Birhythmicity

O. Delcroly and A. Goldbeter, Birhythmicity, chaos, and other patterns of temporal self organization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. USA 79, 6917 6921 (1982). 

G. Houart, G. Dupont, and A. Goldbeter, Bursting, chaos and birhythmicity originating from self-modulation of the inositol 1,4,5-triphosphate signal in a model for intracellular Ca + oscillations. Bull. Math. Biol. 61, 507-530 (1999). 

Fig. 3.1. Model of an enzyme reaction with positive feedback and recycling of product (P) into substrate (S). This model serves as a two-variable prototype for the study of birhythmicity and multiple domains of oscillations (Moran Goldbeter, 1984).

The two-variable model for birhythmicity is built on the basis of eqns (2.7) by incorporating into them a term related to the transformation of product into substrate, in a reaction catalysed by an enzyme whose cooperative kinetics is described by a Hill equation, characterized by a degree of cooperativity n. The kinetic equations of the model thus take the form of eqns (3.1) where the various parameters remain defined as for eqns (2.7) and (2.11) ... 

A conjecture for the origin of birhythmicity, based on phase plane analysis... 

The conjecture for the origin of birhythmicity relies on the creation of a domain of stability within a domain of instability corresponding to the existence of a large-amplitude periodic solution (see section 3.2). The recycling of product into substrate provides a mechanism for the creation of such a zone of stabiUty in the core of an oscillatory domain. 

Birhythmicity coexistence and transition between two simultaneously stable periodic regimes... 

Let us now see how the nullcline deformation due to recycling gives rise to birhythmicity. Shown in fig. 3.6 are the bifurcation diagrams obtained as a function of parameter v for eight increasing values of the maximum rate of recycling, a , for a fixed value of constant K. In each part, the steady-state value of the substrate ao is indicated, as well as the maximum value qim reached by the substrate during oscillations. Solid lines denote stable steady-state or periodic solutions both types of solution are indicated by dashed lines when unstable. The stability properties of the steady state were determined by linear stability analysis of eqns... 

Fig. 3.6. Effect of the recycling reaction on birhythmicity. A series of bifurcation diagrams are represented for increasing values of the maximum rate of product recycling, o- (in s ) (a) 0 (b) 0.5 (c) 0.6 (d) 1.2 (e) 1.3 (f) 1.4 (g) 1.5 (h) 2. Each diagram shows the steady-state concentration of substrate, o, and the mtiximum concentration of substrate in the course of oscillations, < m> s a function of parameter (qvlk ) equal to the steady-state concentration of product. The curve yielding the steady-state level of substrate is therefore identical with the product nullcline. The sohd and dashed lines denote, respectively, stable and unstable branches of periodic or steady-state solutions. Parameter values are <7=10s, L = 5xl0, /iC=10, m = 4, q = l, k = 0.06s. Periodic re mes were obtained by numerical integration of eqns (3.1). The stability properties of the steady state were determined by Unear stabUity analysis. Birhythmicity is apparent in (d)-(f), while in (h) two distinct instabiUty domains appear as a function of parameter v (Moran Goldbeter, 1984).

Further increase in the rate of product recycling gives rise to a supercritical Hopf bifurcation corresponding to the creation of a stable, small-amplitude limit cycle. This phenomenon occurs at the border of the stability domain induced by recycling, inside the domain of existence of the large-amplitude limit cycle (fig. 3.6d). Birhythmicity arises from the coexistence of these two stable limit cycles. 

A typical example of birhythmicity in the phase plane is illustrated in fig. 3.7. The large-amplitude limit cycle encloses a limit cycle of more reduced amplitude these two stable cycles are separated by an unstable limit cycle. Birhythmicity is of interest because it allows the existence of two distinct oscillations in the same conditions, i.e. for a given set of parameter values. Moreover, the passage from one stable rhythm to the other can be effected by means of the same type of perturbation. Thus,... 

Fig. 3.7. Birhythmicity. In the phase plane, two stable limit cycles (solid lines) are separated by an unstable limit cycle (dotted line). The situation is that of the bifurcation diagram of fig. 3.6e, with v = 0.255 s, i.e. (qvlk ) = 4.25. Vertical arrows indicate how to switch from one stable cycle to the other, by adding substrate (Moran Goldbeter, 1984).

The qualitative agreement between the biochemical system and the behaviour of thalamic cells suggests that birhythmicity, predicted by the model, could also occur in these neurons. To demonstrate such a phenomenon, each type of oscillation should be perturbed by depolarization or hyperpolarization, in order to determine whether the perturbation can elicit the switch from one rhythm to the other. Alternatively, an increase followed by a decrease in the applied current could reveal the existence of hysteresis two different rhythms would then be observed at a given value of the membrane potential. Such a procedure, suggestive of birhythmicity, has been followed in another neuronal system (Hoimsgaard et ai, 1988). 

The autocatalytic biochemical model with recycling of product suggests that birhythmicity could occur in neurons of both the inferior olive and the thalamus. A detailed appraisal of the role played by the oscillatory dynamics of the olivary neurons in the coordination of motor control has recently been given by Welsh et al. (1995). 

Oscillatory isozymes another two-variable model for birhythmicity... 

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