# Image segmentation with traveling waves in an exactly solvable recurrent neural network

arXiv: https://arxiv.org/abs/2311.16943 · HTML: https://mlsanity.com/paper/2311.16943/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 45 references that are in the map.
- First posted on arXiv 28 Nov 2023.

## Builds on (most cited first, 7 of 7)

- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Image Segmentation Using Deep Learning: A Survey](https://mlsanity.com/paper/2001.05566/) (2020)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Complex-Valued Autoencoders for Object Discovery](https://mlsanity.com/paper/2204.02075/) (2022)
- [Neuronal Synchrony in Complex-Valued Deep Networks](https://mlsanity.com/paper/1312.6115/) (2013)
- [KuraNet: Systems of Coupled Oscillators that Learn to Synchronize](https://mlsanity.com/paper/2105.02838/) (2021)

## Built on by (most cited first, 8 of 8)

- [Artificial Kuramoto Oscillatory Neurons](https://mlsanity.com/paper/2410.13821/) (2024)
- [Transformers and Cortical Waves: Encoders for Pulling In Context Across Time](https://mlsanity.com/paper/2401.14267/) (2024)
- [Enhancing deep neural networks through complex-valued representations and Kuramoto synchronization dynamics](https://mlsanity.com/paper/2502.21077/) (2025)
- [Spontaneous symmetry breaking and Goldstone modes for deep information propagation](https://mlsanity.com/paper/2605.14685/) (2026)
- [Traveling Waves Integrate Spatial Information Through Time](https://mlsanity.com/paper/2502.06034/) (2025)
- [Binding threshold units with artificial oscillatory neurons](https://mlsanity.com/paper/2505.03648/) (2025)
- [An explicit operator explains end-to-end computation in the modern neural networks used for sequence and language modeling](https://mlsanity.com/paper/2604.20595/) (2026)
- [Frequency-Based Reservoir computing](https://mlsanity.com/paper/2607.24420/) (2026)

## Abstract (arXiv)

We study image segmentation using spatiotemporal dynamics in a recurrent neural network where the state of each unit is given by a complex number. We show that this network generates sophisticated spatiotemporal dynamics that can effectively divide an image into groups according to a scene's structural characteristics. Using an exact solution of the recurrent network's dynamics, we present a precise description of the mechanism underlying object segmentation in this network, providing a clear mathematical interpretation of how the network performs this task. We then demonstrate a simple algorithm for object segmentation that generalizes across inputs ranging from simple geometric objects in grayscale images to natural images. Object segmentation across all images is accomplished with one recurrent neural network that has a single, fixed set of weights. This demonstrates the expressive potential of recurrent neural networks when constructed using a mathematical approach that brings together their structure, dynamics, and computation.