Exploring a regular-spiking limit of autoregressive point-process models with an absolute refractory period

These notes explore a deterministic limit of Autoregressive Point-Process Generalized Linear Model (AR-PPGLM) of a spiking neuron. We consider a Poisson point process that is either strongly driven or strongly suppressed, and take a limit where spiking output becomes deterministic (and instantaneous rates diverge). This deterministic limit is of interest in clarifying how AR-PPGLM models relate to deterministic models of spiking.

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Online Fourier transforms as filter banks and state-space models

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The online Fourier transform

The online Fourier transform tracks a sliding window of the recent history of the time series. This window is updated by advancing one time step, and the FFT is used to get the power spectrum. 

One can speed up computation by making updates recursive: on each time step, first subtract the contribution of the oldest timepoint, then add the contribution from the current timepoint. We can generalize this recursive approach by interpreting the online FT as a bank of band-pass filters.