Hard excitation

Yet another type of complex oscillatory behavior involves the coexistence of multiple attractors. Hard excitation refers to the coexistence of a stable steady state and a stable limit cycle—a situation that might occur in the case of circadian rhythm suppression discussed in Section VI. Two stable limit... 

Let us confirm the correspondence between the relative magnitude of instability and the orders of motions with this < )4 MTRS. Figure 6 shows stroboscopic data of j = 0 mode variables (a0, Po) with sampling period l/fi>o-The left panel presents the data in the time interval 70,000 < t < 75,500, where the leading finite-time Lyapunov exponent is relatively small. A coherent, quasi-periodic motion is clearly seen in these variables. Since almost all energy concentrates in the zeroth mode, the other mode variables j / 0 are hardly excited in this coherent motion. In contrast, the right panel in Fig. 6 presents the data in the time interval 20,000 < t < 25,500, where the leading exponent is... 

The phenomenon of hard excitation (Minorsky, 1962) required in these conditions for the evolution to the limit cycle should be contrasted with the soft excitation that characterizes the spontaneous evolution towards such a cycle, starting from an unstable steady state (see, for example, fig. 2.13). The interest of reversible transitions between a stable steady state and an oscillatory regime stems from the fact that such behaviour is observed in some nerve cells (Best, 1979 Guttman, Lewis Rinzel, 1980). 

An example of such a situation was considered at the end of the preceding chapter the system with two oscillatory isozymes (fig. 3.23) contains two instability mechanisms coupled in parallel. Compared with the model based on a single product-activated enzyme, new behavioural modes may be observed, such as birhythmicity, hard excitation and multiple oscillatory domains as a function of a control parameter. The modes of dynamic behaviour in that model remain, however, limited, because it contains only two variables. For complex oscillations such as bursting or chaos to occur, it is necessary that the system contain at least three variables. 

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. 4.3. 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).

The largest domain in this parameter space is clearly that of simple periodic oscillations. Second in importance is the domain of complex periodic oscillations. Then comes the domain of coexistence between a stable limit cycle and a stable steady state (dotted zone), which situation is associated with the phenomenon of hard excitation. Just below the latter domain are two regions of birhythmicity corresponding to the coexistence of limit cycles LCl and LC2 on the one hand and LC2 and LC3 on the other. These two domains of birhythmicity partly overlap their intersection defines the domain of trirhythmicity, where the three stable limit cycles LCl, LC2 and LC3 coexist. Near the domains of birhythmicity are three distinct regions of chaos (dark zones), whose size is relatively reduced. 

Fig. 10.13. Stability diagram established as a function of the rates of cyclin synthesis, Vj, and degradation, v, in the minimal cascade model of fig. 10.4. The domain of oscillations is determined as in fig. 10.8 parameter values are as in fig. 10.6. A narrow region of hard excitation (not shown) in which a stable limit cycle coexists with a stable steady state is observed just above part of the upper boundary of the instability domain (J.M. Guilmot A. Goldbeter, unpublished results).

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).

Olsen, 1978), chaos (Olsen Degn, 1977 Olsen, 1979, 1983 Aguda Larter, 1991 Geest et al., 1992 Steinmetz et al, 1993), and hard excitation (Aguda, Hofmann-Frisch Olsen, 1990). 

Halobacterium, chaos in, 13 Hard excitation, 6,91,100 domain in parameter space for, 157,158 in model for birhythmicity, 105,107 in model with two instability mechanisms, 120-3... 

Periodic hormone secretion, see Hormone Periodic stimuli efficiency of, 22,304,305 see also Efficiency Pulsatile stimulation Peroxidase reaction bistability, 508 chaos, 13,508 hard excitation, 509... 

Stability diagrams for Ca oscillations, 364 for cAMP oscillations, 203,245 for developmental transitions, 287,290 for glycolytic oscillations, 51 for mitotic oscillator, 431,440,441,443 Start, checkpoint in cell cycle, 413 Steady state, stable or unstable, 49,56,62, 120,121,122,141,203,253-6,288,366, 438 see also Bifurcation diagram Bistability Hard excitation Linear stability analysis Stability diagrams Tristability... 

Suprachiasmatic nuclei (SCN), 7,461,471 Supramolecular organization, 492 Suprathreshold perturbation, 101,104,107, 122,123 see also Excitability Hard excitation Relay of cAMP signals Sustained oscillations, see Limit cycle oscillations Synchronization of glycolytic oscillations, 32,33 of oscillating cell populations, 276... 

It may be noted that the definition of a perfect gas, as given above, does not imply that the heat capacities are independent of temperature. This occurs only in the special case of the monfi tomic gas. However, over small ranges of temperature, especially in regions where the rotational degrees of freedom are fully excited but where the vibrational modes are hardly excited at all, it is often permissible to take the heat capacities as being approximately constant. Under such conditions an important equation may be obtained as follows. 

Some cases of coexistence of a stationary state and a periodic regime lave of course been observed (both being stable simultaneously). Transition rom the former to the latter is called "hard excitation", and is triggered >y a perturbation. An example of tristability, involving two stationary states and a periodic regime, has even been discovered for the Jriggs-Rauscher reaction (30), clearly showing the potentially rich source >rovided by chemical dynamics. 

Simple bistability between two stationary states is but one example of multistability. Other multistable phenomena of interest include hard excitation (Figure 6), i.e., bistability between a... 

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