Complex periodic oscillations bursting

In the phase space, the trajectory associated with bursting oscillations 

That complex periodic behaviour in the model is equivalent to bursting of the type shown in fig. 4.16 becomes particularly clear when we consider the variation of product Pi (or Pj) in the course of time, rather than the variation of the substrate. The pattern of bursting changes, however, as a function of the parameter values of the system. 

The numerical study of eqns (4.1) shows that the behaviour of the biochemical system in the course of bursting can be decomposed into two phases in the first, )8 and y remain close to their steady-state values 

The analysis of the subsystem (4.4b,c) permits us to establish the curve yielding the steady state of variable /3 as a function of parameter a within the rapid system (j8, y). This curve has the form of an S-shaped sigmoid, as indicated in fig. 4.19a. This result can be explained by the analysis developed in chapter 2 for the mono-enzyme model proposed for glycolytic oscillations. In that model, three steady-state values of product y are obtained for a given value of the substrate concentration a when the latter is held constant in an appropriate range (see also fig. 3.3, p. 95). Here, for certain values of parameter a, we observe a similar phenomenon of bistability for jS, the substrate of enzyme E2 in the system (j8, y). The median branch of the hysteresis loop yielding /3q as a function of a is unstable it almost coincides, in its lower part, with the bottom branch of the sigmoid curve. 

At higher values of (fig. 4.19a), the upper and lower branches of the steady-state curve of )8 are stable in the range of a values considered. As indicated above, a behaves in fact as a slow variable of the 

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Fig. 4.3. Different modes of dynamic behaviour observed in the biochemical model with multiple regulation, for increasing values of parameter (in s ) (a) 0.6, simple periodic oscillations (b) 1.2, hard excitation (c) 2, chaos (d) 2.032, complex periodic oscillations (bursting). Only the substrate concentration is represented as a function of time. The curves are obtained by numerical integration of eqns (4.1) for the parameter values of fig. 4.2 (Decroly Goldbeter, 1982).

Table 4.4 shows how the main patterns of bursting occur in the model when parameter moves across the domain of complex periodic oscillations. Bursting occurs after the system has passed a domain of chaotic behaviour, which is itself reached beyond a cascade of period-doubling bifurcations issued from a simple periodic solution. A few narrow windows of chaos separate the first patterns of bursting observed. 

A few of the behavioural modes revealed by the bifurcation diagram of fig. 4.2 are illustrated by fig. 4.3 for four increasing values of k. In fig. 4.3a, the system displays simple periodic behaviour, as in the monoenzyme model studied for glycolytic oscillations. Figure 4.3b illustrates the coexistence between a stable steady state and a limit cycle that the system reaches only after a suprathreshold perturbation (hard excitation). The aperiodic oscillations of fig. 4.3c represent chaotic behaviour, while the complex periodic oscillations shown in fig. 4.3d correspond to the phenomenon of bursting that is associated with series of spikes in product Pi, alternating with phases of quiescence. These various modes of dynamic behaviour, as well additional ones identified by the analysis of the model, are considered in more detail below. 

Fig. 4.17. Phase space trajectory associated with bursting in the biochemical model with multiple regulation. The curve corresponds to the complex periodic oscillations of fig. 4.3d (Decroly Goldbeter, 1982).

Fig. 4.25. Piecewise linear map constructed on the basis of the results of fig. 4.23 to account for complex periodic oscillations of the bursting type. The onedimensional return map x +j=/(x ) is defined by eqns (4.5) which contain the three parameters M, a and b. The arrowed trajectory corresponds to the simple pattern of bursting with three peaks per period Tr(3). Parameter values are a = 6,b = 5,M=ll (Decroly Goldbeter, 1987).

The comparison of the different behavioural domains in parameter space shows that simple periodic oscillations remain, by far, the most common type of dynamic behaviour. Complex periodic oscillations of the bursting type are also rather frequent, but much less than simple oscillations. The coexistence between a steady state and a limit cycle comes third by virtue of the importance of the domain in which such behaviour occurs in the v-k plane. Birhythmicity and chaos come next... 

The complex periodic oscillations of fig. 6.1 were obtained by integration of eqns (6.2). The number of peaks of cAMP in the active phase of bursting depends on parameter values in the model. Thus, a variation in parameter Lj, which measures the relative rates of dephosphorylation and phosphorylation of the receptor in the free state (see chapter 5), controls the number of cAMP spikes over a period. This number can decrease progressively until a single peak subsists the system then recovers its simple periodic behaviour. 

Shown in fig. 6.4 are the different types of bursting obtained in the three-variable model. Besides the type already discussed with regard to fig. 6.3, and exemplified by the pattern Tr(l, 8) in fig. 6.4c, we can also observe qualitatively different patterns of bursting such as those shown in fig. 6.4a and b. The latter two types of complex periodic oscillation occur in the vicinity of point B in fig. 6.2. The origin of such bursting patterns can be comprehended by resorting to a discussion of the dynamics of the fast subsystem pi-y in which variable a is treated as a slowly varying parameter (see below). 

Fig. 12.1. Biochemical models based on various modes of positive feedback in enzyme reactions. All these models admit simple periodic behaviour of the limit cycle type. The coexistence between two stable limit cycles (birhythmidty) or between a stable limit cycle and a stable steady state (hard excitation) is observed in models (b) to (d). Model (d) eilso admits complex periodic oscillations of the bursting type, chaos, as well as the coexistence between three simultaneously stable limit cycles (trirhythmicity) (Goldbeter et oL, 1988).

