Bursting regime

Network Synchronization in Tonic, Chaotic and Bursting Regimes... 

Fig. 7.10 Computer simulation of two gap-junction coupled neurons which originally are operating at different dynamic states, one in the tonic firing regime and the other one in the bursting regime (also indicated in Fig. 6.8b by the points T and B, respectively with the arrow S pointing on the completely synchronized state), (a) Bifurcation diagrams of interspike intervals (ISI) of the originally...

A number of nonsteady polymerization rate techniques can be used to measure ftp [11]. The most widely used method involves pulsed-laser-induced polymerization in the low monomer conversion regime. Briefly, a mixture of monomer and photoinitiator (Section 6.5.3) is illuminated by short laser pulses of about 10 ns (10 sec) duration. The radicals that are created by this burst of ligh propagate for about 1 sec before a second laser pulse produces another crop of radicals. Many of the initially formed radicals will be terminated by the short, mobile radicals created in the second illumination. Analysis of the number molecular weight distribution of the polymer produced permits the estimation of ftp from the relation... 

Thermal Theories. Researchers at Forest Products Laboratory impregnated wood with a metal alloy to determine whether change in thermal conductivity is a mechanism of fire retardants (38). The alloy was selected to melt at 105 °C. The treated and untreated specimens were subjected to a flame on one side and the temperature rise was recorded on the unexposed side. The rise of temperature was slower over the alloy-treated specimen than over the untreated specimen until the melt temperature of the alloy. Above this temperature the treated and untreated specimens then followed the same time-temperature regimes. The untreated specimen burst into... 

For an introduction of the delayed feedback control of collective synchrony we consider suppression of the mean field in an ensemble of A1 = 10000 identical Hindmarsh-Rose neurons (Eqs. (13.5)) in the regime of chaotic bursting. The dynamics of the ensemble is described by the following set of equations,... 

We have used two types of feedback direct control C = X t — t) and differential control C = X t — t) — X t)). The parameters in Eqs. (13.4) are chosen in such a way that individual units are in the regime of chaotic bursting. The efficiency of suppression is quantified by the suppression factor... 

As indicated by the above bifurcation diagrams, the three-variable system (6.3) is capable of displaying different modes of simple or complex oscillatory behaviour. One additional mode is that of birhythmicity for certain values of the parameters, eqns (6.3) admit a coexistence between two simultaneously stable periodic regimes. In the phase plane (pr, a, y), these two regimes correspond to two limit cycles, one of which possesses a smaller amplitude and the second the folded appearance characteristic of bursting (fig. 6.6). 

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. 

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. 

Within this restrictive framework of two-variable models, Albert Goldbeter derives fascinating original results such as birhythmicity, which allows a system to choose between two simultaneously stable oscillatory regimes. With the number of variables, the repertoire of dynamic phenomena increases rapidly. Now, besides simple periodic behaviour we can also predict and observe complex oscillations of the bursting type, the coexistence between more than two rhythms, or the evolution toward chaos. As the author shows, small variations in the values of some control parameters permit the switch from one mode of behaviour to the other. The essential elements, in all cases, are the feedback mechanisms of biochemical reactions and the fact that these reactions occur far from equilibrium. 

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