Spatial Spiking Neural Networks

• Spatial Spiking Neural Networks are spiking neural models that explicitly incorporate spatial structures, such as receptive fields, graphs, and Euclidean neuron layouts, alongside temporal spike dynamics.
• They employ diverse architectures—from adaptive spatial attention in vision backbones to geometry-based recurrent designs—enabling enhanced pattern detection and efficient signal propagation.
• Empirical studies show that pairing explicit spatial modeling with temporal dynamics improves performance and efficiency, though challenges in optimal spatial-temporal integration and hardware mapping remain.

Spatial Spiking Neural Networks (SpSNNs) designate a family of spiking neural network formulations in which spatial structure is treated as a primary computational object alongside spike-driven temporal dynamics. In the recent literature, this spatiality is instantiated in several distinct but related ways: as broad and task-effective spatial receptive fields in dense vision backbones, as explicit graphs over variables in multivariate forecasting, as neuron coordinates in Euclidean space that induce synaptic delays, and as spatial-temporal mappings of neurons, spikes, and states onto hardware memory hierarchies (Zhang et al., 24 Oct 2025, Hu et al., 4 Aug 2025, Landsmeer et al., 10 Dec 2025, Eissa et al., 5 Feb 2025). Earlier work on detecting spatial patterns in rate-encoded spikes, spatially structured recurrent spiking sheets, and modular spatial computing for embodied control provides the broader conceptual background for these uses (Jang et al., 2020, Pyle et al., 2016, Lobov et al., 2021).

1. Conceptual scope of “spatial” in SpSNNs

Across the cited works, “spatial” does not denote a single architecture class. Instead, it refers to several recurring structural commitments. In image and event vision, spatiality means preserving and enhancing the organization of $H \times W$ feature maps, often by enlarging effective receptive fields or by spatial attention over spike feature maps (Zhang et al., 24 Oct 2025, Yao et al., 2022, Cai et al., 2022). In multivariate forecasting, it means explicit graph structure over variables or sensors, with spike-based message passing along learned edges (Hu et al., 4 Aug 2025). In delay-based recurrent models, it means that neurons occupy learnable coordinates and synaptic delays arise from inter-neuron distance rather than from an unconstrained delay matrix (Landsmeer et al., 10 Dec 2025). In neuromorphic systems work, it means the placement of neurons, synapses, spikes, and neuron states across cores, buffers, and off-chip memory, together with scheduling across timesteps (Eissa et al., 5 Feb 2025, Xu et al., 18 May 2025).

This breadth suggests that SpSNNs are best understood as an umbrella concept for SNNs whose performance, interpretability, or efficiency depends on explicit spatial organization rather than on temporal recurrence alone.

Literature strand	Spatial object	Representative formulation
Rate-coded pattern detection	Binary pattern over input neurons	Spatial spike pattern or code word
Vision backbones	$H \times W$ feature maps	ST-ERF, spatial attention, channel mixers
Graph forecasting	Graph over variables/sensors	Adaptive adjacency and spiking GNN layers
Delay-based recurrent SNNs	Euclidean neuron coordinates	Distance-induced synaptic delays
Hardware co-design	Cores, SRAM, DRAM, PE arrays	Spatial-temporal mapping and sparse dispatch

A common misconception is that “spatial” in SNNs refers only to two-dimensional image lattices. The literature is broader: graphs, Euclidean embeddings, modular pathway geometry, and accelerator placement are all treated as spatial structure in this body of work (Hu et al., 4 Aug 2025, Landsmeer et al., 10 Dec 2025, Lobov et al., 2021, Eissa et al., 5 Feb 2025).

2. Formalizations of spatial structure

A foundational formulation appears in work on rate-encoded spatial pattern detection, where the target object is a static spike pattern represented as a binary vector over input neurons. In that setting, an SNN can either detect spatial patterns through ANN-to-SNN conversion or generate them through neural sampling, and the pattern itself contains no temporal structure beyond the firing rates or first-spike transformations used to implement the detector (Jang et al., 2020). A more explicit supervised spatial-pattern setting appears in SpiNNaker-based work, where a pattern is a simultaneous set of spikes from different source neurons and each code word is effectively a binary spatial code over input channels (Davies et al., 2023).

