We consider a piecewise smooth 2 × 2 system, whose solutions locally spirally move around an equilibrium point which lies at the intersection of two discontinuity surfaces. We find a sufficient condition for the stability of this point, in the limit case in which a first-order approximation theory does not give an answer. This condition, depending on the vector field and its Jacobian evaluated at the equilibrium point, is trivially satisfied for piecewise-linear systems, whose first-order part is a diagonal matrix with negative entries. We show how our stability results may be applied to discontinuous recursive neural networks for which the matrix of self-inhibitions of the neurons does not commute with the connection weight matrix. In particular, we find a nonstandard relation between the ratio of the self-inhibition speeds and the structure of the connection weight matrix, which determines the stability.

We propose a system of first-order ordinary differential equations to describe and understand the physiological mechanisms of the interplay between plasma glucose and insulin and their behaviors in diabetes. The proposed model is based on Hill and step functions which are used to simulate the switch-like behavior that occurs in metabolic regulatory variables when some of the threshold parameters are approached. A simplified piecewise-linear system is also proposed to study the possible equilibria and solutions and used to introduce simple theoretical control mechanisms representing the action of an artificial pancreas and regulating exogenous insulin.

In this paper a new data assimilation technique is proposed which is based on the ensemble Kalman filter (EnKF). Such a technique will be effective if few observations of a dynamical system are available and a large model error occurs. The idea is to acquire a fine grid of synthetic observations in two steps: (1) first we interpolate the real observations with suitable polynomial curves; (2) then we estimate the relative measurement errors by means of Brownian bridges.This technique has been tested on the Richards' equation, which governs the water flow in unsaturated soils, where a large model error has been introduced by solving the Richards' equation by means of an explicit numerical scheme. The application of this technique to some synthetic experiments has shown improvements with respect to the classical ensemble Kalman filter, in particular for problems with a large model error.

Here some issues are studied, related to the numerical solution of Richards' equation in a one dimensional spatial domain by a technique based on the Transversal Method of Lines (TMoL). The core idea of TMoL approach is to semi-discretize the time derivative of Richards' equation: afterward a system of second order differential equations in the space variable is derived as an initial value problem.The computational framework of this method requires both Dirichlet and Neumann boundary conditions at the top of the column. The practical motivation for choosing such a condition is argued. We will show that, with the choice of the aforementioned initial conditions, our TMoL approach brings to solutions comparable with the ones obtained by the classical Methods of Lines (hereafter referred to as MoL) with corresponding standard boundary conditions: in particular, an appropriate norm is introduced for effectively comparing numerical tests obtained by MoL and TMoL approach and a sensitivity analysis between the two methods is performed by means of a mass balance point of view. A further algorithm is introduced for deducing in a self-sustaining way the gradient boundary condition on top in the TMoL context.

Liver is crucial in the homeostasis of glycerol, an important metabolic intermediate. Plasma glycerol is imported by hepatocytes mainly through Aquaporin-9 (AQP9), an aquaglyceroporin channel negatively regulated by insulin in rodents. AQP9 is of critical importance in glycerol metabolism since hepatic glycerol utilization is rate-limited at the hepatocyte membrane permeation step. Glycerol kinase catalyzes the initial step for the conversion of the imported glycerol into glycerol-3-phosphate, a major substrate for de novo synthesis of glucose (gluconeogenesis) and/or triacyglycerols (lipogenesis). A model addressing the glucose-insulin system to describe the hepatic glycerol import and metabolism and the correlation with the glucose homeostasis is lacking so far. Here we consider a system of first-order ordinary differential equations delineating the relevance of hepatocyte AQP9 in liver glycerol permeability. Assuming the hepatic glycerol permeability as depending on the protein levels of AQP9, a mathematical function is designed describing the time course of the involvement of AQP9 in mouse hepatic glycerol metabolism in different nutritional states. The resulting theoretical relationship is derived fitting experimental data obtained with murine models at the fed, fasted or re-fed condition. While providing useful insights into the dynamics of liver AQP9 involvement in male rodent glycerol homeostasis our model may be adapted to the human liver serving as an important module of a whole body-model of the glucose metabolism both in health and metabolic diseases.

The infiltration process into the soil is generally modeled by the Richards' partial differential equation (PDE). Inthis paper a new approach for modeling the infiltration process through the interface of two different soils isproposed, where the interface is seen as a discontinuity surface defined by suitable state variables. Thus, theoriginal 1D Richards' PDE, enriched by a particular choice of the boundary conditions, is first approximated bymeans of a time semidiscretization, that is by means of the transversal method of lines (TMOL). In such a way asequence of discontinuous initial value problems, described by a sequence of second order differential systems inthe space variable, is derived. Then, Filippov theory on discontinuous dynamical systems may be applied inorder to study the relevant dynamics of the problem. The numerical integration of the semidiscretized differ-ential system will be performed by using a one-step method, which employs an event driven procedure to locatethe discontinuity surface and to adequately change the vector field.

Here a numerical technique based on the method of lines (MoL) for solving Richards' equation is presented. The Richards' equation deals with modeling infiltration of water into the unsaturated zone. By means of any kind of observations, some values of the state variable are assumed to be available at certain time points, in order to "correct" the numerical solution in the light of these observations. This is done by means of ensemble Kalman filter (EnKF), that is a data assimilation technique based on a Monte Carlo approach. Advantages of this approach are discussed, in the light of existing bibliography.