The Jacobi diffusion process as a neuronal model

Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability

This article establishes cutoff stability also known as abrupt thermalization for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with (possibly degenerate) Lévy noise at fixed noise intensity. The results are based on several ergodicity quantitative lower and upper bounds some of which make use of the recently established shift linearity property of the Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such irregular systems like Jacobi chains and more general networks of coupled harmonic oscillators with a heat bath (including Lévy excitations) at constant temperature on the outer edges and the so-called Brownian gyrator.

Mean-reverting neuronal model based on two alternating patterns

A neuronal action potential model based on the generalised two-state Ornstein-Uhlenbeck process is studied. The model well describes all phases of a neuronal spike cycle, and the intrinsic parameters of the model have clear specification. Laplace transforms of a firing time are obtained explicitly. Formulae for the mean interspike intervals and their variances, as well as for the average duration of the relative refractory period are also obtained.

First-passage times and normal tissue complication probabilities in the limit of large populations

The time of a stochastic process first passing through a boundary is important to many diverse applications. However, we can rarely compute the analytical distribution of these first-passage times. We develop an approximation to the first and second moments of a general first-passage time problem in the limit of large, but finite, populations using Kramers–Moyal expansion techniques. We demonstrate these results by application to a stochastic birth-death model for a population of cells in order to develop several approximations to the normal tissue complication probability (NTCP): a problem arising in the radiation treatment of cancers. We specifically allow for interaction between cells, via a nonlinear logistic growth model, and our approximations capture the effects of intrinsic noise on NTCP. We consider examples of NTCP in both a simple model of normal cells and in a model of normal and damaged cells. Our analytical approximation of NTCP could help optimise radiotherapy planning, for example by estimating the probability of complication-free tumour under different treatment protocols.