Fractional-order Spiking Neural Network

TL;DR

The paper develops fractional-order spiking neural networks (f-SNNs) by embedding Caputo fractional derivatives $D^\alpha$ into neuron dynamics, introducing memory through a power-law kernel and enabling long-range temporal dependencies. The authors derive fractional ABM discretizations, establish theoretical advantages (persistent memory, irreducibility to finite integer-order systems, and improved robustness), and provide an open-source toolbox spikeDE to support end-to-end training across CNNs, Transformers, and GNNs. Empirically, f-SNNs improve accuracy and robustness on neuromorphic datasets and graph benchmarks with energy usage comparable to traditional SNNs, illustrating the practical value of fractional dynamics in neuromorphic computation. The work offers a biologically plausible, mathematically grounded extension of SNNs that broadens temporal modeling capabilities for real-world tasks and provides tooling to foster wider adoption.

Abstract

Spiking Neural Networks (SNNs) draw inspiration from biological neurons to enable brain-like computation, demonstrating effectiveness in processing temporal information with energy efficiency and biological realism. Most existing SNNs are based on neural dynamics such as the (leaky) integrate-and-fire (IF/LIF) models, which are described by first-order ordinary differential equations (ODEs) with Markovian characteristics. This means the potential state at any time depends solely on its immediate past value, potentially limiting network expressiveness. Empirical studies of real neurons, however, reveal long-range correlations and fractal dendritic structures, suggesting non-Markovian behavior better modeled by fractional-order ODEs.Motivated by this, we propose a fractional-order spiking neural network (f-SNN) framework that strictly generalizes integer-order SNNs and captures long-term dependencies in membrane potential and spike trains via fractional dynamics, enabling richer temporal patterns. We also release an open-source toolbox to support the f-SNN framework, applicable to diverse architectures and real-world tasks. Experimentally, fractional adaptations of established SNNs into the f-SNN framework achieve superior accuracy, comparable energy efficiency, and improved robustness to noise, underscoring the promise of f-SNNs as an effective extension of traditional SNNs.

Figures (12)

• Figure 1: Comparison of traditional SNN and f-SNN framework.
• Figure 2: SNN vs f-SNN dynamics. In f-SNNs, past membrane potentials influence the current state via a power-law memory kernel; traditional integer-order SNNs lack this.
• Figure 3: Mittag--Leffler function $E_\alpha(-t^\alpha)$. For $\alpha=1$, LIF shows fast exponential decay ($E_1(-t)=e^{-t}$); for $0<\alpha<1$, f-LIF exhibits slow algebraic decay, reflecting memory.
• Figure 4: Robustness comparison between the proposed f-SNN and two integer-order baselines (LIF in SpikingJelly and LIF in snnTorch). Left: Radar chart aggregating five corruption types (larger is better): Gaussian noise injection, center occlude block, temporal truncate, temporal jitter, and discard frame. Middle: Performance vs. noise level (x-axis: Gaussian noise std). Right: Performance vs. occlusion ratio (x-axis: area ratio of the center block). The f-LIF (f-SNN) shows consistently higher performance and slower degradation under all corruption types.
• Figure 5: Feature map visualizations of LIF and f-LIF in occluded block scenarios.

Theorems & Definitions (3)

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