Selecting parameters for phase space reconstruction of the electrocorticogram (ECoG)

The selection of parameters for phase space reconstruction of empirically observed data has been a source of criticism when estimating the correlation dimension (D2) from observed data rather than from the solution of differential equations, when analyzing noisy and potentially non-stationary signals, such as the electroencephalogram (EEG). The largely arbitrary selection of the time-delay reconstruction (T) of temporal dynamics, and for the embedding (M) of these series, has been widely criticized. This study adopted an analytic and statistical framework within which the scaling behavior of D2 with respect to T and M, could be examined over five data lengths (N = 4096, 8192, 12288, 16384, and 20480) over an 8 x 8 grid of cat EEG. It was found that D2 was invariant over all data lengths only within a very narrow T range (T = 10-16) for M = 4. A statistically significant T by M interaction was found using multiple analysis of variance, with D2 being highly correlated over T as a function of M. Finally, an examination of phase-randomized surrogates indicated that statistically significant differences existed between EEG and phase-randomized surrogates over all data lengths, with time delays (T = 10-16), indicating that the D2 for EEG is phase-dependent when it is invariant with respect to data length. The implications of these findings are discussed with respect to current models of ECoG generation, and their implication with respect to the integration in the brain.