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A Basic Compositional Model for Spiking Neural Networks

12 August 2018
Nancy A. Lynch
Cameron Musco
    LRM
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Abstract

This paper presents a formal, mathematical foundation for modeling and reasoning about the behavior of synchronoussynchronoussynchronous, stochasticstochasticstochastic SpikingSpikingSpiking NeuralNeuralNeural NetworksNetworksNetworks (SNNs)(SNNs)(SNNs). We define a basic SNN model, in which a neuron's only state is a Boolean value indicating whether the neuron is currently firing. We also define the external behaviorexternal\ behaviorexternal behavior of an SNN. We define two operators on SNNs: a composition operatorcomposition\ operatorcomposition operator, which supports modeling of SNNs as combinations of smaller SNNs, and a hiding operatorhiding\ operatorhiding operator, which reclassifies some output behavior of an SNN as internal. We prove results describing how the external behavior of a network built using these operators is related to the external behavior of the component networks. Finally, we give a formal definition of a problemproblemproblem to be solved by an SNN, and give basic results showing how the composition and hiding operators affect the problems that are solved by the networks. We illustrate our definitions with three examples: a Boolean circuit constructed from gates, an AttentionAttentionAttention network constructed from a WinnerWinnerWinner-TakeTakeTake-AllAllAll network and a FilterFilterFilter network, and a toy example involving combining two networks in a cyclic fashion.

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