Complex oscillations

Biological and physiological systems are typical complex systems, which provide examples of aperiodicity and chaos [1-7]. Aperiodic cardiac oscillations are reflected in ECG for different cases of arrhythmia Fig. (12.1). Similarly, chaotic, aperiodic and noisy oscillations are observed in EEC in specific cases as shown in Fig. (12.2). Closely allied with chemical oscillations are membrane oscillations which have considerable relevance in physiological processes including neurological and cardiac disorders in the context of detection and control. 

The term chaotic, as it is now widely used, describes the non-periodic behaviour that arises from the non-linear nature of deterministic systems, not the noisy behaviour arising from random driving behaviour [8], 

Before proceeding further, it would be desirable to discuss the types of complex and chaotic oscillations and the corresponding phase-plane plots. 

Complexity of the above types can also be observed in physical, chemical and biological systems. 

A dynamical system can be described by a set of N non-linear differential equations if n state variables are involved. If X(t) represents a system of state variables 

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Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

We have performed also a reaction field DFT/Molecular Dynamics simulation of this system. We found that after an initial time, when the complex oscillates within the cage at R(N-H) 2.0 a.u. and R(N-C1) 6.0 a.u., a small temperature variation is enough for allowing the complex to overcome the small energetic barrier and, with time, the distance between Cl" and the NH4 fragments starts to increase. Extrapolating to a real solution environment, the two fragments will be completely surrounded by water molecules, i.e. in a solution at infinite dilution the two ions are fully solvated. 

Fig. 12 shows that, within the CNDO approximation, the X-ray conformation is no longer adopted in solution. In fact, the complex oscillates around the S conformation (Fig. 13) from the gauche C to the symmetrical C conformation (Fig. 14). A close look at this motion indicates that it may be decomposed into a simultaneous...

Some of the main examples of biological rhythms of nonelectrical nature are discussed below, among which are glycolytic oscillations (Section III), oscillations and waves of cytosolic Ca + (Section IV), cAMP oscillations that underlie pulsatile intercellular communication in Dictyostelium amoebae (Section V), circadian rhythms (Section VI), and the cell cycle clock (Section VII). Section VIII is devoted to some recently discovered cellular rhythms. The transition from simple periodic behavior to complex oscillations including bursting and chaos is briefly dealt with in Section IX. Concluding remarks are presented in Section X. 

U. Kummer, L. F. Olsen, C. J. Dixon, A. K. Green, E. Bornberg-Bauer, and G. Baier, Switching from simple to complex oscillations in calcium signaling. Biophys. J. 79, 1188-1195 (2000). 

III. COMPLEX OSCILLATIONS IN THE OPEN BELOUSOV-ZHABOTINSKI REACTION... 

A macrolevel cell can be exemplified by the cellular model [231] and [232] when the cell is incorporated into a system of the strongly bonded Pt crystallites applied to a zeolite. The model allows to describe the complex oscillations of the CO oxidation rate. A heavy dependence of the reaction rates to be computed on the way the coupling rules for the neighboring cells are selected is shown. By varying these rules, it is possible to simulate the various experimental conditions. 

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