ChronoSpike: An Adaptive Spiking Graph Neural Network for Dynamic Graphs

arxiv: 2602.01124 · v3 · submitted 2026-02-01 · 💻 cs.LG

ChronoSpike: An Adaptive Spiking Graph Neural Network for Dynamic Graphs

Md Abrar Jahin , Taufikur Rahman Fuad , Jay Pujara , Craig Knoblock This is my paper

Pith reviewed 2026-05-16 08:59 UTC · model grok-4.3

classification 💻 cs.LG

keywords dynamic graphsspiking neural networksgraph neural networkstemporal modelingleaky integrate-and-fireattention mechanismstransformer encoderstability analysis

0 comments p. Extension

The pith

ChronoSpike integrates learnable spiking neurons with spatial attention and a transformer encoder to model dynamic graphs efficiently while maintaining stability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to resolve the trade-off in dynamic graph learning where attention methods scale poorly with time and recurrent ones suffer gradient issues and high memory use. It does so by combining adaptive leaky integrate-and-fire neurons that track continuous membrane states per channel, multi-head attention for spatial relations, and a lightweight transformer for temporal dependencies. This combination is claimed to deliver both local detail and long-range patterns at linear memory cost in sequence length plus a modest attention term. If the claims hold, the result would be models that train several times faster than recurrent alternatives, use a fixed parameter count regardless of graph size, and come with formal bounds on potentials and gradients. The work also supplies interpretability showing varied temporal fields and a primacy bias at high sparsity.

Core claim

ChronoSpike integrates learnable LIF neurons with per-channel membrane dynamics, multi-head spatially-attentive aggregation over continuous features, and a lightweight Transformer temporal encoder. This design enables fine-grained local modeling and long-range dependency capture with O(T · d) activation and state memory plus a small additional per-node attention term, outperforming twelve baselines by 2.0% Macro-F1 and 2.4% Micro-F1 on average across three large benchmarks while training 3-10× faster than recurrent methods under a constant 105K-parameter budget independent of graph size, accompanied by proofs of membrane potential boundedness, gradient flow stability for contraction factor ρ

What carries the argument

learnable LIF neurons with per-channel membrane dynamics combined with multi-head spatially-attentive aggregation and a lightweight Transformer temporal encoder

If this is right

Outperforms twelve state-of-the-art baselines by an average of 2.0% Macro-F1 and 2.4% Micro-F1 on three large benchmarks.
Achieves 3-10× faster training than recurrent methods under a fixed 105K-parameter budget that does not grow with graph size.
Uses O(T · d) activation and state memory together with a small per-node attention cost.
Supplies formal guarantees of membrane potential boundedness, gradient stability under contraction factor ρ<1, and BIBO stability.
Learns heterogeneous temporal receptive fields and a primacy effect while operating at 83-88% sparsity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The fixed parameter budget could support deployment on graphs much larger than those tested without memory scaling issues.
Observed heterogeneity in temporal receptive fields may aid analysis of real temporal networks such as citation or transaction streams.
The stability guarantees under ρ<1 could support safe use in continual online settings where the model updates over time.

Load-bearing premise

That learnable spiking neurons with per-channel dynamics plus spatial attention and a temporal transformer can simultaneously achieve fine local modeling, long-range capture, efficiency, and stability without inheriting the scaling or gradient problems of prior recurrent or attention-only approaches.

What would settle it

Running the reported experiments on the three large benchmarks and finding no average improvement of roughly 2% in Macro-F1 and Micro-F1 or no 3-10× training speedup versus recurrent baselines would falsify the performance claims.

Figures

Figures reproduced from arXiv: 2602.01124 by Craig Knoblock, Jay Pujara, Md Abrar Jahin, Taufikur Rahman Fuad.

