Analysis of bursting and birhythmicity in a two-variable system

The dynamics of the fast subsystem (pj, y), in which a is now a parameter, is governed by  

The situation to be analysed by means of this reduction differs slightly from that of fig. 6.6. While the small limit cycle was contained within the large cycle there, it is located outside the latter cycle in the case considered below. The two oscillatory regimes that coexist in the phase plane in fig. 6.8 are represented as a function of time in fig. 6.9. Here again the large cycle is of the bursting type, with only two peaks per period. 

The curve yielding Jq as a function of a shows the existence of a phenomenon of bistability in the reduced system when the substrate concentration is held constant. Three distinct values of yo are obtained in fig. 6.10 in the range a a a 2. The linear stability analysis of eqns (6.5) reveals that the steady state on the lower branch of the hysteresis curve is always stable, while it is unstable on the median branch, between points Sj and S2. On the upper branch, the steady state is unstable in the domain ai a 2. Two families of periodic solutions, denoted Tj and Fj, appear through a Hopf bifurcation at the points Hj and H2 they disappear at the points H and H 2, of abscissae a l, a 2, when the amplitude of the limit cycle is such that the latter reaches the 

The dynamics of the full, three-variable system (6.3) can be comprehended in terms of the bifurcation diagram obtained for the reduced system, as soon as the substrate is considered as a slow variable rather than as a parameter whose value would remain fixed in the course of time. When the unique steady state admitted by the equations is unstable, the complete system moves along the lower branch of the hysteresis curve in fig. 6.10, starting from a low level of extracellular cAMP. Since adenylate cyclase then operates at a reduced activity, the net rate of substrate input (equal, here, to the difference between the rate of synthesis of ATP and the rate of its utilization in reactions other than that catalysed by adenylate cyclase) exceeds the rate of substrate consumption in the enzyme reaction thus v o- in the evolution equation of a in system (6.3). The substrate therefore slowly accumulates, and the sys- 

When the limit point 2 is reached, an increase in a elicits the abrupt transition towards the upper branch of the hysteresis curve, given that the lower branch has now vanished. But the upper branch of the steady-state curve of the (pr, y) system is unstable, and the analysis of the reduced system predicts that oscillations belonging to the branch of periodic solution F2 should occur when a is close to the value a 2. These oscillations correspond to the active phase of bursting represented in fig. 6.9b. 

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