We investigated the chaotic dynamics underlying the electroencephalogram (EEG) in patients with Alzheimer's disease by nonlinear methods to understand the role of chaos in brain function. In the analysis, we calculated the correlation dimension $D_2$ and the largest Lyapunov exponent $L_1$. A new method, proposed by Kennel et al., for calculating nonlinear invariant measures was used. The method determines the proper minimum embadding dimension by looking at the behavior of nearest neighbors under changes in the embedding dimension d from d to d+1. We showed that it is strikingly faster and more accurate than other algorithms for limited noisy data. We found that, in almost all channels, patients with Alzheimer's disease have significantly lower $D_2$ and $L_1$ than age-approximated non-dimented controls. It is, therefore, inferred that brains injured by Alzheimer's disease have electrophysiologically inactive elements (i.e., neurons and/or synapses) and thus show decreased chaotic behavior. These results support the assumption that chaos plays an important role in brain function, for instance, learning and memory. We suggest that brains can be described by deterministic models. In this paper we show that nonlinear analysis can provide a promising tool for detecting relative changes in the complexity of brain dynamics, which cannot be detected by conventional linear analysis. We propose a nonlinear analysis of the EEG in Alzheimer's disease for diagnosis as a clinical application.
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