Chaos in biological systems

The Abraham-Lorentz force and the time evolution of a chaotic system: The case of charged classical and quantum Duffing oscillators

This paper proves that the Abraham-Lorentz (AL) force can noticeably modify the trajectories of the charged Duffing oscillators over time. The influence of the reaction force on the oscillator evolution is strongly enhanced if the system is considered at the level of quantum mechanics. For example, the AL force examined within the scope of Newtonian description can change the trajectory of the Duffing oscillator only if it has the mass of an electron. However, we showed that when quantum corrections along with the nondeterministic contributions are taken into account, the reaction force of the electromagnetic field affects noticeably even the oscillator with a mass equal to the mass of the Pb ion. The charged Duffing oscillators belong to the class of systems characterized by the chaotic nondeterministic dynamics. In classical terms, the nondeterministic behavior of the discussed systems results from the breaking of the causality principle by the AL force.

Multiple early factors anticipate post-acute COVID-19 sequelae

Post-acute sequelae of COVID-19 (PASC) represent an emerging global crisis. However, quantifiable risk factors for PASC and their biological associations are poorly resolved. We executed a deep multi-omic, longitudinal investigation of 309 COVID-19 patients from initial diagnosis to convalescence (2-3 months later), integrated with clinical data and patient-reported symptoms. We resolved four PASC-anticipating risk factors at the time of initial COVID-19 diagnosis: type 2 diabetes, SARS-CoV-2 RNAemia, Epstein-Barr virus viremia, and specific auto-antibodies. In patients with gastrointestinal PASC, SARS-CoV-2-specific and CMV-specific CD8+ T cells exhibited unique dynamics during recovery from COVID-19. Analysis of symptom-associated immunological signatures revealed coordinated immunity polarization into four endotypes, exhibiting divergent acute severity and PASC. We find that immunological associations between PASC factors diminish over time, leading to distinct convalescent immune states. Detectability of most PASC factors at COVID-19 diagnosis emphasizes the importance of early disease measurements for understanding emergent chronic conditions and suggests PASC treatment strategies.

Self-Organization and Information Processing: From Basic Enzymatic Activities to Complex Adaptive Cellular Behavior

One of the main aims of current biology is to understand the origin of the molecular organization that underlies the complex dynamic architecture of cellular life. Here, we present an overview of the main sources of biomolecular order and complexity spanning from the most elementary levels of molecular activity to the emergence of cellular systemic behaviors. First, we have addressed the dissipative self-organization, the principal source of molecular order in the cell. Intensive studies over the last four decades have demonstrated that self-organization is central to understand enzyme activity under cellular conditions, functional coordination between enzymatic reactions, the emergence of dissipative metabolic networks (DMN), and molecular rhythms. The second fundamental source of order is molecular information processing. Studies on effective connectivity based on transfer entropy (TE) have made possible the quantification in bits of biomolecular information flows in DMN. This information processing enables efficient self-regulatory control of metabolism. As a consequence of both main sources of order, systemic functional structures emerge in the cell; in fact, quantitative analyses with DMN have revealed that the basic units of life display a global enzymatic structure that seems to be an essential characteristic of the systemic functional metabolism. This global metabolic structure has been verified experimentally in both prokaryotic and eukaryotic cells. Here, we also discuss how the study of systemic DMN, using Artificial Intelligence and advanced tools of Statistic Mechanics, has shown the emergence of Hopfield-like dynamics characterized by exhibiting associative memory. We have recently confirmed this thesis by testing associative conditioning behavior in individual amoeba cells. In these Pavlovian-like experiments, several hundreds of cells could learn new systemic migratory behaviors and remember them over long periods relative to their cell cycle, forgetting them later. Such associative process seems to correspond to an epigenetic memory. The cellular capacity of learning new adaptive systemic behaviors represents a fundamental evolutionary mechanism for cell adaptation.

