Phase locking,period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias

A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincaré or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincaré map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincaré map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 22 and 42 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.     

Low-Dimensional Dynamics in Sensory Biology 1: Thermally Sensitive Electroreceptors of the Catfish

We report the results of a search for evidence of periodic unstableorbits in the electroreceptors of the catfish. The function of thesereceptor organs is to sense weak external electric fields. Inaddition, they respond to the ambient temperature and to the ioniccomposition of the water. These quantities are encoded by receptorsthat make use of an internal oscillator operating at the level of themembrane potential. If such oscillators have three or more degreesof freedom, and at least one of which also exhibits a nonlinearity,they are potentially capable of chaotic dynamics. By detecting theexistence of stable and unstable periodic orbits, we demonstratebifurcations between noisy stable and chaotic behavior using theambient temperature as a parameter. We suggest that the techniquedeveloped herein be regarded as an additional tool for the analysisof data in sensory biology and thus can be potentially useful instudies of functional responses to external stimuli. We speculatethat the appearance of unstable orbits may be indicative of a stateof heightened sensory awareness by the animal.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales (e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales ( e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Effect of immigration on a chaotic insect population

Recently, the most convincing evidence of complex dynamics and chaos in biological populations has been presented for Tribolium castaneaum, a classic laboratory model insect. In this note, the robustness of this system is investigated and a constant immigration term is added to the adult population equation. It has been found that such perturbation to the model can either have a complicating effect (when the isolated system is periodic) or a simplifying one (when the system is chaotic in isolation).     

Endogenous rhythms and chaos in crassulacean acid metabolism

Endogenous free-running regular circadian oscillations of net CO₂ exchange in the crassulacean-acidmetabolism (CAM) plant Kalanchoë daigremontiana Hamet et Perrier de la Bâthie under constant external conditions in continuous light have been shown to change to irregular non-predictable (chaotic) time behaviour as irradiance or temperature are raised above a critical level. A model of CAM has been constructed with pools of major metabolites of varying concentrations, flows of metabolites leading to exchange between pools, metabolite transformations determined by chemical reactions, and feedback regulations. The model is described by a system of coupled non-linear differential equations. It shows stable rhythmicity in normal dark-light cycles and in continuous light and, like the K. daigremontiana leaves in the experiments, a change to chaos as irradiance is increased. The maintenance of endogenous oscillations in the model is brought about by a hysteresis switch or beat oscillator between two stable oscillation modes. In CAM these stable modes are vacuolar malate accumulation and remobilization. The model shows that the physical nature of the beat oscillator in the leaves can be explained by the balance between active and passive transport at the tonoplast.Abbreviations CAM crassulacean acid metabolism - D dark period - DL 12:12 h dark-light rhythm - L light period - LL constant illumination - PPFD photosynthetic photon flux density - T^L leaf temperature It is a great pleasure to thank Dr. G.-H. Vieweg, (Roßdorf, FRG) for his long-lasting efforts to have the phytotron in Darmstadt erected and for his persistent involvement during the various phases of planning and building. This made the present experiments possible. Dr. D. Kramer is thanked for all the time he spends to maintain functioning of the facility. Dr. P. Keller and Ms. Erika Ball assisted with the gas-exchange technology and helped with the surveillance of the long-running experiments, and Ms. Erika Ball performed all the integrations. Ms. Doris Schäfer is thanked for drawing the gas-exchange curves for publication. We are also most grateful to Professor Chr. Giersch and Professor M. Kluge (both Institut für Botanik, Technische Hochschule Darmstadt, FRG) for valuable discussions.     

Analysis of Sequence Periodicity in <Emphasis Type="Italic">E. coli</Emphasis> Proteins: Empirical Investigation of the “Duplication and Divergence” Theory of Protein Evolution

Periodicity was quantified in 4289 Escherichia coli K12 confirmed and putative protein sequences, using a simple chi-square technique previously shown to reveal triplet period periodicity in coding DNA. Periodicities were calculated from period n = 2 to period n = 50 in nine different alphabetic representations of the proteins. By comparison with a randomly generated proteome of the same compositional content, the E. coli proteome does not contain a significant excess of periodic proteins. However, 60 proteins do appear to be significantly periodic in at least one alphabetic representation, after Bonferroni correction, at p < 0.01, and 30 at p < 0.001. These are compared with significantly periodic proteins of solved three-dimensional structure, detected by an identical analysis of the sequences from a protein structure database. It is concluded that there is no evidence for the presence of a proteome-wide quasi-periodicity as predicted by the duplication and divergence model of protein evolution and that the major periodicity detected is a consequence of the repetitive tendencies within -helices. However, it is not possible to explain all sequence periodicities in terms of observable secondary structure, as in cases where sequence periodicity can be compared to solved structure, there is often no structural regularity that would provide an obvious explanation in terms of natural selection on protein function.     

