Lars Folke Olsen

Chaos in biological systems

Chaos in Physiology: Health or Disease?.- Patterns of Activity in a Reduced Ionic Model of a Cell from the Rabbit Sinoatrial Node.- Oscillatory Plant Transpiration: Problem of a Mathematically Correct Model.- Bifurcations in a Model of the Platelet Regulatory System.- Chaos in a System of Interacting Nephrons.- Polypeptide Hormones and Receptors: Participants in and Products of a Two-parameter, Dissipative, Measure Preserving, Smooth Dynamical System in Hydrophobic Mass Energy.- Interplay Between Two Periodic Enzyme Reactions as a Source for Complex Oscillatory Behaviour.- Dynamics of Controlled Metabolic Network and Cellular Behaviour.- Periodic Forcing of a Biochemical System with Multiple Modes of Oscillatory Behaviour.- Periodic Behaviour and Chaos in the Mechanism of Intercellular Communication Governing Aggregation of Dictyostelium Amoebae.- Turbulent Morphogenesis of a Prototype Reaction-Diffusion System.- Periodic Solutions and Global Bifurcations for Nerve Impulse Equations.- Homoclinic and Periodic Solutions of Nerve Impulse Equations.- High Sensitivity Chaotic Behaviour in Sinusoidally Driven Hodgkin-Huxley Equations.- Forced Oscillations and Routes to Chaos in the Hodgkin-Huxley Axons and Squid Giant Axons.- Quantification of Chaos from Periodically Forced Squid Axons.- Chaos, Phase Locking and Bifurcation in Normal Squid Axons.- Chaos in Molluscan Neuron.- Pancreatic B-Cell Electrical Activity: Chaotic and Irregular Bursting Pattern.- Multiple Oscillatory States and Chaos in the Endogenous Activity of Excitable Cells: Pancreatic B-Cell as an Example.- Bifurcations in the Rose-Hindmarch and the Chay Model.- Dendritic Branching Patterns.- Chaos and Neural Networks.- Data Requirements for Reliable Estimation of Correlation Dimensions.- Chaotic Dynamics in Biological Information Processing: A Heuristic Outline.- Chaos in Ecology and Epidemiology.- Low Dimensional Storage Attractors in Epidemics of Childhood Diseases in Copenhagen, Denmark.- Periodicity and Chaos in Biological Systems: New Tools for the Study of Attractors.- Populations Under Periodically and Randomly Varying Growth Conditions.- Bi-fractal Basin Boundaries in Invertible Systems.- Homoclinic Bifurcations in Ordinary Differential Equations.- Characterization of Order and Disorder in Spatial Patterns.- Fractals, Intermittency and Morphogenesis.- Temperature Stability of Davydov Solitons.

Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark.

Using traditional spectral analysis and recently developed non-linear methods, we analyze the incidence of six childhood diseases in Copenhagen, Denmark. In three cases, measles, mumps, rubella, the dynamics suggest low dimensional chaos. Outbreaks of chicken pox, on the other hand, conform to an annual cycle with noise superimposed. The remaining diseases, pertussis and scarlet fever, remain problematic. The real epidemics are compared with the output of a Monte Carlo analog of the SEIR model for childhood infections. For measles, mumps, rubella, and chicken pox, we find substantial agreement between the model simulations and the data.

Changing criteria for imposing order

Abstract Theory is most powerful when it provides new approaches that help resolve long-standing problems. We review the origins, the early development, and some recent trends in the study of chaos and show how new methods of non-linear dynamics may be used to infer the underlying order in ‘noisy’ ecological and epidemiological time series. We consider, at some length, the application of these techniques to the study of childhood disease epidemics.

The case for chaos in childhood epidemics. II. Predicting historical epidemics from mathematical models.

The case for chaos in childhood epidemics rests on two observations. The first is that historical epidemics show various ‘fieldmarks’ of chaos, such as positive Lyapunov exponents. Second, phase portraits reconstructed from real-world epidemiological time series bear a striking resemblance to chaotic solutions obtained from certain epidemiological models. Both lines of evidence are subject to dispute: the algorithms used to look for the fieldmarks can be fooled by short, noisy time series, and the same fieldmarks can be generated by stochastic models in which there is demonstrably no chaos at all. In the present paper, we compare the predictive abilities of stochastic models with those of mechanistic scenarios that admit to chaotic solutions. The main results are as follows: (i) the mechanistic models outperform their stochastic counterparts; (ii) forecasting efficacy of the deterministic models is maximized by positing parameter values that induce chaotic behaviour; (iii) simple mechanistic models are equal if not superior to more detailed schemes that include age structure; and (iv) prediction accuracy for monthly notifications declines rapidly with time, so that, from a practical standpoint, the results are of little value. By way of contrast, next amplitude maps can successfully forecast successive changes in maximum incidence one or more years into the future.

