Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory

A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t=±. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.Supported in part by NSF Grant MCS8301724 and by a Biomedical Research Support Grant 2SO7RR0706618 from NIH     

图论在中国慈姑属数量分类研究中的应用

慈姑属(Sagittaria L.)是水生单子叶植物泽泻科(Alismataceae)的一个重要属。近年来,陈家宽等对中国慈姑属进行了系统研究。作者应用聚类分析和主成分分析进行了中国慈姑属的数量分类,取得了良好的效果。由于慈姑属分类性状数目较大,因而在压缩后的三维空间中描述分类群间的距离,     

胡枝子属根瘤菌的多相分类研究

采用了数值分类、全细胞可溶性蛋白电泳分析、ＤＮＡＧ＋Ｃｍｏｌ％和ＤＮＡ相关性的测定以及１６ＳｒＤＮＡＰＣＲＲＦＬＰ分析等多相分类技术对来源于不同地区的１６种寄主的胡枝子根瘤菌进行了系统的分类研究。数值分类的结果表明,在６７％的相似性水平上,全部供试菌可以分为快生型根瘤菌和慢性型根瘤菌两大群,在８０％的相似性水平上又可分为四个亚群。在此基础上,对各亚群的胡枝子根瘤菌进行了ＤＮＡ相关性的测定,以进一步证实和确定它们的分类地位,并通过１６ＳｒＤＮＡＰＣＲＲＦＬＰ分析对各亚群的系统发育关系进行了初步研究。     

A taxometric analysis ofLimonium (Plumbaginaceae) in Western Europe

A taxometric analysis ofLimonium species in Western Europe has provided evidence for the subgeneric groupings proposed by other authors. A cluster analysis suggests that the origin of sexual and agamospermous species has taken place separately. Sexual and agamospermous species are usually clustered separately. The lack of a sexual process in many agamospermous species, as indicated by the distribution of incompatibility morphs and by very low pollen fertility, suggests that the evolution of sexual and agamospermous taxa has been by different processes. It is suggested that following the origin of agamospermous species or species groups by hybridisation, subsequent evolution in the agamospermous taxa has occurred by asexual means. Evidence from the distribution of pollen/stigma incompatibility morphs, especially the existence of self-compatible agamospermous species supports this hypothesis.     

Spatial heterogeneity of wind forcing: application to artificial reef functioning influenced by the circulation in the Bay of Marseilles, France

In the frame of the largest French project of artificial production reefs, initiated by the city of Marseilles in 2001, the present study aimed at describing the hydrodynamic pattern of the coastal area considered, by the use of a 3D numerical modelling. Results were local wind statistics, bottom current fields and drifting particle maps. The knowledge of the hydrodynamic connexions between particle (such as larvae) sources or targeted areas linked to the reefs, allows us to explain the success or failure of the reefs' colonizing. Moreover, the study confirms the wind spatial variability and demonstrates the error resulting from the use of an average but locally absent wind direction.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.     

Numerical simulation of birch pollen dispersion with an operational weather forecast system

We included a parameterisation of the emissions of pollen grains into the comprehensive model system COSMO-ART. In addition, a detailed density distribution of birch trees within Switzerland was derived. Based on these new developments, we carried out numerical simulations of the dispersion of pollen grains for an episode that occurred in April 2006 over Switzerland and the adjacent regions. Since COSMO-ART is based on the operational forecast model of the German Weather Service, we are presenting a feasibility study of daily pollen forecast based on methods which have been developed during the last two decades for the treatment of anthropogenic aerosol. A comparison of the model results and very detailed pollen counts documents the current possibilities and the shortcomings of the method and gives hints for necessary improvements.     

A finite volume method for stochastic integrate-and-fire models

The stochastic integrate and fire neuron is one of the most commonly used stochastic models in neuroscience. Although some cases are analytically tractable, a full analysis typically calls for numerical simulations. We present a fast and accurate finite volume method to approximate the solution of the associated Fokker-Planck equation. The discretization of the boundary conditions offers a particular challenge, as standard operator splitting approaches cannot be applied without modification. We demonstrate the method using stationary and time dependent inputs, and compare them with Monte Carlo simulations. Such simulations are relatively easy to implement, but can suffer from convergence difficulties and long run times. In comparison, our method offers improved accuracy, and decreases computation times by several orders of magnitude. The method can easily be extended to two and three dimensional Fokker-Planck equations.

Spiking neural network simulation: numerical integration with the Parker-Sochacki method

Mathematical neuronal models are normally expressed using differential equations. The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models. Using this method, the solution order can be adapted according to the local conditions at each time step, enabling adaptive error control without changing the integration timestep. The method has been limited to polynomial equations, but we present division and power operations that expand its scope. We apply the Parker-Sochacki method to the Izhikevich ‘simple’ model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods. Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.