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

In the error-backpropagation learning algorithm for spiking neural networks, solving the differentiation of the firing time t(alpha) with respect to the weight w is essential. Bohte et al. see the firing time t(alpha) as a functional of the state variable x(t). But the differentiation of the firing time t(alpha) 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(alpha). 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(alpha) 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.