  Cortical neurons include many sub-cellular processes, operating at multiple
timescales, which may affect their response to stimulation through non-linear
and stochastic interaction with ion channels and ionic concentrations. Since
new processes are constantly being discovered, biophysical neuron models
increasingly become "too complex to be useful" yet "too simple to be
realistic". A fundamental open question in theoretical neuroscience pertains to
how this deadlock may be resolved. In order to tackle this problem, we first
define the notion of a "excitable neuron model". Then we analytically derive
the input-output relation of such neuronal models, relating input spike trains
to output spikes based on known biophysical properties. Thus we obtain
closed-form expressions for the mean firing rates, all second order statistics
(input-state-output correlation and spectra) and construct optimal linear
estimators for the neuronal response and internal state. These results are
guaranteed to hold, given a few generic assumptions, for any stochastic
biophysical neuron model (with an arbitrary number of slow kinetic processes)
under general sparse stimulation. This solution suggests that the common
simplifying approach that ignores much of the complexity of the neuron might
actually be unnecessary and even deleterious in some cases. Specifically, the
stochasticity of ion channels and the temporal sparseness of inputs is exactly
what rendered our analysis tractable, allowing us to incorporate slow kinetics.
