EXISTENCE OF SOLUTIONS FOR A FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATION WITH UNBOUNDED DELAY

In this article, we study the existence of mild solutions for fractional neutral integro-differential equations with infinite delay.

Comments on multiple oscillatory solutions in systems with time-delay feedback

A complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding the disappearance of limit cycle solutions, derive analytical criteria for frequency degeneration, amplitude degeneration, frequency extrema. Furthermore, we discuss the influence of the phase shift parameter and show analytically that the stabilization of the steady state and the decay of all oscillations (amplitude death) cannot happen for global feedback only. Finally, we explain the onset of traveling wave patterns close to the regime of amplitude death.

Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These non-local models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O(1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities.

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

H-infinity control design for time-delay linear systems: a rational transfer function based approach

The aim of this paper is to present new results on H-infinity control synthesis for time-delay linear systems. We extend the use of a finite order LTI system, called comparison system to H-infinity analysis and design. Differently from what can be viewed as a common feature of other control design methods available in the literature to date, the one presented here treats time-delay systems control design with classical numeric routines based on Riccati equations arisen from H-infinity theory. The proposed algorithm is simple, efficient and easy to implement. Some examples illustrating state and output feedback design are solved and discussed in order to put in evidence the most relevant characteristic of the theoretical results. Moreover, a practical application involving a 3-DOF networked control system is presented.

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.

Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Non-alcoholic fatty liver disease : can ultrasound assist in early diagnosis of prediabetes and delay progression to type2 diabetes mellitus

Non Alcoholic Fatty Liver Disease (NAFLD) is a condition that is frequently seen but seldom investigated. Until recently, NAFLD was considered benign, self-limiting and unworthy of further investigation. This opinion is based on retrospective studies with relatively small numbers and scant follow-up of histology data. (1) The prevalence for adults, in the USA is, 30%, and NAFLD is recognized as a common and increasing form of liver disease in the paediatric population (1). Australian data, from New South Wales, suggests the prevalence of NAFLD in “healthy” 15 year olds as being 10%.(2) Non-alcoholic fatty liver disease is a condition where fat progressively invades the liver parenchyma. The degree of infiltration ranges from simple steatosis (fat only) to steatohepatitis (fat and inflammation) steatohepatitis plus fibrosis (fat, inflammation and fibrosis) to cirrhosis (replacement of liver texture by scarred, fibrotic and non functioning tissue).Non-alcoholic fatty liver is diagnosed by exclusion rather than inclusion. None of the currently available diagnostic techniques -liver biopsy, liver function tests (LFT) or Imaging; ultrasound, Computerised tomography (CT) or Magnetic Resonance Imaging (MRI) are specific for non-alcoholic fatty liver. An association exists between NAFLD, Non Alcoholic Steatosis Hepatitis (NASH) and irreversible liver damage, cirrhosis and hepatoma. However, a more pervasive aspect of NAFLD is the association with Metabolic Syndrome. This Syndrome is categorised by increased insulin resistance (IR) and NAFLD is thought to be the hepatic representation. Those with NAFLD have an increased risk of death (3) and it is an independent predictor of atherosclerosis and cardiovascular disease (1). Liver biopsy is considered the gold standard for diagnosis, (4), and grading and staging, of non-alcoholic fatty liver disease. Fatty-liver is diagnosed when there is macrovesicular steatosis with displacement of the nucleus to the edge of the cell and at least 5% of the hepatocytes are seen to contain fat (4).Steatosis represents fat accumulation in liver tissue without inflammation. However, it is only called non-alcoholic fatty liver disease when alcohol - >20gms-30gms per day (5), has been excluded from the diet. Both non-alcoholic and alcoholic fatty liver are identical on histology. (4).LFT’s are indicative, not diagnostic. They indicate that a condition may be present but they are unable to diagnosis what the condition is. When a patient presents with raised fasting blood glucose, low HDL (high density lipoprotein), and elevated fasting triacylglycerols they are likely to have NAFLD. (6) Of the imaging techniques MRI is the least variable and the most reproducible. With CT scanning liver fat content can be semi quantitatively estimated. With increasing hepatic steatosis, liver attenuation values decrease by 1.6 Hounsfield units for every milligram of triglyceride deposited per gram of liver tissue (7). Ultrasound permits early detection of fatty liver, often in the preclinical stages before symptoms are present and serum alterations occur. Earlier, accurate reporting of this condition will allow appropriate intervention resulting in better patient health outcomes. References 1. Chalasami N. Does fat alone cause significant liver disease: It remains unclear whether simple steatosis is truly benign. American Gastroenterological Association Perspectives, February/March 2008 www.gastro.org/wmspage.cfm?parm1=5097 Viewed 20th October, 2008 2. Booth, M. George, J.Denney-Wilson, E: The population prevalence of adverse concentrations with adiposity of liver tests among Australian adolescents. Journal of Paediatrics and Child Health.2008 November 3. Catalano, D, Trovato, GM, Martines, GF, Randazzo, M, Tonzuso, A. Bright liver, body composition and insulin resistance changes with nutritional intervention: a follow-up study .Liver Int.2008; February 1280-9 4. Choudhury, J, Sanysl, A. Clinical aspects of Fatty Liver Disease. Semin in Liver Dis. 2004:24 (4):349-62 5. Dionysus Study Group. Drinking factors as cofactors of risk for alcohol induced liver change. Gut. 1997; 41 845-50 6. Preiss, D, Sattar, N. Non-alcoholic fatty liver disease: an overview of prevalence, diagnosis, pathogenesis and treatment considerations. Clin Sci.2008; 115 141-50 7. American Gastroenterological Association. Technical review on nonalcoholic fatty liver disease. Gastroenterology.2002; 123: 1705-25