Polyhedral Geometry of Time-to-First-Spike Neural Networks
Abstract
We study the expressivity of spiking neural networks, which provide a natural framework for asynchronous, event-driven computation complementary to conventional feedforward neural networks. We consider the time-to-first-spike model in a setting for which the input-output map is continuous and piecewise linear, with affine pieces governed by causal feasibility constraints that determine which presynaptic spikes occur before a neuron fires. We first show that each neuron's firing time admits a maxout-like representation with exponentially many, highly constrained affine pieces. We then formalize causal regions as polyhedral regions with fixed causal sets and derive upper and lower bounds on the maximal number of causal regions in both shallow and multilayer feedforward spiking networks. Our theoretical and experimental results show that spiking networks can generate richer partitions of the input space than conventional feedforward ReLU networks.
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BibTeX
@misc{singh2026polyhedral,
title = {{Polyhedral Geometry of Time-to-First-Spike Neural Networks}},
author = {Manjot Singh and Guido Montúfar and Gitta Kutyniok},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11227},
eprint = {2609.11227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11227},
}
RIS
TY - UNPB
TI - Polyhedral Geometry of Time-to-First-Spike Neural Networks
AU - Manjot Singh
AU - Guido Montúfar
AU - Gitta Kutyniok
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11227
UR - https://arxiv.org/abs/2609.11227
AB - We study the expressivity of spiking neural networks, which provide a natural framework for asynchronous, event-driven computation complementary to conventional feedforward neural networks. We consider the time-to-first-spike model in a setting for which the input-output map is continuous and piecewise linear, with affine pieces governed by causal feasibility constraints that determine which presynaptic spikes occur before a neuron fires. We first show that each neuron's firing time admits a maxout-like representation with exponentially many, highly constrained affine pieces. We then formalize causal regions as polyhedral regions with fixed causal sets and derive upper and lower bounds on the maximal number of causal regions in both shallow and multilayer feedforward spiking networks. Our theoretical and experimental results show that spiking networks can generate richer partitions of the input space than conventional feedforward ReLU networks.
ER -