Uma rede estatística é uma cole,cão de nós representando variáveis aleatórias e um conjunto de arestas que ligam os nós. Um modelo estocástico por isso e chamado um modelo gráfico. Estes modelos, de gráficos e redes, sáo particularmente úteis para examinar as dependéncias estatísticas baseadas em condi,coes do tipo das que ocorrem frequentemente em economia e estatística. Neste artigo as variáveis aleatórias dos nós serão séries temporais ou processos pontuais. Os casos de gráfos direcionados e não-direcionados são apresentados. A statistical network is a collection of nodes representing random variables and a set of edges that connect the nodes. A probabilistic model for such is called a graphi-cal model. These models, graphs and networks are particularly useful for examining statistical dependencies based on conditioning as often occurs in economics and statis-tics. In this paper the nodal random variables will be time series or point proceses. The cases of undirected and directed graphs are focussed on.

Abstract. Through an asymptotic analysis of a simple network, we derive an estimate of the coupling between a pair of units when all other units are unobservable. The analysis is based on a model where the response of each unit is a linear-nonlinear function of a white noise stimulus. The results accurately determine the coupling when all unmeasured units respond to the stimulus differently than the measured pair. To account for the possibility of unmeasured units similar to the measured pair, we cast our results in the framework of “subpopulations, ” which are defined as a group of units who respond to the stimulus similarly. We demonstrate that we can determine when correlations between two units are caused by a connection between their subpopulations, although the precise identity of the units involved in the connection may remain ambiguous. The result is rigorously valid only when the coupling is sufficiently weak to justify a second-order approximation in the coupling strength. We demonstrate through simulations that the results are still valid even with stronger coupling and in the presence of some deviations from the linear-nonlinear model. The analysis is presented in terms of neuronal networks, although the general framework is more widely applicable.

The spectrum and coherency are useful quantities for characterizing the temporal correlations and functional relations within and between point processes. This paper begins with a review of these quantities, their interpretation and how they may be estimated. A discussion of how to assess the statistical significance of features in these measures is included. In addition, new work is presented which builds on the framework established in the review section. This work investigates how the estimates and their error bars are modified by finite sample sizes. Finite sample corrections are derived based on a doubly stochastic inhomogeneous Poisson process model in which the rate functions are drawn from a low variance Gaussian process. It is found that, in contrast to continuous processes, the variance of the estimators cannot be reduced by smoothing beyond a scale which is set by the number of point events in the interval. Alternatively, the degrees of freedom of the estimators can be thought of as bounded from above by the expected number of point events in the interval. Further new work describing and illustrating a method for detecting the presence of a line in a point process spectrum is also presented, corresponding to the detection of a periodic modulation of the underlying rate. This work demonstrates that a known statistical test, applicable to continuous processes, applies, with little modification, to point process spectra, and is of utility in studying a point process driven by a continuous stimulus. While the material discussed is of general applicability to point processes attention will be confined to sequences of neuronal action potentials (spike trains) which were the motivation for this work. 1 Keywords and phrases: Spectrum, coherency, coherence, multitaper, lag window, spike train analysis, point processes, finite size effects in spectra, doubly stochastic Poisson process 1

We present a method to distinguish direct connections between two neurons from common input originating from other, unmeasured neurons. The distinction is computed from the spike times of the two neurons in response to a white noise stimulus. Although the method is based on a highly idealized linear-nonlinear approximation of neural response, we demonstrate via simulation that the approach can work with a more realistic, integrate-and-fire neuron model. We propose that the approach exemplified by this analysis may yield viable tools for reconstructing stimulus-driven neural networks from data gathered in neurophysiology experiments. 1

We discuss cross-spectral analysis and report applications for the investigation of human tremors. For the physiological tremor in healthy subjects, the analysis enables to determine the resonant contribution to the oscillation and allows to test for a contribution of reflexes to this tremor. Comparing the analysis of the relation between the tremor of both hands in normal subjects and subjects with a rare abnormal organization of certain neural pathways proves the involvement of central structures in enhanced physiological tremor. The relation between the left and the right side of the body in pathological tremor shows a specific difference between orthostatic and all other forms of tremor. An investigation of EEG and tremor in patients suffering from Parkinson’s disease reveals the tremor-correlated cortical activity. Finally, the general issue of interpreting the results of methods designed for the analysis of bivariate processes when applied to multivariate processes is considered. We discuss and apply partial cross-spectral analysis in the frame of graphical models as an extention of bivariate cross-spectral analysis for the multivariate case.

Abstract—Electroencephalograms (EEGs) provide a noninvasive way of measuring brainwave activity from sensors placed on the scalp. In this paper we present an approach to measure coupling, or synchrony, between various parts of the brain, critical for motor and cognitive processing, using wavelet coherence of EEG signals. We provide an argument, highlighting the benefits of using this approach as opposed to the regular Fourier based coherence, in the context of localizing short significant bursts of coherence between non-stationary EEG signals, to which regular coherence is insensitive. We further highlight the benefits of the wavelets approach by exploring how a single time-frequency coherence map can be controlled to yield various time and/or frequency resolutions. I.

In this work spike trains of firing times of neurons recorded from various locations in the cat's auditory thalamus are studied. A goal is making inferences concerning connections amongst different regions of the thalamus in both the presence and the absence of a stimulus. Both second-order moment (frequency domain) and full likelihood analyses (a threshold crossing model), are carried through. 1 Introduction The sequence of spikes of a neuron, referred to as a "spike train", may carry important information processed by the brain and thus may underlie cognitive functions and sensory perception [1]. The data studied are recorded stretches of point processes corresponding to the firing times of Statistics Department, University of California, Berkeley y Institute of Physiology, University of Lausanne, Switzerland Pars dorsalis (D) Pars lateralis (PL) Pars magnocellularis (M) Auditory Cortex RE Input Figure 1: A block diagram of the auditory regions of the cat's brain. neurons mea...

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An extended difference of coherence test for comparing and combining several independent coherence estimates: theory and application to the study of motor units and physiological tremor

A review of recent applications of cross-correlation methodologies to human motor unit recording