CNS*2026 Halifax

Roberto Budzinski^1,2,#, Alexandra Busch^2,3,4, Luisa Liboni^2,5, Ján Mináč^2,3, Lyle Muller^2,3,4
1 Department of Neuroscience, University of Lethbridge, Lethbridge AB, Canada
2 Fields Lab for Network Computation, Fields Lab, Toronto ON, Canada
3 Department of Mathematics, Western University, London ON, Canada
4 Western Institute for Neuroscience, Western University, London ON, Canada
5 King's University College at Western University, London ON, Canada
# [email protected]

Introduction
Networks throughout physics and biology use spatiotemporal dynamics for computation [1]. In neural systems, waves of neural activity have recently been shown to shape spiking responses, gate perception, and influence behaviour [2]. However, it remains unclear how network connectivity gives rise to neural dynamics and how these dynamics support computation. To address this question, we introduce a new type of recurrent neural network that admits an exact mathematical solution [3,4], enabling us to directly relate network structure to emergent dynamics and the computations those dynamics perform.

Methods
We introduce a nonlinear recurrent neural network in which each unit is modeled as a complex-valued oscillator. This complex-valued recurrent neural network (cv-RNN) admits a closed-form solution given by an exact propagator. Importantly, this framework introduces a unified matrix representation of the system that encodes the network's connectivity, including connection strengths and delays, and the input. The exact mathematical solution allows us to control the network dynamics, down to the fine-scale pattern of connectivity, allowing us to use the spatiotemporal patterns that emerge for dynamics-based computation in a wide range of tasks [3,4].

Results
We find the cv-RNN can perform a wide range of tasks, including working memory, logic operations, sequence processing, and computer vision, while remaining precise and interpretable mathematically [3,4]. The analytical framework reveals the mechanisms underlying each computation. By exploiting traveling-wave dynamics, the network performs image segmentation and generalizes across different datasets using the same recurrent weights [4]. Further, we create a bio-hybrid version of our cv-RNN, where we leverage patch-clamping techniques to link biological neurons to the recurrent layer, where these neurons can decode the network’s spatiotemporal dynamics and implement computations [3].

Discussion
These results demonstrate that structured spatiotemporal dynamics can serve as a powerful computational substrate in recurrent neural networks. The exact solution links connectivity, input, and emergent dynamics within a unified operator framework. This approach provides a principled way to understand how neural circuits may compute through traveling waves and network dynamics. More broadly, it establishes a general framework for connecting network structure, emergent dynamics, and computation, offering new tools for interpreting biological neural activity and for designing transparent dynamical models in artificial intelligence.

References
[1] Ermentrout et al. (2001), "Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role”, Neuron 29, 33.
[2] Muller et al. (2018), “Cortical travelling waves: mechanisms and computational principles”, Nature Reviews Neuroscience 19.
[3] Budzinski et al. (2024) “An exact mathematical description of computation with transient spatiotemporal dynamics in a complex-valued neural network, Communications Physics 7.
[4] Liboni et al. (2025), “Image segmentation with traveling waves in an exactly solvable recurrent neural network”, Proceedings of the National Academy of Sciences 122.
Acknowledgments
​This work was supported by NSERC, CFREF, NIH, Neuronex NSF, and Canada Research Chairs Program.