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that procedures such exponentiation of a matrix can be carried out automatically using computer algebra programs such as Maple or for data processing available as computational modules in MatLab.

One of the techniques for the computation of hT involves determining the logarithm of the asymptotic growth rate of the major diagonal (“trace”) in the transition matrix symbolically encoding the trajectory which would therefore count the “self visitations” of each indexed boxes as the dynamics proceed. This involves setting up a transition incidence matrix, each box scored for a disallowed, 0, or allowed, 1, transitions and the matrix is exponentiated t times with the logarithm of the asymptotic growth rate of the sum of the diagonal values serving as a (leading eigenvalue) estimate of hT. More technical considerations involving the Frobenius-Perron theorem guaranteeing the existence of such an logarithmic index of new information generation rates, even in random matrices (Seneta, 1981), will not be discussed here.

We have found that computing hT in this way is empirically useful for difficult to obtain or only transiently stationary brain data series. Even with relatively short samples lengths, if one is willing to make the pragmatic assumption of “temporary stationarity” or “things as they are right now will, for the sake of argument, go on forever” (perhaps the best we can do with intrinsically transient brain phenomena) then this “freeze framed” representation of reality yields an asymptotic measure on relatively short sample lengths since they are computationally infinite. A similar approach to hM, requires repeatedly exponentiating a Markoff matrix constructed from relatively short samples and generates the probabilistic (eigenvector) “dual” of hT. hM computed in this way serves as a useful quantity, hM called by some the Kolmogorov entropy in comparisons of control and experimental conditions of the same sample lengths. Systematic decreases in hM (“Kolmogorov entropy”) have been shown to accompany increasing “depth” of sleep using standard sleep staging techniques (Gallez and Babloyantz, 1991) and increases in hM were associated with both positive and negative emotional states induced by movies (Aftanas et al, 1997).

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