An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach

We summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity (an integer multiple of some basic unit size). In particular, we discuss Smoluchowski's equation for aggregation, the Becker-Döring model of simultaneous aggregation and fragmentation, and more general models involving coagulation and fragmentation.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular we are able to treat "patchy'" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs. Article published and (c) American Physical Society 2007

Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modelled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. Here we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded in a simple one-dimensional cable. This system is shown to support saltatory waves as a result of the discrete distribution of spines. Moreover, we demonstrate one of the ways to incorporate noise into the spine-head whilst retaining computational tractability of the model. The SDS model sustains a variety of propagating patterns.

Bayesian alignment of continuous molecular shapes using random fields

Statistical methodology is proposed for comparing molecular shapes. In order to account for the continuous nature of molecules, classical shape analysis methods are combined with techniques used for predicting random fields in spatial statistics. Applying a modification of Procrustes analysis, Bayesian inference is carried out using Markov chain Monte Carlo methods for the pairwise alignment of the resulting molecular fields. Superimposing entire fields rather than the configuration matrices of nuclear positions thereby solves the problem that there is usually no clear one--to--one correspondence between the atoms of the two molecules under consideration. Using a similar concept, we also propose an adaptation of the generalised Procrustes analysis algorithm for the simultaneous alignment of multiple molecular fields. The methodology is applied to a dataset of 31 steroid molecules.

Frequency of elemental events of intracellular Ca²⁺ dynamics

The dynamics of intracellular Ca²⁺ is driven by random events called Ca²⁺ puffs, in which Ca²⁺ is liberated from intracellular stores. We show that the emergence of Ca²⁺ puffs can be mapped to an escape process. The mean first passage times that correspond to the stochastic fraction of puff periods are computed from a novel master equation and two Fokker-Planck equations. Our results demonstrate that the mathematical modeling of Ca²⁺ puffs has to account for the discrete character of the Ca²⁺ release sites and does not permit a continuous description of the number of open channels.

Optimisation of a crossdocking distribution centre simulation model

This paper reports on continuing research into the modelling of an order picking process within a Crossdocking distribution centre using Simulation Optimisation. The aim of this project is to optimise a discrete event simulation model and to understand factors that affect finding its optimal performance. Our initial investigation revealed that the precision of the selected simulation output performance measure and the number of replications required for the evaluation of the optimisation objective function through simulation influences the ability of the optimisation technique. We experimented with Common Random Numbers, in order to improve the precision of our simulation output performance measure, and intended to use the number of replications utilised for this purpose as the initial number of replications for the optimisation of our Crossdocking distribution centre simulation model. Our results demonstrate that we can improve the precision of our selected simulation output performance measure value using Common Random Numbers at various levels of replications. Furthermore, after optimising our Crossdocking distribution centre simulation model, we are able to achieve optimal performance using fewer simulations runs for the simulation model which uses Common Random Numbers as compared to the simulation model which does not use Common Random Numbers.

Investigating output accuracy for a discrete event simulation model and an agent based simulation model

In this paper, we investigate output accuracy for a Discrete Event Simulation (DES) model and Agent Based Simulation (ABS) model. The purpose of this investigation is to find out which of these simulation techniques is the best one for modelling human reactive behaviour in the retail sector. In order to study the output accuracy in both models, we have carried out a validation experiment in which we compared the results from our simulation models to the performance of a real system. Our experiment was carried out using a large UK department store as a case study. We had to determine an efficient implementation of management policy in the store’s fitting room using DES and ABS. Overall, we have found that both simulation models were a good representation of the real system when modelling human reactive behaviour.