**Figure 1.** Figure 1: Overview of the ChronoSpike framework. The dynamic graph is represented as a sequence of snapshots. At each time step, node features are aggregated from sampled neighborhoods using a multi-head attentive spatial aggregator and encoded into spike signals via adaptive LIF neurons. Temporal dependencies across snapshots are captured by a lightweight Transformer-based temporal aggregation module with learnable… view at source ↗

**Figure 2.** Figure 2: Overhead comparison of different methods in terms of model parameter size and average training time per epoch. Models that do not scale to the Patent dataset or do not report overhead statistics are excluded from the comparison. 5.2.2. ABLATION STUDY We conduct ablation studies on DBLP, Tmall, and Patent under different training ratios to assess the contribution of core components of ChronoSpike ( [PITH_F… view at source ↗

**Figure 3.** Figure 3: Parameter sensitivity analysis on DBLP (left) and Tmall (right) datasets (80% training). (a) Learning rate vs. dropout rate heatmap showing optimal regions. (b) SNN parameter α vs. contrastive weight showing robust performance. Darker colors indicate higher Micro-F1 scores. degrade accuracy. SNN-specific parameters. Figure 3b plots the interaction between spike steepness (α) and contrastive weight. Perfor… view at source ↗

**Figure 5.** Figure 5: Training loss curves of ChronoSpike on DBLP, Tmall, and Patent with 40%, 60%, and 80% training splits. ChronoSpike shows stable convergence across all settings, with faster loss reduction at higher supervision. F.3. Interpretability Analysis We conduct an interpretability analysis on the DBLP dataset using 1000 randomly sampled test nodes to understand ChronoSpike’s learned representations and mechanisms. … view at source ↗

**Figure 6.** Figure 6: Learned Temporal Importance. The distribution shows a strong focus on early snapshots (t = 2, 3, 4) rather than the most recent time steps. This shows ChronoSpike’s ability to use long-range historical context. F.3.2. NEUROMORPHIC ACTIVITY ANALYSIS We further analyze spiking behavior to verify ChronoSpike’s event-driven computation [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗

**Figure 7.** Figure 7: Spiking raster plots for two SNN layers. Spike events for Layer 1 (left) and Layer 2 (right) are shown over neurons (y-axis) and input samples (x-axis). Layer 1 Layer 2 Layer 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Firing Rate [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗

**Figure 8.** Figure 8: Distribution of per-sample firing rates across layers. Boxplots summarize firing rates (spikes per neuron per sample) aggregated over neurons for each input sample. Boxes indicate interquartile ranges with medians, whiskers denote full ranges, and triangles indicate means. Layer 2 exhibits higher median firing rates and larger variability across samples than Layer 1. 20 10 0 10 Activation Value 0.0 0.5 1.0… view at source ↗

**Figure 9.** Figure 9: Histograms of membrane potentials measured before spike generation for Layer 1 (left) and Layer 2 (right), aggregated over all neurons, timesteps, and input samples in the evaluation set. Dashed vertical lines indicate mean values (µ), and reported σ denotes standard deviation. Here, “Sparse” denotes the percentage of membrane potentials near zero (|V | < 0.01), indicating quiescent neural activity. 33 [P… view at source ↗

read the original abstract

Dynamic graph representation learning requires capturing both structural relations and temporal evolution, yet existing approaches face a core trade-off: attention-based methods offer expressiveness at $O(T^2)$ complexity, while recurrent architectures suffer from gradient pathologies and dense state storage. Spiking neural networks provide event-driven efficiency but are constrained by sequential propagation, binary information loss, and local aggregation that lacks global context. We propose ChronoSpike, an adaptive spiking graph neural network that integrates learnable LIF neurons with per-channel membrane dynamics, multi-head spatially-attentive aggregation over continuous features, and a lightweight Transformer temporal encoder. This design enables fine-grained local modeling and long-range dependency capture with $O(T \cdot d)$ activation/state memory and an additional $O(T^2)$ per-node attention term that remains small for the horizons evaluated here. ChronoSpike outperforms twelve state-of-the-art baselines on three large benchmarks by $2.0$% Macro-F1 and $2.4$% Micro-F1 on average while achieving $3-10\times$ faster training than recurrent methods with a constant 105K-parameter budget independent of graph size. We provide theoretical guarantees for membrane potential boundedness, gradient flow stability under contraction factor $\rho<1$, and BIBO stability; interpretability analyses reveal heterogeneous temporal receptive fields and a learned primacy effect with $83-88$% sparsity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

ChronoSpike mixes per-channel spiking neurons with spatial attention and a temporal transformer to cut training time on dynamic graphs, but the stability proofs stop short of the full hybrid system.

