Tonic Song's math proves synaptic order breaks fast-synapse limits

In a new preprint, Tonic Song (arXiv:2608.16701) shows that standard fast-synapse limit approximations fail when synaptic signal ordering is ignored. For sparse excitatory-inhibitory threshold-reset networks, the paper constructs two families whose excitatory and inhibitory measures converge weakly to the same delta function, yet reversed arrival orders flip a target neuron's firing outcome precisely in the window x+a-b < θ ≤ x+a. The effect persists under perturbations to the target state, aggregate pulse masses, and bounded drift, proving that componentwise weak convergence alone discards the signed arrival-order information the threshold-reset response needs.

Beyond the core construction, Song demonstrates the discrepancy is macroscopic: on moderately sparse Dale-compatible random block graphs with q_N→∞ and q_N/N→0, the two systems share graph and initial data but their population-averaged firing counts differ by 1/2+o_{L^1}(1) along any deterministic joint scale. A bounded-degree construction and later probe confirm the difference survives reset. Meanwhile, fixed positive-delay kernels with finitely many classes enter a stable regime, where typewise-mixing sparse networks converge to a delayed class mean-field system, and directed Erdos-Renyi graphs yield an O_P(λ_N^{-1/2}+‖π_N-π‖_1) bound. This cleanly separates stable averaging at a fixed delay from singular collapse, giving theorists a sharper handle on when mean-field limits break down in spiking networks.