When Do Spikes Matter? Timing Beyond Rates

When does a spiking neural network (SNN) depend only on how many spikes each neuron fires, and when does it also depend on when they fire? This matters: an SNN that ignores timing is only a time-stepped rate network, and one that uses timing loses information when spikes are reduced to counts. We study this question in wide networks of leaky integrate-and-fire (LIF) neurons with Gaussian weights, which pass on only the mean and the temporal second moment of each layer's spike trains. In wide networks this yields a kernel over spike trains, the spiking analogue of the neural-network Gaussian-process (NNGP) kernel, and rate coding keeps only its time-averaged part. It suffices for exact rate computation that synaptic filtering removes every temporal component except the average, and for approximate rate computation that little passes through. Conversely, we construct two spike populations with identical spike counts and firing rates for which one spiking layer produces different outputs, even when only their rates are read out. In this model timing matters only when structure survives filtering and reaches the readout; for trained networks these quantities are diagnostics.