William Retnaraj | Searcharxiv

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↗

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↗