The Mathematics of Complex Systems:
Theory and Applications

Let $A$ be a standard graded Artinian $\mathbb{K}$-algebra over a field of characteristic zero. We prove that the failure of strong Lefschetz property (SLP) for $A$ is equivalent to the osculating defect of a certain rational variety. Our results extend previous works by eliminating the need for the defining ideal of $A$ to be equigenerated and by extending the study of WLP/SLP in a fixed degree to the study of the SLP in any degree and any range. As a consequence we provide a geometric interpretation of the vanishing of a higher Hessian, extending the classical Gordan-Noether criterion. Moreover, we reobtain some foundational results, including the presence of SLP for codimension 2 Artinian algebras, and the SLP for Artinian Gorenstein algebras with Hilbert function $(1,3,6,6,3,1)$. Joint work with: Charles Almeida (UFMG/Brazil) and Rodrigo Gondim (UFRPE/Brazil)

It is known, due to Buchsbaum-Eisenbud, that any homogeneous Gorenstein ideal $I$ of codimension $n$ in $k[\mathbf{x}]=k[x_1,\ldots,x_n]$ can be obtained as a quotient ideal $(x_1^m,\ldots,x_n^m):\mathfrak{f}$, for some integer $m \geq 1$ and some form $\mathfrak{f}$. In this work, we present how $\mathfrak{f}$ can be retrieved from $I$ by taking the so-called (socle-like) Newton dual of a minimal generator of the Macaulay inverse of $I$. Additionally, we provide conditions under which the Gorenstein ideal $I$ is equigenerated in terms of the exponent $m$ and the form $f$. This work is based on results obtained in collaboration with Aron Simis (UFPE) and Zaqueu Ramos (UFS).

When studying Lefschetz properties on a graded algebra over a field, we are questioning when the products by a degree-1 element induce linear isomorphisms between the graded components of the algebra. In this talk, we will concentrate on such a problem for particular algebras defined over combinatorial structures such as graphs, matroids, and abstract pseudo-varieties, and we are going to present recent advances on the matter.

In this talk we will make a homological discussion of ideals of minors fixing a submatrix in the context of a perfect codimension two ideal. A combinatorial outcome of the results is a proof of the conjecture on the Jacobian ideal of a hyperplane arrangement stated by Burity, Simis and Tohăneanu.

Sub-static manifolds can be viewed as special solutions to the Einstein equations in the presence of stress-momentum fields satisfying a certain dominant energy condition. In this talk, we address the rigidity problem for such manifolds, possibly with a non-empty boundary. First, we establish certain integral inequalities that extend the works of Chrúsciel and Boucher-Gibbons-Horowitz to sub-static manifolds. Even in the static vacuum case, the obtained inequalities improve on the previously known ones. Next, we present local and global splitting theorems under the assumption of the existence of compact minimal hypersurfaces. Finally, we analyze the system arising from static solutions to the Einstein field equations coupled with a σ-model. This is joint work with G. Colombo (Università di Napoli, Italy), L. Mari (Università di Milano, Italy), and M. Rigoli (Università di Milano, Italy).

The main goal of this talk is to prove the uniqueness of the asymptotic planes of complete translating solitons with finite genus, width, and entropy. If time allows, we will also provide some applications of this uniqueness result. This is joint work with Francisco Martin and Niels M. Møller.

In this talk, we will investigate the spectrum of the Laplace–Beltrami operator on a complete, non-compact Riemannian manifold $M^n$, whose Ricci curvature satisfies the inequality $$ \mathrm{Ric} \geq -(n-1)G(r), $$ for some continuous, non-increasing function $G$ such that $G \to 1$ as $r \to \infty$. We will show that if the bottom spectrum attains the maximal value $\frac{(n-1)^2}{4}$ compatible with this curvature condition, then the spectrum of $M$ coincides with the spectrum of hyperbolic space $\mathbb{H}^n$, that is, $$ \sigma(M) = \left[ \frac{(n-1)^2}{4}, \infty \right). $$ Moreover, we show that this result can be localized to an end $E \subset M$ with infinite volume. This is a joint work with L. Mari, M. Ranieri and F. Vitório.

