Spike-Based Winner-Take-All Computation: Fundamental Limits and Order-Optimal Circuits

Abstract

Winner-take-all (WTA) refers to the neural operation that selects a (typically small) group of neurons from a large neuron pool. It is conjectured to underlie many of the brain's fundamental computational abilities. However, not much is known about the robustness of a spike-based WTA network to the inherent randomness of the input spike trains. In this work, we consider a spike-based [Formula: see text]–WTA model wherein [Formula: see text] randomly generated input spike trains compete with each other based on their underlying firing rates and [Formula: see text] winners are supposed to be selected. We slot the time evenly with each time slot of length 1 ms and model the [Formula: see text] input spike trains as [Formula: see text] independent Bernoulli processes. We analytically characterize the minimum waiting time needed so that a target minimax decision accuracy (success probability) can be reached. We first derive an information-theoretic lower bound on the waiting time. We show that to guarantee a (minimax) decision error [Formula: see text] (where [Formula: see text]), the waiting time of any WTA circuit is at least [Formula: see text]where [Formula: see text] is a finite set of rates and [Formula: see text] is a difficulty parameter of a WTA task with respect to set [Formula: see text] for independent input spike trains. Additionally, [Formula: see text] is independent of [Formula: see text], [Formula: see text], and [Formula: see text]. We then design a simple WTA circuit whose waiting time is [Formula: see text]provided that the local memory of each output neuron is sufficiently long. It turns out that for any fixed [Formula: see text], this decision time is order-optimal (i.e., it matches the above lower bound up to a multiplicative constant factor) in terms of its scaling in [Formula: see text], [Formula: see text], and [Formula: see text].