read the letter

The main point is that this paper builds a hybrid model that keeps spiking efficiency while adding attention for global context and a transformer for longer temporal range. The architecture uses learnable LIF neurons with independent per-channel membrane parameters, multi-head attention over continuous node features, and a lightweight transformer encoder. That combination is the actual novelty; prior spiking graph work stayed local and sequential, while pure attention or recurrent baselines paid quadratic or gradient costs. The reported results show a 2% Macro-F1 and 2.4% Micro-F1 lift over twelve baselines on three large datasets, 3-10x faster training than recurrent alternatives, and a fixed 105K-parameter footprint that does not grow with graph size. The added interpretability section on heterogeneous receptive fields and a learned primacy effect with 83-88% sparsity is useful extra material. The theoretical claims cover membrane boundedness, gradient contraction under ρ<1, and BIBO stability, which is more than most empirical papers deliver. The soft spot is exactly the one the stress-test flags: the contraction and stability arguments appear to be shown only for the isolated spiking recurrence. Once multi-head attention and the transformer are folded in, the composite Jacobian includes data-dependent, non-local terms whose contraction is not obviously guaranteed. If the proofs treat attention as an external bounded operator rather than deriving a joint bound, the guarantee does not carry over to the full model. The empirical side also needs the full protocol details to confirm that the baselines were run under matched compute and that the gains survive different random seeds or graph sizes. This work is aimed at researchers who need scalable temporal graph models and are willing to trade some theoretical completeness for practical speed and fixed memory. It is coherent enough on its own terms to deserve a serious referee, though the proofs will require tightening and the experiments will need more controls.

Referee Report

3 major / 2 minor

Summary. The paper proposes ChronoSpike, an adaptive spiking graph neural network for dynamic graphs that integrates learnable LIF neurons with per-channel membrane dynamics, multi-head spatially-attentive aggregation over continuous features, and a lightweight Transformer temporal encoder. It claims to outperform twelve state-of-the-art baselines on three large benchmarks by 2.0% Macro-F1 and 2.4% Micro-F1 on average, achieve 3-10× faster training than recurrent methods with a fixed 105K-parameter budget independent of graph size, provide theoretical guarantees for membrane potential boundedness, gradient flow stability under contraction factor ρ<1, and BIBO stability, and offer interpretability analyses showing heterogeneous temporal receptive fields, a learned primacy effect, and 83-88% sparsity.

Significance. If the empirical gains and stability guarantees hold under the full hybrid architecture, the work would meaningfully advance dynamic graph learning by combining event-driven spiking efficiency with mechanisms for long-range temporal dependencies, addressing key trade-offs in attention-based and recurrent approaches while providing constant memory scaling and formal stability assurances.

major comments (3)

[Theoretical guarantees] Theoretical guarantees section: the contraction mapping argument establishing gradient flow stability under ρ<1 is derived for the isolated per-channel LIF recurrence, but the manuscript does not explicitly bound the composite Jacobian that incorporates the data-dependent multi-head attention weights and Transformer encoder; without this, the ρ<1 guarantee does not necessarily extend to the full system on realistic attention distributions.
[Experiments] Experimental results: the reported average 2.0% Macro-F1 and 2.4% Micro-F1 improvements over twelve baselines are presented without per-benchmark breakdowns, standard deviations across multiple runs, or statistical significance tests, which weakens the ability to assess whether the gains are consistent or driven by particular datasets.
[Model architecture] Architecture description: the claim of O(T·d) activation/state memory with only a small additional O(T²) per-node attention term is stated for the evaluated horizons, but no analysis is given of how the per-channel LIF parameters and attention heads scale with graph size or feature dimension, leaving the constant 105K-parameter budget claim partially unverified.

minor comments (2)

[Methods] Notation for the per-channel membrane time constants and reset potentials is introduced without an explicit table or equation reference in the methods, making it harder to trace the learnable parameters.
[Interpretability] The interpretability section mentions 83-88% sparsity but does not specify the exact sparsity metric (e.g., activation rate or weight sparsity) or provide visualizations of the learned temporal receptive fields.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us identify areas for clarification and improvement. We address each major comment below and have revised the manuscript to incorporate the suggested changes where feasible.

read point-by-point responses

Referee: [Theoretical guarantees] Theoretical guarantees section: the contraction mapping argument establishing gradient flow stability under ρ<1 is derived for the isolated per-channel LIF recurrence, but the manuscript does not explicitly bound the composite Jacobian that incorporates the data-dependent multi-head attention weights and Transformer encoder; without this, the ρ<1 guarantee does not necessarily extend to the full system on realistic attention distributions.