Information Dynamics in Complex Systems Negates a Dichotomy between Chance and Necessity

Entropy increases in the execution of linear physical processes. At equilibrium, all uncertainty about the future is removed and information about the past is lost. Complex systems, on the other hand, can lead to the emergence of order, sustain uncertainty about the future, and generate new information to replace all old information about the system in finite time. The Kolmogorov–Sinai entropy for events and the Kolmogorov–Chaitin complexity for strings of numbers both approximate Shannon’s entropy (an indicator for the removal of uncertainty), indicating that information production is equivalent to the degree of complexity of an event. Thus, in the execution of non-linear processes, information entropy is inseparably tied to thermodynamic entropy. Therein, the critical decision points (bifurcations), which can exert lasting impact on the evolution of the future (the “butterfly effect”), defy the definition of being either born from randomness or from determination. Nevertheless, their information evolution and degree of complexity are amenable to measurement and can meaningfully replace the dichotomy of chance versus necessity. Common anthropomorphic perceptions do not accurately account for the transient durability of information, the potential for major consequences by small actions, or the absence of a discernible opposition between coincidence and inevitability.

An appreciation of the prescience of Don Gilbert (1930-2011): master of the theory and experimental unravelling of biochemical and cellular oscillatory dynamics

We review Don Gilbert's pioneering seminal contributions that both detailed the mathematical principles and the experimental demonstration of several of the key dynamic characteristics of life. Long before it became evident to the wider biochemical community, Gilbert proposed that cellular growth and replication necessitate autodynamic occurrence of cycles of oscillations that initiate, coordinate and terminate the processes of growth, during which all components are duplicated and become spatially re-organised in the progeny. Initiation and suppression of replication exhibit switch-like characteristics, that is, bifurcations in the values of parameters that separate static and autodynamic behaviour. His limit cycle solutions present models developed in a series of papers reported between 1974 and 1984, and these showed that most or even all of the major facets of the cell division cycle could be accommodated. That the cell division cycle may be timed by a multiple of shorter period (ultradian) rhythms, gave further credence to the central importance of oscillatory phenomena and homeodynamics as evident on multiple time scales (seconds to hours). Further application of the concepts inherent in limit cycle operation as hypothesised by Gilbert more than 50 years ago are now validated as being applicable to oscillatory transcript, metabolite and enzyme levels, cellular differentiation, senescence, cancerous states and cell death. Now, we reiterate especially for students and young colleagues, that these early achievements were even more exceptional, as his own lifetime's work on modelling was continued with experimental work in parallel with his predictions of the major current enterprises of biological research.

Electric fields yield chaos in microflows

We present an investigation of chaotic dynamics of a low Reynolds number electrokinetic flow. Electrokinetic flows arise due to couplings of electric fields and electric double layers. In these flows, applied (steady) electric fields can couple with ionic conductivity gradients outside electric double layers to produce flow instabilities. The threshold of these instabilities is controlled by an electric Rayleigh number, Ra(e). As Ra(e) increases monotonically, we show here flow dynamics can transition from steady state to a time-dependent periodic state and then to an aperiodic, chaotic state. Interestingly, further monotonic increase of Ra(e) shows a transition back to a well-ordered state, followed by a second transition to a chaotic state. Temporal power spectra and time-delay phase maps of low dimensional attractors graphically depict the sequence between periodic and chaotic states. To our knowledge, this is a unique report of a low Reynolds number flow with such a sequence of periodic-to-aperiodic transitions. Also unique is a report of strange attractors triggered and sustained through electric fluid body forces.

Nonlinear effects on Turing patterns: time oscillations and chaos

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, the patterns oscillate in time. When varying a single parameter, a series of bifurcations leads to period doubling, quasiperiodic, and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examine the Turing conditions for obtaining a diffusion-driven instability and show that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. These results demonstrate the limitations of the linear analysis for reaction-diffusion systems.

The metabolic core and catalytic switches are fundamental elements in the self-regulation of the systemic metabolic structure of cells

BACKGROUND

Experimental observations and numerical studies with dissipative metabolic networks have shown that cellular enzymatic activity self-organizes spontaneously leading to the emergence of a metabolic core formed by a set of enzymatic reactions which are always active under all environmental conditions, while the rest of catalytic processes are only intermittently active. The reactions of the metabolic core are essential for biomass formation and to assure optimal metabolic performance. The on-off catalytic reactions and the metabolic core are essential elements of a Systemic Metabolic Structure which seems to be a key feature common to all cellular organisms.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the functional importance of the metabolic core we have studied different catalytic patterns of a dissipative metabolic network under different external conditions. The emerging biochemical data have been analysed using information-based dynamic tools, such as Pearson's correlation and Transfer Entropy (which measures effective functionality). Our results show that a functional structure of effective connectivity emerges which is dynamical and characterized by significant variations of bio-molecular information flows.

