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Self-organized neuronal dynamics revealed by cortical alpha-rhythms occur as episodes, which are rarely observed without extraction of the alpha-band from the other spectral components. Three episodes of an unusually long duration of 10 s, two with no signal processing after data recording at the clinic, are described and show evidence of low-dimensional alpha-dynamics. The evidence is gained from an analysis of scaled structures appearing in families of slope curves of the correlation integrals and is checked against time reparametrization. The data for the two unprocessed 10-s episodes are used for a test of the methodology, as well as a re-examination of the adequacy of the model of an autonomous dynamic system in steady state and of the concept of an attractor in brain dynamics investigations. Striking evidence for the model's inadequacy is provided by the episode of subject S1. In this example five consecutive overlapping 6-s sections do show evidence for low-dimensional dynamics, whereas the 10-s section containing those sections does not. The episode of subject 1 provides an example of alpha-activity which may involve self-organized dynamics extending down to low frequencies. The system (the neuronal network) showing episodes of attractor-ruled dynamics, under conditions of blurred and smoothly fading out evidence that it stays on an attractor, is designated as being ruled by a 'shadow-attractor'. This concept is compared with that of a 'quasi-attractor' introduced by H. Haken in studies of physiological systems. One possible mechanism for the observed episodes is proposed, based on a time-dependent number of enslaved sub-systems.

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