Order-Sensitive Fast-Synapse Limits in Sparse Excitatory-Inhibitory Threshold-Reset Networks

Abstract

Componentwise weak convergence of signed synaptic kernels does not, by itself, determine the fast-synapse limit of a sparse threshold-reset network. Within a causal event protocol with clamped refractoriness and smooth positive-delay kernels, we construct two families whose excitatory and inhibitory measures converge weakly to $\delta_0$ while their microscopic arrival orders are reversed. A target fires in the excitatory-first family and not in the inhibitory-first family precisely when $x+a-b<\theta\le x+a$. Strict margins preserve this response under perturbations of the target state, aggregate E/I pulse masses, and bounded drift. The macroscopic effect persists on a moderately sparse Dale-compatible random block graph with $q_N\to\infty$ and $q_N/N\to0$. The two systems share their graph and initial data. Along every deterministic joint scale $\varepsilon_N\downarrow0$, their population-averaged firing counts differ by $1/2+o_{L^1}(1)$. A bounded-degree construction and a later probe show that the discrepancy is macroscopic and can persist through reset. Fixed positive-delay kernels with finitely many classes admit a stable regime. Before grazing, typewise-mixing sparse networks converge to a delayed class mean-field system. Directed Erdos-Renyi graphs yield the bound $O_P(\lambda_N^{-1/2}+\|\pi_N-\pi\|_1)$ when $\lambda_N\to\infty$ and $\lambda_N/N\to0$. This separates stable averaging at a fixed delay from singular collapse. In the latter, componentwise weak convergence discards signed arrival-order information needed by the threshold-reset response.