Doublet-split-scaling of correlation integrals in non-linear dynamics and in neurobiology

The search for the low-dimensional attractor behaviour and the dynamic self-organization of neuronal systems, from an analysis of electroencephalographic (EEG) signals, must be carried out under conditions in which the signals are not stationary for more than a few seconds. We employ a technique that we have introduced for analyzing short signals obeying a differential equation and develop it further. The technique uses the fact that in plots of "slope curves" of d log C(r)/d log r against log C(r), C(r) = correlation integral, for short time sequences, the dynamics may be "trans-embedding-scaled", i.e. a horizontal power-law structure builds up, that is constructed from different slope curves (different embeddings), and appears at the right value of the correlation dimension, although no single slope curve exhibits scaling. Patterns of the family of slope curves are described exhibiting the "doublet-split-scaling" of the correlation integrals. Examples include a solution of the Mackey and Glass delay differential equation and EEG signals. The two components of a doublet differ in the dimensions of the embeddings of which they are formed, i.e. low- and high-dimensions, respectively. The advantages subsequent to recognizing trans-embedding-scaled correlation integrals and doublet-split-scaling are illustrated for EEG delta sleep signals, with emphasis on ideal doublet-split-scaling. Unambiguous evidence of attractor behaviour in delta sleep is presented.