A Simons-Calabi type equation is obtained for Riemannian hypersurfaces immersed in a Riemannian semi-product space. As a consequence, we study stochastically complete hypersurfaces with vanishing mean curvature, as well as those with nonzero constant mean curvature, immersed in Riemannian product spaces, for which the Simons–Calabi equation allows us to establish characterization and classification results. In an analogous way, characterization results are also obtained for the study of hypersurfaces with nonzero maximal constant mean curvature, now immersed in Lorentzian product spaces.

In this talk, we present some applications of a Simons-type formula for spacelike submanifolds in semi-Riemannian warped products. In particular, we consider the cases in which the ambient space is a Robertson-Walker spacetime model, such as the Lorentz–Minkowski, de Sitter, anti–de Sitter, and Einstein de Sitter spacetimes.

We will talk about new area-charge inequalities for the boundary of time-symmetric Einstein-Maxwell initial data sets, in both compact and noncompact cases, under the dominant energy condition. These inequalities lead to novel rigidity theorems with no analogues in the uncharged setting. In the noncompact case, our result is obtained by applying Gromov’s $\mu$-bubble technique in a new geometric context. This is a joint work with Abraão Mendes.

Abstract: In this talk, we will analyze a doubly parabolic Keller-Segel system in the whole space R^d, d ≥ 2, where both cellular and chemical diffusion are governed by fractional Laplacians with distinct exponents. This system generalizes the classical Keller-Segel model by introducing superdiffusion, a form of anomalous diffusion. This extension accounts for nonlocal diffusive effects observed in experimental settings, particularly in environments with sparse targets. We will present results on the local and global well-posedness of mild solutions for this generalized system under suitable conditions on the initial data in L^p(R^d). Additionally, we will characterize the asymptotic behavior of the solutions. This is a joint work with Crystianne Andrade (IMECC, Unicamp).

Abstract: In this talk, we present some of the problems we are currently investigating concerning the equations governing second-grade fluids. In particular, on the temporal decay, we characterize the decay rate of solutions for the second-grade fluid equations in terms of the decay character of the initial data. Furthermore, by imposing slightly more regularity on the initial data, we investigate the large-time behavior of these solutions and compare them to the solutions of the linearized equations.

Abstract: In this talk, we address the existence of solutions for the quasilinear Schrödinger-type problem −Δu − ½Δ(a(x)u²)u + V(x)u = f(u), x ∈ ℜ² where V is a continuous potential and f: ℜ → ℜ is a superlinear function with either exponential subcritical or exponential critical growth. Our main analytical tool is the Nehari manifold method, which we employ to establish the existence of nonnegative solutions as well as nodal solutions. The results extend and complement classical contributions on quasilinear Schrödinger equations obtained via the Nehari method, particularly those of Jia-quan Liu, Ya-qi Wang, and Zhi-Qiang Wang, by considering nonlinearities of an exponential type in two dimensions.

Abstract: In this talk, we present recent results on the existence and multiplicity of semiclassical solutions for nonlinear Dirac equations involving continuous potentials and general nonlinear self-couplings. In the first part, we consider a class of nonlinear Dirac equations with a small parameter ε > 0, where the nonlinearity depends on the modulus of the solution. Under mild assumptions, we prove that the number of solutions is at least the number of global minima of the potential function V, provided ε is sufficiently small. In the second part, assuming that the external potential attains a local minimum, we relate the topological complexity of the set of minimum points to the multiplicity of solutions. The main tools include variational methods and the Lusternik--Schnirelmann category, combined with a penalization technique to handle the strongly indefinite nature of the associated energy functional. Both problems are motivated by the rich structure and analytical challenges posed by the Dirac operator, particularly in the semiclassical regime. Our results contribute to the understanding of nonlinear spinorial models by establishing existence and multiplicity theorems in settings with minimal regularity and without symmetry assumptions.

