Chaotic bursts

Intennittency, in tire context of chaotic dynamical systems, is characterized by long periods of nearly periodic or Taminar motion interspersed by chaotic bursts of random duration [28]. Witliin tliis broad phenomenological... 

The jS-cell model displays chaotic dynamics in the transition intervals between periodic spiking and bursting and between the main states of periodic bursting. A careful description of the bifurcation diagram involves a variety of different transitions, including Hopf and saddle-node bifurcations, period-doubling bifurcations, transitions to inter-mittency, and homoclinic bifurcations. 

Y.-S. Fan andT. R. Chay Generation of periodic and chaotic bursting in an excitable cell model. Biol. Cybern. 1994, 71 417-431. 

X.-J.Wang Genesis of bursting oscillations in the Hindmarsh-Rose model and ho-moclinicity to a chaotic saddle. Physica D 1993, 62 263-274. 

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

Ishizuka, S. and Hayashi, H., Spontaneous epileptiform bursts and longterm potentiation in rat CA3 hippocampal slices induced by chaotic stimulation of mossy fibers, Brain Research, Vol. 790, No. 1-2, 1998, pp. 108-114. 

In experimental systems, intermittency appears as nearly periodic motion interrupted by occasional irregular bursts. The time between bursts is statistically distributed, much like a random variable, even though the system is completely deterministic. As the control parameter is moved farther away from the periodic window, the bursts become more frequent until the system is fully chaotic. This progression is known as the intermittency route to chaos. 

The intensity of the emitted laser light is plotted as a function of time. In the lowest panel of Figure 10.4.5, the laser is pulsing periodically. A bifurcation to intermittency occurs as the system s control parameter (the tilt of the mirror in the laser cavity) is varied. Moving from bottom to top of Figure 10.4.5, we see that the chaotic bursts occur increasingly often. 

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... 

Is sand bursting in conventional wells and shear events in conventional wells chaotic Is it linked to stress redistributions and how do we predict longterm behavior Microseismic monitoring could help quantify these mechanical coupling issues. 

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. 

Table 4.4 shows how the main patterns of bursting occur in the model when parameter moves across the domain of complex periodic oscillations. Bursting occurs after the system has passed a domain of chaotic behaviour, which is itself reached beyond a cascade of period-doubling bifurcations issued from a simple periodic solution. A few narrow windows of chaos separate the first patterns of bursting observed. 

The piecewise linear map does not account, however, for the appearance of chaotic behaviour. A slight modification of the unidimensional map, taking into account some previously neglected details of the Poincare section of the differential system, shows how chaos may appear besides complex periodic oscillations of the bursting type. 

Hayashi, H. S. Ishizuka. 1992. Chaotic nature of bursting discharges in the Onchidium pacemaker neuron. J. Theor. Biol. 156 269-91. 

---