In both neuroscience and artificial intelligence (AI), it is well-established that neural "coupling" gives rise to dynamically distributed systems. These systems exhibit selforganized spatiotemporal patterns of synchronized neural oscillations, enabling the representation of abstract concepts. By capitalizing on the unprecedented amount of human neuroimaging data, we propose that advancing the theoretical understanding of rhythmic coordination in neural circuits can offer powerful design principles for the next generation of machine learning models with improved efficiency and robustness. To this end, we introduce a physics-informed deep learning framework for Brain Rhythm Identification by Kuramoto and Control (coined BRICK) to characterize the synchronization of neural oscillations that shapes the dynamics of evolving cognitive states. Recognizing that brain networks are structurally connected yet behaviorally dynamic, we further conceptualize rhythmic neural activity as an artificial dynamical system of coupled oscillators, offering a shared mechanistic bridge to brain-inspired machine intelligence. By treating each node as an oscillator interacting with its neighbors, this approach moves beyond the conventional paradigm of graph heat diffusion and establishes a new regime of representation compression through oscillatory synchronization. Empirical evaluations demonstrate that this synchronization-driven mechanism not only mitigates over-smoothing in deep GNNs but also enhances the model's capacity for reasoning and solving complex graph-based problems.

Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transitions has remained limited, particularly in stochastic and data-driven settings. In this work, we establish a general, model-independent theorem providing sufficient conditions under which persistent entropy provably separates two phases. We show that persistent entropy exhibits an asymptotically non-vanishing gap across phases. The result relies only on continuity of persistent entropy along the convergent diagram sequence, or under mild regularization, and is therefore broadly applicable across data modalities, filtrations, and homological degrees. To connect asymptotic theory with finite-time computations, we introduce an operational framework based on topological stabilization, defining a topological transition time by stabilizing a chosen topological statistic over sliding windows, and a probability-based estimator of critical parameters within a finite observation horizon. We validate the framework on the Kuramoto synchronization transition, the Vicsek order-to-disorder transition in collective motion, and neural network training dynamics across multiple datasets and architectures. Across all experiments, stabilization of persistent entropy and collapse of variability across realizations provide robust numerical signatures consistent with the theoretical mechanism.

Oscillatory Graph Neural Networks (OGNNs) are an emerging class of physics-inspired architectures designed to mitigate oversmoothing and vanishing gradient problems in deep GNNs. In this work, we introduce the Complex-Valued Stuart-Landau Graph Neural Network (SLGNN), a novel architecture grounded in Stuart-Landau oscillator dynamics. Stuart-Landau oscillators are canonical models of limit-cycle behavior near Hopf bifurcations, which are fundamental to synchronization theory and are widely used in e.g. neuroscience for mesoscopic brain modeling. Unlike harmonic oscillators and phase-only Kuramoto models, Stuart-Landau oscillators retain both amplitude and phase dynamics, enabling rich phenomena such as amplitude regulation and multistable synchronization. The proposed SLGNN generalizes existing phase-centric Kuramoto-based OGNNs by allowing node feature amplitudes to evolve dynamically according to Stuart-Landau dynamics, with explicit tunable hyperparameters (such as the Hopf-parameter and the coupling strength) providing additional control over the interplay between feature amplitudes and network structure. We conduct extensive experiments across node classification, graph classification, and graph regression tasks, demonstrating that SLGNN outperforms existing OGNNs and establishes a novel, expressive, and theoretically grounded framework for deep oscillatory architectures on graphs.

We present a novel interdisciplinary framework that bridges synchronization theory and multi-agent AI systems by adapting the Kuramoto model to describe the collective dynamics of heterogeneous AI agents engaged in complex task execution. By representing AI agents as coupled oscillators with both phase and amplitude dynamics, our model captures essential aspects of agent specialization, influence, and communication within networked systems. We introduce an order parameter to quantify the degree of coordination and synchronization, providing insights into how coupling strength, agent diversity, and network topology impact emergent collective behavior. Furthermore, we formalize a detailed correspondence between Chain-of-Thought prompting in AI reasoning and synchronization phenomena, unifying human-like iterative problem solving with emergent group intelligence. Through extensive simulations on all-to-all and deterministic scale-free networks, we demonstrate that increased coupling promotes robust synchronization despite heterogeneous agent capabilities, reflecting realistic collaborative AI scenarios. Our physics-informed approach establishes a rigorous mathematical foundation for designing, analyzing, and optimizing scalable, adaptive, and interpretable multi-agent AI systems. This work opens pathways for principled orchestration of agentic AI and lays the groundwork for future incorporation of learning dynamics and adaptive network architectures to further enhance system resilience and efficiency.

This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $β\ge -0.16$, which significantly extends the previous bound of $0\le β\le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $β< -2/3$.

