-scale properties, systematically removing irrelevant small-scale details, implemented in the renormalisation group of statistical physics. This analogy is suggestive because the fixed points of the renormalisation group offer an ideal candidate of a truly abstract -- i.e. data independent -- representation. It has been observed that abstraction emerges with depth in neural networks. Deep layers of neural network capture abstract characteristics of data, such as "cat-ness" or "dog-ness" in images, by combining the lower level features encoded in shallow layers (e.g. edges). Yet we argue that depth alone is not enough to develop truly abstract representations. We advocate that the level of abstraction crucially depends on how broad the training set is. We address the issue within a renormalisation group approach where a representation is expanded to encompass a broader set of data. We take the unique fixed point of this transformation -- the Hierarchical Feature Model -- as a candidate for an abstract representation. This theoretical picture is tested in numerical experiments based on Deep Belief Networks trained on data of different breadth. These show that representations in deep layers of neural networks approach the Hierarchical Feature Model as the data gets broader, in agreement with theoretical predictions. 

Extreme orbits—trajectories connecting local extrema in one-dimensional maps—are fundamental in organizing periodic windows within the parameter space of dissipative systems, as cascades of intricate shrimp-like structures. These orbits enable precise localization of periodic sets, reveal their cascading distributions, and link them to complex periodic arrangements, as demonstrated in the circle map and perturbed logistic map. Furthermore, extreme curves, which intersect periodicity cascades at stability centers, define singular fractal boundaries between chaotic and periodic domains in the parameter spaces, deviating from previously assumed universal fractal dimensions. This fractal dimension is observed to be the same in algebraic and transcendental 1D maps with, at least, two control parameters. In strongly dissipative nontwist maps, extreme orbits also govern the global organization of shrimp-like structures, challenging long-held symmetries in fractal dimensions near periodicity cascades. Together, these findings underscore extreme curves as a unifying framework for decoding the interplay between order and chaos in nonlinear dynamics.

A network of coupled time-varying systems, where individual nodes are interconnected through links, is a modeling framework used across many disciplines. In networks of identical nodes exhibiting chaotic behavior, such as Rössler oscillators, clusters of nodes—or even the entire network—can synchronize over a range of coupling strengths [1]. In this study, we demonstrate that small differences among nodes can trigger extreme desynchronization events, known as bubbling [2,3,4], even in regimes where synchronization is expected. This finding highlights the outsized impact of minor unit heterogeneity on network dynamics in real-world systems.

Our study reveals that bubbling is pervasive in networks of Rössler chaotic oscillators, emerging across a broad range of coupling strengths. In network topologies where cluster synchronization is expected, we find that while the hierarchical sequence of cluster formation predicted by the Master Stability Function (MSF) [1,5] remains intact, the parameter space for bubble free synchronization is significantly reduced. To understand these observations, we systematically evaluate four stability criteria based on the transverse stability of the synchronization manifold in networks of identical oscillators. However, none of these criteria accurately predict the bubbling domain. We therefore introduce a novel stability criterion that incorporates these measures along with the duration of time the system spends in transversely unstable regions of the synchronization manifold. This measure—derived from finite-time transverse Lyapunov exponents and the averaging window duration—is the only one that

successfully predicts the bubbling domain, for both global and cluster synchronized states.

Our findings demonstrate that the effective synchronization domain of a network is much smaller than previously assumed and is replaced by epochs of synchronization interspersed with extreme events. Our findings have important implications for real-world systems where synchronized behavior is crucial for the system functionality.

We present a novel two-player game in a chaotic dynamical system where players have opposing objectives regarding the system’s behavior. The game is analyzed using a methodology from the field of chaos control known as partial control. Our aim is to introduce the utility of this methodology in the scope of game theory. These algorithms enable players to devise winning strategies even when they lack complete information about their opponent’s actions. To illustrate the approach, we apply it to a chaotic system, the logistic map. In this scenario, one player aims to maintain the system’s trajectory within a transient chaotic region, while the opposing player seeks to expel the trajectory from this region. The methodology identifies the set of initial conditions that guarantee victory for each player, referred to as the winning sets, along with the corresponding strategies required to achieve their respective objectives. This is a joint work with Gaspar Alfaro and Miguel A.F. Sanjuán from URJC, Spain.

In this talk I will discuss a general procedure to train a stochastic dynamical model for classification and generation tasks. First, stable attractors are planted within the examined model and the weighted matrix of intertangled interactions optimized to steer different items to be classified towards distinct asymptotic attractors. Adding noise improves the robustness of the algorithm and paves the way to a generative version of the scheme. To this end the covariance matrix of the noise is trained and the linear noise approximation employed to analytically estimate the distribution to be sampled for the generation step. The procedure is challenged against different datasets and by employing the continuous Hopfield model as a reference dynamical scheme. 

