  In the discrete Tempotron learning problem a neuron receives time varying
inputs and for a set of such input sequences ($\mathcal S_-$ set) the neuron
must be sub-threshold for all times while for some other sequences ($\mathcal
S_+$ set) the neuron must be super threshold for at least one time. Here we
present a graphical treatment of a slight reformulation of the tempotron
problem. We show that the problem's general form is equivalent to the question
if a polytope, specified by a set of inequalities, is contained in the union of
a set of equally defined polytopes. Using recent results from computational
geometry, we show that the problem is W[1]-hard. This phrasing gives some new
insights into the nature of gradient based learning algorithms. A sampling
based approach can, under certain circumstances provide an approximation in
polynomial time. Other problems, related to hierarchical neural networks may
share some topological structure.
