# An integrate-and-fire model to generate spike trains with long-range dependence

## TL;DR

This paper investigates how integrate-and-fire neuron models can produce spike trains with apparent long-range dependence, distinguishing between true LRD and effects caused by non-stationarity, and introduces a methodology to evaluate stationarity.

## Contribution

It demonstrates that a Markovian integrate-and-fire model with slow adaptation can mimic LRD, but true LRD requires a non-Markovian fractional noise model, and provides a stationarity assessment method.

## Key findings

- Classical IF models can appear to have LRD due to non-stationarity.
- Fractional noise models exhibit genuine LRD in spike trains.
- A methodology for stationarity evaluation helps distinguish true LRD from artifacts.

## Abstract

Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.