Abstract: We establish global multiplicity of positive solutions (existence and nonexistence theory) for a sublinear elliptic equation in RN with mass zero. In order to obtain our results we use a combination of the sub- super solution method and variational techniques. For instance, we need to implement a relevant result of type D^{1,2}(RN) versus X local minimizer for some appropriate space X. This is joint work with Minbo Yang (Zhejiang Normal University), Pedro Ubilla (USACH) and Jiazheng Zhou (UnB).

Abstract: We study a class of elliptic problems, involving a k-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. Na essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular and analyze the intersection number depending on the parameter k and the dimension d. This talk is based on joint work with João Marcos do Ó and Esteban da Silva.

Abstract: This talk presents existence results for positive solutions to a Schrödinger-Poisson system driven by a general class of integrodifferential operators. The system allows for two distinct operators whose parameter orders can be chosen freely under a mild coupling condition. The nonlinearity is of a pure power type, with a growth exponent in a range determined by these orders. Our argument is variational, based on a minimization over a Nehari-Pohozaev manifold. We establish a maximum principle to guarantee the solutions positivity, and show that for the specific case of the fractional Laplacian, this positive solution is also a ground state.

Abstract: We analyze perturbations of the classical Brezis–Nirenberg problem that turn the equation into a supercritical one. While the original critical formulation admits no solutions in certain cases, the perturbed version generates positive radial solutions. In dimensions greater than or equal to three, this mechanism produces solutions in parameter ranges where nonexistence was expected. This is a joint work with Luiz Faria (UFJF-Brazil) and Jeferson Silva (UFOP-Brazil).

Abstract: This will be an introductory talk on how energy is dissipated in celestial systems. I will first explain tides, the main mechanism of energy loss in large bodies. Then I will show the main dynamical consequences of tides in simple two-body problems. Dissipative celestial mechanics aims to describe long transient behavior that leads to simple asymptotic states. These processes can be seen as multi-time-scale singular perturbation problems. However, the correct formulation of the problem is still an open question.

Abstract:We will introduce some concepts related to the Newtonian N-body problem. The relation of particular solutions of this problem with central configurations. Next, we will recall the definition of a stacked central configuration, a concept introduced by Marshall Hampton in 2005. We will then present some results for stacked central configurations with N = 5 and general N.

Abstract:In this talk, it will be considered deformations of Hamiltonian systems, and the average theory will be applied to show the existence of isolated periodic orbits (limit cycles). Special attention will be paid to the coupled harmonic oscillators. In addition, some applications to the principal curvature lines will be explored.

Abstract:We review the classical theory of moments and formulate its implications for problems of equilibrium configurations of systems of interacting particles. Our study assumes only that the interactions are between pairs of particles and are directed along the lines connecting each pair. The equilibrium equations obtained from the theory of moments include celebrated equations from Celestial Mechanics such as Albouy-Chenciner and Dziobek equations for central configurations. All equilibrium equations are geometric in character, as they are expressed only in terms of mutual distances between particles and volumes of subconfigurations. We shall also briefly discuss applications to constraints on mutual distances to ensure a configuration has a given dimension or is co-spherical.

Abstract: In recent years many links between celestial mechanics and billard-type problems have been studied in the research field of mechanical billiards, or billiards in a potential field. Several new integrable mechanical billiards have been identified and in the high energy regime many mechanical billiards in a Kepler potential field in the plane are shown to have chaotic dynamics. In this talk I will survey ideas for establishing integrability and chaos in these systems.

TDA is an applied branch of Algebraic Topology that deals with Data Science. Algebraic Topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. Topology is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations. The fact that TDA ends up being useful to Data Science is rather surprising. For a Computer Scientist, Statistician, or Data Scientist in general to come to grips with the field of TDA seems a long, arduous path to follow. In this brief talk, I aim to demonstrate that there are shortcuts. With examples and very few definitions, I try to convince non-mathematicians they can safely use TDA and are encouraged to follow an introductory course on Algebraic Topology for Data Science.