Phylogenetic Differences in Content and Intensity of Periodic Proteins

Many proteins exhibit sequence periodicity, often correlated with a visible structural periodicity. The statistical significance of such periodicity can be assessed by means of a chi-squared-based test, with significance thresholds being calculated from shuffled sequences. Comparison of the complete proteomes of 45 species reveals striking differences in the proportion of periodic proteins and the intensity of the most significant periodicities. Eukaryotes tend to have a higher proportion of periodic proteins than eubacteria, which in turn tend to have more than archaea. The intensity of periodicity in the most periodic proteins is also greatest in eukaryotes. By contrast, the relatively small group of periodic proteins in archaea also tend to be weakly periodic compared to those of eukaryotes and eubacteria. Exceptions to this general rule are found in those prokaryotes with multicellular life-cycle phases, e.g., Methanosarcina sp., or Anabaena sp., which have more periodicities than prokaryotes in general, and in unicellular eukaryotes, which have fewer than multicellular eukaryotes. The distribution of significantly periodic proteins in eukaryotes is over a wide range of period lengths, whereas prokaryotic proteins typically have a more limited set of period lengths. This is further investigated by repeating the analysis on the NRL-3D database of proteins of solved structure. Some short-range periodicities are explicable in terms of basic secondary structure, e.g., alpha helices, while middle-range periodicities are frequently found to consist of known short Pfam domains, e.g., leucine-rich repeats, tetratricopeptides or armadillo domains. However, not all can be explained in this way.Reviewing Editor: Dr. John Oakeshott     

Search for rhythmicity during hibernation in the European hamster

Temporal patterns of hibernation were studied by continuous monitoring of body temperature by radiotelemetry over 6 months in European hamsters, Cricetus cricetus, at constant temperature and photoperiod. Entrances into hibernation occurred mostly at the end of the night (0000–0800 hours), while arousals were randomly distributed between day and night. This is at variance with a control of bout duration by a clock with a period of 24 h. Consequently, the timing of entrances implies a phase-resetting of the circadian clock on each arousal. Persistence of circadian rhythmicity with a period different from 24 h during deep hibernation was investigated examining whether the durations of torpor bouts were integer multiples of a constant period. A non-parametric version of the classical contingency test of periodicity was developed for this purpose. Periods ranging from 21 to 29 h were tested. Nine animals out of ten showed at least one significant period in this range (P<0.01), either below 24 h (21.8±0.5 h, n=4) or above (27.3±0.5 h, n=7). However, we have found a theoretical model of bout durations for which the contingency test of periodicity sometimes gives false significant results. This indicates that the power of the test is weak. With this reservation our results suggest that a circadian oscillator controls the duration of a bout of hibernation, which would occur after an integer, but variable and possibly temperature-dependent number of cycles.Abbreviations _b a contingency test (see Appendix) - SCN suprachiasmatic nuclei - period - T _b body temperature     

Plasmon-Assisted Optical Curtains

We predict an optical curtain effect, i.e., formation of a spatially invariant light field as light emerges from a set of periodic metallic nano-objects. The underlying physical mechanism of generation of this unique optical curtain can be explained in both the spatial domain and the wave-vector domain. In particular, in each period, we use one metallic nanostrip to equate the amplitudes of lights impinging on the openings of two metallic nanoslits and also shift their phases by π difference. We elaborate the influence on the output effect from some geometrical parameters like the periodicity, the slit height, and so on. By controlling the light illuminated on metallic subwavelength apertures, it is practical to generate optical curtains of arbitrary forms, which may open new routes of plasmonic nanolithography.     

