The correlation dimension as a nonlinear invariant has been applied to EEG and calculated since its appropriate algorithm was suggested by Grassberger and Procaccia. Whenever finite results were obtained by applying GPA, they have been accepted as the strong evidence that EEG is an outcome of a deterministic nonlinear dynamical system or a chaos. But later a filtered noise was known to mimic low-dimensional chaotic attractors, i.e., a filtered noise gives rise to a finite correlation dimension (D2) when examined by the Grassberger-Procaccia algorithm alone.
Here we give an account of how the usual nonlinear invariants such as dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions) are mathematically defined and derived from the differentiable dynamical system. In addition, the relations between them, as well as their experimental determination, are summarized.
Based on mathematically derived nonlinear statistics, a statistical method, named by surrogate test, is suggested and implemented for identifying nonlinearity and determinism in EEG. This method first specifies some linear (stochastic) process as a null hypothesis, and then generates surrogate data sets which are supposed to be consistent with this null hypothesis. After computing discriminating statistics such as the nonlinear invariant measures described above for the original and the surrogate data sets, we compare the value computed for the original data with the ensemble of values computed for the surrogate data sets. A nonlinearity is present in the observed time series if a null hypothesis is statistically rejected.
In this study we obtained a statistically negative result for the nonlinearity of EEG when the characteristic exponent and correlation dimension based surrogate tests were done. It implies that EEG is a stochastic signal. However, there was an explicit limit in th...