A remark on the error-backpropagation learning algorithm for spiking neural networks

In the error-backpropagation learning algorithm for spiking neural networks, one has to differentiate the firing time t α as a functional of the state function x ( t ) . But this differentiation is impossible to perform directly since t α cannot be formulated in a standard form as a functional of x ( t ) . To overcome this difficulty, Bohte et al. (2002) [1] assume that there is a linear relationship between the firing time t α and the state x ( t ) around t = t α . In terms of this assumption, the Frechet derivative of the functional is equal to the derivative of an ordinary function that can be computed directly and easily. Our contribution in this short note is to prove that this equality of differentiations is in fact mathematically correct, without the help of the linearity assumption.