Understanding how network heterogeneity shapes cooperative phenomena remains a central problem in the study of complex systems. Mean-field theories of models on networks provide a fundamental framework to tackle this problem and are a cornerstone of statistical physics, with numerous applications in condensed matter, biology and computer science. In this talk, I will present a novel class of mean-field equations describing the equilibrium properties of Ising spin models on highly connected networks with arbitrary degree distributions. I will show that, even in the high-connectivity regime, moderate degree fluctuations give rise to non-universal behaviour that explicitly depends on the full degree distribution. As a result, standard mean-field models based on fully connected graphs, such as the Curie-Weiss and Sherrington-Kirkpatrick (SK) models, are only recovered when the network degree distribution becomes sharply peaked. I will illustrate how degree heterogeneities modify the replica-symmetric phase diagram of the SK model. Finally, I will briefly discuss how degree fluctuations impact the spectral properties of highly connected networks. These results put forward a novel class of analytically tractable mean-field models that incorporate the effects of structural heterogeneity.

The stabilization problem for linear dynamical systems is well-understood and systematically characterized through the eigenstructure of the system matrices. In contrast, stabilization of nonlinear systems presents significant challenges, as no general framework has yet been established. Existing powerful methods—such as linearization, state feedback stabilization [Krener, 1973], and output feedback stabilization [Isidori, 1985]—are primarily based on the principle of nonlinear cancellation, followed by the application of linear techniques in transformed coordinates. Although these approaches are rigorous and widely applicable, they are typically limited to specific classes of nonlinear systems or yield only local stabilization results.

This presentation proposes a promising possible more general framework based on a particular class of nonlinear systems: polyflow vector fields [Hosler-Meisters, 1989]. Specifically, it aims to: Introduce polyflow vector fields which generalize linear vector fields; Develop a characterization and stability theory for these vector fields using the Koopman operator framework [Zagabe, 2024]; Demonstrate, through illustrative examples, how the inherent linearity of these systems in the

Koopman-lifted coordinates enables the design of a novel (polyflow) stabilization technique; and Explore potential extensions of this approach to generalize existing linear stabilization strategies.

We will focus on two key concepts: polyflow vector fields, in which initial condition is expressed polynomially in the system flow; and the Koopman operator, a linear (but infinite-dimensional) operator that captures the evolution of nonlinear systems [Mauroy et al., 2020].

References

[Hosler-Meisters, 1989] Hosler-Meisters, G. (1989). Polynomial flows on Rn. Banach Center Publications, 1(23):9–24.

[Isidori, 1985] Isidori, A. (1985). Nonlinear control systems: an introduction. Springer. 

[Krener, 1973] Krener, A. J. (1973). On the equivalence of control systems and the linearization of nonlinear systems. SIAM Journal on Control, 11(4):670–676.

[Mauroy et al., 2020] Mauroy, A., Susuki, Y., and Mezić, I. (2020). Koopman operator in systems and control. Springer.

[Zagabe, 2024] Zagabe, M. C. (2024). A Koopman operator approach to stability and stabilization of switched nonlinear systems. PhD thesis, University of Namur, DOI: https://doi.org/10.13140/RG.2.2.25880.81923.

Feature selectivity, the ability of neurons to respond preferentially to specific stimulus configurations, is a fundamental building block of cortical functions. Various mechanisms have been proposed to explain its origins, differing primarily in their assumptions about the connectivity between neurons. Some models attribute selectivity to structured, tuning-dependent feedforward or recurrent connections, whereas others suggest it can emerge within randomly connected networks when interactions are sufficiently strong. This range of plausible explanations makes it challenging to identify the core mechanisms of feature selectivity in the cortex. We developed a novel, data-driven approach to construct mechanistic models by utilizing connectomic data-synaptic wiring diagrams obtained through electron microscopy-to minimize preconceived assumptions about the underlying connectivity. With this approach, leveraging the MICrONS dataset, we investigate the mechanisms governing selectivity to oriented visual stimuli in layer 2/3 of mouse primary visual cortex. We show that connectome-constrained network models replicate experimental neural responses and point to connectivity heterogeneity as the dominant factor shaping selectivity, with structured recurrent and feedforward connections having a noticeable but secondary effect in its amplification. These findings provide novel insights on the mechanisms underlying feature selectivity in cortex and highlight the potential of connectome-based models for exploring the mechanistic basis of cortical functions.

