First-passage times in integrate-and-fire neurons with stochastic thresholds.

We consider a leaky integrate-and-fire neuron with deterministic subthreshold dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process. The formulation of this minimal model is motivated by the experimentally observed widespread variation of neural firing thresholds. We show numerically that the mean first-passage time can depend nonmonotonically on the noise amplitude. For sufficiently large values of the correlation time of the stochastic threshold the mean first-passage time is maximal for nonvanishing noise. We provide an explanation for this effect by analytically transforming the original model into a first-passage-time problem for Brownian motion. This transformation also allows for a perturbative calculation of the first-passage-time histograms. In turn this provides quantitative insights into the mechanisms that lead to the nonmonotonic behavior of the mean first-passage time. The perturbation expansion is in excellent agreement with direct numerical simulations. The approach developed here can be applied to any deterministic subthreshold dynamics and any Gauss-Markov processes for the firing threshold. This opens up the possibility to incorporate biophysically detailed components into the subthreshold dynamics, rendering our approach a powerful framework that sits between traditional integrate-and-fire models and complex mechanistic descriptions of neural dynamics.

A continuum model of cell proliferation and nutrient transport in a perfusion bioreactor.

Tissue engineering aims to regenerate, repair or replace organs or defective tissues. This tissue regeneration often occurs in a bioreactor. Important challenges in tissue engineering include ensuring adequate nutrient supply, maintaining the desired cell distribution and achieving sufficiently high cell yield. To put laboratory experiments into a theoretical framework, mathematical modelling of the physical and biochemical processes involved in tissue growth is a useful tool. In this work, we derive and solve a model for a cell-seeded porous scaffold placed in a perfusion bioreactor in which fluid delivers nutrients to the cells. The model describes the key features, including fluid flow, nutrient delivery, cell proliferation and consequent variation of scaffold porosity. Fluid flow through the porous scaffold is modelled by Darcy's law, and nutrient delivery is described by a reaction-advection-diffusion equation. A reaction-diffusion equation describes the evolution of cell density, in which cell proliferation is modelled via logistic growth and cell spreading via non-linear diffusion, which depends on cell density. The effect of shear stress on nutrient consumption and cell proliferation is also included in the model. COMSOL (a commercial finite element solver) is used to solve the model numerically. The results reveal the dependence of the cell distribution and total cell yield on the initial cell density and scaffold porosity. We suggest various seeding strategies and scaffold designs to improve the cell distribution and total cell yield in the engineered tissue construct.

Nikolaevskiy equation with dispersion.

The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral "Goldstone" mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilize some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram ("Busse balloon") for the traveling waves can be remarkably complicated.