Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion

We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller’s boundary classification. We compare the frequently used Euler–Maruyama and Milstein methods, two Lie–Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong–Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler–Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler–Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie–Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features

Variability Measures of Positive Random Variables

During the stationary part of neuronal spiking response, the stimulus can be encoded in the firing rate, but also in the statistical structure of the interspike intervals. We propose and discuss two information-based measures of statistical dispersion of the interspike interval distribution, the entropy-based dispersion and Fisher information-based dispersion. The measures are compared with the frequently used concept of standard deviation. It is shown, that standard deviation is not well suited to quantify some aspects of dispersion that are often expected intuitively, such as the degree of randomness. The proposed dispersion measures are not entirely independent, although each describes the interspike intervals from a different point of view. The new methods are applied to common models of neuronal firing and to both simulated and experimental data

Efficient Olfactory Coding in the Pheromone Receptor Neuron of a Moth

The concept of coding efficiency holds that sensory neurons are adapted, through both evolutionary and developmental processes, to the statistical characteristics of their natural stimulus. Encouraged by the successful invocation of this principle to predict how neurons encode natural auditory and visual stimuli, we attempted its application to olfactory neurons. The pheromone receptor neuron of the male moth Antheraea polyphemus, for which quantitative properties of both the natural stimulus and the reception processes are available, was selected. We predicted several characteristics that the pheromone plume should possess under the hypothesis that the receptors perform optimally, i.e., transfer as much information on the stimulus per unit time as possible. Our results demonstrate that the statistical characteristics of the predicted stimulus, e.g., the probability distribution function of the stimulus concentration, the spectral density function of the stimulation course, and the intermittency, are in good agreement with those measured experimentally in the field. These results should stimulate further quantitative studies on the evolutionary adaptation of olfactory nervous systems to odorant plumes and on the plume characteristics that are most informative for the ‘sniffer’. Both aspects are relevant to the design of olfactory sensors for odour-tracking robots