The training of multilayer spiking neural networks (SNNs) using the error backpropagation algorithm has made significant progress in recent years. Among the various training schemes, the error backpropagation method that directly uses the firing time of neurons has attracted considerable attention because it can realize ideal temporal coding. This method uses time-to-first spike (TTFS) coding, in which each neuron fires at most once, and this restriction on the number of firings enables information to be processed at a very low firing frequency. This low firing frequency increases the energy efficiency of information processing in SNNs, which is important not only because of its similarity with information processing in the brain, but also from an engineering point of view. However, only an upper limit has been provided for TTFS-coded SNNs, and the information-processing capability of SNNs at lower firing frequencies has not been fully investigated. In this paper, we propose two spike timing-based sparse-firing (SSR) regularization methods to further reduce the firing frequency of TTFS-coded SNNs. The first is the membrane potential-aware SSR (M-SSR) method, which has been derived as an extreme form of the loss function of the membrane potential value. The second is the firing condition-aware SSR (F-SSR) method, which is a regularization function obtained from the firing conditions. Both methods are characterized by the fact that they only require information about the firing timing and associated weights. The effects of these regularization methods were investigated on the MNIST, Fashion-MNIST, and CIFAR-10 datasets using multilayer perceptron networks and convolutional neural network structures.

翻译：近年来，利用误差反向传播算法训练多层脉冲神经网络（SNNs）取得了显著进展。在多种训练方案中，直接利用神经元触发时间的误差反向传播方法因其能实现理想的时间编码而备受关注。该方法采用首次脉冲时间（TTFS）编码，每个神经元最多触发一次，这种对触发次数的限制使得信息能以极低触发频率进行处理。这种低触发频率提升了SNNs信息处理的能效，其重要性不仅在于与大脑信息处理的相似性，更在于工程应用价值。然而，现有研究仅给出了TTFS编码SNNs的上限，尚未充分探索SNNs在更低触发频率下的信息处理能力。本文提出两种基于脉冲时序的稀疏触发（SSR）正则化方法，旨在进一步降低TTFS编码SNNs的触发频率。第一种是膜电位感知SSR（M-SSR）方法，该方法推导自膜电位值损失函数的极端形式；第二种是触发条件感知SSR（F-SSR）方法，这是一种基于触发条件得到的正则化函数。两种方法均仅需利用触发时序及相关权值信息。我们在MNIST、Fashion-MNIST和CIFAR-10数据集上，采用多层感知机网络和卷积神经网络结构验证了这些正则化方法的效果。