Develop a polytope calculus for compositions of TTFS neurons

Develop an analogous polytope calculus for compositions of time-to-first-spike neurons with constrained parameters, and determine whether the combinatorics of the resulting structured polytopes can yield sharper linear-region counting results for shallow and deep spiking neural networks.

Background

The paper represents the firing-time map of a single positive-weight TTFS neuron through a lifted polytope whose vertices correspond to nonempty causal sets. This provides a precise polyhedral description of the neuron's affine pieces, but the paper does not extend that construction to network compositions.

The authors explicitly identify the absence of a corresponding calculus for composing these constrained maxout-like representations as a route toward improving the upper and lower bounds on causal-region complexity in shallow and deep SNNs.

References

Since TTFS neurons admit a maxout-like representation with constrained parameters, it is natural to ask whether an analogous polytope calculus can be developed for compositions of TTFS neurons.

— Polyhedral Geometry of Time-to-First-Spike Neural Networks  (2609.11227 - Singh et al., 10 Sep 2026) in Section 1, Future work; Section 7, Future work

It may nevertheless be possible to develop a robust version of the construction, in which the zero-cross pair weights are replaced by sufficiently small positive weights while preserving the folding behavior on suitable subregions.

— Polyhedral Geometry of Time-to-First-Spike Neural Networks  (2609.11227 - Singh et al., 10 Sep 2026) in Section 5, Lower bounds, immediately after Theorem 5.5