Combining frequency and time domain approaches to systems with multiple spike train input and output

A frequency domain approach and a time domain approach have been combined in an investigation of the behaviour of the primary and secondary endings of an isolated muscle spindle in response to the activity of two static fusimotor axons when the parent muscle is held at a fixed length and when it is subjected to random length changes. The frequency domain analysis has an associated error process which provides a measure of how well the input processes can be used to predict the output processes and is also used to specify how the interactions between the recorded processes contribute to this error. Without assuming stationarity of the input, the time domain approach uses a sequence of probability models of increasing complexity in which the number of input processes to the model is progressively increased. This feature of the time domain approach was used to identify a preferred direction of interaction between the processes underlying the generation of the activity of the primary and secondary endings. In the presence of fusimotor activity and dynamic length changes imposed on the muscle, it was shown that the activity of the primary and secondary endings carried different information about the effects of the inputs imposed on the muscle spindle. The results presented in this work emphasise that the analysis of the behaviour of complex systems benefits from a combination of frequency and time domain methods

Polynomial Cointegration among Stationary Processes with Long Memory

n this paper we consider polynomial cointegrating relationships among stationary processes with long range dependence. We express the regression functions in terms of Hermite polynomials and we consider a form of spectral regression around frequency zero. For these estimates, we establish consistency by means of a more general result on continuously averaged estimates of the spectral density matrix at frequency zeroComment: 25 pages, 7 figures. Submitted in August 200

Nonparametric directionality measures for time series and point process data

The need to determine the directionality of interactions between neural signals is a key requirement for analysis of multichannel recordings. Approaches most commonly used are parametric, typically relying on autoregressive models. A number of concerns have been expressed regarding parametric approaches, thus there is a need to consider alternatives. We present an alternative nonparametric approach for construction of directionality measures for bivariate random processes. The method combines time and frequency domain representations of bivariate data to decompose the correlation by direction. Our framework generates two sets of complementary measures, a set of scalar measures, which decompose the total product moment correlation coefficient summatively into three terms by direction and a set of functions which decompose the coherence summatively at each frequency into three terms by direction: forward direction, reverse direction and instantaneous interaction. It can be undertaken as an addition to a standard bivariate spectral and coherence analysis, and applied to either time series or point-process (spike train) data or mixtures of the two (hybrid data). In this paper, we demonstrate application to spike train data using simulated cortical neurone networks and application to experimental data from isolated muscle spindle sensory endings subject to random efferent stimulation

Feasibility study of parameter estimation of random sampling jitter using the bispectrum

An actual sampling process can be modeled as a random process, which consists of the regular (uniform) deterministic sampling process plus an error in the sampling times which constitutes a zero-mean noise (the jitter). In this paper we discuss the problem of estimating the jitter process. By assuming that the jitter process is an i.i.d. one, with standard deviation that is small compared to the regular sampling time, we show that the variance of the jitter process can be estimated from the n th order spectrum of the sampled data, n =2, 3, i.e., the jitter variance can be extracted from the 2nd-order spectrum or the 3rd-order spectrum (the bispectrum) of the sampled data, provided the continuous signal spectrum is known. However when the signal skewness exceeds a certain level, the potential performance of the bispectrum-based estimation is better than that of the spectrum-based estimation. Moreover, the former can also provide jitter variance estimates when the continuous signal spectrum is unknown while the latter cannot. This suggests that the bispectrum of the sampled data is potentially better for estimating any parameter of the sampling jitter process, once the signal skewness is sufficiently large.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43577/1/34_2005_Article_BF01183740.pd

Gridded and direct Epoch of Reionisation bispectrum estimates using the Murchison Widefield Array

We apply two methods to estimate the 21~cm bispectrum from data taken within the Epoch of Reionisation (EoR) project of the Murchison Widefield Array (MWA). Using data acquired with the Phase II compact array allows a direct bispectrum estimate to be undertaken on the multiple redundantly-spaced triangles of antenna tiles, as well as an estimate based on data gridded to the $uv$-plane. The direct and gridded bispectrum estimators are applied to 21 hours of high-band (167--197~MHz; $z$=6.2--7.5) data from the 2016 and 2017 observing seasons. Analytic predictions for the bispectrum bias and variance for point source foregrounds are derived. We compare the output of these approaches, the foreground contribution to the signal, and future prospects for measuring the bispectra with redundant and non-redundant arrays. We find that some triangle configurations yield bispectrum estimates that are consistent with the expected noise level after 10 hours, while equilateral configurations are strongly foreground-dominated. Careful choice of triangle configurations may be made to reduce foreground bias that hinders power spectrum estimators, and the 21~cm bispectrum may be accessible in less time than the 21~cm power spectrum for some wave modes, with detections in hundreds of hours.Comment: 19 pages, 10 figures, accepted for publication in PAS

Random line tessellations of the plane: statistical properties of many-sided cells

We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter \alpha\geq 1. For \alpha=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for \alpha=2 it coincides with the typical Poisson-Voronoi cell. Let p_n(\alpha) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of \log p_n(\alpha) up to terms that vanish as n\to\infty. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when \alpha>1, but gets delocalized for the Crofton cell, \alpha=1, which is a singular point of the parameter range. The large-n expansion of \log p_n(1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the {\it typical} n-sided cell of a Poisson line tessellation.Comment: 26 pages, 3 figure

On directed information theory and Granger causality graphs

Directed information theory deals with communication channels with feedback. When applied to networks, a natural extension based on causal conditioning is needed. We show here that measures built from directed information theory in networks can be used to assess Granger causality graphs of stochastic processes. We show that directed information theory includes measures such as the transfer entropy, and that it is the adequate information theoretic framework needed for neuroscience applications, such as connectivity inference problems.Comment: accepted for publications, Journal of Computational Neuroscienc