Testing the determinism of EEG and MEG

NEURAL MASS ACTIVITY produces irregular time series such as the EEG and MEG. It is already apparent, through visual inspection, that these time series cannot simply result ...J from an uncoordinated arbitrary firing of neurons. Indeed, we expect that neurons must cooperate and partially synchronize their firing patterns in order to produce meaningful output. Although we may believe that we have visually identified patterns within these irregular time series, attempts to systematically track the code with linear and stochastic statistical techniques have left us in frustration. More recently it has been suggested that the ability to trace the dynamics of a system (e.g., to freeze them in a state space, a space that is spanned by the system's variables) might decode more of the brain's cryptic and enigmatic language (Etbert, et aI., 1994). If the dynamics of the underlying system can be reduced to a set of deterministic laws, then the phase space trajectory will converge toward a subset of the phase-space. This invariant subset is referred to as an attractor. Given a particular time series, the initial question one may ask is if one can identify an attractor. If the answer is yes, then it is possible to view the series as a manifestation of a deterministic dynamic system (albeit possibly a very complex one). How can we gain information about the deterministic processes governing a particular nonlinear system? As a first step, it is possible to estimate the determinism inherent in a given time series. Kaplan & Glass (1992) developed a direct test for determinism in a given time series. This article introduces this method for analyzing EEG and MEG and presents comparisons with other nonlinear measures such as the fractal dimension. The ~ estimations of the fractal dimension of the EEG has received considerable attention starting with the studies of Agnes Babloyantz (Babloyantz, 1985; Babloyantz & Destexhe, 1986; Babloyantz, et aI., 1985. See Elbert, et aI., 1994, for a recent review). In principle, such an estimation of the fractal dimension of the system generating a time series requires a "sufficient number" of data points. Estimations of d-dimensional systems reach up to more than 10d points. When dimensions are estimated from physiological time series containing less than 10,000-20,000 points, the resulting values cannot be accepted as absolute terms. Results can only be interpreted when comparisons between different time series are made, such as those between conditions or groups. Furthermore, data are nonstationary, and the measures presume that the data generator does not change. Therefore, relative dif-