Erik Mosekilde

Chaotic synchronization : applications to living systems

Coupled Nonlinear Oscillators Transverse Stability of Coupled Maps Unfolding the Riddling Bifurcation Time-Continuous Systems Coupled Pancreatic Cells Chaotic Phase Synchronization Population Dynamic Systems Clustering of Globally Maps Interacting Nephrons Coherence Resonance Oscillators.

Modeling absorption kinetics of subcutaneous injected soluble insulin

Absorption of subcutaneously injected soluble insulin deviates markedly from simple first-order kinetics and depends both on the volume and concentration of the injected solution. This paper presents a model of the absorption process in which insulin is presumed to be present in subcutis in a low molecular weight form, a high molecular weight form, and an immobile form where the molecules are bound to the tissue. The model describes how diffusion and absorption gradually reduce the insulin concentrations in the subcutaneous depot and thereby shift the balance between the three forms in accordance with usual laws of chemical kinetics. By presuming that primarily low molecular weight insulin penetrates the capillary walls, the model can account for experimentally observed variations in the absorption rate over a wide range of volumes and of concentrations. The model is used to determine the effective diffusion constant D for insulin in subcutis, the absorption rate constant B for low molecular weight insulin, the equilibrium constant Q between high and low molecular weight insulin, the binding capacity C for insulin in the tissue, and the average life time T for insulin in its bound state. Typical values for a bolus injection in the thigh of fasting type I diabetic patients are D=0.9 × 10−4 cm2/min, B=1.3 × 10−2/min, and Q=0.13 (ml/IU)2. Binding of insulin in the tissue is significant only at small concentrations. The binding capacity is of the order of C=0.05 IU/cm3 with a typical average life time in the bound state of T=800 min. Combined with a simplified model for distribution and degradation of insulin in the body, the absorption model is used to simulate variations in plasma free insulin concentrations with different delivery schedules, i.e., bolus injection and dosage by means of an infusion pump. The simulations show that a pump repetition frequency of 1–2 per hr is sufficient to secure an almost constant plasma insulin concentration.

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.

Synchronization phenomena in nephron–nephron interaction

Experimental data for tubular pressure oscillations in rat kidneys are analyzed in order to examine the different types of synchronization that can arise between neighboring functional units. For rats with normal blood pressure, the individual unit (the nephron) typically exhibits regular oscillations in its tubular pressure and flow variations. For such rats, both in-phase and antiphase synchronization can be demonstrated in the experimental data. For spontaneously hypertensive rats, where the pressure variations in the individual nephrons are highly irregular, signs of chaotic phase and frequency synchronization can be observed. Accounting for a hemodynamic as well as for a vascular coupling between nephrons that share a common interlobular artery, we develop a mathematical model of the pressure and flow regulation in a pair of adjacent nephrons. We show that this model, for appropriate values of the parameters, can reproduce the different types of experimentally observed synchronization. (c) 2001 American Institute of Physics.

Bifurcation analysis of nephron pressure and flow regulation

One- and two-dimensional continuation techniques are applied to study the bifurcation structure of a model of renal flow and pressure control. Integrating the main physiological mechanisms by which the individual nephron regulates the incoming blood flow, the model describes the interaction between the tubuloglomerular feedback and the response of the afferent arteriole. It is shown how a Hopf bifurcation leads the system to perform self-sustained oscillations if the feedback gain becomes sufficiently strong, and how a further increase of this parameter produces a folded structure of overlapping period-doubling cascades. Similar phenomena arise in response to increasing blood pressure. The numerical analyses are supported by existing experimental results on anesthetized rats. (c) 1996 American Institute of Physics.

Absolute and convective instabilities in a one-dimensional Brusselator flow model

The paper considers a one-dimensional Brusselator model with a uniform flow of the mixture of reaction components. An absolute as well as a convective instability can arise for both the Hopf and the Turing modes. The corresponding linear stability analysis is presented and supported by the results of computer simulations of the nonlinear equations. Finally, the condition for spatially undamped tails (the Cherenkov condition) is obtained. This represents a new mechanism for pattern formation in chemical reaction-diffusion systems.

