Spectral properties of neuronal pulse interval modulation

We determine the power spectrum of an ideal neuron which encodes information using a pulse interval modulation scheme in continuous time. We develop this by considering the rigorous derivation of the Digital Pulse Interval Modulation (DPIM) coding scheme spectra of L. Vangelista et al. in the limit of the coding slot size approaching zero. We show in this limit the spectrum is identical to that of a filtered renewal process frequently used to model neuroscience time series data. Using this renewal theory equivalence we then use the `Fundamental Isometry Theorem´ developed by Win & Ridolfi to show that introducing firing time jitter (as a simple model for noise effects) removes non-Poisson structure and reduces the utility of spectral feature selection. Lastly we show with sufficient jittering that the Bartlett spectrum of any renewal process reduces to that of a Poisson process, with a spectral density consistent with Carson´s theorem for shot noise.