Neuron dynamics in the presence of1/fnoise

Abstract

Interest in understanding the interplay between noise and the response of a non-linear device cuts across disciplinary boundaries. It is as relevant for unmasking the dynamics of neurons in noisy environments as it is for designing reliable nanoscale logic circuit elements and sensors. Most studies of noise in non-linear devices are limited to either time-correlated noise with a Lorentzian spectrum (of which the white noise is a limiting case) or just white noise. We use analytical theory and numerical simulations to study the impact of the more ubiquitous "natural" noise with a 1/f frequency spectrum. Specifically, we study the impact of the 1/f noise on a leaky integrate and fire model of a neuron. The impact of noise is considered on two quantities of interest to neuron function: The spike count Fano factor and the speed of neuron response to a small step-like stimulus. For the perfect (non-leaky) integrate and fire model, we show that the Fano factor can be expressed as an integral over noise spectrum weighted by a (low pass) filter function given by F(t, f ) = sinc 2 (πf t). This result elucidates the connection between low frequency noise and disorder in neuron dynamics. Under 1/f noise, spike dynamics lacks a characteristic correlation time, inducing the leaky and non-leaky models to exhibit non-ergodic behavior and Fano factor increasing logarithmically as a function of time. We compare our results to experimental data of single neurons in vivo [M.C. Teich, C. Heneghan, S.B. Lowen, T. Ozaki, and E. Kaplan, Journal of the Optical Society of America A 14, 529 (1997)], and show how the 1/f noise model provides much better agreement than the usual approximations based on Lorentzian noise. The low frequency noise, however, complicates the case for information coding scheme based on interspike intervals by introducing variability in the neuron response time. On a positive note, the neuron response time to a step stimulus is, remarkably, nearly optimal in the presence of 1/f noise. An explanation of this effect elucidates how the brain can take advantage of noise to prime a subset of the neurons to respond almost instantly to sudden stimuli.

Statistical structure of neural spiking under non-Poissonian or other non-white stimulation

“…As shown in the Appendix A.2, the leading-order approximations for the CV and the SCC, Eqs. (9) and (13), are consistent with exact analytical results for the variability of the spike count on large time scales, which have been obtained previously (Middleton et al 2003;Sobie et al 2011;Moreno-Bote et al 2014). Indeed, the Fano factor F (t), defined as the variance to mean ratio of the spike count in a time window of length t, can be approximated for general t > 0 as…”

“…2.2, we show that even power spectra with an algebraic decay proportional to 1/f γ are captured by our approach because such noise arises from infinitely many exponentially correlated processes with a continuum of correlation times (see e.g. Sobie et al (2011)). …”

How noise contributes to time-scale invariance of interval timing

“…Many sources of noise (variability) may affect the functioning of an interval timing network, such as small fluctuations in the intrinsic frequencies of the inputs and in the encoding and retrieving the weights w i (T ) by the output neuron(s) [23,24,77,78]. Here we show analytically that one noise source is sufficient to produce time-scale invariance [23].…”

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

“…(Abry et al, 1995). 1/f noise has been successfully applied to describe many phenomena, from heartbeat (Peng et al, 1993) to internet traffic (Willinger et al, 1997), including neuronal fractal dynamics (Lowen et al, 2001;Sobie et al, 2011).…”

“…The second approach to model LRD is an Integrateand-Fire model with 1/f noise proposed by Sobie et al (2011), strongly related to our model. The link between fractional Brownian motion and 1/f noise is explained in (Abry et al, 1995), although there is no definition of 1/f noise as clear and universally accepted as the definition of fBm can be.…”