A Basic Compositional Model for Spiking Neural Networks

This paper presents a formal, mathematical foundation for modeling and reasoning about the behavior of $synchronous$, $stochastic$ $Spiking$ $Neural$ $Networks$ $(SNNs)$. 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 $external\ behavior$ of an SNN. 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 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 $problem$ 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 $Attention$ network constructed from a $Winner$-$Take$-$All$ network and a $Filter$ network, and a toy example involving combining two networks in a cyclic fashion.