Oscillation wave patterns, modeling

The experimental study of Ca waves in fertilized eggs (Gilkey et al., 1978 Jaffe, 1983 Busa Nuccitelli, 1985) preceded the observation of temporal oscillations of Ca " in these and other cells. Likewise, the theoretical study of spatiotemporal Ca patterns was initially uncoupled from the study of models for Ca oscillations. Thus, empirical models were proposed to account for the propagation of Ca waves in amphib-... 

Various traveling wave patterns are found when g is less than g, a value that depends on the initial concentrations of reagents. When the feedback coefficient exceeds g, cluster patterns can occur. Clusters consist of sets of domains in which nearly all of the elements in a domain oscillate with the same amplitude and phase (18-19). Clusters have been observed in model studies of arrays of coupled neurons (20), but they are rare in chemical systems. In the simplest case, a system consists of two clusters that oscillate 180° out of phase each cluster can consist of several spatial domains. 

As Fig. 3.13 shows, the population trends for both populations exhibit a continuous wave pattern. For the given initial conditions these oscillations have a constant period and ampUtude. There is an interdependence between the two population sizes increase of one of them impedes the growth of other. In the case of some chemical process described by the Lotka—Volterra model, the concentratimis of the intermediates K and B would be oscillating. 

In this chapter we shall examine some of the effects of molecular fluctuations on chemical oscillations, waves and patterns. There are many ways one can attempt to study fluctuation dynamics in reacting systems, the most familiar of which are master equation models [ 1 ]. Here we present results obtained using a specific class of cellular automaton models, termed lattice-gas cellular automata [2-4]. These cellular automaton models provide a mesoscopic description of the spatially-distributed reacting system and are constructed to model the microscopic collision dynamics. The modeling strategy and rule construction are different from those for traditional cellular automata and are based on lattice-gas cellular automaton models for hydrodynamics [5]. However, reactive lattice-gas models differ from the corresponding hydrodynamics models in a number of important respects and are closely related to master equation descriptions of the reactive dynamics. 

With certain critical Pco/Poi ratios, structural oscillations can be observed [306]. Patterns of stationary and/or traveling waves can actually be seen by means of photoemission electron microscopy (see Ref. 313, and note Section XVIII-7B. Such behavior can be modeled mathematically (e.g.. Refs. 214, 314). 

An experimental confirmation of such patterns in systems whose dynamics seem to be well described by the prototype N-NDR model (i.e. where the negative feedback arises from a delayed transport of the electroactive species) is still missing. This is not really astonishing because the predicted parameter region for complex patterns is quite small in the model [31, 34], It also depends on the parameters entering the reaction term such that probably not all N-NDR oscillators exhibit these wave phenomena. Hence, the requirements for spatial instabilities of limit cycles are much more restrictive than for temporal oscillations where any system with an N-NDR (independent of the detailed kinetic) possesses also an experimentally accessible parameter range that exhibits oscillations. 

The discussion of the experimental results in Section in.2 shows that a fundamental understanding of pattern formation in electrochemical systems has been achieved. However, it also demonstrates that the present state represents just a first step toward a complete picture of possible dynamic behaviors. There are many observations that cannot yet be explained, for example, the spatiotemporal period-doubling bifurcation detected during the electrodissolution of iron, the occurrence of antiphase oscillations during Ni electrodissolution, or the emergence of modulated waves during the electrodissolution of Co. Nevertheless, these phenomena seem to be understandable through an extension of the models introduced in Section III.l. 

In conclusion, the study of models for Ca signalling indicates that the phenomena of Ca oscillations and waves are closely intertwined. The spatiotemporal patterns correspond to the propagation of a Ca front in a biochemically excitable or oscillatory medium, at a rate much higher than that associated with simple diffusion. Such a property could also underlie a possible role of Ca waves in intercellular communication (Charles et al., 1991). The results presented in figs. 9.30 and 9.31 show that a unique mechanism, based on CICR, can account for the... 

To simulate the pattern formation observed in our experiments, we employ a model of the BZ reaction (21). We add a global linear feedback term to account for the bromide ion production that results from the actinic illumination, vqf = naxC av ss% where q> is the quantum yield. The results of our simulations mimic those of the experiments. Bulk oscillations and travelling waves are observed in die model for smaller values of g. At higher g values, standing, irregular and localized clusters are observed in the same sequence and with the same patterns of hysteresis as in the experiments... 

Linear stability analysis provides one, rather abstract, approach to seeing where spatial patterns and waves come from. Another way to look at the problem has been suggested by Fife (1984), whose method is a bit less general but applies to a number of real systems. In Chapter 4, we used phase-plane analysis to examine a general two variable model, eqs. (4.1), from the point of view of temporal oscillations and excitability. Here, we consider the same system, augmented with diffusion terms a la Fife, as the basis for chemical wave generation ... 

In addition to temporal oscillations, solutions of tubulin and GTP can generate spatial patterns, including traveling waves of microtubule assembly and polygonal networks (Mandelkow et al., 1989). This system may provide a useful experimental model for understanding pattern formation in cells. 

Spectrum, the excitation patterns, i.e., the vertical aspects of the spectrum were quite well described in terms of the harmonic-oscillator model [4]. More meson-nucleon resonances have been identified meanwhile [5], all fitting quite well into the picture of the three-quark model with spin, radial, or orbital excitations [6]. These early baryons approximately obey SU(3)f flavour symmetry the wave function, including colour, spin, space, and flavour degrees of freedom, is antisymmetric under the exchange of any pair of quarks. The changes induced by the mass difference between the d and the u quarks or between the ordinary and the strange quarks can be treated as small corrections. 

In addition to excitability, the FHN model can also possess oscillatory solutions. Figure 5 a) shows the configuration of the nullclines for a different choice of parameters. Now the system has an imstable fixed point or steady state and the stable attracting state is a limit cycle oscillation shown as the closed loop surrounding the unstable fixed point in the (Mt )-plane of the figure. Chemical patterns such as spiral waves can form in oscillatory as well as excitable systems and we shall have occasion to discuss some aspects of patterns in oscillatory media below. 

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