In dense vision SNNs, the most explicit spatial formalism is the Spatio-Temporal Effective Receptive Field (ST-ERF). For a spiking layer with output spike $y_{(m,n)}[t]$ and input spike $x_{(i,j)}[t-\tau]$, the ST-ERF is defined as

$$\mathrm{ERF}^{(\mathcal{S},\mathcal{T})}_{(i,j)}[\,y_{(m,n)}[t], \tau;\mathbf{x}\,] = \frac{\partial y_{(m,n)}[t]}{\partial x_{(i,j)}[t-\tau]}.$$

This turns receptive-field analysis from a purely spatial ANN tool into a joint space-time diagnostic for SNNs, and spatial ERF and temporal ERF become marginalizations of the same object (Zhang et al., 24 Oct 2025).

In graph-based SpSNNs, the central formal object is an adaptive adjacency matrix learned from node embeddings,

$$\mathbf{A} = \sigma(\mathbf{E}\mathbf{E}^{\top}) + \lambda \mathbf{I},$$

where nodes correspond to variables or sensors and edges encode learned spatial dependency. Spatial propagation is then realized by graph attention and spike-based neighborhood aggregation over $\mathbf{A}$ (Hu et al., 4 Aug 2025).

In geometry-based recurrent SpSNNs, spatial structure is formalized by learnable neuron coordinates $\mathbf{r}_i \in \mathbb{R}^D$, with delays induced by Euclidean distance,

$d_{ij} = \|\mathbf{r}_i - \mathbf{r}_j\|_2.$

This replaces $N^2$ independently trained delays with $H \times W$0 position parameters and imposes a low-dimensional geometric constraint on temporal propagation (Landsmeer et al., 10 Dec 2025).

A different analytic viewpoint is provided by Graphical Neural Activity Threads (GNATs), where spikes become vertices in a directed acyclic graph and directed edges encode estimated causal influence. The weakly connected components of this graph define spatiotemporal threads of activity, and the construction is explicitly adapted to spatially structured networks with distance-dependent connectivity and delays (Theilman et al., 2023). At a larger dynamical scale, spatially extended recurrent E–I SNNs on a two-dimensional torus are analyzed through Fourier modes of distance-dependent connectivity kernels; broad inhibition relative to excitation can destabilize the spatially uniform state and induce symmetry-breaking spatiotemporal patterns (Pyle et al., 2016).

3. Architectural mechanisms for spatial processing

The dense-vision line of work identifies a specific failure mode of Transformer-based SNNs: early convolution-based modules can be useful for local feature extraction but can also impede the formation of a robust global spatial ST-ERF across timesteps. To address this, the Meta-SDT backbone replaces early channel mixers with two alternatives. MLPixer is a fully MLP-based mixer, while SRB is a splash-and-reconstruct block that combines a $H \times W$1 convolution with an MLP. Both are inserted only in the first two stages, leaving later spike-driven self-attention stages unchanged; the stated effect is stronger global spatial ERF formation at shallow depth, which is particularly relevant for dense prediction (Zhang et al., 24 Oct 2025).

Several papers pursue spatial selectivity through explicit attention over spike feature maps. SCTFA computes spatial attention over $H \times W$2 maps, channel attention over feature channels, fuses them into a three-dimensional attention tensor, and injects this tensor into the membrane-potential update at the next timestep. The effect is a temporally persistent spatial-channel modulation rather than a one-step reweighting of outputs (Cai et al., 2022). MA-SNN generalizes this approach with temporal-wise, channel-wise, and spatial-wise attention branches and uses the resulting weights to optimize membrane potentials directly; in the spatial branch, channel-pooled membrane tensors are convolved to produce a two-dimensional spatial mask that gates $H \times W$3 position-wise (Yao et al., 2022). FSTA-SNN approaches the same problem from a frequency-domain perspective: its spatial attention branch uses DCT-based frequency features derived from temporally averaged spike maps, while its temporal branch reweights steps whose spectra differ mainly in amplitude rather than shape (Yu et al., 2024). STAA-SNN combines a spike-driven self-attention mechanism, temporal position encoding, step attention, and time-step random dropout; it treats spatial self-attention over current features and previous membrane states as a core aggregator inside conv-based SNN layers (Zhang et al., 4 Mar 2025).

Graph-based SpSNNs use a different architectural vocabulary. SpikeSTAG first refines node-wise temporal features in an Observation Block, then performs multi-hop spatial propagation with a Multi-Scale Spike Aggregation module described as a spiking GraphSAGE, and finally fuses graph-derived spike features with long-range temporal dynamics in a Dual-Path Spike Fusion block that combines an LSTM path and a spiking self-attention path (Hu et al., 4 Aug 2025).

