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

*(English)*Zbl 1402.92110Summary: 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.

##### MSC:

92C20 | Neural biology |

##### Keywords:

interspike interval statistics; stochastic integrate-and-fire model; long-range dependence; stationarity
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\textit{A. Richard} et al., J. Comput. Neurosci. 44, No. 3, 297--312 (2018; Zbl 1402.92110)

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