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# Fluctuation scaling in neural spike trains

*BMC Neuroscience*
**volume 15**, Article number: P119 (2014)

The fluctuation scaling (FS) law has been observed in a wide variety of phenomena. It states that the variance of a quantity has a power function relation with the mean. Since Taylor found it in ecological systems [1], the FS law has been demonstrated in many natural and social systems, showing a universality of the law [2]. In this study, The FS law for neural spike trains is formulated using the framework of renewal point processes.

In order to quantify the variability of neural firing, two quantities: inter-spike interval (ISI) and spike count, are often measured from spike trains. We first consider the ISI statistics. Let ${t}_{1},{t}_{2},\dots ,{t}_{n}$ be a sequence of spike times, and ${x}_{i}={t}_{i}-{t}_{i-1}\left(i=2,\dots ,n\right)$ be ISIs. Let$\mu =E\left(X\right)$ and ${\sigma}^{2}=Var\left(X\right)$ denote the mean and variance of ISI, respectively. We consider spike trains such that under a stationary condition ${\sigma}^{2}$ has a power function relation with $\phantom{\rule{0.5em}{0ex}}\mu $ as

The scaling exponent $\phantom{\rule{0.5em}{0ex}}\alpha $ characterizes the 'intrinsic' dispersion of neuronal firing. For a Poisson (random) process, $\alpha =2$. On the other hand, $\alpha >2\left(\alpha <2\right)$ implies the tendency for the timing of spikes to be over (under) dispersed for large means, and under (over) dispersed for small means.

Consider next the counting statistics. Let ${N}_{\left(t,t+\text{\Delta}\right]}$ be the number of spikes occurred in the counting window $\left(t,t+\text{\Delta}\right]$. We prove that if the spike train is a renewal process and the interval statistics has the scaling property (1), then the variance of ${N}_{\left(t,t+\text{\Delta}\right]}$ per unit time is asymptotically scaled by the mean of ${N}_{\left(t,t+\text{\Delta}\right]}$ per unit time (i.e., the rate) as

with $\beta =3-\alpha $ for $\text{\Delta}>>1$.

In the presentation, I show the following two results:

The fluctuation scaling law emerges in the first-passage time to a threshold of certain diffusion processes (i.e., integrate-and-fire models).

The likelihood function of spike trains is constructed, based on which I propose a method for extracting the scaling exponent from nonstationary spike trains. This method is applied to biological spike train data to characterize the variability of neuronal firing.

Possible implications of these results are discussed in terms of characterizing intrinsic dynamics of neuronal discharge.

## References

- 1.
Taylor LR: Aggregation, variance and the mean. Nature. 1961, 189: 732-735. 10.1038/189732a0.

- 2.
Eisler Z, Bartos I, Kertesz J: Fluctuation scaling in complex systems: Taylor's law and beyond. Advances in Physics. 2008, 57: 89-142. 10.1080/00018730801893043.

## Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 24700287.

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

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Koyama, S. Fluctuation scaling in neural spike trains.
*BMC Neurosci* **15, **P119 (2014). https://doi.org/10.1186/1471-2202-15-S1-P119

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### Keywords

- Spike Train
- Renewal Process
- Neuronal Firing
- Spike Time
- Interval Statistic