Week 05.05.2025 – 11.05.2025

Neural networks with multiple discrete attractors are a common memory model, previously built under specific connectivity and dynamics assumptions. We analyse local stability of discrete fixed points in a broad class of networks with graded neural activities, and show that all fixed points are stable below a critical load that is distinct from the classical "storage capacity". This critical value depends on the statistics of neural activities in the fixed points as well as the single-neuron activation function. We derive a theory of this stability for the case of dense patterns, expected in the presence of noise, by analysing the bulk and the outliers of the Jacobian spectrum. Our analysis highlights the computational benefits of noisy threshold-linear activation and sparse-like patterns.