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Geometric models, fiber bundles and biomedical applications. (English) Zbl 1093.92048

Nikitin, A.G.(ed.) et al., Proceedings of the fifth international conference on symmetry in nonlinear mathematical physics, Kyïv, Ukraine, June 23–29, 2003. Part 3. Kyïv: Institute of Mathematics of NAS of Ukraine (ISBN 966-02-3227-6). Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Mathematics and its Applications. 50(3), 1518-1525 (2004).
A geometric approach to studying the physiological disturbances that occur in human epilepsy is proposed. The generic situation of the brain dynamics is instability of the trajectories in the Lyapunov sense. At first, to characterize the dynamical instability, the authors examine some properties that characterize the EEG signal, including the spectrum of Lyapunov exponents, the energy, geodesics and fiber bundles. Then the geometric description of dynamical instability is given as the instability of geodesics of a suitable manifold. Such a geodesic flow is described by the Jacobi-Levi-Civita (JLC) equation for geodesic spread. Then a geometric approach is sequentially applied for characterizing dynamical instabilities, based on nonlinear estimation of dynamical characteristics. In addition to characterizing epileptic seizures, the authors discuss other diagnostically useful topics including the \(T\)-index of the short-term maximum Lyapunov exponents (STLmax) among critical sites of the cerebral cortex. The methods are illustrated using EEG data previously recorded from transgenic epileptic mouse.
For the entire collection see [Zbl 1088.17002].

MSC:

92C50 Medical applications (general)
92C20 Neural biology
58A99 General theory of differentiable manifolds
37N25 Dynamical systems in biology
92C05 Biophysics
92C55 Biomedical imaging and signal processing
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