Neuron dynamics in the presence of 1/f noise

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. This result elucidates the connection between low frequency noise and disorder in neuron dynamics. We compare our results to experimental data of single neurons in vivo, 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.Comment: Phys. Rev. E in pres

Scaling of Horizontal and Vertical Fixational Eye Movements

Eye movements during fixation of a stationary target prevent the adaptation of the photoreceptors to continuous illumination and inhibit fading of the image. These random, involuntary, small, movements are restricted at long time scales so as to keep the target at the center of the field of view. Here we use the Detrended Fluctuation Analysis (DFA) in order to study the properties of fixational eye movements at different time scales. Results show different scaling behavior between horizontal and vertical movements. When the small ballistics movements, i.e. micro-saccades, are removed, the scaling exponents in both directions become similar. Our findings suggest that micro-saccades enhance the persistence at short time scales mostly in the horizontal component and much less in the vertical component. This difference may be due to the need of continuously moving the eyes in the horizontal plane, in order to match the stereoscopic image for different viewing distance.Comment: 5 pages, 4 figure

Interpretations of Presburger Arithmetic in Itself

Presburger arithmetic PrA is the true theory of natural numbers with addition. We study interpretations of PrA in itself. We prove that all one-dimensional self-interpretations are definably isomorphic to the identity self-interpretation. In order to prove the results we show that all linear orders that are interpretable in (N,+) are scattered orders with the finite Hausdorff rank and that the ranks are bounded in terms of the dimension of the respective interpretations. From our result about self-interpretations of PrA it follows that PrA isn't one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201

Investigating the correlation between paediatric stride interval persistence and gross energy expenditure

<p>Abstract</p> <p>Background</p> <p>Stride interval persistence, a term used to describe the correlation structure of stride interval time series, is thought to provide insight into neuromotor control, though its exact clinical meaning has not yet been realized. Since human locomotion is shaped by energy efficient movements, it has been hypothesized that stride interval dynamics and energy expenditure may be inherently tied, both having demonstrated similar sensitivities to age, disease, and pace-constrained walking.</p> <p>Findings</p> <p>This study tested for correlations between stride interval persistence and measures of energy expenditure including mass-specific gross oxygen consumption per minute (<inline-formula><graphic file="1756-0500-3-47-i1.gif"/></inline-formula>), mass-specific gross oxygen cost per meter (<it>VO</it>₂) and heart rate (HR). Metabolic and stride interval data were collected from 30 asymptomatic children who completed one 10-minute walking trial under each of the following conditions: (i) overground walking, (ii) hands-free treadmill walking, and (iii) handrail-supported treadmill walking. Stride interval persistence was not significantly correlated with <inline-formula><graphic file="1756-0500-3-47-i1.gif"/></inline-formula> (p > 0.32), <it>VO</it>₂(p > 0.18) or HR (p > 0.56).</p> <p>Conclusions</p> <p>No simple linear dependence exists between stride interval persistence and measures of gross energy expenditure in asymptomatic children when walking overground and on a treadmill.</p

The Visibility Graph: a new method for estimating the Hurst exponent of fractional Brownian motion

Fractional Brownian motion (fBm) has been used as a theoretical framework to study real time series appearing in diverse scientific fields. Because its intrinsic non-stationarity and long range dependence, its characterization via the Hurst parameter H requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic f^(-b) noises. Taking advantage of these facts, we propose a brand new methodology to quantify long range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.Comment: 5 pages, submitted for publicatio

Pedestrian Solution of the Two-Dimensional Ising Model

The partition function of the two-dimensional Ising model with zero magnetic field on a square lattice with m x n sites wrapped on a torus is computed within the transfer matrix formalism in an explicit step-by-step approach inspired by Kaufman's work. However, working with two commuting representations of the complex rotation group SO(2n,C) helps us avoid a number of unnecessary complications. We find all eigenvalues of the transfer matrix and therefore the partition function in a straightforward way.Comment: 10 pages, 2 figures; eqs. (101) and (102) corrected, files for fig. 2 fixed, minor beautification

Fractal analyses reveal independent complexity and predictability of gait

Locomotion is a natural task that has been assessed for decades and used as a proxy to highlight impairments of various origins. So far, most studies adopted classical linear analyses of spatio-temporal gait parameters. Here, we use more advanced, yet not less practical, non-linear techniques to analyse gait time series of healthy subjects. We aimed at finding more sensitive indexes related to spatio-temporal gait parameters than those previously used, with the hope to better identify abnormal locomotion. We analysed large-scale stride interval time series and mean step width in 34 participants while altering walking direction (forward vs. backward walking) and with or without galvanic vestibular stimulation. The Hurst exponent α and the Minkowski fractal dimension D were computed and interpreted as indexes expressing predictability and complexity of stride interval time series, respectively. These holistic indexes can easily be interpreted in the framework of optimal movement complexity. We show that α and D accurately capture stride interval changes in function of the experimental condition. Walking forward exhibited maximal complexity (D) and hence, adaptability. In contrast, walking backward and/or stimulation of the vestibular system decreased D. Furthermore, walking backward increased predictability (α) through a more stereotyped pattern of the stride interval and galvanic vestibular stimulation reduced predictability. The present study demonstrates the complementary power of the Hurst exponent and the fractal dimension to improve walking classification. Our developments may have immediate applications in rehabilitation, diagnosis, and classification procedures

The universal Glivenko-Cantelli property

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are equivalent: 1. F is a universal Glivenko-Cantelli class. 2. N_{[]}(F,\epsilon,\mu)0 and every probability measure \mu. 3. F is totally bounded in L^1(\mu) for every probability measure \mu. 4. F does not contain a Boolean \sigma-independent sequence. It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.Comment: 26 page

Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector

In our companion paper, we focussed on the quantisation of the Rarita-Schwinger sector of Supergravity theories in various dimensions by using an extension of Loop Quantum Gravity to all spacetime dimensions. In this paper, we extend this analysis by considering the quantisation of additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton and gravitino in various dimensions. As a generic example, we study concretely the quantisation of the 3-index photon of 11d SUGRA, but our methods easily extend to more general p-form fields. Due to the presence of a Chern-Simons term for the 3-index photon, which is due to local SUSY, the theory is self-interacting and its quantisation far from straightforward. Nevertheless, we show that a reduced phase space quantisation with respect to the 3-index photon Gauss constraint is possible. Specifically, the Weyl algebra of observables, which deviates from the usual CCR Weyl algebras by an interesting twist contribution proportional to the level of the Chern-Simons theory, admits a background independent state of the Narnhofer-Thirring type.Comment: 12 pages. v2: Journal version. Minor clarifications and correction

Pulse-driven quantum dynamics beyond the impulsive regime

We review various unitary time-dependent perturbation theories and compare them formally and numerically. We show that the Kolmogorov-Arnold-Moser technique performs better owing to both the superexponential character of correction terms and the possibility to optimize the accuracy of a given level of approximation which is explored in details here. As an illustration, we consider a two-level system driven by short pulses beyond the sudden limit.Comment: 15 pages, 5 color figure