Neural architecture search has also been extended into explicitly spatial-temporal design space. “Spatial-Temporal Search for Spiking Neural Networks” argues that successful ANN architectures are not optimal for SNNs and jointly searches cell structure, layer topology, surrogate gradient functions, and temporal parameters. Spatial choices are optimized with spike-based differentiable hierarchical search, while temporal behavior is optimized through surrogate-gradient search, temporal parameter search, and hybrid SNN–ANN search (Che et al., 2024).

4. Learning rules and computational regimes

One line of work keeps the spatial task simple and the learning rule local. On SpiNNaker, supervised spatial-pattern learning is implemented with pair-based nearest-neighbour STDP and a teacher spike that determines whether each pre-synaptic event participates in potentiation or depression. The output neuron learns a weight pattern aligned with a code word, and a subsequent homeostatic factor rescales all learned weights so that the output neuron fires exactly one spike for trained patterns. For a single trained pattern, the reported behavior is “the ideal detector,” with 100% accuracy (Davies et al., 2023).

A different local-learning perspective appears in modular embodied SNNs with unidirectional subnet couplings. There, spatial computation is implemented through the “shortest path rule”: on the network scale, STDP potentiates shorter neural pathways and depresses longer alternatives. Synaptic competition and neuronal competition, together with modular topology and lateral inhibition, enable Pavlovian conditioning in a robotic obstacle-avoidance setting (Lobov et al., 2021).

Directly trained SpSNNs instead rely on backpropagation through time with surrogate gradients. This is explicit in ST-ERF analysis, which uses a rectangular surrogate so that spatial-temporal receptive fields can be computed via gradients and visualized across layers (Zhang et al., 24 Oct 2025). SpikeSTAG uses surrogate gradients with an arctangent-shaped derivative for LIF neurons in both the graph aggregation and spiking self-attention modules (Hu et al., 4 Aug 2025). STAA-SNN, SCTFA-SNN, MA-SNN, and FSTA-SNN all place their spatial or spatio-temporal attention mechanisms around LIF membrane updates and train them end-to-end with surrogate-gradient methods (Zhang et al., 4 Mar 2025, Cai et al., 2022, Yao et al., 2022, Yu et al., 2024).

Geometry-based SpSNNs extend direct training to delay parameters derived from position. Their methodological novelty is that delays are not free parameters; exact gradients with respect to delays and neuron positions are computed through custom-derived rules combined with automatic differentiation, and the framework is stated to support arbitrary neuron models and architectures (Landsmeer et al., 10 Dec 2025).

At the recurrent population level, spatially extended E–I networks show that spatial computation can emerge from a dynamical regime change rather than from explicit attention or graph modules. When inhibitory projections become broader than excitatory projections, the spatially uniform fixed point becomes unstable through a symmetry-breaking bifurcation, and the network generates spatiotemporal patterns that can serve as a reliable dynamical reservoir when the input itself contains spatial heterogeneity (Pyle et al., 2016).

5. Empirical regimes and reported results

The empirical literature covers dense vision, event classification, multivariate forecasting, speech and toy temporal tasks, event-based stereo, and hardware-level efficiency studies. The reported gains are heterogeneous because they reflect different definitions of spatial structure, but a common theme is that explicit spatial modeling improves performance only when it is paired with temporal dynamics rather than substituted for them.

Domain	Representative reported result	Citation
Object detection / segmentation	On COCO 2017 with Mask R-CNN, SRB reaches $H \times W$4 $H \times W$5 for Tiny and $H \times W$6 for Base; on ADE20K with Semantic FPN, SRB reaches $H \times W$7 for Tiny and $H \times W$8 for Base	(Zhang et al., 24 Oct 2025)
Multivariate forecasting	On Electricity with $H \times W$9, SpikeSTAG reports 1.566M parameters, 3.75 Ops, 4.39 mJ, and $y_{(m,n)}[t]$0	(Hu et al., 4 Aug 2025)
Event / image classification	STAA-SNN reports 97.14% on CIFAR-10, 82.05% on CIFAR-100, and 70.40% on ImageNet; MA-SNN reports spike-count reductions of 84.9% on Gesture and 81.6% on Gait with energy-efficiency improvements of $y_{(m,n)}[t]$1 and $y_{(m,n)}[t]$2	(Zhang et al., 4 Mar 2025, Yao et al., 2022)
Sparse geometric delays	Dynamically sparsified SpSNNs maintain full accuracy at 90% sparsity and use up to 18x fewer parameters	(Landsmeer et al., 10 Dec 2025)
Mapping / accelerators	STEMS reports up to 12x reduction in off-chip data movement and 5x reduction in energy; SpikeX reports 15.1x–150.87x reduction in EDP	(Eissa et al., 5 Feb 2025, Xu et al., 18 May 2025)

