A Proof of a Key Formula in the Error-Backpropagation Learning Algorithm for Multiple Spiking Neural Networks

摘要

In the error-backpropagation learning algorithm for spiking neural networks, solving the differentiation of the firing time t~α with respect to the weight w is essential. Bohte et al. see the firing time t~α as a functional of the state variable x(t). But the differentiation of the firing time t~α with respect to the state variable x(t) is impossible to perform directly. To overcome this problem, Bohte et al. assume that the state variable x(t) is a linear function of the time t around t = t~α. Then, it seems that the solution of Bohte et al. is used by all related Literatures. In particular, Ghosh-Dastidar and Adeli offer another explanation. In this paper, we consider the firing time t~α as a function of the time t and the weight w and prove that the key formula for multiple spiking neural networks is in fact mathematically correct through the implicit function theorem.