CONCLUSIONS/SIGNIFICANCE

We have quantified essential aspects of the metabolic core functionality. The always active enzymatic reactions form a hub--with a high degree of effective connectivity--exhibiting a wide range of functional information values being able to act either as a source or as a sink of bio-molecular causal interactions. Likewise, we have found that the metabolic core is an essential part of an emergent functional structure characterized by catalytic modules and metabolic switches which allow critical transitions in enzymatic activity. Both, the metabolic core and the catalytic switches in which also intermittently-active enzymes are involved seem to be fundamental elements in the self-regulation of the Systemic Metabolic Structure.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Peak-to-peak dynamics in food chain models

We show in this paper that the chaotic regimes of many food chain models often enjoy a very peculiar property, known as peak-to-peak dynamics. This means that the maximum (peak) density of the populations of any trophic level can be easily forecasted provided the last two peaks of the same population are known. Moreover, extensive simulation shows that only the last peak is needed if the forecast concerns the population at the top of the food chain and that peaks variability often increases from bottom to top. All these findings bring naturally to the conclusion that top populations should be sampled in order to have higher chances to detect peak-to-peak dynamics. The analysis is carried out by studying ditrophic food chain models with seasonally varying parameters, tritrophic food chain models with constant parameters, and more complex food chain and food web models.

Computational approaches to cellular rhythms

Oscillations arise in genetic and metabolic networks as a result of various modes of cellular regulation. In view of the large number of variables involved and of the complexity of feedback processes that generate oscillations, mathematical models and numerical simulations are needed to fully grasp the molecular mechanisms and functions of biological rhythms. Models are also necessary to comprehend the transition from simple to complex oscillatory behaviour and to delineate the conditions under which they arise. Examples ranging from calcium oscillations to pulsatile intercellular communication and circadian rhythms illustrate how computational biology contributes to clarify the molecular and dynamical bases of cellular rhythms.

On the aperiodic locomotor behavior of Halobacterium salinarium under periodic light stimuli

Two long time series of swimming intervals of a bacterium inverting its motion under periodic light pulses are analysed. The associated next-period plots reveal, through their filiform structure, that the underlying dynamics are low-dimensional. Using recently described properties of such dynamics, a simple second-order black-box model for the swimming intervals is derived and validated. The model reinforces the conjecture that this bacterium is endowed with an oscillator controlling the switching of the flagellar motor.

Oscillations and noise: inherent instability of pressure support ventilation?

Pressure support ventilation (PSV) is almost universally employed in the management of actively breathing ventilated patients with acute respiratory failure. In this partial support mode of ventilation, a fixed pressure is applied to the airway opening, and flow delivery is monitored by the ventilator. Inspiration is terminated when measured inspiratory flow falls below a set fraction of the peak flow rate (flow cutoff); the ventilator then cycles to a lower pressure and expiration commences. We used linear and nonlinear mathematical models to investigate the dynamic behavior of pressure support ventilation and confirmed the predicted behavior using a test lung. Our mathematical and laboratory analyses indicate that pressure support ventilation in the setting of airflow obstruction can be accompanied by marked variations in tidal volume and end-expiratory alveolar pressure, even when subject effort is unvarying. Unstable behavior was observed in the simplest plausible linear mathematical model and is an inherent consequence of the underlying dynamics of this mode of ventilation. The mechanism underlying the observed instability is "feed forward" behavior mediated by oscillatory elevation in end-expiratory pressure. In both mathematical and mechanical models, unstable behavior occurred at impedance values and ventilator settings that are clinically realistic.

Dynamic behavior during noninvasive ventilation: chaotic support?

