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1986 | Buch

Oscillations and Chaos in the Pancreatic β-Cell Kapitel lesen Erstes Kapitel lesen

Nonlinear Oscillations in Biology and Chemistry

Proceedings of a meeting held at the University of Utah, May 9–11, 1985

herausgegeben von: Hans G. Othmer

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Biomathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

This volume contains the proceedings of a meeting entitled 'Nonlinear Oscillations in Biology and Chemistry', which was held at the University of Utah May 9-11,1985. The papers fall into four major categories: (i) those that deal with biological problems, particularly problems arising in cell biology, (ii) those that deal with chemical systems, (iii) those that treat problems which arise in neurophysiology, and (iv), those whose primary emphasis is on more general models and the mathematical techniques involved in their analysis. Except for the paper by Auchmuty, all are based on talks given at the meeting. The diversity of papers gives some indication of the scope of the meeting, but the printed word conveys neither the degree of interaction between the participants nor the intellectual sparks generated by that interaction. The meeting was made possible by the financial support of the Department of Mathe­ matics of the University of Utah. I am indebted to Ms. Toni Bunker of the Department of Mathematics for her very able assistance on all manner of details associated with the organization of the meeting. Finally, a word of thanks to all participants for their con­ tributions to the success of the meeting, and to the contributors to this volume for their efforts in preparing their manuscripts.

Inhaltsverzeichnis

Frontmatter

Biological Systems

Frontmatter
Oscillations and Chaos in the Pancreatic β-Cell
Abstract
The voltage across the plasma membrane of the pancreatic 3-cell displays burst activity when glucose between ~8.5 mM and 16.5 mM is added to the perfusion medium (Dean and Mathews, 1970; Ribalet and Beigleman, 1979; Atwater et al., 1980). This consists of an active phase during which spikes are generated and a silent phase during which the membrane is hyperpolarized. In high glucose concentrations above 16.6 mM, the burst pattern disappears and spike activity becomes continuous. The active phase has been associated with the stimulus for the release of insulin (Meissner, 1976; Scott et al., 1981).
Teresa Ree Chay
On Different Mechanisms for Membrane Potential Bursting
Abstract
A number of mathematical models have been proposed to describe the electrical bursting activity of biological excitable membrane systems. Many of these models have been formulated for specific applications [4,5,14]. One of our goals has been to understand the basic underlying qualitative structure of these models and to distinguish, possibly different, classes of models for bursting. In this paper we contrast two examples which illustrate different mathematical mechanisms.
John Rinzel, Young Seek Lee
A Mitotic Oscillator with a Strange Attractor and Distributions of Cell Cycle Times
Abstract
The concept of the existence of the cell cycle appeared shortly after the advent of light microscopy in the last century when natural scientists first described the intranuclear events involved in mitosis and cytokinesis. It was not long before significant intercellular variation in the cell generation time (the elapsed time between cell birth and the production of two daughter cells, also called the intermitotic time) was described in a variety of cell types.
Michael C. Mackey, Martin Santavy, Pavla Selepova
An Analysis of One- and Two-Dimensional Patterns in a Mechanical Model for Morphogenesis
Abstract
In early embryonic development, fibroblast cells move through an extracellular matrix (ECM) exerting large traction forces which deform the ECM. We model these mechanical interactions mathematically and show that the various effects involved can combine to produce pattern in cell density. A linear analysis exhibits a wide selection of dispersion relations, suggesting a richness in pattern forming capability of the model. A nonlinear bifurcation analysis is presented for a simple version of the governing field equations. The one-dimensional analysis requires a non-standard element. The two-dimensional analysis shows the possibility of roll and hexagon pattern formation. A realistic biological application to the formation of feather germ primordia is briefly discussed.
P. K. Maini, J. D. Murray, G. F. Oster

