A nonautonomous phenomenological model for On and Off responses of cells in sensory system.

Many neurons in mammalian sensory systems exhibit On and Off responses when given appropriate excitatory and inhibitory stimuli. In some instances, such neurons can also exhibit a Mixed response where diminished On and Off responses are both present. In this manuscript, we present a simple single cell model for these ubiquitous stimulus-response patterns. The model is nonautonomous consisting of two fast variables (one being the voltage), one slow recovery variable, and a time dependent stimuli current I(t). For piecewise constant I(t), On and Off responses can be reproduced and it is shown that their dependence on both the duration and the intensity of the input can be derived using singular perturbation techniques. Furthermore, we show that for certain stimuli I(t) the voltage has spike trains both during and immediately after the stimuli is presented. Such Mixed responses have also been measured experimentally, and the current model reproduces all three responses robustly for different net synaptic currents I(t).

Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species.

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.

Controllability of excitable systems.

Mathematical models of cell electrical activity typically consist of a current balance equation, channel activation (or inactivation) variables and concentrations of regulatory agents. These models can be thought of as nonlinear filters whose input is some applied current I (possibly zero) and output is a membrane potential V. A natural question to ask is if the applied current I can be deduced from the potential V. For a surprisingly large class of models the answer to this question is yes. To show this, we first demonstrate how many models can be embedded into higher dimensional quasilinear systems. For quasilinear models, a procedure for determining the inverse of the nonlinear filter is then described and demonstrated on two models: (1) the FitzHugh-Nagumo model and (2) the Sherman-Rinzel-Keizer (SRK) [Sherman et al., (1988, Biophysics Journal, 54, 411-425)] model of bursting electrical activity in pancreatic beta-cells. For the latter example, the inverse problem is then used to deduce model parameter values for which the model and experimental data agree in some measure. An advantage of the correlation technique is that experimental values for activation (and/or regulatory) variables need not be known to make the estimates for these parameter values.

Fast subsystem bifurcations in strongly coupled heterogeneous collections of excitable cells.

A continuum model for a heterogeneous collection of excitable cells electrically coupled through gap junctions is introduced and analysed using spatial averaging, asymptotic and numerical techniques. Heterogeneity is modelled by imposing a spatial dependence on parameters which define the single cell model and a diffusion term is used to model the gap junction coupling. For different parameter values, single cell models can exhibit bursting, beating and a myriad of other complex oscillations. A procedure for finding asymptotic estimates of the thresholds between these (synchronous) behaviors in the cellular aggregates is described for the heterogeneous case where the coupling strength is strong. This procedure is tested on a model of a strongly coupled heterogeneous collection of bursting and beating cells. Since isolated pancreatic beta-cells have been observed to both burst and beat, this test of the spatial averaging techniques provides a possible explanation to measured discrepancies between the electrical activities of isolated beta-cells and coupled collections (islets) of beta-cells.

Glucose diffusion in pancreatic islets of Langerhans.

We investigate the time required for glucose to diffuse through an isolated pancreatic islet of Langerhans and reach an equilibrium. This question is relevant in the context of in vitro electrophysiological studies of the response of an islet to step changes in the bath glucose concentration. Islet cells are electrically coupled by gap junctions, so nonuniformities in islet glucose concentration may be reflected in the activity of cells on the islet periphery, where electrical recordings are made. Using a mathematical model of hindered glucose diffusion, we investigate the effects of the islet porosity and the permeability of a surrounding layer of acinar cells. A major factor in the determination of the equilibrium time is the transport of glucose into islet beta-cells, which removes glucose from the interstitial spaces where diffusion occurs. This transport is incorporated by using a model of the GLUT-2 glucose transporter. We find that several minutes are required for the islet to equilibrate to a 10 mM change in bath glucose, a typical protocol in islet experiments. It is therefore likely that in electrophysiological islet experiments the glucose distribution is nonuniform for several minutes after a step change in bath glucose. The delay in glucose penetration to the inner portions of the islet may be a major contributing factor to the 1-2-min delay in islet electrical activity typically observed after bath application of a stimulatory concentration of glucose.

Correlations of rates of insulin release from islets and plateau fractions for beta-cells.

Pancreatic beta-cells in intact islets of Langerhans perfused with various glucose concentrations exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. The fraction of the time spent in the active phase is called the plateau fraction and appears to be strongly correlated with the rate of release of insulin from islets as glucose concentration is varied. Here this correlation is quantified and a theoretical development is presented in detail. Experimental rates of insulin release are correlated with "effective" plateau fractions over a range of glucose concentrations. There are a number of different models for BEA in pancreatic beta-cells and a method is developed here to quantify the dependence of a glucose dependent parameter on glucose concentration. As an example, the plateau fractions computed from the Sherman-Rinzel-Keizer model are matched with experimental plateau fractions to obtain a relationship between the model's glucose-dependent parameter, beta, and glucose concentration. Knowledge of the relationships between beta and glucose concentration and between experimental measurements of rates of insulin release and plateau fractions permits the determination of theoretical rates of insulin release from the model.

The biopharmaceutical properties of solid dosage forms. I. An evaluation of 23 brands of phenylbutazone tablets.

The potency, disintegration and dissolution characteristics of 23 brands of phenylbutazone tablets were determined. Five (21.7%) of the 23 brands failed to comply with the minimum requirements of the compendia or the regulations appended to the Food and Drugs Act. The in vitro characteristics of four brands were substantially different from those that disintegrated and released the drug satisfactorily. The in vivo characteristics of three of the four brands were compared with those observed for a pharmaceutically acceptable product. The latter product released the drug to the blood quickly, but the former products released the drug only after the tablets had been in the body for six to eight hours and, in the case of one product, released quantities much below those that would be acceptable to the physician. These results show that different products containing the same drug are not necessarily equivalent. This is contrary to the generic equivalency hypothesis which assumes that all products comply with specifications and, therefore, must be clinically effective.