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 δ₀ 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 < θ ≤ 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 qₙ → ∞ and qₙ/N → 0. The two systems share their graph and initial data. Along every deterministic joint scale εₙ ↓ 0, their population-averaged firing counts differ by 1/2 + oₗ₁(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 Erdős–Rényi graphs yield the bound Oₚ(λₙ⁻¹ᐟ² + ‖πₙ − π‖₁) when λₙ → ∞ and λₙ/N → 0. 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.