Within event and image recognition, several attention-based SpSNNs report that stronger spatial selectivity is compatible with lower spike rates rather than requiring denser firing. FSTA-SNN reports a 33.99% overall firing-rate reduction and gives accuracy improvements on CIFAR-10, CIFAR-100, ImageNet, and CIFAR10-DVS, including $y_{(m,n)}[t]$3 on ImageNet with ResNet34 and $y_{(m,n)}[t]$4 on CIFAR10-DVS at $y_{(m,n)}[t]$5 (Yu et al., 2024). SCTFA-SNN reports 97.33% on DVS Gesture, 86.55% on SL-Animals-DVS, and 98.72% on MNIST-DVS, with consistent gains over degenerated spatial-temporal or channel-temporal variants (Cai et al., 2022). The NAS-based spatial-temporal search framework reports 96.43% on CIFAR-10, 78.96% on CIFAR-100, and 70.21% on ImageNet, while also surpassing a specially designed ANN on event-based deep stereo with 26$y_{(m,n)}[t]$6 lower computational cost, reported as $y_{(m,n)}[t]$7 (Che et al., 2024).

This body of evidence does not support a single dominant recipe. Instead, it indicates that “spatial strength” can arise from larger receptive fields, explicit graph propagation, geometric delay constraints, structured attention, or architecture search, and that the relevant benchmark depends on which notion of spatiality is being optimized.

6. Systems implications, misconceptions, and open problems

A persistent misconception is that spike sparsity alone guarantees efficiency. Systems work argues the opposite. STEMS shows that stateful neuron dynamics can dominate data movement because membrane states must be read, updated, and written across timesteps; the tool therefore treats neuron states as first-class operands and jointly explores layer fusion, time batching, and spatial partitioning (Eissa et al., 5 Feb 2025). SpikeX makes a similar point at the accelerator level: unstructured spatial and temporal firing sparsity must be explicitly mapped onto a systolic-array dataflow, with Neuro-Temporal Work Units, activation-induced weight tailoring, and co-optimization of network training and hardware configuration (Xu et al., 18 May 2025). In this literature, a spatial SNN is not merely a model architecture; it is also a placement and scheduling problem.

Another misconception is that longer temporal depth automatically compensates for weak spatial design. FSTA-SNN reports that there is not a significant difference in feature learning across different time steps and that increasing the time steps has limited effect on feature learning (Yu et al., 2024). The ST-ERF work likewise argues that one should not rely solely on stacking more early convolutions or on longer temporal depth to grow receptive fields; instead, early global spatial ERF should be designed explicitly (Zhang et al., 24 Oct 2025). SpikeSTAG makes a complementary point in forecasting: purely temporal SNN models either ignore spatial structure or treat spatial correlations implicitly, whereas explicit graph learning and multi-hop spike aggregation materially improve long-horizon performance (Hu et al., 4 Aug 2025).

Open problems recur across the surveyed papers. ST-ERF analysis depends strongly on neuron parameters such as $y_{(m,n)}[t]$8, threshold, and reset type, and the precise interaction between neurodynamics and receptive-field shape remains analytically incomplete (Zhang et al., 24 Oct 2025). Graph-based SpSNNs still rely on static learned adjacency and hybrid ANN–SNN temporal paths, leaving dynamic graphs and fully spike-based temporal fusion as open directions (Hu et al., 4 Aug 2025). Geometry-based SpSNNs have been validated on Yin-Yang and SHD, but broader evaluation on vision and larger sequence tasks remains to be done, as does event-driven rather than time-driven simulation (Landsmeer et al., 10 Dec 2025). Spatial-temporal NAS has so far explored limited operation spaces and only a subset of neuron types, while broader search spaces involving recurrence, feedback, and excitatory/inhibitory structure remain open (Che et al., 2024). GNAT analysis suggests a further unresolved question: whether reusable spatiotemporal threads can become a design primitive, not just a post hoc analysis tool, for large spatially structured SNNs (Theilman et al., 2023).

Taken together, these works indicate that SpSNNs are not a single architecture but a research program organized around a common claim: temporal spike processing becomes more effective, interpretable, and often more efficient when spatial organization is made explicit and optimized rather than inherited passively from ANN templates or hardware defaults.