Permutation entropy (PE) is a powerful and widely adopted method for assessing the predictability and complexity of time series. While it is parameter-free, classical PE treats all ordinal patterns equally, which can obscure subtle dynamical structures by discarding information carried by amplitude variations. Weighted Permutation Entropy (WPE) addresses this limitation by introducing weights derived from the data’s amplitude, improving sensitivity to regime changes and outlier behavior. In this talk, I will describe the recently introduced Generalized Weighted Permutation Entropy (GWPE) method, a unified extension of both PE and WPE. GWPE broadens the framework by offering a tunable balance between ordinal structure and amplitude information, allowing flexible weighting schemes to enhance the effect of both small and large fluctuations. This generalization not only preserves the interpretability and computational efficiency of PE but also provides a richer lens to analyze predictability and complexity in nonlinear systems. I will outline the theoretical foundation of GWPE, demonstrate its computation, and present illustrative applications on synthetic systems and real-world datasets. The aim is to show how GWPE captures fluctuation dynamics that standard PE and WPE may miss and to highlight open questions for applying GWPE in multi-scale and high-frequency data analysis.

Differential equations can be used to model many scientific and engineering problems. However, for many physical systems of practical interest, these differential equations are analytically intractable. Consequently, there is great interest in developing computational techniques to solve differential equations numerically. An artificial intelligence approach to solving differential equations is based on Neural Networks. A Neural Network is a computational model inspired by the structure and function of the human brain. At its core, it’s a system designed to recognize patterns and learn to process data in a way that mimics human cognition. The basic principle for solving differential equations with Neural Networks is to look at the problem as an optimization problem. Defining a differential equation generically as D(u) = F, where D is the differential equation of interest, u is a possible solution to D, and F is a known function. Let u_trial be the output of a Neural Network. If u_trial is a trial solution for D, then the residual of this trial solution can be expressed as R = D(u_trial) - F. The core idea is to utilize the squared residual, R^2, as the loss function during the Neural Network’s training process, thereby reducing the problem of solving the differential equation to a minimization problem.

This presentation explores the morphological and spatial complexity of Sargassum spp. across individual to regional scales through an integrated computational approach. At the individual level, we employ rule-based modeling via L-systems to simulate realistic growth architectures of different Sargassum morphotypes, capturing topological constraints, apical dominance, and branching variability. These simulations provide a generative framework for testing hypotheses about developmental plasticity and environmental modulation of form. At the regional scale, we apply multifractal analysis to segmented remote sensing products of Sargassum strandings in the tropical Atlantic. The results reveal distinct multifractal signatures among morphological patch types, suggesting scale-invariant properties and heterogeneity in aggregation patterns potentially linked to hydrodynamic or biological drivers. Together, these modeling and analysis strategies offer complementary perspectives on the emergence of structural complexity in Sargassum systems, contributing to a deeper understanding of active matter dynamics in coastal and oceanic environments.

Ocean color satellite observations provide access to a wide range of key biogeochemical descriptors (phytoplankton biomass, particulate and dissolved matter, and associated carbon stocks). These data, available continuously since 1997, offer spatial resolutions ranging from 20 m to 1 km and quasi-daily temporal coverage, making it possible to study marine ecosystems at multiple scales—from synoptic to local, from extreme events to long-term changes. Interpreting the ocean color signal requires the development of specific algorithms, relying on various approaches (simple empirical models or machine learning-based methods). Similarly, exploiting the maps derived from satellite observations requires the implementation of adapted analytical tools (time-series decomposition, classifications, Self-Organizing Maps). This presentation will provide an overview of the potential of ocean color observations for studying the impacts of climatic and anthropogenic forcings on the marine environment, as well as the tools used.

