Distributed-Order Fractional Spiking Neural Network

Most existing Spiking Neural Networks (SNNs) are based on *first-order* ordinary differential equations (ODEs) to model neuron potential dynamics. Recently, fractional-order spiking neural networks (*f*-SNNs) extend conventional SNNs by incorporating fractional-order ODEs, enabling the modeling of long-range temporal dependencies in neuron potential states. However, existing first-order SNNs and *f*-SNNs are typically restricted to a single derivative order, which limits their capacity to represent heterogeneous temporal dynamics across different time scales. In this work, we propose Distributed-Order Fractional Spiking Neural Networks (*DF*-SNNs), which generalize single-order neuron dynamics by integrating multiple fractional orders through learnable distributed weights. This formulation enables neurons to accumulate historical information across *multiple temporal scales* within a unified framework. We develop practical numerical implementations based on the Grünwald–Letnikov discretization, allowing efficient training of *DF*-SNNs within standard SNN architectures. We show that the proposed distributed-order formulation induces intrinsic multi-scale temporal memory, providing a richer class of temporal representations compared to single-order fractional dynamics. Extensive experiments demonstrate consistent performance gains of *DF*-SNNs over conventional SNNs and single-order *f*-SNNs.