Polyhedral Geometry of Time-to-First-Spike Neural Networks

• URL: http://arxiv.org/abs/2609.11227v1
• Date: Thu, 10 Sep 2026 08:27:38 GMT
• Title: Polyhedral Geometry of Time-to-First-Spike Neural Networks
• Abstract summary: We study the expressivity of spiking neural networks, which provide a framework for asynchronous, event-driven computation.<n>We consider the time-to-first-spike model in a setting for which the input-output map is continuous and piecewise linear.<n>We first show that each neuron's firing time admits a maxout-like representation with exponentially many, highly constrained affine pieces.<n>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.
• Score: 28.641313300035975
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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