Low-dimensional dynamic self-organization in delta-sleep: effect of partial sleep deprivation.

Studies on extended data including 37 electroencephalographic (EEC) records of delta-sleep, each 10(3) s long (six subjects: up to seven nights per subject, comprising normal sleep, partial deprivation and recovery), confirmed earlier conclusions that rare episodes of low-dimensional dynamic self-organization, with life-times between 10 and 20 s, are present in stage 4 sleep. Particular care was taken of the Theiler correction which, in some delta-sleep signals, required the deletion of trajectory points covering nearly one pseudo-period. The percentage of segments showing an episode, i.e. the attractor probability, decreased with a change in sleep conditions either deprivation or recovery prior to the next deprivation. Repetition of deprivation over three nights resulted in an adaptation process, manifested by an increase in attractor probability. After the sharp decrease in probability observed when recovery was established prior to the next deprivation, and on return to normal conditions of sleep at 2200 hours, the probability was immediately close to that observed in normal baseline sleep conditions free of any interference. The observation of a definite effect of sleep deprivation and recovery upon the number of stage 4 attractors observed provides a line of approach to the physiological significance of the probability of such attractors.

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.