Torus birth bifurcations in a DC/DC converter

Considering a pulsewidth modulated dc/dc converter as an example, this paper describes a border-collision bifurcation that can lead to the appearance of quasi-periodicity in piecewise-smooth dynamical systems. We demonstrate how a two-dimensional torus can arise from a periodic orbit through a bifurcation in which two complex-conjugate Poincare characteristic multipliers jump abruptly from the inside to the outside of the unit circle. The torus may be ergodic or resonant. However, in both cases the diameter of the torus develops approximately linearly with the distance to the bifurcation point as opposed to the characteristic parabolic form of the well-known Neimark-Sacker bifurcation. The paper also considers the birth of a torus via a subcritical Neimark-Sacker bifurcation in the piecewise-smooth system. Particular emphasis is given to the development of resonance zones via border-collision bifurcations

Absorption kinetics of insulin after subcutaneous administration

Many diabetic patients depend on regular and well-controlled administration of insulin to avoid unacceptable excursions in plasma glucose. A complicating factor is that the absorption of insulin shows a considerable variability, both between patients, and from administration to administration for the same patient. To understand the mechanisms that influence this variability we present a quantitative description of the absorption kinetics for both soluble insulin and insulin crystals. The concentration dependent distribution of insulin between different oligomers is first analysed and described. Next, the disappearance of soluble and crystalline insulin from subcutis is described and explained as a function of the administered dose, the insulin concentration and crystal specific parameters, but without diffusion. The effect of diffusion is then included, and the appearance of insulin in plasma following subcutaneous administration is simulated and discussed. Our results not only explain the observed variability, but they also explain how dose size, insulin concentration, insulin crystals etc. influence the absorption kinetics.

Topics in Nonlinear Dynamics: Applications to Physics, Biology and Economic Systems

A deterministic model of die tossing (to discuss stochasticity vs. determinism) the forced Duffings equation (chaos in a simple nonlinear system, bifurcations) coupled period-doubling systems coupled thermostatically controlled radiators (frequency locking and chaos in technical control systems) kidney pressure and flow regulation (period-doubling and chaos in a biological system) insulin-glucose metabolism (frequency-locking in experiments on human subjects) environmental and microbiological population models (higher order chaos) the beer production distribution system (chaos in human decision making behaviour) coupled economic sectors (entrainment in the macroeconomic system) coupled map lattices (transition to spatially extended systems) pattern formation in chemical reaction-diffusion systems (Turing structures).

Stochastic simulation of vertebral trabecular bone remodeling

Bone remodeling changes bone mass, architecture, and thereby bone strength, during normal aging. These changes seem to be accelerated during the menopause. Several therapeutic agents have been used in order to delay the onset of the menopause-related changes. The effects of these agents on the remodeling process have been determined histomorphometrically in several short-term clinical studies, but data from long-term clinical studies are difficult to achieve, as are data on the influence on bone strength. The aim of this study was to develop a computer stimulation model that could assist in predicting the long-term effects of changes in the remodeling process on bone mass, trabecular thickness, and perforations. The paper presents such a stochastic model of the remodeling process in human vertebral trabecular bone. The computer model is based on histomorphometric and structural data from human studies. It is presented in terms of flow charts, and simulations performed with the model are discussed in relation to measurements on human vertebral bone samples. The results show that a menopause-related doubling of the activation frequency causes a transient, mainly reversible bone loss. If the menopause is accompanied by an increase in both activation frequency and resorption depth, then the resulting bone loss will be more pronounced and with a larger part being irreversible bone loss (perforations). The two antiresorptive agents. Etidronate and estrogen both cause a slight increase in bone mass (reducing remodeling space), and Etidronate also seems capable of preventing perforations. During fluoride therapy, an initial increase in remodeling space followed by a reduction is seen. Very few perforations are found to take place during fluoride therapy. The present model has been validated by assessing the effects of the menopause and treatment with antiresorptive or anabolic agents. It was found that the results mirrored or anabolic agents. It was found that the results mirrored very closely the results (bone mass measurements) from short-term clinical studies. It is therefore concluded that the model provides a tool for evaluating existing and new therapeutic regimens.