This presentation is situated within the school of mathematics of complex systems and demonstrates how physical phenomena from two seemingly distinct domains—the quantum dynamics in billiards and the propagation of light through turbulent media—reveal common underlying principles when analyzed with stochastic tools. In particular, the generalization of the Rayleigh stochastic process serves as a unifying framework that enables the description of transitions between order and chaos through non-Gaussian probability density functions and power spectral densities characterized by 1/f^α-type laws. In the context of quantum billiards, this approach provides a description of the transition from integrable dynamics to chaotic regimes by extending Dyson’s Brownian motion model. This extension captures phenomena such as energy level repulsion, the emergence of composite distributions, and the appearance of freezing regimes associated with multifractal structures. Similarly, in the study of optical turbulence, it is observed that the propagation of light beams through heterogeneous flows generates erratic displacements that do not conform to the Gaussian paradigm. Here, modeling via Weibull processes and Burr Type XII distributions accounts for the presence of heavy tails and extreme events, as well as the emergence of power spectra exhibiting 1/f^α scaling under conditions of strong spatial correlation. Both scenarios—quantum chaos and atmospheric turbulence—demonstrate that the mathematics of stochastic processes not only provides a common language for diverse phenomena but also offers predictive tools to understand complex transitions. The connection between probability distributions and power spectra reveals a shared structure in which clustering intermittency and amplitude intermittency coexist, illustrating the power of the stochastic approach as a bridge between theory and experiment in complex systems.

Ecologists have long investigated how demographic and movement parameters determine the spatial distribution and critical habitat size of a population. However, most models oversimplify movement behavior, neglecting how landscape heterogeneity influences individual movement. We relax this assumption and introduce a reaction-advection-diffusion equation that describes population dynamics when individuals exhibit space-dependent movement bias toward preferred regions. Our model incorporates two types of these preferred regions: a high-quality habitat patch, termed ‘habitat’, which is included to model avoidance of degraded habitats like deforested regions; and a preferred location, such as a chemoattractant source or a watering hole, that we allow to be asymmetrically located with respect to habitat edges. In this scenario, the critical habitat size depends on both the relative position of the preferred location and the movement bias intensities. When preferred locations are near habitat edges, the critical habitat size can decrease when diffusion increases, a phenomenon called the drift paradox. Also, ecological traps arise when the habitat overcrowds due to excessive attractiveness or the preferred location is near a low-quality region. Our results highlight the importance of species-specific movement behavior and habitat preference as drivers of population dynamics in fragmented landscapes and, therefore, in the design of protected areas [1]. [1] Dornelas, V., de Castro, P., Calabrese, J. M., Fagan, W. F., & Martinez‐Garcia, R. (2024), Movement bias in asymmetric landscapes and its impact on population distribution and critical habitat size, Proceedings of the Royal Society A, 2024, 0185.

Although rapid or recurrent fluctuations in resource availability are typical of natural habitats, their demographic consequences for small populations have yet to be thoroughly assessed. The present study aims to clarify how seasonal fluctuations in resource availability and demographic stochasticity interact to influence the persistence of small populations. Our analysis begins with a deterministic consumer–resource model driven by seasonal forcing, from which we develop an individual-based stochastic simulation with discrete generations to account for demographic and mutational variability. We demonstrate how phase lags among multiple substitutable resources, or adaptive shifts in metabolic allocation, can buffer extinction risk without changing average resource inputs.

Understanding how species adapt and survive amid the rapid environmental changes driven by climate change has never been more urgent. We will introduce the foundations of evolutionary theory and the concept of fitness landscapes before turning to a critical challenge: the idea of evolutionary rescue. We will examine the possibilities and limits of adaptation in avoiding extinction through rapid evolutionary change, drawing on insights from my recent work in mathematical modeling.

Understanding the complexity of climate dynamics is essential for improving predictive models and resilience strategies in the face of increasing hydroclimatic extremes. This talk presents advanced methods used to assess the nonlinear and scale-dependent behavior of climate variables: Multifractal Detrended Fluctuation Analysis (MFDFA), the Fisher–Shannon Information Plane, and the Complexity–Entropy Causality Plane (CECP). MFDFA is used to quantify long-range correlations, multifractality, and the contribution of small/large fluctuations in time series. The Fisher–Shannon plane provides a joint assessment of information content and disorder, enabling the identification of regimes with high unpredictability and low structural complexity. CECP is used to characterize the dynamical nature of the time series by mapping their position in the entropy–complexity space, distinguishing stochastic and chaotic behaviors. Together, these methods offer a robust framework for diagnosing the intrinsic variability and emergent patterns in climate systems.

