Statistical problems related to excitation threshold and reset value of membrane potentials

Die vorliegende Arbeit ist motiviert durch biologische Fragestellungen bezüglich des Verhaltens von Membranpotentialen in Neuronen. Ein vielfach betrachtetes Modell für spikende Neuronen ist das Folgende. Zwischen den Spikes verhält sich das Membranpotential wie ein Diffusionsprozess X der durch die SDGL dX_t= beta(X_t) dt+ sigma(X_t) dB_t gegeben ist, wobei (B_t) eine Standard-Brown'sche Bewegung bezeichnet. Spikes erklärt man wie folgt. Sobald das Potential X eine gewisse Exzitationsschwelle S überschreitet entsteht ein Spike. Danach wird das Potential wieder auf einen bestimmten Wert x_0 zurückgesetzt. In Anwendungen ist es manchmal möglich, einen Diffusionsprozess X zwischen den Spikes zu beobachten und die Koeffizienten der SDGL beta() und sigma() zu schätzen. Dennoch ist es nötig, die Schwellen x_0 und S zu bestimmen um das Modell festzulegen. Eine Möglichkeit, dieses Problem anzugehen, ist x_0 und S als Parameter eines statistischen Modells aufzufassen und diese zu schätzen. In der vorliegenden Arbeit werden vier verschiedene Fälle diskutiert, in denen wir jeweils annehmen, dass das Membranpotential X zwischen den Spikes eine Brown'sche Bewegung mit Drift, eine geometrische Brown'sche Bewegung, ein Ornstein-Uhlenbeck Prozess oder ein Cox-Ingersoll-Ross Prozess ist. Darüber hinaus beobachten wir die Zeiten zwischen aufeinander folgenden Spikes, die wir als iid Treffzeiten der Schwelle S von X gestartet in x_0 auffassen. Die ersten beiden Fälle ähneln sich sehr und man kann jeweils den Maximum-Likelihood-Schätzer explizit angeben. Darüber hinaus wird, unter Verwendung der LAN-Theorie, die Optimalität dieser Schätzer gezeigt. In den Fällen OU- und CIR-Prozess wählen wir eine Minimum-Distanz-Methode, die auf dem Vergleich von empirischer und wahrer Laplace-Transformation bezüglich einer Hilbertraumnorm beruht. Wir werden beweisen, dass alle Schätzer stark konsistent und asymptotisch normalverteilt sind. Im letzten Kapitel werden wir die Effizienz der Minimum-Distanz-Schätzer anhand simulierter Daten überprüfen. Ferner, werden Anwendungen auf reale Datensätze und deren Resultate ausführlich diskutiert.

Maximum principle preserving high order schemes for convection-dominated diffusion equations

The maximum principle is an important property of solutions to PDE. Correspondingly, it's of great interest for people to design a high order numerical scheme solving PDE with this property maintained. In this thesis, our particular interest is solving convection-dominated diffusion equation. We first review a nonconventional maximum principle preserving(MPP) high order finite volume(FV) WENO scheme, and then propose a new parametrized MPP high order finite difference(FD) WENO framework, which is generalized from the one solving hyperbolic conservation laws. A formal analysis is presented to show that a third order finite difference scheme with this parametrized MPP flux limiters maintains the third order accuracy without extra CFL constraint when the low order monotone flux is chosen appropriately. Numerical tests in both one and two dimensional cases are performed on the simulation of the incompressible Navier-Stokes equations in vorticity stream-function formulation and several other problems to show the effectiveness of the proposed method.

A proposed parameterization of interface discontinuity factors depending on neighborhood for pin-by-pin diffusion computations for LWR

There exists an interest in performing full core pin-by-pin computations for present nuclear reactors. In such type of problems the use of a transport approximation like the diffusion equation requires the introduction of correction parameters. Interface discontinuity factors can improve the diffusion solution to nearly reproduce a transport solution. Nevertheless, calculating accurate pin-by-pin IDF requires the knowledge of the heterogeneous neutron flux distribution, which depends on the boundary conditions of the pin-cell as well as the local variables along the nuclear reactor operation. As a consequence, it is impractical to compute them for each possible configuration. An alternative to generate accurate pin-by-pin interface discontinuity factors is to calculate reference values using zero-net-current boundary conditions and to synthesize afterwards their dependencies on the main neighborhood variables. In such way the factors can be accurately computed during fine-mesh diffusion calculations by correcting the reference values as a function of the actual environment of the pin-cell in the core. In this paper we propose a parameterization of the pin-by-pin interface discontinuity factors allowing the implementation of a cross sections library able to treat the neighborhood effect. First results are presented for typical PWR configurations.

