# Equivalence of approximation by networks of single- and multi-spike neurons

arXiv: https://arxiv.org/abs/2603.13478 · HTML: https://mlsanity.com/paper/2603.13478/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 28 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 18 of 18)

- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [Training Spiking Neural Networks Using Lessons From Deep Learning](https://mlsanity.com/paper/2109.12894/) (2021)
- [The Heidelberg spiking datasets for the systematic evaluation of spiking neural networks](https://mlsanity.com/paper/1910.07407/) (2019)
- [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)
- [Temporal Coding in Spiking Neural Networks with Alpha Synaptic Function: Learning with Backpropagation](https://mlsanity.com/paper/1907.13223/) (2019)
- [Equivalence of approximation by convolutional neural networks and fully-connected networks](https://mlsanity.com/paper/1809.00973/) (2018)
- [BS4NN: Binarized Spiking Neural Networks with Temporal Coding and Learning](https://mlsanity.com/paper/2007.04039/) (2020)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [High-performance deep spiking neural networks with 0.3 spikes per neuron](https://mlsanity.com/paper/2306.08744/) (2023)
- [An Exact Mapping From ReLU Networks to Spiking Neural Networks](https://mlsanity.com/paper/2212.12522/) (2022)
- [Neuromorphic Computing and Sensing in Space](https://mlsanity.com/paper/2212.05236/) (2022)
- [Smooth Exact Gradient Descent Learning in Spiking Neural Networks](https://mlsanity.com/paper/2309.14523/) (2023)
- [DelGrad: Exact event-based gradients for training delays and weights on spiking neuromorphic hardware](https://mlsanity.com/paper/2404.19165/) (2024)
- [Energy efficiency analysis of Spiking Neural Networks for space applications](https://mlsanity.com/paper/2505.11418/) (2025)
- [Stable Learning Using Spiking Neural Networks Equipped With Affine Encoders and Decoders](https://mlsanity.com/paper/2404.04549/) (2024)
- [Robust and accelerated single-spike spiking neural network training with applicability to challenging temporal tasks](https://mlsanity.com/paper/2205.15286/) (2022)
- [Causal pieces: analysing and improving spiking neural networks piece by piece](https://mlsanity.com/paper/2504.14015/) (2025)

## Built on by (most cited first, 1 of 1)

- [Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learning](https://mlsanity.com/paper/2605.04995/) (2026)

## Abstract (arXiv)

In a spiking neural network, is it enough for each neuron to spike at most once? In recent work, approximation bounds for spiking neural networks have been derived, quantifying how well they can fit target functions. However, these results are only valid for neurons that spike at most once, which is commonly thought to be a strong limitation. Here, we show that the opposite is true for a large class of spiking neuron models, including the commonly used leaky integrate-and-fire model with subtractive reset: for every approximation bound that is valid for a set of multi-spike neural networks, there is an equivalent set of single-spike neural networks with only linearly more (or less) neurons, in the maximum number of spikes, for which the bound holds. The same is true for the reverse direction too, showing that regarding their approximation capabilities in general machine learning tasks, single-spike and multi-spike neural networks are equivalent. Consequently, many approximation results in the literature for single-spike neural networks also hold for the multi-spike case.