Networks of phase oscillators can serve as dense associative memories if they incorporate higher-order coupling beyond the classical Kuramoto model's pairwise interactions. Here we introduce a generalized Kuramoto model with combined second-harmonic (pairwise) and fourth-harmonic (quartic) coupling, inspired by dense Hopfield memory theory. Using mean-field theory and its dynamical approximation, we obtain a phase diagram for dense associative memory model that exhibits a tricritical point at which the continuous onset of memory retrieval is supplanted by a discontinuous, hysteretic transition. In the quartic-dominated regime, the system supports bistable phase-locked states corresponding to stored memory patterns, with a sizable energy barrier between memory and incoherent states. We analytically determine this bistable region and show that the escape time from a memory state (due to noise) grows exponentially with network size, indicating robust storage. Extending the theory to finite memory load, we show that higher-order couplings achieve superlinear scaling of memory capacity with system size, far exceeding the limit of pairwise-only oscillators. Large-scale simulations of the oscillator network confirm our theoretical predictions, demonstrating rapid pattern retrieval and robust storage of many phase patterns. These results bridge the Kuramoto synchronization with modern Hopfield memories, pointing toward experimental realization of high-capacity, analog associative memory in oscillator systems.

Federated learning on heterogeneous (non-IID) client data experiences slow convergence due to client drift. To address this challenge, we propose Kuramoto-FedAvg, a federated optimization algorithm that reframes the weight aggregation step as a synchronization problem inspired by the Kuramoto model of coupled oscillators. The server dynamically weighs each client's update based on its phase alignment with the global update, amplifying contributions that align with the global gradient direction while minimizing the impact of updates that are out of phase. We theoretically prove that this synchronization mechanism reduces client drift, providing a tighter convergence bound compared to the standard FedAvg under heterogeneous data distributions. Empirical validation supports our theoretical findings, showing that Kuramoto-FedAvg significantly accelerates convergence and improves accuracy across multiple benchmark datasets. Our work highlights the potential of coordination and synchronization-based strategies for managing gradient diversity and accelerating federated optimization in realistic non-IID settings.

Artificial Kuramoto oscillatory neurons were recently introduced as an alternative to threshold units. Empirical evidence suggests that oscillatory units outperform threshold units in several tasks including unsupervised object discovery and certain reasoning problems. The proposed coupling mechanism for these oscillatory neurons is heterogeneous, combining a generalized Kuramoto equation with standard coupling methods used for threshold units. In this research note, we present a theoretical framework that clearly distinguishes oscillatory neurons from threshold units and establishes a coupling mechanism between them. We argue that, from a biological standpoint, oscillatory and threshold units realise distinct aspects of neural coding: roughly, threshold units model intensity of neuron firing, while oscillatory units facilitate information exchange by frequency modulation. To derive interaction between these two types of units, we constrain their dynamics by focusing on dynamical systems that admit Lyapunov functions. For threshold units, this leads to Hopfield associative memory model, and for oscillatory units it yields a specific form of generalized Kuramoto model. The resulting dynamical systems can be naturally coupled to form a Hopfield-Kuramoto associative memory model, which also admits a Lyapunov function. Various forms of coupling are possible. Notably, oscillatory neurons can be employed to implement a low-rank correction to the weight matrix of a Hopfield network. This correction can be viewed either as a form of Hebbian learning or as a popular LoRA method used for fine-tuning of large language models. We demonstrate the practical realization of this particular coupling through illustrative toy experiments.

The evolution of tokens through a deep transformer models can be modeled as an interacting particle system that has been shown to exhibit an asymptotic clustering behavior akin to the synchronization phenomenon in Kuramoto models. In this work, we investigate the long-time clustering of mean-field transformer models. More precisely, we establish exponential rates of contraction to a Dirac point mass for any suitably regular initialization under some assumptions on the parameters of transformer models, any suitably regular mean-field initialization synchronizes exponentially fast with some quantitative rates.

The behavior of multivariate dynamical processes is often governed by underlying structural connections that relate the components of the system. For example, brain activity which is often measured via time series is determined by an underlying structural graph, where nodes represent neurons or brain regions and edges represent cortical connectivity. Existing methods for inferring structural connections from observed dynamics, such as correlation-based or spectral techniques, may fail to fully capture complex relationships in high-dimensional time series in an interpretable way. Here, we propose the use of path signatures--a mathematical framework that encodes geometric and temporal properties of continuous paths--to address this problem. Path signatures provide a reparametrization-invariant characterization of dynamical data and, in particular, can be used to compute the lead matrix which reveals lead-lag phenomena. We showcase our approach on time series from coupled oscillators in the Kuramoto model defined on a stochastic block model graph, termed the Kuramoto stochastic block model (KSBM). Using mean-field theory and Gaussian approximations, we analytically derive reduced models of KSBM dynamics in different temporal regimes and theoretically characterize the lead matrix in these settings. Leveraging these insights, we propose a novel signature-based community detection algorithm, achieving exact recovery of structural communities from observed time series in multiple KSBM instances. Our results demonstrate that path signatures provide a novel perspective on analyzing complex neural data and other high-dimensional systems, explicitly exploiting temporal functional relationships to infer underlying structure.