Authors: We acknowledge that the contraction mapping is presented for the per-channel LIF recurrence in isolation. However, because the multi-head attention weights are obtained via softmax (hence elementwise bounded in [0,1] with row-stochastic property) and the lightweight Transformer employs residual connections and layer normalization (both Lipschitz with constant 1), the composite Jacobian norm remains bounded by a factor that preserves overall contractivity. In the revised manuscript we add a supporting lemma that explicitly bounds the spectral norm of the attention and Transformer Jacobians under the same assumptions used for the LIF analysis, yielding a composite contraction factor ρ' < 1 that continues to guarantee gradient flow stability. revision: yes
Referee: [Experiments] Experimental results: the reported average 2.0% Macro-F1 and 2.4% Micro-F1 improvements over twelve baselines are presented without per-benchmark breakdowns, standard deviations across multiple runs, or statistical significance tests, which weakens the ability to assess whether the gains are consistent or driven by particular datasets.

Authors: We agree that the aggregate averages alone are insufficient for rigorous evaluation. The revised manuscript now includes a detailed table reporting Macro-F1 and Micro-F1 for each of the three benchmarks separately, together with standard deviations computed over five independent random seeds and p-values from paired t-tests against the strongest baseline on each dataset. These additions confirm that the reported gains are consistent and statistically significant across all benchmarks. revision: yes
Referee: [Model architecture] Architecture description: the claim of O(T·d) activation/state memory with only a small additional O(T²) per-node attention term is stated for the evaluated horizons, but no analysis is given of how the per-channel LIF parameters and attention heads scale with graph size or feature dimension, leaving the constant 105K-parameter budget claim partially unverified.

Authors: The per-channel LIF module uses exactly three learnable scalars per channel (decay rate, threshold, and reset value), while each attention head employs fixed-size projection matrices whose dimensions depend only on the chosen feature dimension d (set to 64) and the number of heads (set to 4). Consequently the total parameter count is independent of the number of nodes or edges and remains fixed at 105K for the reported configuration. The revised manuscript adds an explicit parameter-count table and a short scaling paragraph confirming that the budget stays constant for any graph size under fixed d and T. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on external benchmarks and separate theoretical guarantees

full rationale

The paper reports empirical outperformance on three external large benchmarks against twelve baselines, with performance numbers presented as measured results rather than derived by construction from fitted parameters. Stability properties (membrane potential boundedness, gradient flow under ρ<1, BIBO stability) are listed as independent theoretical guarantees without equations that reduce the composite hybrid Jacobian (including attention) back to the isolated LIF recurrence by definition. No self-citation chains, ansatz smuggling, or renaming of known results appear as load-bearing steps in the abstract or described architecture. The derivation chain therefore remains self-contained against external data and stated assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the effectiveness of the hybrid architecture whose learnable parameters are fitted during training and on standard domain assumptions about spiking neuron stability and gradient behavior.

free parameters (2)

per-channel LIF membrane parameters
Learnable parameters controlling membrane dynamics for each feature channel that are adapted to data during training.
multi-head attention weights
Parameters in the spatially-attentive aggregation layers fitted to graph structure and features.

axioms (2)

domain assumption Membrane potential remains bounded
Invoked to support theoretical stability guarantees for the spiking neurons.
domain assumption Gradient flow contracts with factor ρ<1
Assumed to ensure stable training of the recurrent-like temporal components.

pith-pipeline@v0.9.0 · 5557 in / 1575 out tokens · 45602 ms · 2026-05-16T08:59:20.510874+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Neuromorphic Graph Anomaly Detection via Adaptive STDP and Spiking Graph Neural Networks
cs.NE 2026-04 unverdicted novelty 5.0

ASTDP-GAD unifies spiking neural computation, STDP learning, and graph anomaly detection with claimed theoretical guarantees on encoding, convergence, and score calibration.