Chemical Systems

Frontmatter
Electrically Coupled Belousov-Zhabotinskii Oscillators: Experimental Observation of Chaos in a Chemical System and Identification of its Source in the Field-Noyes Equations
Abstract
The Belousov-Zhabotinskii (BZ) reaction is by far the chemical oscillator that is best characterized experimentally and best understood mechanistically (Field et al. 1972; Field 1985; Field and Boyd, 1985). It is the metal-ion-catalyzed oxidation by bromate ion (BrO 3 - ) of any of a large class of organic materials in a strongly acidic, aqueous medium. In the experiments reported here, the Ce(IV)/Ce(III) couple is used as the metal-ion catalyst and acetylacetone (CH3COCH2-COCH3), which we will refer to as AA, is used as the organic material to avoid bubble formation. All experiments were carried out in 2.73 M H2SO4 and in continuous-flow, stirred tank reactors (CSTR) driven by peristaltic pumps. The general behavior of oscillating chemical reactions in CSTR experiments has been reviewed by DeKepper and Boissonade (1985), and the detailed CSTR behavior of the BZ reaction with malonic acid has been described by DeKepper and Bar-Eli (1983).
Michael F. Crowley, Richard J. Field
Losing Amplitude and Saving Phase
Abstract
Many chemical and biological phenomena are modeled as systems of coupled limit cycle oscillators. These models are inherently complex in that they involve often large numbers of coupled ordinary and partial differential equations. To understand any of the behavior of these systems, simplifying assumptions are made. One such assumption is that the individual oscillators are nearly identical and weakly coupled. In this case only the phase of the individual oscillators matters and so the coupled system becomes a smaller system on a k-torus. This technique has been applied to discrete systems [1–3] as well as to reaction-diffusion equation [4,5]. Many interesting aspects of chemical and biological systems can be understood by studying the simple phase-models [6–8]. For example, see the paper by Kopell in this volume.
Bard Ermentrout
Spiral Waves in Excitable Media
Abstract
The beautiful spiral waves of oxidation in the Belousov-Zhabotinskii reaction [20] are the source of many interesting questions about periodic structures in excitable media. Because they are easy to produce and photograph, pictures of these spirals have appeared in a number of popular science oriented magazines. The study of spirals takes on a more personal interest when one realizes that fibrillation and sudden cardiac arrest from heart attacks may also be due to the appearance of rotating spiral waves of electrical activity on the ventricular myocardium [1], [14]. Immediately questions like “How do spirals form?” and “Can Spirals be prevented?” or “Can one predict if a heart attack will be fatal?” spring to mind. Going beyond questions of self preservation, we may also ask about the properties of spirals, such as their wavelength and frequency, or the conditions necessary to. sustain spiral activity in a given medium.
James P. Keener

Neurophysiology

Frontmatter
Biomechanical and Neuromotor Factors in Mammalian Locomotor-Respiratory Coupling
Abstract
As a non-mathematical biologist I imagine that one of the great joys of those persons engaged in applied mathematics might be that of describing and predicting the behavior of complex living systems through abstractions that incorporate a relatively few key parameters. When this works, we can be a bit more confident about the rules underlying that behavior. Even when the biology does not conform, there may still be much of value in interesting, clever or otherwise novel mathematical models. But I presume (perhaps naively) that a principal objective in such ventures is that of uncovering the rules, interactions and constraints that govern the behavior of real biological systems. To the extent that this is so, the actual behavior of such systems will serve to both test and to set limits on such models.
D. M. Bramble
Analysis of a VCON Neuromime
Abstract
The voltage-controlled oscillator neuromime (VCON) described below is motivated by three things: First, van der Pols equation [1] and similar “flush-and-fill” models have been used since the 1920s to study neural activity. Subsequent work on van der Pols equation resulted in a map of parameter space that describes phase-locking of the oscillator to external forcing [2,3] Second, R. Guttman, et. al., (See 4] used experimental procedures developed by Hodgkin and Huxley for studying squid axon membranes, and they obtained a similar phase-locking portrait for these membranes. This showed that neuron membranes have rich phase-locking, or synchronization properties. Third, I developed and studied a model of a neuron that emphasizes frequency encoded information in [5]. This model is based on a voltage controlled oscillator circuit, and it is called VCON. A VCON provides a straightforward method for building circuit analogs for neural networks, and its associated mathematical model is in phase-amplitude coordinates, so it avoids a major difficulty in dealing with nonlinear oscillators.
F. C. Hoppensteadt
Coupled Oscillators and Locomotion by Fish
Abstract
Fish of many species propel themselves through water by rhythmic undulations; fins are used for stabilization and change of direction, but not for stereotypic straight line movements [1], The undulations are caused by contractions which pass down the muscles along the spinal cord (with muscles on the opposite sides of the fish 180° out of phase). These contractions are in turn directed by motoneurons which emerge from special positions in the spinal cord having a spatial periodicity that matches the segmentation of the backbone. Measurements from these positions (“ventral roots”) show rhythmic voltage changes (bursts of activity) at each such point, with a uniform frequency and a phase lag between any two points that are proportional to the distance between the points; i.e., the neural activity is a constant speed travelling wave. For technical reasons, much of this data has been gathered for dogfish and lamprey [1,2]; some of the observations have been corroborated for other species.
N. Kopell
Experimental Studies of Chaotic Neural Behavior: Cellular Activity and Electroencephalographic Signals
Abstract
Deterministic systems can display a form of highly irregular, quasirandom behavior called chaos. Even though the observed behavior is very complex, the systems which generate it can be very simple. Thus, in at least some instances, irregular biological systems may obey a simple, potentially discoverable, deterministic dynamical law. These systems can undergo reversible transitions to and from chaotic dynamics in response to small changes in parameter values. As a long term goal, this form of analysis may suggest more effective responses to disordered behavior in physiological control systems.
This contribution is concerned with chaos in neural systems and its possible role in epileptogenesis. Calculation of information dimension provides a procedure for distinguishing between chaotic and random behavior. This technique is applied to experimental data from two preparations: spontaneous activity of cortical neurons in the pre- and post-central gyri of the squirrel monkey and human electroen-cephalographic signals. In each case the results suggest that these systems can display low dimensional chaotic behavior.
P. E. Rapp, I. D. Zimmerman, A. M. Albano, G. C. deGuzman, N. N. Greenbaun, T. R. Bashore

