Derivation of a field equation of brain activity

We present a nonlinear field theory of the brain under realistic anatomical connectivity conditions describing the interaction between functional units within the brain. This macroscopic field theory is derived from the quasi-microscopic conversion properties of neural populations occurring at synapses and somas. Functional units are treated as inhomogeneities within a nonlinear neural tissue.     

Aerial Sorocarp Development By the Aggregative Ciliate,Sorogena Stoianovitchae*

SYNOPSIS. Sorogena stoianovitchae Bradbury & Olive, an epiphytic ciliate found in various parts of the world, has a trophic stage that feeds on members of the ciliate genus Colpoda. When grown in the presence of the food ciliate, it multiplies rapidly. When the cells become abundant they aggregate at the water surface on inserted plant fragments or floating pollen grains, the sides of culture dishes, or on floating films such as those deposited by bacteria or pollen grains. an aggregate mounds up and becomes ensheathed above the water level, after which the mass of cells called a sorogen rises aerially at the apex of a stalk deposited at its base. the tapering, noncellular stalk consists of a conspicuously furrowed sheath that encloses a mucilaginous matrix. At completion of stalk development the cells of the sorogen become encysted. the sorocysts are commonly discharged by fracturing of the drying sorus. Alternating light and dark conditions are required for sorocarp development.     

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z < 0 and a boundary layer for z > 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions. Supported by a National Science Foundation graduate fellowship. This work was performed under National Science Foundation grant DMS-9024963 and Air Force Office of Scientific Research grant AFOSR-F49620-94-1-0044.     

A new mathematical model for the homeostatic effects of sleep loss on neurobehavioral performance

The two-process model of sleep regulation makes accurate predictions of sleep timing and duration for a variety of experimental sleep deprivation and nap sleep scenarios. Upon extending its application to waking neurobehavioral performance, however, the model fails to predict the effects of chronic sleep restriction. Here we show that the two-process model belongs to a broader class of models formulated in terms of coupled non-homogeneous first-order ordinary differential equations, which have a dynamic repertoire capturing waking neurobehavioral functions across a wide range of wake/sleep schedules. We examine a specific case of this new model class, and demonstrate the existence of a bifurcation: for daily amounts of wakefulness less than a critical threshold, neurobehavioral performance is predicted to converge to an asymptotically stable state of equilibrium; whereas for daily wakefulness extended beyond the critical threshold, neurobehavioral performance is predicted to diverge from an unstable state of equilibrium. Comparison of model simulations to laboratory observations of lapses of attention on a psychomotor vigilance test (PVT), in experiments on the effects of chronic sleep restriction and acute total sleep deprivation, suggests that this bifurcation is an essential feature of performance impairment due to sleep loss. We present three new predictions that may be experimentally verified to validate the model. These predictions, if confirmed, challenge conventional notions about the effects of sleep and sleep loss on neurobehavioral performance. The new model class implicates a biological system analogous to two connected compartments containing interacting compounds with time-varying concentrations as being a key mechanism for the regulation of psychomotor vigilance as a function of sleep loss. We suggest that the adenosinergic neuromodulator/receptor system may provide the underlying neurobiology.     

Aggregative Root Placement: A Feature During Interspecific Competition in Inland Sand-Dune Habitats

Segregation of roots is frequently observed in competing root systems. However, recently, intensified root growth in response to a neighbouring plant has been described in pot experiments [Gersani M, Brown J S, O'Brien E E, Maina G M and Abramsky Z 2001. J. Ecol. 89, 660–669]. This paper examines whether intense root growth towards a neighbour (aggregation) plays a role in competitive interactions between plant species from open nutrient-poor mid-European sand ecosystems. In a controlled field-competition experiment, root distribution patterns of intra- and interspecific pairs as well as single control plants of Corynephorus canescens, Festuca psammophila, Hieracium pilosella, Hypochoeris radicata and Conyza canadensis were investigated after one growing season. Under intraspecific competition plants tended to segregate their root systems, while under interspecific competition most species tended to aggregate roots towards their neighbours even at the expense of root development at the opposite competition-free side of the target. Preference of a root aggregation strategy over the occupation of competition-free soil in interspecific competition emphasizes the importance of contesting between individuals in relation to mere resource acquisition. It is suggested that in the presence of a competitor the plants might use root aggregation as a defensive reaction to maintain a strong competitive response and exclusive access to the resources of already occupied soil volumes.     

A population balance equation model of aggregation dynamics in Taxus suspension cell cultures

The nature of plant cells to grow as multicellular aggregates in suspension culture has profound effects on bioprocess performance. Recent advances in the measurement of plant cell aggregate size allow for routine process monitoring of this property. We have exploited this capability to develop a conceptual model to describe changes in the aggregate size distribution that are observed over the course of a Taxus cell suspension batch culture. We utilized the population balance equation framework to describe plant cell aggregates as a particulate system, accounting for the relevant phenomenological processes underlying aggregation, such as growth and breakage. We compared model predictions to experimental data to select appropriate kernel functions, and found that larger aggregates had a higher breakage rate, biomass was partitioned asymmetrically following a breakage event, and aggregates grew exponentially. Our model was then validated against several datasets with different initial aggregate size distributions and was able to quantitatively predict changes in total biomass and mean aggregate size, as well as actual size distributions. We proposed a breakage mechanism where a fraction of biomass was lost upon each breakage event, and demonstrated that even though smaller aggregates have been shown to produce more paclitaxel, an optimum breakage rate was predicted for maximum paclitaxel accumulation. We believe this is the first model to use a segregated, corpuscular approach to describe changes in the size distribution of plant cell aggregates, and represents an important first step in the design of rational strategies to control aggregation and optimize process performance.     

