Applying an error-backpropagation algorithm to spiking neural networks frequently needs to employ fictive derivatives of spike functions (popularly referred to as surrogate gradients) because the spike function is considered non-differentiable. The non-differentiability comes into play given that the spike function is viewed as a numeric function, most popularly, the Heaviside step function of membrane potential. To get back to basics, the spike function is not a numeric but a Boolean function that outputs True or False upon the comparison of the current potential and threshold. In this regard, we propose a method to evaluate the gradient of spike function viewed as a Boolean function for fixed- and floating-point data formats. For both formats, the gradient is considerably similar to a delta function that peaks at the threshold for spiking, which justifies the approximation of the spike function to the Heaviside step function. Unfortunately, the error-backpropagation algorithm with this gradient function fails to outperform popularly employed surrogate gradients, which may arise from the narrow peak of the gradient function and consequent potential undershoot and overshoot around the spiking threshold with coarse timesteps. We provide theoretical grounds of this hypothesis.

将误差反向传播算法应用于脉冲神经网络时，通常需要采用脉冲函数的虚构导数（通常称为代理梯度），因为脉冲函数被认为是不可微分的。鉴于 spike 函数被视为数字函数，最常见的是膜电位的 Heaviside 阶跃函数，因此不可微性开始发挥作用。回到基础，spike 函数不是一个数字，而是一个布尔函数，它在比较电流电位和阈值时输出 True 或 False。在这方面，我们提出了一种方法来评估被视为定点和浮点数据格式的布尔函数的尖峰函数的梯度。对于这两种格式，梯度与在尖峰阈值处达到峰值的 delta 函数非常相似，这证明了尖峰函数近似于 Heaviside 阶跃函数的合理性。不幸的是，具有此梯度函数的误差反向传播算法无法胜过普遍使用的代理梯度，这可能是由于梯度函数的窄峰以及随之而来的粗略时间步长在尖峰阈值附近的潜在下冲和过冲引起的。我们提供了这一假设的理论依据。