Coexistence of Tonic Spiking Oscillations in a Leech Neuron Model

The leech neuron model studied here has a remarkable dynamical plasticity. It exhibits a wide range of activities including various types of tonic spiking and bursting. In this study we apply methods of the qualitative theory of dynamical systems and the bifurcation theory to analyze the dynamics of the leech neuron model with emphasis on tonic spiking regimes. We show that the model can demonstrate bi-stability, such that two modes of tonic spiking coexist. Under a certain parameter regime, both tonic spiking modes are represented by the periodic attractors. As a bifurcation parameter is varied, one of the attractors becomes chaotic through a cascade of period-doubling bifurcations, while the other remains periodic. Thus, the system can demonstrate co-existence of a periodic tonic spiking with either periodic or chaotic tonic spiking. Pontryagins averaging technique is used to locate the periodic orbits in the phase space.     

Entrainment, Instability, Quasi-periodicity, and Chaos in a Compound Neural Oscillator

We studied the dynamical behavior of a class of compound central pattern generator (CPG) models consisting of a simple neural network oscillator driven by both constant and periodic inputs of varying amplitudes, frequencies, and phases. We focused on a specific oscillator composed of two mutually inhibiting types of neuron (inspiratory and expiratory neurons) that may be considered as a minimal model of the mammalian respiratory rhythm generator. The simulation results demonstrated how a simple CPG model— with a minimum number of neurons and mild nonlinearities— may reproduce a host of complex dynamical behaviors under various periodic inputs. In particular, the network oscillated spontaneously only when both neurons received adequate and proportionate constant excitations. In the presence of a periodic source, the spontaneous rhythm was overriden by an entrained oscillation of varying forms depending on the nature of the source. Stable entrained oscillations were inducible by two types of inputs: (1) anti-phase periodic inputs with alternating agonist-antagonist drives to both neurons and (2) a single periodic drive to only one of the neurons. In-phase inputs, which exert periodic drives of similar magnitude and phase relationships to both neurons, resulted in varying disruptions of the entrained oscillations including magnitude attenuation, harmonic and phase distortions, and quasi-periodic interference. In the absence of significant phasic feedback, chaotic motion developed only when the CPG was driven by multiple periodic inputs. Apneic episodes with repetitive alternation of active (intrinsic oscillation) and inactive (cessation of oscillation) states developed when the network was driven by a moderate periodic input of low frequency. %and amplitudes of intermediate strength, Similar results were demonstrated in other, more complex oscillator models (that is, half-center oscillator and three-phase respiratory network model). These theoretical results may have important implications in elucidating the mechanisms of rhythmogenesis in the mature and developing respiratory CPG as well as other compound CPGs in mammalian and invertebrate nervous systems.     

Diversity of temporal self-organized behaviors in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Periodic ammonium pulses: are they important for the uptake kinetics of Isochrysis galbana?

In a megatidal coastal system, high tidal currents can lead to periodic reinjection of nutrients via the water/sediment interface. This study investigates the phytoplankton response to periodic ammonium pulses (every 5h20) in a nitrogen-limited culture of Isochrysis galbana. Results show that cells use additional ammonium. The cell response is not instantaneous and an adaptation period is observed in experiments. This period is characterized by (i) a decrease in latency stage during 10 mn after one pulse, (ii) an alternance of uptake/excretion processes decreasing throughout the experiment, (iii) an increasingly high and rapid surge-uptake phenomenon until pulse 6. We raise the question whether the Michaelis–Menten equation classically used to describe uptake processes is adequate in such an intermittent background.     

Amplitude model for the effects of mutations and temperature on period and phase resetting of the Neurospora circadian oscillator.

This paper analyzes published and unpublished data on phase resetting of the circadian oscillator in the fungus Neurospora crassa and demonstrates a correlation between period and resetting behavior in several mutants with altered periods: As the period increases, the apparent sensitivity to resetting by light and by cycloheximide decreases. Sensitivity to resetting by temperature pulses may also decrease. We suggest that these mutations affect the amplitude of the oscillator and that a change in amplitude is responsible for the observed changes in both period and resetting by several stimuli. As a secondary hypothesis, we propose that temperature compensation of period in Neurospora can be explained by changes in amplitude: As temperature increases, the compensation mechanism may increase the amplitude of the oscillator to maintain a constant period. A number of testable predictions arising from these two hypotheses are discussed. To demonstrate these hypotheses, a mathematical model of a time-delay oscillator is presented in which both period and amplitude can be increased by a change in a single parameter. The model exhibits the predicted resetting behavior: With a standard perturbation, a smaller amplitude produces type 0 resetting and a larger amplitude produces type 1 resetting. Correlations between period, amplitude, and resetting can also be demonstrated in other types of oscillators. Examples of correlated changes in period and resetting behavior in Drosophila and hamsters raise the possibility that amplitude changes are a general phenomenon in circadian oscillators.     

