arXiv subjects

Simone Betteti

Publications and source records attributed to Simone Betteti.

8 recordsLinked to original sources

arXiv2026

Timescale Limits of Linear-Threshold Networks

Linear-threshold networks (LTNs) capture the mesoscale behavior of interacting populations of neurons and are of particular interest to control theorists due to their dynamical richness and relative ease of analysis. The aim of this paper is to advance the study of global asymptotic stability in LTNs with asymmetric neural interactions and heterogeneous dissipation under the structural Lyapunov diagonal stability (LDS) condition. To this end, we introduce a one-parameter family of LTNs that preserves the LDS condition and has a parameter-independent equilibrium set. In the fast limit, this family converges to a projected dynamical system (PDS), while in the slow limit, it converges to a discontinuous hard-selector system (HSS). Under LDS, we prove that the fast PDS limit is globally exponentially stable and that the HSS limit is globally asymptotically stable. This alignment suggests that the limiting systems capture essential mechanisms governing stability across the entire LTN family. Together with numerical evidence, these findings indicate that resolving stability at the fast and slow endpoints provides a promising and structurally grounded path toward establishing global stability for LTNs with biologically plausible recurrence and diagonal dissipation.

eess.SY↗

arXiv2026

Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics

Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

eess.SY↗

arXiv2026

A Dynamical Theory of Sequential Retrieval in Input-Driven Hopfield Networks

Reasoning is the ability to integrate internal states and external inputs in a meaningful and semantically consistent flow. Contemporary machine learning (ML) systems increasingly rely on such sequential reasoning, from language understanding to multi-modal generation, often operating over dictionaries of prototypical patterns reminiscent of associative memory models. Understanding retrieval and sequentiality in associative memory models provides a powerful bridge to gain insight into ML reasoning. While the static retrieval properties of associative memory models are well understood, the theoretical foundations of sequential retrieval and multi-memory integration remain limited, with existing studies largely relying on numerical evidence. This work develops a dynamical theory of sequential reasoning in Hopfield networks. We consider the recently proposed input-driven plasticity (IDP) Hopfield network and analyze a two-timescale architecture coupling fast associative retrieval with slow reasoning dynamics. We derive explicit conditions for self-sustained memory transitions, including gain thresholds, escape times, and collapse regimes. Together, these results provide a principled mathematical account of sequentiality in associative memory models, bridging classical Hopfield dynamics and modern reasoning architectures.

cs.NE↗

arXiv2026

Incremental Input-to-State Stability and Equilibrium Tracking for Stochastic Contracting Dynamics

In this paper, we study the contractivity of nonlinear stochastic differential equations (SDEs) driven by deterministic inputs and Brownian motions. Given a weighted $\ell_2$-norm for the state space, we show that an SDE is incrementally noise- and input-to-state stable if its vector field is uniformly contracting in the state and uniformly Lipschitz in the input. This result is applied to error estimation for time-varying equilibrium tracking in the presence of noise affecting both the system dynamics and the input signals. We consider both Ornstein-Uhlenbeck processes modeling unbounded noise and Jacobi diffusion processes modeling bounded noise. Finally, we turn our attention to the associated Fokker-Planck equation of an SDE. For this context, we prove incremental input-to-state stability with respect to an arbitrary $p$-Wasserstein metric when the drift vector field is uniformly contracting in the state and uniformly Lipschitz in the input with respect to an arbitrary norm.

eess.SY↗

arXiv2025

Competition, stability, and functionality in excitatory-inhibitory neural circuits

Energy-based models have become a central paradigm for understanding computation and stability in both theoretical neuroscience and machine learning. However, the energetic framework typically relies on symmetry in synaptic or weight matrices - a constraint that excludes biologically realistic systems such as excitatory-inhibitory (E-I) networks. When symmetry is relaxed, the classical notion of a global energy landscape fails, leaving the dynamics of asymmetric neural systems conceptually unanchored. In this work, we extend the energetic framework to asymmetric firing rate networks, revealing an underlying game-theoretic structure for the neural dynamics in which each neuron is an agent that seeks to minimize its own energy. In addition, we exploit rigorous stability principles from network theory to study regulation and balancing of neural activity in E-I networks. We combine the novel game-energetic interpretation and the stability results to revisit standard frameworks in theoretical neuroscience, such as the Wilson-Cowan and lateral inhibition models. These insights allow us to study cortical columns of lateral inhibition microcircuits as contrast enhancer - with the ability to selectively sharpen subtle differences in the environment through hierarchical excitation-inhibition interplay. Our results bridge energetic and game-theoretic views of neural computation, offering a pathway toward the systematic engineering of biologically grounded, dynamically stable neural architectures.

q-bio.NC↗

arXiv2025

Contraction and concentration of measures with applications to theoretical neuroscience

We investigate the asymptotic behavior of probability measures associated with stochastic dynamical systems featuring either globally contracting or $B_{r}$-contracting drift terms. While classical results often assume constant diffusion and gradient-based drifts, we extend the analysis to spatially inhomogeneous diffusion and non-integrable vector fields. We establish sufficient conditions for the existence and uniqueness of stationary measures under global contraction, showing that convergence is preserved when the contraction rate dominates diffusion inhomogeneity. For systems contracting only outside of a compact set and with constant diffusion, we demonstrate mass concentration near the minima of an associated non-convex potential, like in multistable regimes. The theoretical findings are illustrated through Hopfield networks, highlighting implications for memory retrieval dynamics in noisy environments.

math.DS↗

arXiv2024

Firing Rate Models as Associative Memory: Excitatory-Inhibitory Balance for Robust Retrieval

Firing rate models are dynamical systems widely used in applied and theoretical neuroscience to describe local cortical dynamics in neuronal populations. By providing a macroscopic perspective of neuronal activity, these models are essential for investigating oscillatory phenomena, chaotic behavior, and associative memory processes. Despite their widespread use, the application of firing rate models to associative memory networks has received limited mathematical exploration, and most existing studies are focused on specific models. Conversely, well-established associative memory designs, such as Hopfield networks, lack key biologically-relevant features intrinsic to firing rate models, including positivity and interpretable synaptic matrices that reflect excitatory and inhibitory interactions. To address this gap, we propose a general framework that ensures the emergence of re-scaled memory patterns as stable equilibria in the firing rate dynamics. Furthermore, we analyze the conditions under which the memories are locally and globally asymptotically stable, providing insights into constructing biologically-plausible and robust systems for associative memory retrieval.

q-bio.NC↗

arXiv2024

Input-Driven Dynamics for Robust Memory Retrieval in Hopfield Networks

The Hopfield model provides a mathematically idealized yet insightful framework for understanding the mechanisms of memory storage and retrieval in the human brain. This model has inspired four decades of extensive research on learning and retrieval dynamics, capacity estimates, and sequential transitions among memories. Notably, the role and impact of external inputs has been largely underexplored, from their effects on neural dynamics to how they facilitate effective memory retrieval. To bridge this gap, we propose a novel dynamical system framework in which the external input directly influences the neural synapses and shapes the energy landscape of the Hopfield model. This plasticity-based mechanism provides a clear energetic interpretation of the memory retrieval process and proves effective at correctly classifying highly mixed inputs. Furthermore, we integrate this model within the framework of modern Hopfield architectures, using this connection to elucidate how current and past information are combined during the retrieval process. Finally, we embed both the classic and the new model in an environment disrupted by noise and compare their robustness during memory retrieval.

q-bio.NC↗