In this talk, I present two tools for the quantitative analysis of active collective systems—illustrated with experimental data from schools of Danio rerio. The goal is to provide participants with a practical and conceptual introduction to methods widely used in statistical physics and complex systems modeling. 1. Mean Square Displacement (MSD): I show how this classic tool allows us to classify diffusion regimes—normal, superdiffusive, or subdiffusive—and reveal dynamic transitions induced by factors such as population density. I apply MSD to demonstrate how individual fish behavior changes as group size increases, exhibiting superdiffusion at low densities and transitioning to more restricted diffusion in larger groups. 2. Levy Flight and Power Laws: I explain how to identify efficient search patterns through the distribution of step lengths. I show that, under certain conditions, individual movements follow truncated power laws, compatible with Lévy flights—a strategy optimized for exploration in uncertain environments. I discuss statistical criteria for distinguishing these patterns from Brownian motion and their biological significance. Throughout the presentation, I emphasize best practices in data analysis, methodological challenges, and physical interpretation of results.

Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory, these systems are simple prototypes. The conservative dynamics of a billiard may vary from regular to chaotic, depending only on the shape of the boundary. In this work, we experimentally investigated active particle-environment interactions in Lemon Billiards. To achieve this, we developed an Arduino robot interacting with the billiard table, which acts as our active particle, and controlled the shape of the Lemon Billiard with a parameter γ ∈ [0, 0.5]. We experimentally confirm that the system’s dynamics continuously change from regular (γ = 0) to fully chaotic (γ = 0.5), passing through cases of mixed-phase space for intermediate values of γ. We also analyze the robot’s sensitivity to the initial conditions using the phase space as a function of time and the Lyapunov exponent to characterize the dynamics observed through the robot-environment interaction. Furthermore, we introduced a methodology based on recurrence analysis to compare numerical data with experimental data on a dynamical system. Numerical analyses complement the experimental results, showing a significant concordance. Our experimental setup presents an alternative method for measuring mixed-phase spaces in a billiard system outside the scope of optical physics. The robot-environment interaction provides an excellent alternative platform for experimental studies in physical systems, including additional aspects of chaos and dynamical systems such as confined self-propelled particles and self-avoiding active particles.

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The presumed proximity to a critical point is believed to endow the brain with scale-invariant dynamics that enhance information processing, storage, and transmission. We discuss how such collective behavior can emerge from neuronal interactions, focusing on how populations encode sensory stimulus intensity. Using simplified computational models, we find that this task is most effectively performed near a phase transition, consistent with the critical brain hypothesis. From the experimental point of view, although hallmarks of criticality, such as scale invariance and non-Gaussian dynamics, have been observed in brain activity, a direct connection between criticality and task outcome has remained elusive for decades. To bridge this gap, we explore neuronal activity in the visual cortex of mice engaged in a visual recognition task using a phenomenological renormalization group (PRG) approach. Our findings point to intriguing relationships between neuronal scaling behavior and task performance under various conditions, offering new perspectives on the potential role of critical dynamics in cortical processing.

An important working hypothesis to investigate brain activity is whether it operates in a critical regime. Recently, maximum-entropy phenomenological models have emerged as an alternative way of identifying critical behavior in neuronal data sets. In this talk we will discuss the signatures of criticality from a firing rate-based maximum-entropy approach on data sets generated by computational models, and we compare them to experimental results. We will show that the maximum entropy approach consistently identifies critical behavior around the phase transition in models and rules out criticality in models without phase transition. The maximum-entropy-model results are compatible with results for cortical data from urethane-anesthetized rats data, providing further support for criticality in the brain.

