Conservation laws for explainability in spiking neural networks

Abstract

Abstract Explainability in event-driven dynamical models such as Spiking Neural Networks (SNNs) is often adapted from methods developed for static networks, even though the explanation itself is defined over a spatio-temporal computation graph. We present a conservation-law interpretation of Layer-wise Relevance Propagation (LRP) in SNNs as a discrete transport scheme on the unrolled graph, where relevance is the transported quantity and the redistribution rule defines a flux on graph edges. We derive a graph-based spatio-temporal continuity equation, make residual terms and stabilizer-induced sinks explicit, and analyze numerical failure modes associated with thresholding, gating, and signed relevance. We further formulate a flux-corrected threshold redistribution rule that normalizes backward messages at active spike events when the normalization is well posed. We evaluate the method on cislunar telemetry and DVS128 Gesture against Spiking Integrated Gradients and SNN-DeepLIFT across 10 independent runs. Conservation-LRP achieves AOPC values of 0.42 ± 0.03 on CARD and 0.38 ± 0.04 on DVS128 Gesture, with a normalized completeness residual of 0.62 ± 0.18% and a top-layer randomization correlation of 0.12 ± 0.04. In a component-level ablation, disabling flux correction increases the threshold-node residual from 0.11 ± 0.05% to 6.27 ± 1.10% and the completeness residual from 0.62 ± 0.18% to 3.84 ± 0.72%, while perturbation AOPC changes only modestly. These results support conservation diagnostics as a practical complement to perturbation-based evaluation for event-driven neural explanations. The article substantially extends a prior ICLR workshop version with fuller derivations, broader experiments, and expanded implementation-level analysis.