# UltraLIF: Fully Differentiable Spiking Neural Networks via Ultradiscretization and Max-Plus Algebra

arXiv: https://arxiv.org/abs/2602.11206 · HTML: https://mlsanity.com/paper/2602.11206/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 10 Feb 2026.

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

- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [Incorporating Learnable Membrane Time Constant to Enhance Learning of Spiking Neural Networks](https://mlsanity.com/paper/2007.05785/) (2020)
- [SuperSpike: Supervised learning in multi-layer spiking neural networks](https://mlsanity.com/paper/1705.11146/) (2017)
- [Long short-term memory and learning-to-learn in networks of spiking neurons](https://mlsanity.com/paper/1803.09574/) (2018)
- [Optimal ANN-SNN Conversion for High-accuracy and Ultra-low-latency Spiking Neural Networks](https://mlsanity.com/paper/2303.04347/) (2023)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Supervised learning based on temporal coding in spiking neural networks](https://mlsanity.com/paper/1606.08165/) (2016)
- [Fast and energy-efficient neuromorphic deep learning with first-spike times](https://mlsanity.com/paper/1912.11443/) (2019)
- [S4NN: temporal backpropagation for spiking neural networks with one spike per neuron](https://mlsanity.com/paper/1910.09495/) (2019)
- [An Analytical Estimation of Spiking Neural Networks Energy Efficiency](https://mlsanity.com/paper/2210.13107/) (2022)
- [Reconsidering the Energy Efficiency of Spiking Neural Networks Inference from Analytical Perspectives](https://mlsanity.com/paper/2409.08290/) (2024)
- [TropNNC: Structured Neural Network Compression Using Tropical Geometry](https://mlsanity.com/paper/2409.03945/) (2024)

## Abstract (arXiv)

Spiking Neural Networks (SNNs) offer energy-efficient, biologically plausible computation but suffer from non-differentiable spike generation, necessitating reliance on heuristic surrogate gradients. This paper introduces UltraLIF, a principled framework that replaces surrogate gradients with ultradiscretization, a mathematical formalism from tropical geometry providing continuous relaxations of discrete dynamics. The central insight is that the max-plus semiring underlying ultradiscretization naturally models neural threshold dynamics: the log-sum-exp function serves as a differentiable soft-maximum that converges to hard thresholding as a learnable temperature parameter eps → 0. Two neuron models are derived from distinct dynamical systems: UltraLIF from the LIF ordinary differential equation (temporal dynamics) and UltraDLIF from the diffusion equation modeling gap junction coupling across neuronal populations (spatial dynamics). Both yield fully differentiable SNNs trainable via standard backpropagation with no forward-backward mismatch. Theoretical analysis establishes pointwise convergence to classical LIF dynamics with quantitative error bounds and bounded non-vanishing gradients. Experiments on six benchmarks spanning static images, neuromorphic vision, and audio demonstrate improvements over surrogate gradient baselines, with gains most pronounced in single-timestep (T=1) settings on neuromorphic and temporal datasets. An optional sparsity penalty enables significant energy reduction while maintaining competitive accuracy.