Finding the model that best captures patterns hidden within noisy data is a central problem in science. In this context, the Ising model has been widely used to infer pairwise patterns in binary data. In recent years, attention has been brought to high-order patterns of data and the question of how to detect them. We will discuss the use of (classical) spin models with interactions of order higher than two to extract such patterns in binary data, and why this problem is challenging. By analyzing the information-theoretic complexity of spin models, we will see that, despite their appearance, models with high-order interactions are not necessarily more complex than pairwise models. 

We will then focus on a sub-family of spin models with minimal information-theoretic complexity, which we call Minimally Complex Models (MCMs). These models have interactions organized in a community-like manner, and we will see how this can be used to identify groups of highly correlated variables in binary data. Uncovering such community structures in noisy data is crucial to understanding emergent phenomena in many complex systems, such as the brain, or health or social systems. So far, existing techniques rely solely on the pairwise correlation patterns of the data; in contrast, our approach takes into account all higher-order data patterns. We will demonstrate the capabilities of our approach against pairwise community detection on artificial data with built-in high-order community structures, and discuss the consequence of using a pairwise approach when the data is inherently high-order. Finally, we will discuss possible applications of MCMs in the context of neuroscience. Using Minimally Complex Models opens up new ways to tackle high-dimensional data modeling.

Synchronization (self-organization in time) and swarming (self-organization in space) are universal phenomena that co-occur in many biological and physical systems. Swarmalators are entities that swarm through space and synchronize in time and are potentially considered to replicate the complex dynamics of many real-world systems. We will discuss minimal model of swarmalators which are solvable and give rich dynamics. Previously, the internal dynamics of swarmalators have been taken as a phase oscillator inspired by the Kuramoto model. Here we examine the internal dynamics utilizing an amplitude oscillator capable of exhibiting periodic and chaotic behaviors. We discuss the effect of a predator-like agent in the swarmalators model. The collective behaviors of swarmalators with higher-order interactions is also discuss.

Keywords: Swarming, synchronization, stability analysis.

References:

[1] Gourab Kumar Sar, Dibakar Ghosh, and Kevin O’Keeffe, “Solvable model of driven matter with pinning”, Phys. Rev. E, 109, 044603 (2024).

[2] Samali Ghosh, Suvam Pal, Gourab Kumar Sar, and Dibakar Ghosh, “Amplitude responses of swarmalators”, Phys. Rev. E, 109, 054205 (2024).

[3] Md Sayeed Anwar, Gourab Kumar Sar, Matjaž Perc and Dibakar Ghosh, “Collective dynamics of swarmalators with higher-order interactions”, Communications Physics, 7, 59 (2024).

[4] Kevin O’Keeffe, Gourab Kumar Sar, Md Sayeed Anwar, Joao U. F. Lizárraga, Marcus A. M. de Aguiar and Dibakar Ghosh, “A solvable two-dimensional swarmalator model”, Proceedings of the Royal Society A, 480, 20240448 (2024).

[5] Gourab Kumar Sar, and Dibakar Ghosh, “Flocking and swarming in a multi-agent dynamical system”, Chaos, 33, 123126 (2023).

[6] Samali Ghosh, Gourab Kumar Sar, Soumen Majhi, Dibakar Ghosh, “Antiphase synchronization in a population of swarmalators”, Phys. Rev. E, 108, 034217 (2023).

[7] Md Sayeed Anwar, Gourab Kumar Sar, Timoteo Carletti, and Dibakar Ghosh, “A two-dimensional swarmalator model with higher-order interactions ”, SIAM Journal on Applied Mathematics (2025).

Intelligent systems must navigate a fundamental tradeoff: they require internal representations that are both structured enough to generalize and selective enough to preserve individual distinctions. In this talk, I will present a unifying theoretical framework that formalizes this generalization–identification tradeoff as an emergent property of finite semantic resolution. We derive closed-form, universal Pareto curves that constrain the accuracy of generalization and identification tasks across diverse stimulus spaces, architectures, and noise conditions. These curves reveal sharp scaling laws—most notably a 1/n collapse in multi-item processing—and define optimal semantic resolutions for different tasks. Empirical results from minimal ReLU models, convolutional networks, and state-of-the-art vision-language models validate the theory, showing how resolution boundaries spontaneously self-organize during learning. I will conclude by discussing implications for representational bottlenecks in cognitive systems and large models, and how these limits intersect with topological and renormalization-based approaches to abstraction and multiscale processing.