Softmax-Free Attention in Spiking Transformers as the Zero-Temperature Limit of Two-Level Systems

Spiking transformers achieve competitive accuracy with conventional transformers while offering $38$-$57\times$ energy efficiency on neuromorphic hardware, yet no theoretical framework guides their design. This paper establishes the first comprehensive expressivity theory for spiking self-attention. We prove that spiking attention with Leaky Integrate-and-Fire neurons is a universal approximator of continuous permutation-equivariant functions, providing explicit spike circuit constructions including a novel lateral inhibition network for softmax normalization with proven $O(1/\sqrt{T})$ convergence. We derive tight spike-count lower bounds via rate-distortion theory: $\varepsilon$-approximation requires $Ω(L_f^2 nd/\varepsilon^2)$ spikes, with rigorous information-theoretic derivation. Our key insight is input-dependent bounds using measured effective dimensions ($d_{\text{eff}}=47$--$89$ for CIFAR/ImageNet), explaining why $T=4$ timesteps suffice despite worst-case $T \geq 10{,}000$ predictions. We provide concrete design rules with calibrated constants ($C=2.3$, 95% CI: $[1.9, 2.7]$). Experiments on Spikformer, QKFormer, and SpikingResformer across vision and language benchmarks validate predictions with $R^2=0.97$ ($p<0.001$). Our framework provides the first principled foundation for neuromorphic transformer design.

We revisit the statistical mechanics of Transformer attention via modern Hopfield models. We present a free-energy formalism to characterize parameter regimes and critical thresholds for robust attention inference. The resulting free-energy landscape incorporates randomness in query states and attention weights. Attention then emerges as the equation of motion of the corresponding statistical-mechanical system (i.e., modern Hopfield model) through the standard free-energy minimization principle. This allows us to identify first-order phase transitions in attention between glassy and robust-inference phases. We further extend to attentions with stochastic queries and keys. Together, our results provide a rigorous statistical-mechanics framework for characterizing phase structure and inference reliability across a family of Transformer attention mechanisms.

Integrating Spiking Neural Networks (SNNs) with Transformer architectures offers a promising pathway to balance energy efficiency and performance, particularly for edge vision applications. However, existing Spiking Transformers face two critical challenges: (i) a substantial performance gap compared to their Artificial Neural Networks (ANNs) counterparts and (ii) high memory overhead during inference. Through theoretical analysis, we attribute both limitations to the Spiking Self-Attention (SSA) mechanism: the lack of locality bias and the need to store large attention matrices. Inspired by the localized receptive fields (LRF) and membrane-potential dynamics of biological visual neurons, we propose LRF-Dyn, which uses spiking neurons with localized receptive fields to compute attention while reducing memory requirements. Specifically, we introduce a LRF method into SSA to assign higher weights to neighboring regions, strengthening local modeling and improving performance. Building on this, we approximate the resulting attention computation via charge-fire-reset dynamics, eliminating explicit attention-matrix storage and reducing inference-time memory. Extensive experiments on visual tasks confirm that our method reduces memory overhead while delivering significant performance improvements. These results establish it as a key unit for achieving energy-efficient Spiking Transformers.