Reference graph

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, Gt−1) with current temporal context (edges from Gt)

Dynamic Graph Sampling:For each target node v at timestep t, we perform hybrid neighborhood sampling that combines historical structure (cumulative edges from G1, . . . , Gt−1) with current temporal context (edges from Gt). This dual-sampling strategy balances long-term structural memory with recent temporal dynamics

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Each layer k applies learnable transformations W(k) s ,W (k) n ∈R d×dh to aggregate neighbor features adaptively, producing synaptic inputsI (t) for the spiking neurons

Spatial Feature Encoding:Sampled neighborhoods undergo K-layer graph aggregation using multi-head attention mechanisms. Each layer k applies learnable transformations W(k) s ,W (k) n ∈R d×dh to aggregate neighbor features adaptively, producing synaptic inputsI (t) for the spiking neurons. 19 ChronoSpike: An Adaptive Spiking Graph Neural Network for Dynami...

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Each neuron i maintains a membrane potential u(t) i governed by learnable time constants τi and thresholds Vth,i, enabling heterogeneous temporal dynamics across feature channels

Spike-Based Temporal Modeling:Aggregated spatial features are converted into sparse spike trains via adaptive LIF neurons. Each neuron i maintains a membrane potential u(t) i governed by learnable time constants τi and thresholds Vth,i, enabling heterogeneous temporal dynamics across feature channels. Spikes s(t) i ∈ {0,1} are emitted when u(t) i exceeds ...

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,s(T) } across all timesteps are aggregated using a lightweight Transformer encoder with learned positional encodings

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Hence, all conditions of Theorem D.1 hold for each neuron independently. Therefore, after at most one spike, |u(t) v,i| ≤max n Vth,i,|u reset|+ M+|u reset| τv,i 1− 1− 1 τv,i o ,∀t≥1(25) Taking the maximum over all(v, i)yields a uniform boundBfor the entire network. (b) BIBO stability.Let external inputs be bounded. Then, synaptic inputs are bounded by Ass...

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Numerical computations were performed using NumPy 1.26.4 (Harris et al., 2020) and SciPy 1.15.3 (Virtanen et al., 2020)

support for GPU acceleration. Numerical computations were performed using NumPy 1.26.4 (Harris et al., 2020) and SciPy 1.15.3 (Virtanen et al., 2020). For machine learning utilities including preprocessing, evaluation metrics, and data splitting, we employed scikit-learn 1.7.2 (Pedregosa et al., 2011). Additional utilities included tqdm 4.67.1 for progres...

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Cong, W., Zhang, S., Kang, J., Yuan, B., Wu, H., Zhou, X., Tong, H., and Mahdavi, M

IEEE, 2021. Cong, W., Zhang, S., Kang, J., Yuan, B., Wu, H., Zhou, X., Tong, H., and Mahdavi, M. Do we re- ally need complicated model architectures for tem- poral networks? InThe eleventh international conference on learning representations, ICLR 2023, kigali, rwanda, may 1-5, 2023. OpenReview.net,

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Number: 04. Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. Pytorch: An imperative style, high-performance deep learning library. InAdvances in Ne...

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work page doi:10.1038/s41586-019-1677-2 2017

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URL https://openreview.net/forum? id=skoBTs4ke4. Wen, Z. and Fang, Y . Trend: Temporal event and node dy- namics for graph representation learning. InProceedings of the ACM web conference 2022, pp. 1159–1169, 2022. Wu, H., Zhang, Y ., Weng, W., Zhang, Y ., Xiong, Z., Zha, Z.- J., Sun, X., and Wu, F. Training spiking neural networks with accumulated spikin...

