On the Solvability of the {XOR} Problem by Spiking Neural Networks

TL;DR

The work addresses the classic XOR solvability problem within spiking neural networks by recasting linear separability in a temporal encoding framework. Using a fixed input encoder, a randomly initialized LIF reservoir, and a trainable linear decoder, solvability is framed as a convex-hull/linear-programming problem over augmented feature vectors. A key finding is that zero refractory time with graded temporal spikes and the reset-to-mod mechanism enables a minimal two-neuron hidden layer to solve all binary gate constellations, including XOR, with sparse solutions and measurable success across random initializations. These results provide design principles for efficient SNN implementations of logic tasks and offer a framework for analyzing weight-distribution-driven solvability in temporally encoded networks.

Abstract

The linearly inseparable XOR problem and the related problem of representing binary logical gates is revisited from the point of view of temporal encoding and its solvability by spiking neural networks with minimal configurations of leaky integrate-and-fire (LIF) neurons. We use this problem as an example to study the effect of different hyper parameters such as information encoding, the number of hidden units in a fully connected reservoir, the choice of the leaky parameter and the reset mechanism in terms of reset-to-zero and reset-by-subtraction based on different refractory times. The distributions of the weight matrices give insight into the difficulty, respectively the probability, to find a solution. This leads to the observation that zero refractory time together with graded spikes and an adapted reset mechanism, reset-to-mod, makes it possible to realize sparse solutions of a minimal configuration with only two neurons in the hidden layer to resolve all binary logic gate constellations with XOR as a special case.

Figures (3)

• Figure 1: Architecture of SNN considered in the paper, consisting of a reservoir $R$ of fully connected hidden leaky integrate-and-fire (LIF) neurons of the same type (same threshold and leaky parameter) with weight matrix $W$ and a linear output classifier $L$. The encoder weights $E = (1, 0, \ldots, 0)$ are fixed, $W$ is generated randomly and the decoder weights $D$ are learned.
• Figure 2: Enumeration of all binary logical gates where XOR is represented by case $6$.
• Figure 3: Variants $A$, $A'$, $B$ and $C$ for temporal encoding of logical gates by means of mixed positive and negative graded spikes.