Acute noninvasive ventilation is generally applied via face mask, with modified pressure support used as the initial mode to assist ventilation. Although an adequate seal can usually be obtained, leaks frequently develop between the mask and the patient's face. This leakage presents a theoretical problem, since the inspiratory phase of pressure support terminates when flow falls to a predetermined fraction of peak inspiratory flow. To explore the issue of mask leakage and machine performance, we used a mathematical model to investigate the dynamic behavior of pressure-supported noninvasive ventilation, and confirmed the predicted behavior through use of a test lung. Our mathematical and laboratory analyses indicate that even when subject effort is unvarying, pressure-support ventilation applied in the presence of an inspiratory leak proximal to the airway opening can be accompanied by marked variations in duration of the inspiratory phase and in autoPEEP. The unstable behavior was observed in the simplest plausible mathematical models, and occurred at impedance values and ventilator settings that are clinically realistic.

Exhibition of intrinsic properties of certain systems in response to external disturbances

Two systems, which are models of biological processes, are considered. These systems are notable for responding to different external actions nearly alike. This is associated with the fact that an external action excites only their natural intrinsic motions. Two kinds of external actions, harmonic and random, are studied. It is shown that each of them induces a transition to a new state that can be treated as a peculiar kind of phase transitions. Characteristics of these phase transitions are found.

The temporal dynamics of tics in Gilles de la Tourette syndrome

BACKGROUND

Statistical characterization of tic behavior in Gilles de la Tourette syndrome (GTS) may provide insight into the dynamic functioning of the human central nervous system, as well as improve the quantitative assessment of tic symptom severity.

METHODS

Twenty-two medication-free GTS subjects underwent videotaping of their tics. The intervals between temporally adjacent tics were measured, and the statistical properties of these intervals were assessed through graphical representation of frequency distributions, autoregressive integrated moving average (ARIMA) modeling, spectral analysis, and construction of first return maps.

RESULTS

The frequency distribution of tic interval durations followed an inverse power law of temporal scaling. Spectral analyses similarly demonstrated that the spectral power density of tic interval duration scales inversely with frequency. ARIMA modeling suggested that the time series for tics are nonstationary as well as moving average processes. The first return maps demonstrated "burstlike" behavior and short-term periodicity in tics, and proved that successive tic intervals are not statistically independent. Graphic display of the time series confirmed shortterm periodicity, and in addition suggested the presence of period doubling.

CONCLUSIONS

These findings are suggestive though not conclusive evidence for the presence of a fractal, deterministic, and possibly chaotic process in the tic time series. These analytic methods provide insight into the temporal features of tics that commonly are described clinically (such as short-term bouts or bursting, and longer term waxing and waning), and they reveal certain important temporal features of tics that have not been clinically described. The methods may also prove useful in the improved characterization of tic symptom severity.

Fractal organization of the pointwise correlation dimension of the heart rate

OBJECTIVE

To depict and quantify the degree of organization of the heart rate variability (HRV) in normal subjects.

METHODS

A modified algorithm was created to estimate series of 'point-dimensions' (PD2) from interbeat (R-R) interval series of 10 healthy subjects (21-56 years). Our innovation is twofold: (i) we quantified instances of low-dimensional chaos, random fluctuations, and those for which our method failed to provide either (due to poor statistics); (ii) consecutive subepochs of PD2s underwent a relative dispersion (RD) analysis, yielding an index (D) which quantifies the dynamical organization of the heart rate generator.

RESULTS

The mean values of PD2 series varied between 4.58 and 5.88 (mean+/-SD= 5.21+/-0.41, n = 10). For group 1 (21-30 years, n = 6) we found an averaged PD2 of 5.49+/-0.27, while for group 2 (47-56 years, n = 4) PD2 averaged 4.79+/-0.17. The RD analysis performed for subepochs of PD2s yielded both instances obeying fractal scaling (D < 1.5) and stochasticity (D > 1.5). The average D for group 1 was 1.39+/-0.04 (14 subepochs) and for group 2, 1.20+/-0.008 (8 subepochs). Paired t-test and Hartley F-max test for comparison between D values and homogeneity of variance between the two groups were performed, yielding P-values 0.004 and 0.02, respectively.

CONCLUSIONS

The complexity of the HRV seems to be modulated by a non-random fractal mechanism of a 'hyperchaotic' system, i.e. it can be hypothesized to contain more than one attractor. Also, our results support the 'chaos hypothesis' put forth recently, namely, the complexity of the cardiovascular dynamics is reduced with aging. The index of relative dispersion of the dimensional complexity has to be tested in various clinico-pathological settings, in order to corroborate its value as a potential new physiological measure.

Sexual reproduction and population dynamics: the role of polygyny and demographic sex differences

Most models of population dynamics do not take sexual reproduction into account (i.e., they do not consider the role of males). However, assumptions behind this practice--that no demographic sex differences exist and males are always abundant enough to fertilize all the females--are usually not justified in natural populations. On the contrary, demographic sex differences are common, especially in polygynous species. Previous models that consider sexual reproduction report a stabilizing effect through mixing of different genotypes, thus suggesting a decrease in the propensity for complex of dynamics in sexually reproducing populations. Here we show that considering the direct role of males in reproduction and density dependence leads to the conclusion that a two-sex model is not necessarily more stable compared with the corresponding one-sex model. Although solutions exist where sexual reproduction has a stabilizing effect even when no genotypic variability is included (primarily when associated with monogamy), factors like polygyny, sex differences in survival or density dependence, and possible alterations of the primary sex ratio (the Trivers-Willard mechanism), may enlarge the parametric region of complex dynamics. Sexual reproduction therefore does not necessarily increase the stability of population dynamics and can have destabilizing effects, at least in species with complicated mating systems and sexual dimorphism.

Steady state approximation in the minimal model of the alternative pathway of complement

Complement is a response mechanism of the immune system. Two initiation pathways have been characterized for complement. The classical pathway is antibody mediated while the alternative pathway is not. Since the alternative pathway is independent of antibodies, it is always active. For the alternative pathway we have previously developed a minimal model. Using parameters within physiological bounds, the model showed complex behavior also within physiological bounds. Thus the model seems to be an appropriate representation of the alternative pathway response. By applying a steady state assumption to the Michaelis Menten step of the minimal model, we reduce the number of variables from six to five. A comparison between the dynamics of the minimal and contracted models reveals that the two descriptions may not be compatible. Although both systems show chaotic behavior it occurs in different regions of parameter space.

Chaos in a minimal model of the alternative pathway of the complement system

In previous work, we introduced a minimal model of the alternative pathway of the complement. We also limited our analysis to a reduced set of parameter values because, for some parameters, experimentally supported estimates were not found. On the other hand, changes in value of some parameters may be a result of a pathological condition. Therefore, here we extend our analysis and include a wider range of values of five of the physiologically relevant parameters. For all the parameters considered, we observe chaotic oscillations, and we construct bifurcation diagrams using Poincaré sections of local maxima.

Chaos and the dynamics of biological populations

As first emphasized in the early 1970s, the nonlinearities that are inherent in simple models for the regulation of plant and animal populations can lead to chaotic dynamics. This review deals with a variety of instances where chaotic phenomena can arise, particularly in interactions between prey and predators (including hosts and pathogens, hosts and parasitic insects, and harvested populations). Some of the complications in disentangling deterministic chaos from environmental noise will be discussed. The combination of population biology with population genetics leads to an even richer assortment of nonlinear phenomena and to the suggestion that many genetic polymorphisms may vary cyclically or chaotically (rather than being steady, as usually is assumed implicitly). I argue that complex dynamics - including chaos - is likely to be pervasive in population biology and population genetics, even in seemingly simple situations. But superimposed environmental noise, in heterogeneous natural settings, will usually complicate the dynamics, making it unlikely that population data will exhibit elegant properties (such as universalities in period doubling) associated with the underlying maps. The existence of chaotic régimes of dynamical behaviour can, however, invalidate standard techniques for analysing population data to reveal density-dependent mechanisms; this, I believe, may currently be the most significant implication of dynamical chaos for population biology.

The size control of fission yeast revisited

An analysis was made of cell length and cycle time in time-lapse films of the fission yeast Schizosaccharomyces pombe using wild-type (WT) cells and those of various mutants. The more important conclusions about ‘size controls’ are: (1) there is a marker in G2 in WT cells provided by a rate change point (RCP) where the linear rate of length growth increases by approximately 30%. The period before this RCP is dependent on size and can be called a ‘sizer’. The period after the RCP is nearly independent of size and can be called a ‘timer’. The achievement of a critical threshold size is at or near the RCP which is on average at about 0.3 of the cycle (halfway through G2). This is much earlier than was previously believed. (2) The RCP is at about the time when H1 histone kinase activity and the B type cyclin cdc13 start to rise in preparation for mitosis. The RCP is also associated with other metabolic changes. (3) In wee1 mutants, the mitotic size control is replaced by a G1/S size control which is as strong as the mitotic control. As in WT cells, there is a sizer which precedes the RCP followed by a timer but the RCP is at about the G1/S boundary and has a larger increase (approximately 100%) in rate. (4) cdc25 is not an essential part of the size control at mitosis or at the G1/S boundary. (5) Three further situations have been examined in which the mitotic size control has been abolished. First, induction synchronisation by block and release of cdc2 and cdc10. In the largest oversize-cells which are produced, the RCP is pushed back to the beginning of the cycle. There is no sizer period but only a timer. Second, when both the antagonists wee1 and cdc25 are absent in the double mutant wee1-50 cdc25 delta. In this interesting situation there is apparently no mitotic size control and the cycle times are quantised. Third, in rum1 delta wee1-50 where the normal long G1 in wee1 is much reduced, there is probably no size control either in G1 or in G2 causing a continuous shortening of division length from cycle to cycle.

Intermittency and transient chaos from simple frequency-dependent selection

Frequency-dependent selection is an important determinant of the evolution of gametophytic self-incompatibility systems in plants, aposematic (warning) and cryptic coloration, systems of mimicry, competitive interactions among members of a population, mating preferences, predator-prey and host-parasite interactions, aggression and other behavioural traits. Past theoretical studies of frequency-dependent selection have shown it to be a plausible mechanism for the maintenance of genetic variability in natural populations. Here, through an analysis of a simple deterministic model for frequency-dependent selection, we demonstrate that complex dynamic behaviour is possible under a broad range of parameter values. In particular we show that the model exhibits not only cycles and chaos but also, for a more restricted set of parameters, transient chaos and intermittency: alterations between an apparently deterministic behaviour and apparently chaotic fluctuations. This behaviour, which has not been stressed within the population genetics literature, provides an explanation for erratic dynamics of gene frequencies.

A nonlinear dynamical (chaos) approach to the analysis of broiler growth

Mathematical chaos has been observed in a number of biological areas, suggesting that order can be found in systems previously described as random. Nonlinear analyses were conducted to determine whether periodicity or chaos was evident in the growth responses of broiler chickens. Analyses of the absolute growth rate and growth rate acceleration were conducted for four lines of broilers selected at 14 or 42 d for high or low growth rates (Experiment 1) and for a commercial broiler strain (Experiment 2). Resulting Lyapunov exponents (LE) and correlation dimensions (CD) were statistically evaluated. Time series and return map graphics were analyzed. In both experiments, independence of day-to-day growth responses was indicated by low r2 values. In Experiment 1, there were significant differences between lines in growth rate (low, 9.1 +/- .3; high, 12.9 +/- .5 g/d) and the standard deviation of growth rate (low, 5.8 +/- .2; high 7.3 +/- .3 g/d). There were no significant differences for LE or CD values between lines or day of selection. In general, the positive LE, noninteger values of CD, and return map graphics in both experiments suggested the presence of chaotic dynamics. Evaluation of mathematical chaos in broiler growth may give insight into the dynamics and modeling of growth and diseases associated with growth.

Temporal heterogeneity of the blood flow velocity at the middle cerebral artery in the normal human characterized by fractal analysis

The objective of this study is to characterize the temporal fluctuation of the axial blood flow velocity (BFV) at the middle cerebral artery (MCA). Biological observables such as BFV present complex oscillations. The irregularity of physiological systems may be assessed by fractal analysis by computing the fractal dimension (D gamma) and the corresponding temporal correlation (r gamma). The BFV at the MCA was registered with transcranial Doppler ultrasonography (TCD) in four adult volunteers. As fractal processes are assumed to have no absolute time scale, two time scales were compared. The digitized signal was averaged respectively at 1-s intervals and for each heart beat. D gamma and r gamma were determined using relative dispersion analysis. The results were D gamma = 1.24 +/- 0.09 and r gamma = 0.45 +/- 0.19 (mean +/- SD) for the 1-s based time scale and D(r) = 1.17 +/- 0.09 and r gamma = 0.57 +/- 0.20 for the heart-beat scale. We conclude that the temporal heterogeneity of the BFV at the MCA in the normal human has fractal properties. Fractal analysis of TCD data may become useful in clinical diagnosis because loss of complexity in physiological systems has been linked to senescence or disease conditions. Wide variations of the so called normal values of BFV measured by TCD have been reported. The physiological BFV fluctuations may explain, in part, the variability of values recorded during routine TCD diagnostic examinations. Our observations may also be of value for understanding the interaction of the vascular endothelium and the blood flow stream (shear stress).

Insulin stimulation of high frequency phosphorylation dynamics in murine erythroleukemia cells

The phosphorylation potentials of two proteins (M(r) 81 kDa and 63 kDa) in extracts of murine erythroleukemic (MEL) cells both vary in an oscillatory manner, sometimes changing by as much as 100-fold in 10 min. Direct analysis of the temporal changes indicates the existence of periodic modulation of the frequencies, amplitudes and mean levels of the two rhythms. In both cases, periodogram analyses, by two methods, confirm the presence of several oscillations having periods in the range 20-100 min which tend to occur in (pseudo) periodic bursts. Insulin has been found to enhance these oscillations in a manner comparable with its effect on rhythmic variations in cell morphology. Despite the marked similarity in the behaviour of the two proteins, no particular phase relationship existed between the two temporal variations, suggesting differences between the underlying driving forces.

Fractal analysis of steady-state-flicker visual evoked potentials: feasibility

Electrophysiological models of visual evoked potential recording have assumed that response variability is caused predominantly by random noise added to a true steady signal. Since neuronal geometry has a fractal structure, neural activity may demonstrate deterministic nonlinear dynamics, i.e., chaos. We recorded several-minute time-series traces of the visual evoked potential magnitude in response to full-field flicker from three glaucoma patients and one normal subject. When plotted in phase space, the steady-state response derived from a lock-in amplifier shows an apparent so-called strange attractor (extended nonrepeating loops) rather than the pattern expected from a signal-plus-noise model (a fuzzy dot). The fractal dimension of this attractor may be a more sensitive indicator of early optic-nerve damage than are visual evoked potential latency or amplitude measures.

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three-variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five-variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three-variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small-amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.

Suppression of chaos by periodic oscillations in a model for cyclic AMP signalling in Dictyostelium cells

We investigate how the introduction of cells oscillating periodically affects the behaviour of a suspension of Dictyostelium discoideum amoebae undergoing chaotic oscillations of cyclic AMP. The analysis of a model indicates that a tiny proportion of periodic cells suffices to transform chaos into periodic oscillations in such suspensions. A similar result is obtained by forcing the aperiodic oscillations by a small-amplitude, periodic input of cyclic AMP. The results provide an explanation for the observation of regular oscillations in suspensions of a putatively chaotic mutant of Dictyostelium discoideum. More generally, the results show how chaos in biological systems may disappear through the coupling with periodic oscillations.

Nonrandom structures in the locomotor behavior of Halobacterium: a bifurcation route to chaos?

Halobacteria spontaneously reverse their swimming direction about every 10-15 s. They respond to light stimuli by a transient perturbation of this rhythm. During periodic stimulation the system shows features that are known from nonlinear oscillators. Increasing stimulation frequencies cause the following phenomena: (i) the frequency of reversals follows the stimulation frequency, (ii) transition to a state where a long and a short interval occur alternatingly and further transition to four interval lengths, (iii) appearance of irregular interval sequences, which, in a two-dimensional plot of successive intervals, reveal clearly discernible structures and suggest chaotic motion. A similar series of events can be induced in the absence of periodic stimulation, when a control parameter is changed to various constant levels. The data suggest that the system is governed by deterministic dynamical laws.

The estimation of the Kolmogorov entropy from a time series and its limitations when performed on EEG

A method to estimate a lower bound of the Kolmogorov entropy-the so called K2-entropy-from a time series is presented which avoids use of the generalized correlation integral. The influence of the norm is studied. The method is demonstrated on some standard examples. The entropy of the attractor apparent in the EEG of the foetal sheep is estimated and the results are compared with results obtained from synthesized data featuring some basic properties of EEG. This gives an insight into the limitations of the procedure.