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URL https: //doi.org/10.1609/aaai.v39i21.34382

doi: 10.1609/AAAI.V39I21.34382. URL https: //doi.org/10.1609/aaai.v39i21.34382. Yun, C., Bhojanapalli, S., Rawat, A. S., Reddi, S. J., and Kumar, S. Are transformers universal approximators of sequence-to-sequence functions? In8th International Conference on Learning Representations, ICLR 2020, Ad- dis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net,

work page doi:10.1609/aaai.v39i21.34382 2020

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3D-LLaV A: Towards generalist 3D LMMs with omni superpoint transformer

URL https://openreview.net/forum? id=ByxRM0Ntvr. Zhang, H., Li, J., Zhu, Y ., Chen, L., and Kuang, L. SGNNBench: A Holistic Evaluation of Spiking Graph Neural Network on Large-scale Graph, Septem- ber 2025a. URL http://arxiv.org/abs/2509. 21342. arXiv:2509.21342 [cs]. Zhang, T., Yu, K., Zhang, J., and Wang, H. DA- LIF: Dual Adaptive Leaky Integrate-and-Fi...

work page doi:10.1109/cvpr52734.2025.01303 2025

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URL https: //doi.org/10.1609/aaai.v32i1.11257

doi: 10.1609/AAAI.V32I1.11257. URL https: //doi.org/10.1609/aaai.v32i1.11257. Zhu, Z., Peng, J., Li, J., Chen, L., Yu, Q., and Luo, S. Spiking graph convolutional networks. In Raedt, L. D. (ed.),Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI 2022, Vienna, Austria, 23-29 July 2022, pp. 2434–2440. ijcai.org,...

work page doi:10.1609/aaai.v32i1.11257 2022

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, Gt−1) with current temporal context (edges from Gt)

Dynamic Graph Sampling:For each target node v at timestep t, we perform hybrid neighborhood sampling that combines historical structure (cumulative edges from G1, . . . , Gt−1) with current temporal context (edges from Gt). This dual-sampling strategy balances long-term structural memory with recent temporal dynamics

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Each layer k applies learnable transformations W(k) s ,W (k) n ∈R d×dh to aggregate neighbor features adaptively, producing synaptic inputsI (t) for the spiking neurons

Spatial Feature Encoding:Sampled neighborhoods undergo K-layer graph aggregation using multi-head attention mechanisms. Each layer k applies learnable transformations W(k) s ,W (k) n ∈R d×dh to aggregate neighbor features adaptively, producing synaptic inputsI (t) for the spiking neurons. 19 ChronoSpike: An Adaptive Spiking Graph Neural Network for Dynami...

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[13] [13]

Each neuron i maintains a membrane potential u(t) i governed by learnable time constants τi and thresholds Vth,i, enabling heterogeneous temporal dynamics across feature channels

Spike-Based Temporal Modeling:Aggregated spatial features are converted into sparse spike trains via adaptive LIF neurons. Each neuron i maintains a membrane potential u(t) i governed by learnable time constants τi and thresholds Vth,i, enabling heterogeneous temporal dynamics across feature channels. Spikes s(t) i ∈ {0,1} are emitted when u(t) i exceeds ...

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,s(T) } across all timesteps are aggregated using a lightweight Transformer encoder with learned positional encodings

Temporal Aggregation and Prediction:Spike representations {s(1), . . . ,s(T) } across all timesteps are aggregated using a lightweight Transformer encoder with learned positional encodings. This global temporal aggregation captures 20 ChronoSpike: An Adaptive Spiking Graph Neural Network for Dynamic Graphs Algorithm 3ChronoSpike Inference Require:Test nod...

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Hence, all conditions of Theorem D.1 hold for each neuron independently. Therefore, after at most one spike, |u(t) v,i| ≤max n Vth,i,|u reset|+ M+|u reset| τv,i 1− 1− 1 τv,i o ,∀t≥1(25) Taking the maximum over all(v, i)yields a uniform boundBfor the entire network. (b) BIBO stability.Let external inputs be bounded. Then, synaptic inputs are bounded by Ass...

work page 2019

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Numerical computations were performed using NumPy 1.26.4 (Harris et al., 2020) and SciPy 1.15.3 (Virtanen et al., 2020)

support for GPU acceleration. Numerical computations were performed using NumPy 1.26.4 (Harris et al., 2020) and SciPy 1.15.3 (Virtanen et al., 2020). For machine learning utilities including preprocessing, evaluation metrics, and data splitting, we employed scikit-learn 1.7.2 (Pedregosa et al., 2011). Additional utilities included tqdm 4.67.1 for progres...

work page 2020