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Four dimensional chaos and intermittency in a mesoscopic model of the electroencephalogram
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Image of FIG. 1.
FIG. 1.

Time series ((a) and (b)) and spectra ((c) and (d)) for and for the parameter set under investigation. The broadband nature of the power spectrum is evident in panels (c) and (d), which are clearly indicative of chaotic dynamics. Note the presence of dominant rhythms and higher harmonics alongside the broadband power spectral activity, as well as the significant amount of power at high frequencies, especially in the band 80–100 Hz.

Image of FIG. 2.
FIG. 2.

Phase plot for and for the parameter set under investigation, with . The shape is peculiarly blurred, with a persistent shadow of points originating from the supporting orbit around which the chaotic activity develops.

Image of FIG. 3.
FIG. 3.

Relevant section of the one parameter bifurcation plot in . In (a), two Hopf points (triangles) are present in the bottom right corner of the figure, from which orbits with different properties emerge. Different colors indicate different branches, stable (unstable) branches are continuous (dashed), whereas orbit bifurcations are indicated by circles (period doubling, PD), squares (saddle-nodes on limit cycle, SNLC), and diamonds (torus, TR). Both period doubling cascades are in black, PD points are in red, and generations (up to three) are indicated by subscripts. The ordinate is given by the maxima of , hence only the top halves of the familiar pitchfork shapes of period doubling cascades is presented. The first, second, and third generations of these cascades are displayed. In (b), a closer detailed depiction of the two inverse period doubling cascades is provided. The stable attractor that participates in the intermittency phenomenon described in this paper belongs to the stable branch in green in the interval and , between the two SNLC points.

Image of FIG. 4.
FIG. 4.

In (a) and (b), the stable orbits from the fourth generation branch in the period doubling cascade PDC1 are shown, for and . Orbits are scaled so that the intrinsic period of the oscillation is  = 1. Notice the similarity with part of the chaotic time series in Fig. 1 . In (c) and (d), examples of intermittency between the chaotic attractor and the stable orbit, for are illustrated. The intervals of small amplitude oscillations correspond to the system being on the stable attractor, and large amplitude bursts are due to the chaotic attractor.


Generic image for table
Table I.

Lyapunov spectrum for the dynamics under investigation for the Liley model. The mean and standard deviation of each Lyapunov exponent was obtained from 25 independent simulation runs each started from different, random, initial conditions. For further details refer to the Methods section.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Four dimensional chaos and intermittency in a mesoscopic model of the electroencephalogram