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We consider the model of random evolution on the real line consisting in a Brownian motion perturbed by alternating jumps. We give the probability density of the process and pinpoint a connection with the limit density of a telegraph process subject to alternating jumps. We study the first-crossing-time probability in two special cases, in the presence of a constant upper boundary.

We propose the empirical cumulative entropy as a variability measure suitable to describe the information content in neuronal firing data. Some useful characteristics and an application to a real dataset are also discussed.

A stochastic model for the firing activity of a neuronal unit has been recently proposed in [4]. It includes the decay effect of the membrane potential in the absence of stimuli, and the occurrence of excitatory inputs driven by a Poisson process. In order to add the effects of inhibitory stimuli, we now propose a Stein-type model based on a suitable exponential transformation of a bilateral birth-death process on ${\mathbb Z}$ and characterized by state-dependent nonlinear birth and death rates. We perform an analysis of the probability distribution of the stochastic process describing the membrane potential and make use of a simulation-based approach to obtain some results on the firing density.

The cumulative entropy is a new measure of information, alternative to the classical differential entropy. It has been recently proposed in analogy with the cumulative residual entropy studied by Wang et al. (2003a) and (2003b). After recalling its main properties, including a connection to reliability theory, we discuss estimates of random lifetimes based on the empirical cumulative entropy, which is suitably expressed in terms of the dual normalized sample spacings.

Motivated by applications in mathematical biology concerning randomly alternatingmotion ofmicro-organisms, we analyze a generalized integrated telegraph process. The random times between consecutive velocity reversals are gamma-distributed, and perform an alternating renewal process. We obtain the probability law and the mean of the process.