A Formal Tool for Verification of Probabilistic Spiking Neural Networks Based on Quotient Abstractions

Deploying spiking neural networks (SNNs) on neuromorphic hardware demands aggressive synaptic pruning while preserving temporal computation integrity. Existing strategies either neglect neuronal criticality or rely on convex relaxations of the inherently combinatorial pruning problem whose fractional masks, upon binarisation, destroy accuracy at moderate-to-high sparsity. We present Criticality-Constrained Quadratic Pruning (CQP), a native PyTorch pipeline that fuses weight magnitude with surrogate-gradient criticality into an analytically exact importance metric, eliminating the rounding artefacts endemic to solver-based approaches. We formally characterise a continuous-relaxation trap wherein OSQP-solver fractional masks overshoot the intended sparsity by up to 12 percentage points (pp), precipitating a 44 pp accuracy collapse. We identify and remediate a zombie-weight failure mode in which Adam's first-moment tensors resurrect pruned synapses, violating the binary sparsity guarantee. An iterative schedule - prune, fine-tune with gradient masking, recompute criticality, and repeat - eliminates gradient staleness at high sparsity. A KL-divergence temporal analysis identifies a redundant simulation timestep, enabling a free 10% theoretical energy reduction without weight modification. On MNIST (60,000 training examples), CQP yields 95.6% accuracy at 90% sparsity versus 93.4% for magnitude pruning (+2.2 pp). A criticality-threshold sweep reveals an empirical criticality cliff: accuracy falls from 87.0% to 14.4% as the threshold reaches tau = 0.9, constituting a quantitative SNN-level analogue of the Critical Brain Hypothesis. Combined weight sparsification and temporal truncation yield a compound 73% reduction in per-inference energy at 70% sparsity, confirming the practical value of the proposed pipeline for neuromorphic deployment.

Spiking Neural Networks (SNNs) have garnered increasing attention as one of bio-inspired models due to their great potential in neuromorphic computing and sparse computation. Many practical algorithms and techniques have been developed; however, theoretical understandings of the generalization, that is, the extent to which SNNs perform well on unseen data, are far from clear. Recent advances disclosed an excitation-dependent and architecture-related generalization bound such that the Rademacher complexity of SNNs with stochastic firing can be upper bounded by an exponential function relative to the excitation probability and the architecture depth. In this paper, we theoretically investigate the generalization bounds of SNNs with several integration-and-fire schemes via Rademacher complexity. We recognize that the empirical Rademacher complexity of SNNs is close to the SNN configurations, which is exponential to the network depth and the maximum time duration of received spike sequences, superlinear and subquadratic to the network width, polynomial to the parameter norm, inverse-linear to the number of training samples, and independent of the computations within spiking neurons, achieving a more precise rate than conventional studies. Our theoretical results may support the scope of SNN theories and shed some insight into the development of SNNs.

Spiking neural networks (SNNs) enable sparse and event-driven computation, but their low-bit deployment remains incomplete because recurrent membrane states are commonly retained in floating point even after weight quantization. Quantizing these states is challenging because their distributions differ across channels and from the preceding weights, while small perturbations near the firing threshold may alter spike decisions and accumulate over time. We propose PTQ4SNN, a membrane-aware post-training quantization framework that jointly quantizes weights and recurrent membrane states using only a small calibration set. First, a channel-wise Unified Scale Bridge constrains the membrane scale as s_mem,c = s_w,c * 2^k_c, adapting to membrane distributions while enabling shift-compatible scale conversion. Second, Mixed-Precision Bit Allocation assigns 2/4/8-bit precision to membrane channels according to firing activity and quantization sensitivity under an average-bit budget. The framework operates on reusable projection-LIF pairs and supports both convolutional SNNs and spike-driven Transformers without backbone retraining. Experiments on static and event-based classification and semantic segmentation show that PTQ4SNN effectively preserves model accuracy under W4 quantization and approximately 4-bit membrane precision.