# Causal pieces: analysing and improving spiking neural networks piece by piece

arXiv: https://arxiv.org/abs/2504.14015 · HTML: https://mlsanity.com/paper/2504.14015/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 57 references that are in the map.
- First posted on arXiv 18 Apr 2025.

## Builds on (most cited first, 30 of 31)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [EuroSAT: A Novel Dataset and Deep Learning Benchmark for Land Use and Land Cover Classification](https://mlsanity.com/paper/1709.00029/) (2017)
- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Training Spiking Neural Networks Using Lessons From Deep Learning](https://mlsanity.com/paper/2109.12894/) (2021)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Training Deep Spiking Neural Networks using Backpropagation](https://mlsanity.com/paper/1608.08782/) (2016)
- [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)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Accurate and efficient time-domain classification with adaptive spiking recurrent neural networks](https://mlsanity.com/paper/2103.12593/) (2021)
- [Efficient Neuromorphic Signal Processing with Loihi 2](https://mlsanity.com/paper/2111.03746/) (2021)
- [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)
- [Surrogate gradients for analog neuromorphic computing](https://mlsanity.com/paper/2006.07239/) (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)
- [Bottom-up and top-down approaches for the design of neuromorphic processing systems: Tradeoffs and synergies between natural and artificial intelligence](https://mlsanity.com/paper/2106.01288/) (2021)
- [Fluctuation-driven initialization for spiking neural network training](https://mlsanity.com/paper/2206.10226/) (2022)
- [An Exact Mapping From ReLU Networks to Spiking Neural Networks](https://mlsanity.com/paper/2212.12522/) (2022)
- [The Yin-Yang dataset](https://mlsanity.com/paper/2102.08211/) (2021)
- [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)
- [Performance Evaluation of Neuromorphic Hardware for Onboard Satellite Communication Applications](https://mlsanity.com/paper/2401.06911/) (2024)
- [Stable Learning Using Spiking Neural Networks Equipped With Affine Encoders and Decoders](https://mlsanity.com/paper/2404.04549/) (2024)
- [hxtorch.snn: Machine-learning-inspired Spiking Neural Network Modeling on BrainScaleS-2](https://mlsanity.com/paper/2212.12210/) (2022)
- [Scalable Network Emulation on Analog Neuromorphic Hardware](https://mlsanity.com/paper/2401.16840/) (2024)

## Built on by (most cited first, 2 of 2)

- [Quadratic integrate-and-fire neurons exhibit less fragmented loss landscapes and outperform leaky integrate-and-fire neurons in spike-based gradient descent](https://mlsanity.com/paper/2606.03935/) (2026)
- [Equivalence of approximation by networks of single- and multi-spike neurons](https://mlsanity.com/paper/2603.13478/) (2026)

## Abstract (arXiv)

We introduce "causal pieces", a novel concept for analysing spiking neural networks (SNNs), inspired by "linear pieces" used to study expressivity and trainability in artificial neural networks (ANNs). Causal pieces partition the input and parameter space of a feedforward SNN with single-spike coding into distinct regions where the same subnetwork causes the output spikes. For networks of current-based leaky integrate-and-fire (LIF) neurons with large membrane time constants, we show that within each causal piece, output spike times are locally Lipschitz continuous with respect to inputs and network parameters. We further prove a lower bound on the approximation error that depends on the number of causal pieces. Thus, the number of causal pieces is a measure of the approximation capabilities of SNNs, which is valid despite spike-time discontinuities and applies to networks with both excitatory and inhibitory synapses. Empirically, we find that parameter initialisations yielding more causal pieces on the training set strongly correlate with SNN training success across multiple benchmarks, including Yin-Yang, Fashion-MNIST, and EuroSAT. Moreover, simulations with standard single-spike LIF neurons indicate that our findings extend beyond the theoretically analysed regime. These results establish causal pieces as a powerful and principled tool for analysing and improving the computational capabilities of SNNs.