FRACTAL: SSM with Fractional Recurrent Architecture for Computational Temporal Analysis of Long Sequences

Spiking Neural Networks (SNNs) are well-regarded for their biological plausibility and energy efficiency in processing sequential data. However, dominant SNN architectures typically rely on first-order Ordinary Differential Equations (ODEs) to govern neuronal state transitions. This first-order assumption imposes a "memoryless" bottleneck, limiting the model's capacity to capture the complex, long-range dependencies inherent in long-sequence tasks. In this work, we propose LongSpike, a novel SNN framework that integrates fractional-order State-Space Modeling, or f-SSM, from control theory into the spiking domain. By extending traditional integer-order SSMs to the fractional-calculus regime, LongSpike enables the hierarchical integration of neuronal dynamics with long-memory kernels. To mitigate the computational overhead and parallelization challenges typically associated with fractional operators, we leverage a state-space formulation that supports efficient, parallel training. Empirical evaluations on challenging benchmarks, including Long Range Arena (LRA), large-scale WikiText-103, and Speech Commands, demonstrate that LongSpike outperforms state-of-the-art SNNs in accuracy while preserving sparse synaptic computation. The code is available at https://github.com/xinruihe389-commits/LongSpike.

State space models (SSMs) have emerged as efficient linear-time alternatives to attention for long-sequence modeling. However, existing SSMs often suffer from instability and memory degradation over extended horizons due to poorly conditioned first-order updates and unbalanced update geometry. We introduce MuonSSM, a general framework that stabilizes SSM training by explicitly conditioning the geometry of memory updates rather than the recurrent transition matrix. MuonSSM augments SSMs with a momentum-based pathway and a lightweight Newton Schulz transformation on low-rank input injections, yielding bounded and spectrally conditioned updates while preserving parallel scan complexity. Theory shows that MuonSSM improves gradient propagation, mitigates spectral amplification, and enriches memory representations over long horizons. Extensive experiments across language, vision, and time-series benchmarks show consistent gains in accuracy, robustness, and long-context performance when integrated into diverse SSM backbones. These results establish geometric conditioning of updates as a principled pathway to stable, scalable sequence modeling.

Modern sequence modeling is dominated by two families: Transformers, whose self-attention can access arbitrary elements of the visible sequence, and structured state-space models, which propagate information through an explicit recurrent state. These mechanisms face different limitations on long contexts: when attention is diffuse, the influence of individual tokens is diluted across the effective support, while recurrent state propagation can lose long-range sensitivity unless information is actively preserved. As a result, both mechanisms face challenges in preserving and selectively retrieving information over long contexts. We propose Sessa, a decoder that places attention inside a recurrent feedback path. This creates many attention-based paths through which past tokens can influence future states, rather than relying on a single attention read or a single recurrent chain. We prove that, under explicit assumptions and matched regimes, Sessa admits power-law memory tails $O(\ell^{-β})$ for $0 < β< 1$, with slower decay than in the corresponding Transformer and Mamba-style baselines. We further give an explicit construction that achieves this power-law rate. Under the same assumptions, Sessa is the only model class among those considered that realizes flexible selective retrieval, including profiles whose influence does not decay with distance. Consistent with this theoretical advantage, across matched experiments, Sessa achieves the strongest performance on long-context benchmarks while remaining competitive with Transformer and Mamba-style baselines on short-context language modeling.