一类高阶非线性中立型时滞差分方程正解的存在性定理

研究了一类高阶非线性中立型时滞差分方程正解的存在性,给出了该类方程存在有界最终正解的一个充要条件。     

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfs-hading effect.We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation.     

Non-local models for the formation of hepatocyte-stellate cell aggregates

Liver cell aggregates may be grown in vitro by co-culturing hepatocytes with stellate cells. This method results in more rapid aggregation than hepatocyte-only culture, and appears to enhance cell viability and the expression of markers of liver-specific functions. We consider the early stages of aggregate formation, and develop a new mathematical model to investigate two alternative hypotheses (based on evidence in the experimental literature) for the role of stellate cells in promoting aggregate formation. Under Hypothesis 1, each population produces a chemical signal which affects the other, and enhanced aggregation is due to chemotaxis. Hypothesis 2 asserts that the interaction between the two cell types is by direct physical contact: the stellates extend long cellular processes which pull the hepatocytes into the aggregates. Under both hypotheses, hepatocytes are attracted to a chemical they themselves produce, and the cells can experience repulsive forces due to overcrowding. We formulate non-local (integro-partial differential) equations to describe the densities of cells, which are coupled to reaction-diffusion equations for the chemical concentrations. The behaviour of the model under each hypothesis is studied using a combination of linear stability analysis and numerical simulations. Our results show how the initial rate of aggregation depends upon the cell seeding ratio, and how the distribution of cells within aggregates depends on the relative strengths of attraction and repulsion between the cell types. Guided by our results, we suggest experiments which could be performed to distinguish between the two hypotheses.     

Activation waves in a model of platelet aggregation: existence of solutions and stability of travelling fronts

Platelets cohere to one another to form platelet aggregates as part of the blood's clotting response. The ability of a platelet to participate in this process depends on its prior activation by chemicals released into the blood plasma by other activated platelets. We study the piecewise-linear system of reaction-diffusion equations which, in one spatial dimension, describe the chemically-mediated spread of platelet activation. We establish the existence of classical solutions to this system of equations, and show that these solutions do not blow up in finite time. We also explicitly construct travelling front solutions and discuss their stability. Finally, we present numerical evidence which suggests that for a broad range of initial data with the correct limiting values at ± , the solution to the initial value problem rapidly evolves into the travelling front solution provided the front is linearly stable.     

Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread

Summary In this paper we use Aronson's and Weinberger's [1–4] concept of asymptotic speed to estimate the asymptotic behaviour of the solution of a nonlinear integral equation (with the nonlinearity not being monotone), which describes the development of a spatially distributed population.     

Averaging methods in the delayed-logistic equation

The delayed logistic equation is analyzed using the averaging method. Using the transformation of coordinates v=ln N/K it is shown that the first order term in perturbation theory yields N=K exp(r ^* cos t/2) when the delay time T exceeds some critical value T _c. The amplitude r^* is equal to (40/3 – 2)^1/2 and is an expansion parameter that is proportional to (T – T_c). Comparison of the exponential solution of N and numerical results for the ratio N _maximum/N _minimum provides a good fit for values of larger than the results using the N coordinate as the perturbed coordinate.     

Discontinuous Galerkin method for piecewise regular solutions to the nonlinear age-structured population model

A discontinuous Galerkin approximation of the nonlinear Lotka-McKendrick equation is considered in the frequent case when the solution is only piecewise regular. An O(h(r+1/2)) error estimate for rth order polynomial finite elements is proved, as well as piecewise H(1)-regularity of the exact solution which guarantees the error estimate for r=0. Certain implementational details which improve the robustness of the method are also addressed.     

Derivation and analysis of a system modeling the chemotactic movement of hematopoietic stem cells

It has been shown that hematopoietic stem cells migrate in vitro and in vivo following the gradient of a chemotactic factor produced by stroma cells. In this paper, a quantitative model for this process is presented. The model consists of chemotaxis equations coupled with an ordinary differential equation on the boundary of the domain and subjected to nonlinear boundary conditions. The existence and uniqueness of a local solution is proved and the model is simulated numerically. It turns out that for adequate parameter ranges, the qualitative behavior of the stem cells observed in the experiment is in good agreement with the numerical results. Our investigations represent a first step in the process of elucidating the mechanism underlying the homing of hematopoietic stem cells quantitatively.      

The spread of a parasitic infection in a spatially distributed host population

Starting from a stochastic model for the spread of a parasitic infection in a spatially distributed host population we describe the way to a continuum formulation by a deterministic model in terms of a nonlinear partial differential equation and an integro-differential equation. The hosts are assumed to occupy fixed spatial positions, whereas the parasites are mobile, however can propagate only within the hosts. To perform the continuum limit we suppose that the size N _h of the host population, the size N _p of the parasite population, and the ratio N _p /N _h tend to infinity. Accordingly, the parameters determining the time evolution of the host and parasite populations are rescaled suitably.Parts of this work have been elaborated during a stay at the Institute of Applied Mathematics at the University of Zürich