Mathematical Methods

Frontmatter
A Period-Doubling Bubble in the Dynamics of Two Coupled Oscillators
Abstract
A period-doubling cascade in the bifurcation diagram of two Brusselators coupled by diffusion is continued to a particular parameter regime, where it is seen numerically to be associated with other bifurcation branches, and in particular, “decascades;” we call the resulting bifurcation effect a period-doubling bubble. Moreover the dynamics of the bubble formation can be described. The emphasis in this note in on describing the phenomenon, although the (strong) possibility of describing it analytically in terms of unfolding a singularity which comes from interactions of singularities of the single oscillators is discussed, as well as a discussion of possibly similar behavior in other coupled oscillators.
J. C. Alexander
Bistable Behavior in Coupled Oscillators
Abstract
We consider a very simple model of two identical nonlinear oscillators, each with an asymptotically stable limit cycle, coupled together by a linear diffusion path. The system depends on two parameters: the natural frequency of the individual oscillators and the intensity of the coupling. Our main result Is that the coupled system exhibits bistable behavior for an open set of parameter values which includes moderate values of the parameters rather than just very large or very small values.
D. G. Aronson, E. J. Doedel, H. G. Othmer
Continuation of Arnold Tongues in Mathematical Models of Periodically Forced Biological Oscillators
Abstract
Periodic stimulation of spontaneously oscillating physiological rhythms has powerful effects on the intrinsic rhythm. As the frequency and amplitude of the periodic stimulus are varied, a large number of different coupling patterns are set up between the stimulus and the spontaneous oscillator. In one class of rhythms, there are a fixed number, N, of cycles of the stimulus for each M cycles of the spontaneous rhythm, and the spontaneous oscillation occurs at fixed phase (or phases) of the periodic stimulus. We call such rhythms N:M phase locking. In addition to phase-locked rhythms, it is also possible to observe irregular or aperiodic rhythms in which fixed phase relationships and regular repeating cyclic patterns are not observed. The following generalizations are applicable to a large number of experiments on periodic forcing of biological oscillators (Guevara et al., 1981; Glass et al., 1984; Petrillo and Glass, 1984).
Leon Glass, Jacques Bélair
Averaging and Synchronization of Weakly Coupled Systems
Abstract
The averaging method for ordinary differential equations is briefly reviewed and new results on the uniform validity of its approximations over infinite intervals of time are given. As an application a synchronization theorem for weakly coupled systems is proved.
Humberto Carrillo Calvet
Variational Principles for Periodic Solutions of Autonomous Ordinary Differential Equations
Abstract
In many chemical and biological oscillators, one would like to find approximations to the observed periodic solutions. Usually this is done by trying to show that the initial value problem for the differential equation has a stable limit cycle and trying to compute the resulting solution. When the equation is an autonomous ordinary differential equation in more than two variables, this is often very difficult.
Giles Auchmuty
A Numerical Analysis of Wave Phenomena in a Reaction Diffusion Model
Abstract
A numerical analysis of wave phenomena in a reaction diffusion model is given. These take place in the neighborhood of a singularity in the underlying ordinary differential equations governing the spatially uniform solutions. Many of the states, whether traveling waves, stationary waves (patterns) or uniform states, coexist and are asymptotically stable. The number of states and their complexity increases as a size parameter becomes larger.
E. J. Doedel, J. P. Kernevez
On a Nonlinear Hyperbolic Equation Describing Transmission Lines, Cell Movement, and Branching Random Walks
Abstract
This is a preliminary and expository report on a nonlinear hyperbolic equation that arises from a variety of distinct phenomena. We derive the equation
$$ {\varepsilon ^2}{v_{tt}} + \left( {1 + g\left( v \right)} \right){v_t} = {k^2}{v_{xx}} + f\left( v \right) $$
(1)
as the equation for the voltage along a transmission line with nonlinear shunt conductance and a series inductance along the length of the line, from simple models of movement and reproduction in tissue cells and one celled organisms, and from a mathematical treatment of a branching random walk. In addition, with the proper scaling and choice of f and g this nonlinear hyperbolic equation can be viewed as the equation that describes a continuum of coupled van der Pol oscillators. In equation (1) the value of ε 2 need not be small, but the choice of the notation ε 2 suggests analogies with other well known nonlinear partial differential equations, and we will mention some of these analogies below. The purpose of this report is to briefly explain and motivate all of these derivations and to present some basic results about the solutions of this equation. In addition, the purpose is to show how the probabilistic interpretation of the equation arising out of the branching random walk helps in the understanding and motivation of the results. Detailed proofs of the new results will be presented elsewhere.
Steven R. Dunbar, Hans G. Othmer
Backmatter
Metadaten
Titel
Nonlinear Oscillations in Biology and Chemistry
herausgegeben von
Hans G. Othmer
Copyright-Jahr
1986
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-93318-9
Print ISBN
978-3-540-16481-4
DOI
https://doi.org/10.1007/978-3-642-93318-9