  Many systems are modulated by unknown slow processes. This hinders analysis
in highly non-linear systems, such as excitable systems. We show that for such
systems, if the input matches the sparse `spiky' nature of the output, the
spiking input-output relation can be derived. We use this relation to reproduce
and interpret the irregular and complex 1/f response observed in isolated
neurons stimulated over days. We decompose the neuronal response into
contributions from its long history of internal noise and its short (few
minutes) history of inputs, quantifying memory, noise and stability.
