A Basic Compositional Model for Spiking Neural Networks

Abstract

AbstractWe present a formal, mathematical foundation for modeling and reasoning about the behavior of synchronous, stochastic Spiking Neural Networks (SNNs), which have been widely used in studies of neural computation. Our approach follows paradigms established in the field of concurrency theory.Our SNN model is based on directed graphs of neurons, classified as input, output, and internal neurons. We focus here on basic SNNs, in which a neuron’s only state is a Boolean value indicating whether or not the neuron is currently firing. We also define the external behavior of an SNN, in terms of probability distributions on its external firing patterns. We define two operators on SNNs: a composition operator, which supports modeling of SNNs as combinations of smaller SNNs, and a hiding operator, which reclassifies some output behavior of an SNN as internal. We prove results showing how the external behavior of a network built using these operators is related to the external behavior of its component networks. Finally, we definition the notion of a problem to be solved by an SNN, and show 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 Attention network constructed from a Winner-Take-All network and a Filter network, and a toy example involving combining two networks in a cyclic fashion.KeywordsSpiking Neural NetworksComposition of networksCompositionality