Theorem. If F is a df on [0,1], that is strictly increasing, and such that F(x) > x if O < x < ?, and F(x) < x if < x < 1, then F is not n-divisible for any n. There are similar, less simple results for df's with unbounded support. As a special case it follows that the uniform distribution is not n-divisible for any n though it is decomposable in many ways (see e.g., [2]). Kudina [1] uses a similar technique to prove the indecomposability of the arcsine df. It is well known that id df's with bounded support have zero variance, i.e., are degenerate. Similarly a df on [0,a] with variance exceeding a/(4n) is not ndivisible. This argument is extended to prove the following theorem (cf. [4], also for references).

Stochastic models for single neuron firing trains: a survey.

Changes in membrane potential of a neuron are, under certain experimental conditions, manifested by spike discharges. The number of spikes recorded from the neuron within a given period represents the information which the neuron conveys; and the message, thus transmitted, is carried by the pattern of spikes and the intervals between spikes. This can be modeled by a stochastic point process. This article reviews in three parts recent developments of the statistical analysis of discharge patterns of spike trains recorded from individual nerve cells, and suggests several methods for identifying certain types of spike trains. The first part reviews those stochastic processes Y ( t ) used as models for the membrane potential changes. The occurence time of a spike is defined as a first-passage time of Y ( t ) upcrossing a threshold value. These upcrossing times form a point process describing the occurences of spikes. Evidently, the occurence rate depends on the underlying assumption of Y ( t ). Presently, a majority of the point process models adopt the renewal-process assumptions. The second part deals with statistical inferences on the models and the validation methods. The research in this area is scarce; the inference on renewal processes is based mostly on Cox and Lewis's book The Statistical Analysis of Series of Events . We have used the contiguity theory due to Le Cam [30] to derive several asymptotically optimal statistical procedures. The last part discusses some popular statistical techniques employed and some areas needed for further investigation.

On statistical methods in neuronal spike-train analysis

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Stochastic analysis of multivariate point processes

A class of stationary multivariate point processes is considered in which the events of one of the point processes act as regeneration points for the entire multivariate process. Some important properties of such processes are derived including the joint probability generating function for numbers of events in an interval of fixed length and the asymptotic behaviour of such processes. The general theory is then applied in three bivariate examples. Finally, some simple monotonicity results for stationary and renewal point processes (which are used in the second example) are proved in two appendices.

Regenerative multivariate point processes

Stochastic processes in neurophysiology: transformation from point to continuous processes.

This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.

A. J. Lawrance

Papers

Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses