A Traveling Wave Model for Invasion by Precursor and Differentiated Cells

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher’s equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.     

Travelling Waves in Hybrid Chemotaxis Models

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.     

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.     

Stability switches,oscillatory multistability,and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.     

Mathematical analysis of a model describing evolution of an asexual population in a changing environment

We investigate a mathematical model for an asexual population with non-overlapping (discrete) generations, that exists in a changing environment. Sexual populations are also briefly discussed at the end of the paper. It is assumed that selection occurs on the value of a single polygenic trait, which is controlled by a finite number of loci with discrete-effect alleles. The environmental change results in a moving fitness optimum, causing the trait to be subject to a combination of stabilising and directional selection.This model is different from that investigated by Waxman and Peck [Genetics 153 (1999) 1041] where overlapping generations and continuous effect alleles were considered. In this paper, we consider non-overlapping generations and discrete effect alleles. However in [Genetics 153 (1999) 1041] and the present work, there is the same pattern of environmental change, namely a constant rate of change of the optimum.From [Genetics 153 (1999) 1041], no rigorous theoretical conclusion can be drawn about the form of the solutions as t grows large. Numerical work carried out in [Genetics 153 (1999) 1041] suggests that the solution is a lagged travelling wave solution, but no mathematical proof exists for the continuous model. Only partial results, regarding existence of travelling wave solutions and perturbed solutions, have been established (see [Nonlin. Anal. 53 (2003) 683; An integral equation describing an asexual population in a changing environment, Preprint]). For the discrete case of this paper, under the assumption that the ratio between the unit of genotypic value and the speed of environment change is a rational number, we are able to give rigorous proof of the following conclusion: the population follows the environmental change with a small lag behind, moreover, the lag is represented using a calculable quantity.     

A continuum model for root growth

This paper presents a study of a simple one-dimensional continuum model for growth of the plant root. A fundamental constitutive equation is derived. The model is studied by means of various special cases of increasing complexity. Asymptotic expansions are used to derive approximate solutions to the equation of the model under the fundamental assumption that cell wall thickness is small in comparison with the diameter of the cell. The basic results of the study may be summarized as follows. The observed growth pattern of the root cannot be modelled by a mechanical system whose properties are independent of position on the root. The observed pattern can be modelled by a simple mechanical system in which, for example, cell wall yield stress first decreases and then increases. Two fundamental observations are made based on the modelling study. The first is that any mechanical model must take into account the convective displacement from the tip of points along the root. The second is that in describing growth, data on cell wall mechanical properties are meaningless without corresponding data on cell water potential, and vice versa.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

Simulation of the water distribution in soil

Summary A mathematical model is described which enables the flow of water, the water content and the matric potential to be calculated for any depth in an uncropped soil. It is based on a numerical solution to the flow equation and requires data describing the weather and the soil hydraulic properties. The measurement of these properties is described and the simulated water contents and matric potentials obtained from their use in the model are compared with those measured in a field experiment.     

一个造血模型的概周期正解

应用Ｓｃｈａｕｄｅｒ不动点定理,研究了一个造血模型概周期正解的存在性及唯一性。     

Inhibited enzyme electrodes. Part 1: Theoretical model

A theoretical model is developed for an electrochemical sensor for toxic substances which works by measuring the inhibition of the enzyme activity. The enzyme is assumed to follow Michaelis-Menten kinetics and the diffusion kinetic equation describing the concentration profile of the enzyme's substrate in the electrolyte layer between the electrode and the membrane covering the electrode is solved. A complete set of analytical solutions is found which corresponds to a number of different rate limiting processes. The set of solutions is described in a case diagram. The use of cytochrome oxidase in particular is discussed.     

Current flow and potential in a three-dimensional syncytium

Analytical solutions are obtained describing the spread of potential in a continuous syncytium which is the three-dimensional analogue of the “cable” and “thin sheet” models. Two discrete (finite-element) models are also developed and their behaviour shown to agree with that of the continuous model. All three models are able to account for the peculiarities of passive responses to intracellular current injection in tissues such as cardiac and smooth muscle. The time course and spread of junction potentials in electrically coupled tissues is discussed.     

One-dimensional food chain equations and their generalization

Two equations describing one-dimensional food chains are known to possess soliton solutions. It is demonstrated that both equations are embraced within another equation, which arises in the theory of chains of enzymic reactions. We find an elliptic function solution to this equation. We obtain a one-soliton solution from it and re-derive the elliptic function solutions of the two ecological equations.     

Continuous Modeling of Arterial Platelet Thrombus Formation Using a Spatial Adsorption Equation

In this study, we considered a continuous model of platelet thrombus growth in an arteriole. A special model describing the adhesion of platelets in terms of their concentration was derived. The applications of the derived model are not restricted to only describing arterial platelet thrombus formation; the model can also be applied to other similar adhesion processes. The model reproduces an auto-wave solution in the one-dimensional case; in the two-dimensional case, in which the surrounding flow is taken into account, the typical torch-like thrombus is reproduced. The thrombus shape and the growth velocity are determined by the model parameters. We demonstrate that the model captures the main properties of the thrombus growth behavior and provides us a better understanding of which mechanisms are important in the mechanical nature of the arterial thrombus growth.     

Diffusion approximation and first passage time problem for a model neuron

A diffusion equation for the transition p.d.f. describing the time evolution of the membrane potential for a model neuron, subjected to a Poisson input, is obtained, without breaking up the continuity of the underlying random function. The transition p.d.f. is calculated in a closed form and the average firing interval is determined by using the steady-state limiting expression of the transition p.d.f. The Laplace transform of the first passage time p.d.f. is then obtained in terms of Parabolic Cylinder Functions as solution of a Weber equation, satisfying suitable boundary conditions. A continuous input model is finally investigated.     

Growth of yeast Candida utilis in crotonaldehyde containing media

Summary The inhibitory influence of the higher concentration of 20butenal, crotonaldehyde was followed during the batch and long-term continuous fermentation of Candida utilis growing on synthetic ethanol. Most crotonaldehyde is removed from the medium by biotransformation. Crotonaldehyde inhibits the growth, lengthens the lag phase and decreases the biomass yield and the content of crude proteins in the biomass. The yeast C. utilis is capable of growing on media containing very high concentrations of inhibitor in the in-flow during continuous cultivation. Uncharacteristic transport oscillations of the content of crotonaldehyde were observed for which acidic groups on the cell membrane are probably responsible. A sensitive method which is suitable for measuring very low concentration of crotonaldehyde in aqueous solutions is described. Crotonaldehyde acts as an uncompetitive inhibitor with slight mixed type of inhibition. An equation describing the kinetics of inhibition was derived.     

A one locus, biased mutation model and its equivalence to an unbiased model

Experimental data suggests that for some continuously-varying characters under stabilising selection, mutation may cause a mean change in the value of the character. A one locus, mathematical model of a continuously-varying biological character with this property of biased mutation is investigated. Via a mathematical transformation, the equilibrium equation describing a large population of individuals is reduced to the equilibrium equation describing a mutationally unbiased problem. Knowledge of an unbiased problem is thus sufficient to determine all equilibrium properties of the corresponding biased problem. In the biased mutation problem, the dependence of the mean equilibrium value of the character, as a function of the mutational bias, is non-monotonic and remains small, for all levels of mutational bias. The analysis presented in this work sheds new light on Turelli's House of Cards Approximation.     

Equilibrium solutions of age-specific population dynamics of several species

Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L ¹ of the dynamics and for existence as well as for stability of equilibrium solutions are given.     

Cable Theory for Finite Length Dendritic Cylinders with Initial and Boundary Conditions

The cable equation is solved in the Laplace transform domain for arbitrary initial and boundary conditions. The cable potential is expressed directly in terms of the impedance of the terminations and the cable electrotonic length. A computer program is given to invert the transform. Numerical solutions may be obtained for any particular model by inserting expressions describing the terminations and parameter values into the program, without further computation by the modeler. For a finite length cable, sealed at one end, the solution is expressed in terms of the ratio of the termination impedance to the impedance of the finite length cable, a generalization of the steady-state conductance ratio. Analysis of a model of a soma with several primary dendrites shows that the dendrites may be lumped into one equivalent cylinder if they have the same electrotonic length, even though they may vary in diameter. Responses obtained under voltage clamp are conceptually predictable from measurements made under current clamp, and vice versa. The equalizing time constants of an infinite series expression of the solution are the negative reciprocals of the roots of the characteristic equation. Examination of computed solutions shows that solutions which differ theoretically may be indistinguishable experimentally.     

A stochastic model of bacterial spore germination

A stochastic approach is utilized to develop a model equation capable of describing the time course of germination in a sample of bacterial spores. The time required by a spore to complete the change characteristic of germination consists of an initial interval of no change followed immediately by the duration of the change itself. The experimental basis of the proposed model is the observation that each of these time intervals is distributed over a range of values in a spore sample. Mixed continuous and discrete probabilities are employed in arriving at an average single-spore germination curve which, to a different scale, describes the sample in time.