Spatial Spiking Neural Networks Enable Efficient and Robust Temporal Computation

The efficiency of modern machine intelligence depends on high accuracy with minimal computational cost. In spiking neural networks (SNNs), synaptic delays are crucial for encoding temporal structure, yet existing models treat them as fully trainable, unconstrained parameters, leading to large memory footprints, higher computational demand, and a departure from biological plausibility. In the brain, however, delays arise from physical distances between neurons embedded in space. Building on this principle, we introduce Spatial Spiking Neural Networks (SpSNNs), a framework in which neurons learn coordinates in a finite-dimensional Euclidean space and delays emerge from inter-neuron distances. This replaces per-synapse delay learning with position learning, substantially reducing parameter count while retaining temporal expressiveness. Across the Yin-Yang and Spiking Heidelberg Digits benchmarks, SpSNNs outperform SNNs with unconstrained delays despite using far fewer parameters. Performance consistently peaks in 2D and 3D networks rather than infinite-dimensional delay spaces, revealing a geometric regularization effect. Moreover, dynamically sparsified SpSNNs maintain full accuracy even at 90% sparsity, matching standard delay-trained SNNs while using up to 18x fewer parameters. Because learned spatial layouts map naturally onto hardware geometries, SpSNNs lend themselves to efficient neuromorphic implementation. Methodologically, SpSNNs compute exact delay gradients via automatic differentiation with custom-derived rules, supporting arbitrary neuron models and architectures. Altogether, SpSNNs provide a principled platform for exploring spatial structure in temporal computation and offer a hardware-friendly substrate for scalable, energy-efficient neuromorphic intelligence.

💡 Research Summary

The paper introduces Spatial Spiking Neural Networks (SpSNN), a novel framework that replaces per‑synapse trainable delays with learnable neuron positions in a finite‑dimensional Euclidean space. In biological brains, transmission latency is determined by the physical distance between neurons; the authors capture this principle mathematically by defining the synaptic delay d_ij as the Euclidean distance between neuron i and neuron j divided by a fixed propagation speed. Consequently, a network of N neurons in D dimensions requires only D·N additional parameters (the coordinates) instead of the N² delay parameters required by conventional delay‑trainable SNNs. This reduction dramatically lowers memory footprint and computational load while imposing a natural geometric regularization on the delay matrix.

Training is performed using exact gradients obtained through automatic differentiation augmented with custom gradient rules that handle the non‑differentiable spike‑induced updates. The forward pass runs a time‑discretized simulation with a spike queue that respects the calculated delays; the backward pass propagates gradients through the same simulation graph, allowing any spiking neuron model (e.g., LIF, Izhikevich) and any architecture (feed‑forward or recurrent) to be used without modification.

The authors evaluate SpSNN on two widely adopted neuromorphic benchmarks: Yin‑Yang (YY) and Spiking Heidelberg Digits (SHD). YY is a simple 3‑class classification task where each input point is encoded by a single time‑to‑first‑spike (TTFS) event; SHD is a more demanding audio‑digit classification problem using rate‑coded spike trains. For each task they vary the hidden neuron count (10–300) and the spatial dimensionality D (0‑dimensional – no delays, ∞‑dimensional – unconstrained per‑synapse delays, 2‑dimensional, and 3‑dimensional).

Results show that low‑dimensional SpSNNs consistently outperform the unconstrained delay baseline despite using far fewer trainable parameters. On YY, a 2‑D SpSNN reaches 98.1 % ± 0.1 test accuracy, surpassing the ∞‑dimensional model with the same parameter budget. On SHD, a 3‑D SpSNN achieves 83.6 % ± 0.9, again beating the infinite‑dimensional counterpart. Plotting accuracy versus total trainable parameters reveals that the best performance is achieved at intermediate dimensions (2‑D for YY, 3‑D for SHD), confirming hypothesis H2: a finite spatial embedding acts as a regularizer that prevents over‑fitting and guides the network toward simpler solutions.

To isolate the source of this regularization, the authors construct a “tortuous” variant that allows non‑straight connections, effectively breaking the triangle inequality. Accuracy remains unchanged, indicating that the performance gain stems from parameter reduction rather than geometric constraints per se.

Sparsification experiments further demonstrate robustness: dynamically pruning up to 90 % of the synaptic weights leaves test accuracy virtually intact, highlighting the potential for extreme memory and energy savings on neuromorphic hardware. Because learned coordinates map directly onto physical locations, SpSNNs can be placed on 2‑D or 3‑D neuromorphic chips with minimal routing overhead, simplifying delay management and reducing power consumption.

In summary, Spatial Spiking Neural Networks provide (1) a principled, biologically inspired way to encode temporal information via spatial embeddings, (2) a dramatic reduction in trainable parameters while improving or matching accuracy, (3) inherent regularization that yields better generalization, (4) strong resilience to aggressive sparsification, and (5) a natural pathway to efficient hardware implementation. The work positions SpSNN as a promising substrate for scalable, energy‑efficient spiking AI and for exploring the interplay between space and time in neural computation.