On a comparison method to reaction-diffusion systems and its applications to chemotaxis

In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations.

Intermediation for technology diffusion and user innovation in a developing rural economy:a social learning perspective

Technology intermediaries are seen as potent vehicles for addressing perennial problems in transferring technology from university to industry in developed and developing countries. This paper examines what constitutes effective user-end intermediation in a low-technology, developing economy context, which is an under-researched topic. The social learning in technological innovation framework is extended using situated learning theory in a longitudinal instrumental case study of an exemplar technology intermediation programme. The paper documents the role that academic-related research and advisory centres can play as intermediaries in brokering, facilitating and configuring technology, against the backdrop of a group of small-scale pisciculture businesses in a rural area of Colombia. In doing so, it demonstrates how technology intermediation activities can be optimized in the domestication and innofusion of technology amongst end-users. The design components featured in this instrumental case of intermediation can inform policy making and practice relating to technology transfer from university to rural industry. Future research on this subject should consider the intermediation components put forward, as well as the impact of such interventions, in different countries and industrial sectors. Such research would allow for theoretical replication and help improve technology domestication and innofusion in different contexts, especially in less-developed countries.

Numerical Approximation of a Fractional-In-Space Diffusion Equation, I

2000 Mathematics Subject Classification: 26A33 (primary), 35S15 (secondary)

Solving Fractional Diffusion-Wave Equations Using a New Iterative Method

Mathematics Subject Classification: 26A33, 31B10

Special Functions and Pathways for Problems in Astrophysics: An Essay in Honor of A.M. Mathai

The paper provides a review of A.M. Mathai's applications of the theory of special functions, particularly generalized hypergeometric functions, to problems in stellar physics and formation of structure in the Universe and to questions related to reaction, diffusion, and reaction-diffusion models. The essay also highlights Mathai's recent work on entropic, distributional, and differential pathways to basic concepts in statistical mechanics, making use of his earlier research results in information and statistical distribution theory. The results presented in the essay cover a period of time in Mathai's research from 1982 to 2008 and are all related to the thematic area of the gravitationally stabilized solar fusion reactor and fractional reaction-diffusion, taking into account concepts of non-extensive statistical mechanics. The time period referred to above coincides also with Mathai's exceptional contributions to the establishment and operation of the Centre for Mathematical Sciences, India, as well as the holding of the United Nations (UN)/European Space Agency (ESA)/National Aeronautics and Space Administration (NASA) of the United States/ Japanese Aerospace Exploration Agency (JAXA) Workshops on basic space science and the International Heliophysical Year 2007, around the world. Professor Mathai's contributions to the latter, since 1991, are a testimony for his social con-science applied to international scientific activity.

Numerical Solution of Fractional Diffusion-Wave Equation with two Space Variables by Matrix Method

Mathematics Subject Classi¯cation 2010: 26A33, 65D25, 65M06, 65Z05.

Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary

Increasing the Calculation Speed of an Acceleration Scheme for an Age-Structured Diffusion Model

Дойчин Бояджиев, Галена Пеловска - В статията се предлага оптимизиран алгоритъм, който е по-бърз в сравнение с по- рано описаната ускорена (модифицирана STS) диференчна схема за възрастово структуриран популационен модел с дифузия. Запазвайки апроксимацията на модифицирания STS алгоритъм, изчислителното времето се намаля почти два пъти. Това прави оптимизирания метод по-предпочитан за задачи с нелинейност или с по-висока размерност.

Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12

Remarks on blow up time for solutions of a nonlinear diffusion system with time dependent coefficients

2000 Mathematics Subject Classification: 35K55, 35K60.