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We present a qualitative analysis of a generic model structure that can simulate the bursting and spiking dynamics of many biological cells. Four different scenarios for the emergence of bursting are described. In this connection a number of theorems are stated concerning the relation between the phase portraits of the fast subsystem and the global behavior of the full model. It is emphasized that the onset of bursting involves the formation of a homoclinic orbit that travels along the route of the bursting oscillations and, hence, cannot be explained in terms of bifurcations in the fast subsystem. In one of the scenarios, the bursting oscillations arise in a homoclinic bifurcation in which the one-dimensional (1D) stable manifold of a saddle point becomes attracting to its whole 2D unstable manifold. This type of homoclinic bifurcation, and the complex behavior that it can produce, have not previously been examined in detail. We derive a 2D flow-defined map for this situation and show how the map transforms a disk-shaped cross-section of the flow into an annulus. Preliminary investigations of the stable dynamics of this map show that it produces an interesting cascade of alternating pitchfork and boundary collision bifurcations.PACS. 05.45.-a Nonlinear dynamics and nonlinear dynamical systems

The present study investigates the interaction between the tubuloglomerular feedback (TGF) response and the myogenic mechanism by use of a mathematical model. The two control mechanisms are implemented in a spatially distributed model of the rat renal juxtamedullary afferent arteriole. The model of the afferent arteriole is based on in vivo measurements of the stress-strain relation in muscle strips. Analysis of experimental data shows that the myogenic response can be modeled by a linear relation between the transmural pressure and the level of activation of the vascular smooth muscle cells. The contribution of TGF to smooth muscle activity is assumed to be a linear function of the glomerular capillary pressure. The results show that the myogenic response plays an important role in renal blood flow autoregulation. Without a myogenic response, mechanisms such as TGF that are localized in the distal segments of the microvasculature would not be able to achieve autoregulation because of passive, pressure-mediated effects in the upstream vascular segments. In addition, it is shown that a strong myogenic response may lead to both propagation and enhancement of vascular effects mediated through mechanisms located in the distal part of the afferent arteriole. An ascending myogenic response could enhance the regulatory efficiency of the TGF mechanism by increasing the open-loop gain of the system. However, such a synergistic interaction will only be observed when the two mechanisms operate on more or less separate segments of the afferent arteriole. In the case where they operate on common segments of the arteriole, the outcome of the interaction may well be antagonistic.

To improve the understanding of the mechanisms underlying the behavior of plasma non-esterified fatty acids (NEFA) in the postprandial state, we have developed a physiology-based mathematical model of plasma NEFA dynamics. Known physiological mechanisms are quantified and used to describe NEFA dynamics. Insulin is the major regulator of NEFA metabolism in the postprandial state. Plasma NEFA levels are thus highly dependent on the insulin concentration, the insulin sensitivity of adipose tissue, and the maximal lipolytic rate. In the postabsorptive state, e.g., at low insulin, adipose tissue lipolysis results in a net export of NEFA from adipose tissue to other tissues. Postprandially, the rise in insulin results in: Decreased lipolysis; a higher rate of lipoprotein lipase (LPL) activity; and decreased NEFA uptake and reesterification by adipose tissue stimulation of reesterification. The result is a drop in plasma NEFA after a carbohydrate containing meal. When insulin returns to postabsorptive levels, a rebound in plasma NEFA often occurs. This rebound is due to a restoration of lipolysis, a decrease in NEFA reesterification by adipose tissue and an increased LPL-as insulin activates LPL with a delay of several hours. In conclusion, movements of NEFA depend strongly on insulin-with postprandial plasma NEFA being almost inversely related to the insulin concentration in healthy humans. The model provides an integrative view of NEFA dynamics and a framework for quantitative and conceptual understanding of plasma NEFA fluxes.

A new method, the heat-washout method, for measuring total cutaneous blood flow rate is introduced. The measurements were performed with a transcutaneous (tc) PO2-electrode that is capable of heating and measuring local temperature, and it is constructed with a thermostatically controlled cap. The probe was heated electrically to a selected temperature 2-10 degrees above normal skin temperature. When the temperature was stable, the heating element was turned off, and the temperature was registered every 10 s until a stable baseline temperature, Tb, was obtained. Tb was subtracted from the registered temperatures giving deltaTs that were plotted in a semilogarithmic diagram. The heat-washout was monoexponential, and the slope was used for calculating blood flow rate in accordance with the principle of Kety, using a known partition coefficient. The method was applied to the forearm in two subjects, and the results were compared to blood flow rates obtained simultaneously by the 133Xe-washout method in the same area. The equation of the regression line was y = 2.5 + 0 x 968X and the correlation coefficient was 0 x 986 at temperature levels of 37-45 degrees C. In the pulp of the thumb, blood flow rates, in arteriovenous anastomoses, were estimated in two subjects by subtracting the capillary blood flow rate, measured by 133Xe-washout, from the total cutaneous blood flow rate, measured by heat-washout. Due to a relatively low diffusions coefficient for 133Xe compared to heat, 133Xe cannot be used for measurement of blood flow rate in arteriovenous anastomoses.