An enzyme reaction with a strange attractor

Abstract A simple model of the oscillating peroxidase-oxidase reaction is presented. The model is capable of reproducing all previously observed dynamic patterns exhibited by the experimental system including its chaotic behaviour.

Mitochondria regulate the amplitude of simple and complex calcium oscillations

In a mathematical model for simple calcium oscillations [Biophys. Chem. 71 (1998) 125], it has been shown that mitochondria play an important role in the maintenance of constant amplitudes of cytosolic Ca(2+) oscillations. Simple plausible rate laws for Ca(2+) fluxes across the inner mitochondrial membrane have been used in this model. Here we show that it is possible to use the same rate laws as a plug-in element in other existing mathematical models and obtain the same effect on amplitude regulation. This result appears to be universal, independent of the type of model and the type of Ca(2+) oscillations. We demonstrate this on two models for spiking Ca(2+) oscillations [J. Biol. Chem. 266 (1991) 11068; Cell Calcium 14 (1993) 311] and on two recent models for bursting Ca(2+) oscillations; one of them being a receptor-operated model [Biophys. J. 79 (2000) 1188] and the other one being a store-operated model [BioSystems 57 (2000) 75].

Oscillations in peroxidase-catalyzed reactions and their potential function in vivo

The peroxidase-oxidase reaction has become a model system for the study of oscillations and complex dynamics in biochemical systems. In the present paper we give an overview of previous experimental and theoretical studies of the peroxidase-oxidase reaction. Recent in vitro experiments have raised the question whether the reaction also exhibits oscillations and complex dynamics in vivo. To investigate this possibility further we have undertaken new experimental studies of the reaction, using horseradish extracts and phenols which are widely distributed in plants. The results are discussed in light of the occurrence and a possible functional role of oscillations and complex dynamics of the peroxidase-oxidase reaction in vivo.

Oscillations in the peroxidase-oxidase reaction: a comparison of different peroxidases

The nonlinear behavior of the peroxidase-oxidase reaction was studied using structurally different peroxidases. For the first time sustained oscillations with peroxidases other than horseradish peroxidase in a single-enzyme system were observed. All peroxidases that showed significant oxidase activity were able to generate sustained oscillations. When adjusting the overall reaction rate, either of the two modifiers 2,4-dichlorophenol or Methylene blue could be omitted from the reaction. Due to the observation of different enzyme intermediates when using different peroxidases, we conclude that the mechanisms responsible for oscillatory kinetics may vary from one peroxidase to the other.

Transient kinetics of the reaction between cytochrome c-552 or plastocyanin and P-700 in subchloroplast particles

We have investigated the kinetics of reduction of P-700 in Photosystem I-enriched subchlroplast particles by cytochrome c-552 from Euglena and by plastocyanin, and the effects of cations on these reactions. In both cases, the results can be explained in terms of a bimolecular reaction between P-700+ and the electron donor. Addition of low concentrations of mono-, di- or polyvalent cations stimulate, whereas addition of high concentrations of cations inhibit these reactions. The effects of cations can be interpreted as effects on the membrane surface potential of the Photosystem I particles, as shown previously (Tamura, N., Yamamoto, Y. and Nishimura, M. (1980) Biochim. Biophys. Acta 592, 536–545). The maximal second-order rate constants obtained for the reduction of P-700+ are 9 · 107 M−1 · s−1 if cytochrome c-552 is the electron donor, and 1.5 · 108 M−1 · s−1 when the donor is plastocyanin. Using the latter rate constant, we are able to demonstrate that the apparently biphasic kinetics of reduction of P-700+ observed in intact chloroplasts show a good fit to those expected for a bimolecular reaction between P-700+ and plastocyanin.

Prediction Analysis for Measles Epidemics

A newly devised procedure of prediction analysis, which is a linearized version of the nonlinear least squares method combined with the maximum entropy spectral analysis method, was proposed. This method was applied to time series data of measles case notification in several communities in the UK, USA and Denmark. The dominant spectral lines observed in each power spectral density (PSD) can be safely assigned as fundamental periods. The optimum least squares fitting (LSF) curve calculated using these fundamental periods can essentially reproduce the underlying variation of the measles data. An extension of the LSF curve can be used to predict measles case notification quantitatively. Some discussions including a predictability of chaotic time series are presented.