While the coexistence between two limit cycles or between a limit cycle and a stable steady state is also shared by the two-variable models of fig. 12.1b and c, new modes of complex dynamic behaviour arise because of the presence of a third variable in the multiply regulated system. The coexistence between three simultaneously stable limit cycles, i.e. trirhythmicity, is the first of these. Moreover, the interaction between two instability-generating mechanisms allows the appearance of complex periodic oscillations, of the bursting type, as well as chaos. The system also displays the property of final state sensitivity (Grebogi et ai, 1983a) when two stable limit cycles are separated by a regime of unstable chaos. 

The piecewise linear map does not account, however, for the appearance of chaotic behaviour. A slight modification of the unidimensional map, taking into account some previously neglected details of the Poincare section of the differential system, shows how chaos may appear besides complex periodic oscillations of the bursting type. 

Probably the first examples of complex periodic oscillations in chemistry are found in studies of the BZ reaction in a CSTR in the mid-1970s. In these experiments, a number of investigators (Zaikin and Zhabotinsky, 1973 Sorensen, 1974 Marek and Svobodova, 1975 De Kepper et al., 1976 Graziani et al., 1976) observed bursting, a form of oscillation commonly seen in neurons, but previously unobserved in simple chemical reactions. Bursting consists of periods of relatively... 

In addition to the compound oscillation depicted in Figures 12.16 and 12.17, the collision of two limit cycles may lead to other scenarios. One possibility is complex periodic oscillation in which one cycle of one type is followed by several of another type. This type of behavior is analogous to the bursting mode of oscillation of neural oscillators, in which a period of relative quiescence is followed by a series of action potentials. In Figure 12.18, we compare a membrane potential trace from a crab neuron with potential oscillations in a pair of physically coupled chlorine dioxide-iodide oscillators. 

Fig. 1.19. Complex, but strictly periodic, oscillations in a chemical reaction showing bursting in a model of the Belousov-Zhabotinskii reaction. (Reprinted with permission from Bar-Eli, K and Noyes, R. M. (1988). J. Chem. Phys., 88, 3636-54. American Institute of Physics.)...

Fig. 4.16. Examples of bursting oscillations in neurophysiology, (a) Variation of the membrane potential in the neuron R15 from Aplysia, at two different time scales (Adams Benson, 1985). (b) Complex periodic variation of the membrane potential (V ) of pancreatic 3-cells (upper curve), associated with a simple periodic variation of the extracellular potassium concentration (V ) (Lebrun Atwater, 1985).

Fig. 4.18. Different types of simple or complex periodic behaviour observed in the model for the multiply regulated biochemical system, as a function of parameter k, (a) 15 s", simple periodic oscillations (b) 8 s bursting with a series of small, rapid oscillations on the top of slower oscillations of larger amplitude ...

The first indication of complex oscillations was that of the complex periodic behaviour of the bursting type shown in fig. 6.1 (Martiel Goldbeter, 1985). These oscillations resemble those obtained in the model for the multiply regulated enzyme system analysed in chapter 4. 

Similar complex oscillatory phenomena have been observed in a closely related model containing two regulated enzyme reactions coupled in a different manner (Li, Ding Xu, 1984). An additional indication of the generality of the results obtained in the multiply regulated biochemical system is given by the study of the model for the synthesis of cAMP in Dictyostelium amoebae. In addition to simple periodic oscillations and excitability (see chapter 5), this realistic model based on experimental observations also predicts the appearance of more complex oscillatory phenomena in the form of birhythmicity, bursting and chaos (chapter 6). 

The pattern of activity in many neurons consists not of single action potentials or of simple periodic firing, but of the more complex temporal mode known as bursting. We came across bursting, which consists of relatively quiescent, hyper-polarized periods alternating with periods in which a series or burst of action potentials occurs, in the previous chapter on coupled oscillators. Bursting oscillations have the potential to carry significantly more information than simple periodic (e.g., sinusoidal) oscillations, because, in addition to the frequency and amplitude, there is information content in the number of action potentials per cycle, the time between aetion potentials, and the duty cycle, that is, the frac-... 

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. 

Other complex oscillatory waveforms include bursting , in which large-amplitude oscillations are interspersed with periods of non-oscillatory evolution, or by small oscillations (Fig. 1.19). 

Simple periodic behaviour is far from being the only mode of oscillation observed in chemical and, even more, biological systems. For many nerve cells, indeed, particularly in molluscs, oscillations take the form of bursts of action potentials, recurring at regular intervals representing a phase of quiescence. The best-characterized example of this mode of oscillatory behaviour known as bursting is provided by the R15 neuron of Aplysia (Alving, 1968 Adams Benson, 1985). Neurons of the central nervous system of mammals (Johnston Brown, 1984) also present this type of oscillations. In addition, complex oscillations have been observed and modelled in chemical systems (see, for example, Janz, Vanacek Field, 1980 Rinzel Troy, 1982, 1983 Petrov, Scott Sho waiter, 1992). 

From simple periodic behaviour to complex oscillations, including bursting and chaos... 

Complex oscillations such as bursting or chaos arise when these two effects acquire comparable importance, so that an interplay occurs between the two, simultaneously active, oscillatory mechanisms. In contrast, when parameter values are such that one of the mechanisms is active while the other remains silent , oscillations have a simple periodic character. 

The dynamics of cytosolic Ca does not always possess a simple periodic nature. Thus, upon stimulation, some cells exhibit complex transients that resemble bursting oscillations (Woods et al, 1987 Berridge et ai, 1988 Cuthbertson, 1989). It is not possible to generate such complex behaviour when the models schematized in fig. 9.17 contain only two variables, namely Y and Z. 

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