The structure function as new integral measure of spatial and temporal properties of multichannel EEG

The first-order temporal structure functions (SFs), i.e., the first-order statistical moment of absolute increments of scaled multichannel resting state EEG signals in healthy children and teenagers over a wide range of temporal separation (time lags) are computed. Our research shows that the sill level (asymptote) of the SF is mainly defined by a determinant of EEG correlation matrix reflecting the EEG spatial structure. The temporal structure of EEG is found to be characterized by power-law scaling or statistical-scale invariance over time scales less than 0.028 s and at least by two dominant frequencies differing by less than 0.3 Hz. These frequencies define the oscillation behavior of the SF and are mainly distributed within the range of 7.5-12.0 Hz. In this paper, we propose the combined Bessel and exponential model that fits well the empirical SF. It provides a good fit with the mean relative error fitting of 2.8 % over the time lag range of 1 s, using a sampling interval of 4 ms, for all cases under analysis. We also show that the hyper gamma distribution (HGD) fits to the empirical probability density functions (PDFs) of absolute increments of scaled multichannel resting state EEG signals at any given time lag. It means that only two parameters (sample mean of absolute increments and relevant coefficient of variation) may approximately define the empirical PDFs for a given number of channels. A three-dimensional feature vector constructed from the shape and scale parameters of the HGD and the sill level may be used to estimate the closeness of the real EEG to the "random" EEG characterized by the absence of temporal and spatial correlation.