Consider the transform from a discrete neuronal spike train to a continuous neurophysiological response such as postsynaptic membrane voltage or muscle contraction. Often, we can record only the response. Here we therefore ask about the inverse of this transform: given the response, how can we estimate from it the spike train that produced it? In previous work, we have developed a kernel-based "decoding" method for system identification of the forward transform. Using several variant approaches based on this method, here we demonstrate how to identify the spike train, in the minimal case from just the response, with synthetic as well as real synaptic and neuromuscular data.