Variable-Order Spiking Neural Networks

Most existing spiking neural networks (SNNs) describe membrane dynamics with first-order ordinary differential equations, so earlier activity affects the membrane only through an exponential decay with a fixed time constant. Fractional-order SNNs (*f*-SNNs) address this limitation by replacing the first-order derivative with a fractional one, whose power-law kernel lets the membrane potential depend on long-range history. However, *f*-SNNs fix the derivative order for the entire input sequence, so the balance between recent and distant history cannot adapt as the input evolves. To address this problem, we propose Variable-Order Spiking Neural Networks (*VO*-SNNs). Specifically, we introduce variable-order leaky integrate-and-fire (*VO*-LIF) neurons, whose fractional order varies over time and depends on the current input and the accumulated membrane memory. Based on a variable-order er Grünwald–Letn discretization, *VO*-LIF neurons admit a direct charge–spike–reset update that can be trained end to end with surrogate gradients and adds only a small number of order parameters. *VO*-SNNs recover first-order SNNs and fixed-order *f*-SNNs as special cases, and we prove that their discrete dynamics are causal, remain bounded under an explicit step-size condition, and that time-varying orders produce non-stationary history kernels that no fixed order can reproduce. Extensive experiments show that *VO*-SNNs perform well on event-based recognition with multiple architectures, event-based object detection, graph-spike node classification, and neural response prediction from mouse visual cortex recordings, improving accuracy over fixed-order *f*-SNNs by up to 7.44 percentage points while consuming energy comparable to *f*-SNNs on neuromorphic hardware.