Major depressive disorder MDD with anhedonia—a reduced ability to experience pleasure—is linked to elevated cortisol and glutamate-related brain damage, particularly affecting the frontal cortex. We investigated how this condition alters brain activity dynamics using a mouse model of depression induced by chronic corticosterone exposure. Adult male mice N27 were genetically modified to express the calcium indicator jRGECO1a in the anterior cingulate/medial prefrontal cortex. Using high-speed, high-resolution two-photon imaging, we tracked the activity of 200 neurons in each mouse for 30 minutes across 50 days. Mice were split into three groups: one received normal drinking water NO CORT, while the other two received corticosterone in their water CORT. Anhedonia was assessed using the sucrose preference test and was significantly elevated in the CORT group after three weeks (p<0.001). Following this period, half of the CORT group received a single subanesthetic injection of ketamine CORTKET, while the other half received saline CORTSAL. Throughout the experiment, the number of active neurons, spike rates, and background noise remained stable across groups. However, pairwise neuronal correlations declined significantly in the CORT group during peak anhedonia, while the NO CORT group showed no such change. Both CORTKET and CORTSAL groups exhibited gradual recovery of these correlations over two weeks after corticosterone was stopped. We also observed disruptions in higher-order brain activity patterns known as neuronal avalanches. In the anhedonic state, the power-law scaling relationship between avalanche size and duration was significantly altered but recovered post-treatment. These findings demonstrate that chronic stress-induced anhedonia disrupts both pairwise and network-level brain dynamics in the frontal cortex. Our results suggest thatDeviation of Parabolic Avalanches dynamics in Mouse Frontal Cortex during Anhedonia1 depressive states interfere with the brainʼs critical state, potentially impairing cognitive and emotional function.

Optoretinography is an emerging method for detecting and measuring functional responses from neurons in the living human retina. Its potential applications are compelling and broad, spanning clinical assessment of retinal disease, investigation of fundamental scientific questions, and rapid evaluation of experimental therapeutics for blinding retinal diseases. Progress in all these domains hinges on the development of robust methods for quantifying observed responses in relation to visible stimuli. In this talk, we will describe an optoretinographic imaging platform: full-field swept-source optical coherence tomography with adaptive optics, measure cone responses in two healthy volunteers to a variety of stimulus patterns, and propose a simple model for predicting and quantifying responses to those stimuli.

Cocaine dependence continues to affect millions of people worldwide. In Brazil, data from the Lenad III survey indicate that approximately 5.4% of the population over 14 years of age (about 9.3 million people) have used cocaine at least once in their lifetime; among these users, a substantial fraction report frequent use, perceive loss of control, and simultaneously express a desire to stop. This scenario highlights the urgency of understanding how cocaine alters brain circuits, particularly reward-related pathways such as the nucleus accumbens (NAc), to identify new therapeutic targets, and these epidemiological data help shape the experimental design of our study. In this talk, I will present data combining state-of-the-art of electrophysiological techniques (silicon probes, patch clamp, miniscope imaging, optogenetics), and deep-learning–based extraction of behavioral features, to characterize neural dynamics in the NAc and in cholinergic projections from the laterodorsal tegmentum (LDT) to the NAc. I will show that cocaine induces persistent changes in population dynamics in both LDT and NAc. Specifically, cocaine increases the excitability of LDT cholinergic neurons that project to the NAc while reducing the excitability of non-projecting cholinergic neurons. Functionally, optogenetic activation of LDT–NAc cholinergic projections enhances cocaine conditioning, whereas their inhibition diminishes the reinforcing effects of the drug. Finally, I will discuss future directions aimed at extracting nonlinear dynamic signatures from these circuits, with the potential to serve as transdiagnostic biomarkers of motivational states and vulnerability to addiction. This approach seeks to bridge rodent and human data and to pave the way for innovative diagnostic tools in computational psychiatry.

How the human brain processes information during different cognitive tasks is one of the central questions in contemporary neuroscience. Understanding the statistical properties of brain signals during specific activities is a promising way to address this issue. Here, we analyze invasive and non-invasive brain signals across a diversity of cortical states in the light of information-theoretic quantifiers. We employ a symbolic information approach to determine the probability distribution function associated with time series from different cortical areas. These probabilities are then used to calculate the corresponding Shannon entropy and a statistical complexity measure based on the disequilibrium between the actual time series and one with a uniform probability distribution function. We show that the Euclidean distance in the complexity–entropy plane and an asymmetry index for complexity are useful for comparing different conditions. Our results demonstrate that this method can distinguish sleep stages and cognitive tasks, and it may serve as a potential biomarker for neurological diseases.

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