Timescale Limits of Linear-Threshold Networks

TL;DR

This paper investigates the stability limits of linear-threshold networks (LTNs), demonstrating that their fast and slow dynamical limits are globally stable, which advances understanding of neural population models.

Contribution

It introduces a family of LTNs preserving the LDS condition, connecting their stability properties to limiting projected and hard-selector systems.

Findings

Fast limit (PDS) is globally exponentially stable.

Slow limit (HSS) is globally asymptotically stable.

Stability at limits suggests potential for global stability in LTNs.

Abstract

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.