Aperiodic oscillations of cAMP chaos

As indicated by the bifurcation diagrams established as a function of parameters v (- pv ) and k, chaos can occur in the three-variable model for cAMP signalling. The phenomenon is also seen in the seven-variable version of the model for parameter values close to those producing bursting in fig. 6.1, the numerical integration of eqns (6.2) indeed shows the existence of aperiodic oscillations (fig. 6.11). The irregularity of these sustained oscillations shows up in the varying amplitude of the cAMP peaks, but mostly in the intervals between successive peaks. As 

In the phase plane (p, a, 13), the projection of the trajectory followed by the seven-variable system (6.2) takes the form of a strange attractor of which fom successive states are shown in fig. 6.12. The system remains confined within a portion of the phase space, but the curve it follows in the course of oscillations never passes twice through any given point. This is one way by which this behaviour differs from complex periodic oscillations. Some cycles on the strange attractor are 

When the maximum of a peak of intracellular cAMP in fig. 6.11 is plotted as a function of the maximum of the preceding peak, we obtain a return map yielding )8 +i as a function of /3 . This one-dimensional 

Another indication of the occurrence of chaos is given by the route leading to this irregular oscillatory behaviour in parameter space. Aperiodic oscillations indeed arise in the model after a cascade of period-doubling bifurcations as a function of parameter v, which measures the net rate of ATP supply to the adenylate cyclase reaction site. The successive values of parameter v corresponding to these bifurcations obey the universal scheme described by Feigenbaum (1978) for the onset of chaos. Thus, the values of v (in s ) associated with the first three period doublings, from period 1 to period 8, are as follows  

Whereas the above data were obtained in the seven-variable version of the model, chaos has also been foimd in the reduced version (6.3) containing only three variables, namely pr, a and y. More details about these results are given at the end of this chapter. 

---