AUTHOR=Soudry Daniel , Meir Ron TITLE=The neuronal response at extended timescales: a linearized spiking input–output relation JOURNAL=Frontiers in Computational Neuroscience VOLUME=8 YEAR=2014 URL=https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2014.00029 DOI=10.3389/fncom.2014.00029 ISSN=1662-5188 ABSTRACT=

Many biological systems are modulated by unknown slow processes. This can severely hinder analysis – especially in excitable neurons, which are highly non-linear and stochastic systems. We show the analysis simplifies considerably if the input matches the sparse “spiky” nature of the output. In this case, a linearized spiking Input–Output (I/O) relation can be derived semi-analytically, relating input spike trains to output spikes based on known biophysical properties. Using this I/O relation we obtain closed-form expressions for all second order statistics (input – internal state – output correlations and spectra), construct optimal linear estimators for the neuronal response and internal state and perform parameter identification. These results are guaranteed to hold, for a general stochastic biophysical neuron model, with only a few assumptions (mainly, timescale separation). We numerically test the resulting expressions for various models, and show that they hold well, even in cases where our assumptions fail to hold. In a companion paper we demonstrate how this approach enables us to fit a biophysical neuron model so it reproduces experimentally observed temporal firing statistics on days-long experiments.