The input space of single‑spike neural networks can be decomposed into causal pieces, and the number of such pieces has been shown to give a lower bound on the approximation error for certain networks. The maximal number of realizable causal pieces, however, remains unknown. This work provides two complementary upper bounds. The first is a combinatorial bound expressed via poly‑Bernoulli numbers and applies to a wide class of spiking neuron models. Its order is $$e^{2(L-1)N\log N+\mathcal{O}(N)}$$ where $L$ denotes network depth and $N$ the maximum layer width. The second bound is derived from hyperplane arrangements for specific neuron models. For the non‑leaky integrate‑and‑fire model with exponential synapses and the ReLU‑type spike response model, the bound scales as $$\mathcal{O}\bigl(N^{(4L-3)(N_0-1)}\bigr)$$; for the leaky integrate‑and‑fire model with Dirac‑δ synapses, it scales as $$\mathcal{O}\bigl(N^{L(N_0-1)}\bigr)$$, where $N_0$ is the input dimension. Moreover, for the non‑leaky integrate‑and‑fire model with Dirac‑δ synapses, additional temporal structure allows the general combinatorial bound to be sharpened to depend only on $N_0$, independent of depth and width.
Review