# Dynamic Manifold Hopfield Networks for Context-Dependent Associative Memory

arXiv: https://arxiv.org/abs/2506.01303 · HTML: https://mlsanity.com/paper/2506.01303/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 53 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## Builds on (most cited first, 12 of 12)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [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)
- [LDMNet: Low Dimensional Manifold Regularized Neural Networks](https://mlsanity.com/paper/1711.06246/) (2017)
- [Neuron-Astrocyte Associative Memory](https://mlsanity.com/paper/2311.08135/) (2023)
- [Back to the Continuous Attractor](https://mlsanity.com/paper/2408.00109/) (2024)
- [Hopfield Neural Network Flow: A Geometric Viewpoint](https://mlsanity.com/paper/1908.01270/) (2019)
- [Input-Driven Dynamics for Robust Memory Retrieval in Hopfield Networks](https://mlsanity.com/paper/2411.05849/) (2024)
- [Emergence of robust memory manifolds](https://mlsanity.com/paper/2109.03879/) (2021)

## Abstract (arXiv)

Neural population activity in cortical and hippocampal circuits can be flexibly reorganized by context, suggesting that cognition relies on dynamic manifolds rather than static representations. However, how such dynamic organization can be realized mechanistically within a unified dynamical system remains unclear. Continuous Hopfield networks provide a classical attractor framework in which neural dynamics follow gradient descent on a fixed energy landscape, constraining retrieval within a static attractor manifold geometry. Extending this approach, we introduce Dynamic Manifold Hopfield Networks (DMHN), continuous dynamical models in which contextual modulation dynamically reshapes attractor geometry, transforming a static attractor manifold into a context-dependent family of neural manifolds. In DMHN, network interactions are learned in a data-driven manner, to intrinsically deform the geometry of its attractor manifold across cues without explicit context-specific parameterization. As a result, in associative retrieval, DMHN achieve substantially higher capacity and robustness than classical and modern Hopfield networks: when storing 2N patterns in a network of N neurons, DMHN attain reliable retrieval with an average accuracy of 64%, compared with 1% and 13% for classical and modern variants, respectively. Together, these results establish dynamic reorganization of attractor manifold geometry as a principled mechanism for context-dependent remapping in neural associative memory.