On the Nonlearnability of a Single Spiking Neuron

We study the computational complexity of training a single spiking neuron N with binary coded inputs and output that, in addition to adaptive weights and a threshold, has adjustable synaptic delays. A synchronization technique is introduced so that the results concerning the nonlearn-ability of spiking neurons with binary delays are generalized to arbitrary real-valued delays. In particular, the consistency problem for N with programmable weights, a threshold, and delays, and its approximation version are proven to be NP-complete. It follows that the spiking neurons with arbitrary synaptic delays are not properly PAC learnable and do not allow robust learning unless RP = NP. In addition, the representation problem for N, a question whether an n-variable Boolean function given in DNF (or as a disjunction of O(n) threshold gates) can be computed by a spiking neuron, is shown to be coNP-hard.