David R. Brillinger

Time series: data analysis and theory

This book will be most useful to applied mathematicians, communication engineers, signal processors, statisticians, and time series researchers, both applied and theoretical. Readers should have some background in complex function theory and matrix algebra and should have successfully completed the equivalent of an upper division course in statistics.

The Fourier approach to the identification of functional coupling between neuronal spike trains

I. I N T R O D U C T I O N The study of the behaviour of small networks of neurones frequently requires the determination of measures of the strength of association between component neurones, an assessment of their timing relations, and the identification of which neurones may interact directly or are influenced by common inputs. In many of these studies the principal quantities available for analysis are the sequences of extracellularly recorded action potentials (neuronal spike trains). The subsequent analytical work is then based entirely on the relations between the times of occurrence of the action potentials recorded from different neurones. In these circumstances neuronal spike trains are frequently represented as the mathematical entity known as a stochastic point process. These processes are described by providing a probability law for a set of ordered times

Maximum Likelihood Analysis of Spike Trains of Interacting Nerve Cells

Suppose that a neuron is firing spontaneously or that it is firing under the influence of other neurons. Suppose that the data available are the firing times of the neurons present. An “integrate several inputs and fire” model is developed and studied empirically. For the model a neurons firing occurs when an internal state variable crosses a random threshold. This conceptual model leads to maximum likelihood estimates of internal quantities, such as the postsynaptic potentials of the measured influencing neurons, the membrane potential, the absolute threshold and also estimates of derived quantities such as the strength-duration curve and the recovery process of the threshold. The models validity is examined via an estimate of the conditional firing probability. The approach appears useful for estimating biologically meaningful parameters, for examining hypotheses re these parameters, for understanding the connections present in neural networks and for aiding description and classification of neurons and synapses. Analyses are presented for a number of data sets collected for the sea hare,Aplysia californica, by J. P. Segundo. Both excitatory and inhibitory examples are provided. The computations were carried out via the Glim statistical package. An example of a Glim program realizing the work is presented in the Appendix.

A Biometrics Invited Paper with Discussion: The Natural Variability of Vital Rates and Associated Statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.

The Spectral Analysis of Stationary Interval Functions

We consider stationary. additive. interval functions X(Δ). These are vector valued stochastic processes having real intervals Δ = (α, β] as domain, having finite dimensional distributions invariant under time translation and satisfying

Identification of synaptic interactions

This paper studies the influence exerted by the presynaptic spike train on the postsynaptic one. It applies to synaptic exploration a novel method for characterization of point-process systems (Brillinger, 1974, 1975a), and draws from it physiologically meaningful conclusions. The departure point was a large data set of action potential trains from an Aplysia network whose neurons are connected by monosynaptic inhibitory or excitatory PSPs, and either discharged spontaneously or were driven by intracellular pulses. First, a sequence of “kernels” is estimated, each with a physiological connotation relevant to synaptic transmission. The kernel independent of time — of zero-order — measures the postsynaptic rate with no presynaptic discharge. That of a single time argument — of first-order — relates to the rate effect of the average PSP. Those of two, three, or more time arguments — of second, third or higher-order — relate to interactions between two, three, or more postsynaptic potentials (e.g. to facilitation) and/or spikes (e.g. to refractoriness). Then successive models are constructed recursively and based on the kernel of zero-order, on the kernels of zero and first order, on those of zero, first and second order, and so forth, until a desired approximation is achieved. The plausibilities of each kernel estimate and of each model are evaluated separately by way of spectra and coherences. The “linear” model based upon the zero and first-order kernel was tested (after that based exclusively on the zero-order one was proven inadequate). When presynaptic discharges are very irregular and at intermediate or low rates, it provides satisfactory description and prediction, and the first-order kernel is an uncontaminated display of the rate effects of the average presynaptic spike: this constitutes the “linear” domain. When presynaptic discharges are bursty, regular or very fast, the linear model is unsatisfyctory: this is referred to as “non-linear” domain. Reasons for non-linearity lie in PSP facilitation and anti-facilitation, conversion of membrane current into firing rate, after-spike excitability oscillations, and special pacemaker interactions. The model can be extended to three-neuron networks where partial coherences exract interactions between followers, even while submitted to a common driver. The basic and ubiquitous issues of spike train description and stability were discussed. The counting and the interval statistic of spike trains provide equivalent descriptions and their current opposition is conceptually meaningless. Concomitant short-term fluctuations in spike generation intensity at preand postsynaptic levels have funciional significance beyond changes in the overall average rate or interval: they are made precise by parameters whose definition, estimation and physiological interpretation are presented here. Some stability of the experimental preparation is presupposed by investigators, but variations (e.g. from cycles or deterioration) always exist. Hence, decisions as to the preparations evolution and as to tolerable changes must be made, and based upon pre-existing knowledge, educated guesses and practical considerations. This study provided basic knowledge of the individual synapse considered the elementary building block of the nervous system when viewed as a network of interacting nerve cells. It also contributed generally applicable mathematical techniques which were illustrated by application to relatively well studied and simple networks.

The Identification of Point Process Systems

and say that the point process N is the output of the system &`operating on the input process M. We write M(A) to denote the measure of the time interval A for a realization of the input process and N(A) the corresponding measure for N. In practice M(A) refers to the number of occurrences in A of some phenomenon of interest and N(A) to the corresponding number of occurrences of some second phenomenon. We illustrate with two examples, one specific, the other more vague.

Nerve Cell Spike Train Data Analysis: A Progression of Technique

Abstract Collections of occurrence times of events taking place irregularly in time provide a fairly common, but not broadly discussed, data type. This article is concerned with the particular circumstance of firing times in nerve cells that interact and form networks. The article reviews a progression of statistical analysis techniques: description, association as measured by moments and correlation, regression, and finally likelihood. The data is point process, but may be seen as that of regression and of multivariate analysis in standard parlance. A simple description of data collected simultaneously for one or more cells is provided.

Statistical Inference for Stationary Point Processes

This work is divided into three principal sections which also correspond to the three lectures given at Bloomington. The topics cover, some useful point process parameters and their properties, estimation of time domain parameters and the estimation of freq1.1ence domain parameters.

Fourier analysis of stationary processes

This paper begins with a description of some of the important procedures of the Fourier analysis of real-valued stationary discrete time series. These procedures include the estimation of the power spectrum, the fitting of finite parameter models, and the identification of linear time invariant systems. Among the results emphasized is the one that the large sample statistical properties of the Fourier transform are simpler than those of the series itself. The procedures are next generalized to apply to the cases of vector-valued series, multidimensional time series or spatial series, point processes, random measures, and finally to stationary random Schwartz distributions. It is seen that the relevant Fourier transforms are evaluated by different formulas in these further cases, but that the same constructions are carried out after their evaluation and the same statistical results hold. Such generalizations are of interest because of current work in the fields of picture processing and pulse-code modulation.