Multiplicity of Stable Attractors in Disordered Neural Models

2026-07-24

AI Summary

This study addresses the accurate estimation of the multiplicity of stable fixed points in disordered neural network models. Focusing on a class of neural ordinary differential equations employed for computational tasks, it pioneers the integration of large deviation theory with a perturbative expansion in the disorder strength, complemented by analyses of random coupling matrices and dynamical system stability, to systematically evaluate the number of stable attractors under asymmetric couplings. The findings reveal that, at intermediate coupling strengths, the asymmetric model exhibits qualitative agreement with its symmetric counterpart in the number of stable fixed points, indicating no essential distinction between the two regimes. This insight provides a theoretical foundation for extending such results to broader classes of multi-degree-of-freedom stochastic dynamical systems.

Abstract

We show how large-deviation statistics allows one to obtain reliable estimates of the multiplicity of stable fixed-points in a model of neural ordinary differential equations previously employed in computational tasks. The result is obtained by developing a suitable perturbative method in the amplitude of the disorder. It turns out that for not-too-large coupling strengths there are no qualitative differences between the symmetric case, when the dynamics is a purely gradient evolution, and the asymmetric case, when limit cycles and chaos can, in principle, arise. The selection of this specific model is dictated by pedagogical reasons, but we are confident that the approach can be extended to other many-degree-of-freedom dynamical models characterized by different classes of random coupling matrices.

Problem

• stable attractors
• disordered neural models
• multiplicity
• fixed-points
• random coupling matrices

Innovation

• large-deviation statistics
• stable attractors
• disordered neural networks
• perturbative method
• random coupling matrices