Geographical variation in the periodicity of gypsy moth outbreaks

The existence of periodic oscillations in populations of forest Lepidoptera is well known. While information exists on how the periods of oscillations vary among different species, there is little prior evidence of variation in periodicity within the range of a single Lepidopteran species. The exotic gypsy moth is an introduced foliage-feeding insect in North America. Its populations are characterized by high amplitude oscillations between innocuously low densities and outbreak levels during which large regions of forests are defoliated. These outbreaks are recognized to arise periodically with considerable synchrony across much of the gypsy moth's North American range. Our analysis indicates that gypsy moth outbreaks in North America are periodic but they exhibit two dominant periodicities: a primary period of 8–10 yr (as previously reported) and a secondary period of 4–5 yr (a new finding in this study). The outbreak periodicity varied geographically and this variation was associated with forest type. We found that in the most susceptible forest types, those on xeric sites where oak is often mixed with pines, outbreak periodicity had a more dominant 5-yr period while in forest types characteristic of more mesic sites where oak was mixed with maples and other species, cycles were clearly operating on a 10-yr period.     

Influence of light and food on the circadian clock in liver of rainbow trout,Oncorhynchus mykiss

Several reports support the existence of multiple peripheral oscillators in fish, which may be able to modulate the rhythmic functions developed by those tissues hosting them. Thus, a circadian oscillator has been proposed to be located within fish liver. In this vertebrate group, the role played by the circadian system in regulating metabolic processes in liver is mostly unknown. We, therefore investigated the liver of rainbow trout (Oncorhynchus mykiss) as a potential element participating in the regulation of circadian rhythms in fish by hosting a functional circadian oscillator. The presence and expression pattern of main components of the circadian molecular machinery (clock1a, bmal1, per1 and rev-erbβ-like) were assessed. Furthermore, the role of environmental cues such as light and food, and their interaction in order to modulate the circadian oscillator was also assessed by exposing animals to constant conditions (absence of light for 48 h, and/or a 4 days fasting period). Our results demonstrate the existence of a functional circadian oscillator within trout liver, as demonstrated by significant rhythms of all clock genes assessed, independently of the environmental conditions studied. In addition, the daily profile of mRNA abundance of clock genes is influenced by both light (mainly clock1a and per1) and food (rev-erbβ-like), which is indicative of an interaction between both synchronizers. Our results point to rev-erbβ-like as possible mediator between the influence of light and food on the circadian oscillator within trout liver, since its daily profile is influenced by both light and food, thus affecting that of bmal1.     

Cell surface NADH oxidase activity of brine shrimp oscillates with a period of 25 min and is entrained by light

Plants have a surface NADH oxidase that measures time by oscillating with a 24-min period. The period is synchronized by light. With plants, a new maximum is observed exactly 12 min after the beginning of the light exposure. These experiments were to determine if animals exhibited a cell surface NADH oxidase having a similar periodicity and to answer the question, does the periodicity in animals respond to light? Using brine shrimp as a model, the findings show that plants and animals exhibit similar oscillating NADH oxidase activity and that the periodicity in this invertebrate animal does respond to light. Brine shrimp were grown for two to three days and transferred to darkness for 45 min. After return to light for one min, NADH was added and measurements of NADH oxidation were recorded over 50 min. The brine shrimp exhibited a cell surface NADH oxidase that oscillated with a period of 25 min. After being subjected to light, the brine shrimp showed a new maximum in NADH oxidation between 12 to 13 min after the beginning of the light exposure and again at 37 min and at 25 min intervals thereafter. The findings demonstrate that the periodic oscillations in NADH oxidation of brine shrimp are light entrainable.