The neuron's response at extended timescales

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.