Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Uncovering the dynamics of cardiac systems using stochastic pacing and frequency domain analyses

Alternans of cardiac action potential duration (APD) is a well-known arrhythmogenic mechanism which results from dynamical instabilities. The propensity to alternans is classically investigated by examining APD restitution and by deriving APD restitution slopes as predictive markers. However, experiments have shown that such markers are not always accurate for the prediction of alternans. Using a mathematical ventricular cell model known to exhibit unstable dynamics of both membrane potential and Ca2+ cycling, we demonstrate that an accurate marker can be obtained by pacing at cycle lengths (CLs) varying randomly around a basic CL (BCL) and by evaluating the transfer function between the time series of CLs and APDs using an autoregressive-moving-average (ARMA) model. The first pole of this transfer function corresponds to the eigenvalue (λalt) of the dominant eigenmode of the cardiac system, which predicts that alternans occurs when λalt≤−1. For different BCLs, control values of λalt were obtained using eigenmode analysis and compared to the first pole of the transfer function estimated using ARMA model fitting in simulations of random pacing protocols. In all versions of the cell model, this pole provided an accurate estimation of λalt. Furthermore, during slow ramp decreases of BCL or simulated drug application, this approach predicted the onset of alternans by extrapolating the time course of the estimated λalt. In conclusion, stochastic pacing and ARMA model identification represents a novel approach to predict alternans without making any assumptions about its ionic mechanisms. It should therefore be applicable experimentally for any type of myocardial cell.

Eluding oblivion with smart stochastic selection of synaptic updates

The variables involved in the equations that describe realistic synaptic dynamics always vary in a limited range. Their boundedness makes the synapses forgetful, not for the mere passage of time, but because new experiences overwrite old memories. The forgetting rate depends on how many synapses are modified by each new experience: many changes means fast learning and fast forgetting, whereas few changes means slow learning and long memory retention. Reducing the average number of modified synapses can extend the memory span at the price of a reduced amount of information stored when a new experience is memorized. Every trick which allows to slow down the learning process in a smart way can improve the memory performance. We review some of the tricks that allow to elude fast forgetting (oblivion). They are based on the stochastic selection of the synapses whose modifications are actually consolidated following each new experience. In practice only a randomly selected, small fraction of the synapses eligible for an update are actually modified. This allows to acquire the amount of information necessary to retrieve the memory without compromising the retention of old experiences. The fraction of modified synapses can be further reduced in a smart way by changing synapses only when it is really necessary, i.e. when the post-synaptic neuron does not respond as desired. Finally we show that such a stochastic selection emerges naturally from spike driven synaptic dynamics which read noisy pre and post-synaptic neural activities. These activities can actually be generated by a chaotic system.

Do males matter? The role of males in population dynamics

Models of population dynamics generally neglect the presence of males. While this assumption holds under many circumstances, behavioural ecology increasingly tells us that the presence (or absence) of males may have an impact on female fitness, and hence population sizes. Here we ask the question of whether males matter to population dynamics, operationally defined as a dependency of population growth on the relative density of males. We provide simple models, and evaluate the current empirical evidence for them, that illustrate various mechanisms of such male influence: mate searching behavior, male resource use (including effects of sexual dimorphism), sexual harassment and sexual segregation. In each case, theory predicts that males can have an effect on population densities, and in some extreme cases a positive feedback between an increasingly male-biased sex ratio and the effects on female harassment may theoretically even bring about population extinction. The results of this study, and the literature reviewed, show that the males can have a substantial effect on population dynamics, particularly so when human influences result in biased sex ratios.