# Dynamical stability for dense patterns in attractor neural networks

arXiv: https://arxiv.org/abs/2507.10383 · HTML: https://mlsanity.com/paper/2507.10383/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 69 references that are in the map.
- First posted on arXiv 14 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [Attractor and integrator networks in the brain](https://mlsanity.com/paper/2112.03978/) (2021)
- [Linking connectivity, dynamics and computations in low-rank recurrent neural networks](https://mlsanity.com/paper/1711.09672/) (2017)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [Transition to chaos in random neuronal networks](https://mlsanity.com/paper/1508.06486/) (2015)
- [Dynamics of random recurrent networks with correlated low-rank structure](https://mlsanity.com/paper/1909.04358/) (2019)
- [What is the dynamical regime of cerebral cortex?](https://mlsanity.com/paper/1908.10101/) (2019)
- [Efficiency of local learning rules in threshold-linear associative networks](https://mlsanity.com/paper/2007.12584/) (2020)
- [Back to the Continuous Attractor](https://mlsanity.com/paper/2408.00109/) (2024)
- [Local Dynamics in Trained Recurrent Neural Networks](https://mlsanity.com/paper/1511.05222/) (2015)
- [Dynamics of Random Neural Networks with Bistable Units](https://mlsanity.com/paper/1409.2535/) (2014)
- [Properties of networks with partially structured and partially random connectivity](https://mlsanity.com/paper/1311.4672/) (2013)
- [Soft-margin classification of object manifolds](https://mlsanity.com/paper/2203.07040/) (2022)
- [Simultaneous embedding of multiple attractor manifolds in a recurrent neural network using constrained gradient optimization](https://mlsanity.com/paper/2310.18708/) (2023)
- [Cross-talk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: phase diagram (I)](https://mlsanity.com/paper/1304.1457/) (2013)
- [Mapping low-dimensional dynamics to high-dimensional neural activity: A derivation of the ring model from the neural engineering framework](https://mlsanity.com/paper/2002.03420/) (2020)

## Abstract (arXiv)

Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neural activity that are stored in the recurrent connections between neurons, such that they become attractors of the network's dynamics. During memory recall, network dynamics thus converge toward one of these memory patterns when started from a noisy or partial cue. Therefore, memory performance critically hinges on the dynamical stability of the stored patterns. However, previous theoretical approaches only studied dynamical stability under highly restrictive conditions that do not readily apply to biological neural circuits. Here, we develop a theory of the local stability of discrete fixed points in a broad class of networks with graded neural activities and in the presence of noise. Using methods from random matrix theory, we analyze the bulk and outliers of the eigenvalue spectra of the Jacobians that characterize network dynamics around fixed points. We show that either all fixed points are stable or all of them are unstable, depending on whether their number is below a ``critical load for stability'', which is distinct from the classical critical capacity that measures the maximal number of achievable fixed points regardless of their stability. We further analyze the dependence of this critical load for stability on experimentally measurable quantities characterizing the statistics of memory patterns and the activation functions of neurons. Our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory.