Rarity and chaos

This paper gives counterexamples to previously developed analyses which attempted to link rarity (recurrently small densities) with irregular dynamics in one dimensional difference equation models for population growth. The problem lies in a technical definition of rarity, which turns out to be irrelevant. Nevertheless, certain numerical evidence (bifurcation diagrams for parametrized families) suggests that there are easily calculated upper and lower bounds to densities, which are closely approximated in the chaotic regime.     

Neural chaos and schizophrenia

Recent data indicate that random-like processes are related to the defects in the organization of semantic memory in schizophrenia which is more disorganized and less definable than those of controls with more semantic links and more bizarre and atypical associations. These aspects of schizophrenic cognition are similar to characteristics of chaotic nonlinear dynamical systems. In this context, the hypothesis tested in this study is that dynamic changes of electrodermal activity (EDA) as a measure of brain and autonomic activity may serve as a characteristic which can be used as an indicator of possible neural chaotic process in schizophrenia. In the present study, bilateral EDA in rest conditions were measured in 40 schizophrenic patients and 40 healthy subjects. Results of nonlinear and statistical analysis indicate left-side significant differences of positive largest Lyapunov exponents in schizophrenia patients compared to the control group. This might be interpreted that the neural activity during rest in schizophrenic patients is significantly more chaotic than in the control group. The relationship was confirmed by surrogate data testing. These data suggest that increased neural chaos in patients with schizophrenia may influence brain processes that can cause random-like disorganization of mental processes.     

Vasomotion: the case for chaos

Fluctuations in vascular calibre, a phenomenon known as vasomotion, are ubiquitous in the microcirculation and represent emergent behaviour that involves synchronisation of Ca²⁺ oscillations in individual vascular cells. Ideally, coordinated interactions between locally generated vasomotion and neuro-humoral control mechanisms will allow optimal sensing of flow and pressure within vascular networks and thereby facilitate synergistic readjustments in local vascular conductance and flow under conditions of dynamically changing metabolic demand. Indeed, many studies have reported that vasomotion becomes more prominent under pathophysiological conditions, suggesting that it may serve as an adaptive homeodynamic response that maintains or re-establishes flow when perfusion is compromised. We here summarise evidence that the apparent irregular nature of vasomotion reflects deterministic interactions between a small number of dominant control variables, rather than random events, and may therefore be formally classified as chaotic. We also discuss the potential physiological benefits of chaos in the microcirculation and the key roles of signalling via gap junctions and nitric oxide.     

Correlations in one-dimensional fully developed chaos

Correlations in the baker map and the tent map as examples of one-dimensional, fully developed chaos are considered. It is shown, utilizing symbolic dynamical systems derived from these maps, that the vanishing second-order correlation function is not sufficient to guarantee uncorrelatedness. Importance of the higher-order, especially third-order, correlation functions is emphasized for chaotic systems. In search of the quantities that grasp correlational behaviors as a whole in chaotic systems, it is proposed to use the fixed-separation correlation integral, which is a modified quantity of the usual correlation integral devised to calculate the fractal dimension of strange attractors, for these maps. It is shown that the new quantity contains all the even-number orders of autocorrelation function that are commonly considered.     

Coloured noise or low-dimensional chaos?

Devising a method capable of distinguishing a low-dimensional chaotic signal that might be embedded in a noisy stochastic process has become a major challenge for those involved in time-series analysis. Here a null hypothesis approach is used in conjunction with a known nonlinear predictive test, to probe for the presence of chaos in epidemiological data. A probabilistic set of rules is used to stimulate a historic record of New York City measles outbreaks, generally understood to be governed by a chaotic attractor. The simulated runs of 'surrogate data' are carefully constructed so as to be free from any underlying low-dimensional chaotic process. They therefore serve as a useful null model against which to test the observed time series. However, despite the assumed differences between the dynamics of measles outbreaks and the null model, a nonlinear predictive scheme is found to be unable to differentiate between their characteristic time series. The methodology confirms that, if there is in fact a chaotic signal in the measles data, it is extremely difficult to detect in time series of such limited length. The results have general relevance to the analysis of physical, ecological and environmental time series.     

The controlled chaos of shifty pathogens

Bacterial pathogens use novel mechanisms to vary their surface structures. Three new genome sequences provide a perspective on these mechanisms in Borrelia burgdorferi, Neisseria meningitidis, and Campylobacter jejuni, which cause lyme disease, meningitis and gastroenteritis, respectively.     

Terminal chaos for information processing in neurodynamics

New nonlinear phenomenon — terminal chaos caused by failure of the Lipschitz condition at equilibrium points of dynamical systems is introduced. It is shown that terminal chaos has a well organized probabilistic structure which can be predicted and controlled. This gives an opportunity to exploit this phenomenon for information processing. It appears that chaotic states of neurons activity are associated with higher level of cognitive processes such as generalization and abstraction.