Chris SACKELLARES,  Panos PARDALOS, Paul CARNEY
University of Florida, Gainesville,
FL 32611-6595, USA
E-mail: sackellares@epilepsy.health.ufl.edu, pardalos@ufl.edu, carnepr@peds.ufl.edu

Geometric models, fiber bundles and biomedical applications

Abstract:
The existence of complex chaotic, unstable, noisy and nonlinear dynamics in brain electrical activity requires new approaches to the study of brain dynamics. One approach is the combination of certain geometric concepts, control approach, and optimization. In this report we discuss the use of a differential geometric approach to the control of the Lyapunov exponents and the characterization of statistical information to predict and "correct" brain dynamics. This approach assumes that information about the physiological state comes in the form of a nonlinear time series with noise (the electrocencephalogram). The approach involves a geometric description of the Lyapunov exponents for the purpose of correcting of the nonlinear process that provides adaptive dynamic control. We separate the Lyapunov exponents into tangent space (fiber bundle) and its functional space. Control involves signal processing, calculation of information characteristics, measurement of the Lyapunov exponents, and feed-back to the system. We demonstrate the computational aspects of the proposed geometric approach on the base of different mathematical models in the presence of noise of various origins. We also review the EEG signal, and outline a typical application of the geometrical representation: three dimensional reconstruction of the Lyapunov exponents and correlation dimension obtained from EEG data. The novelty in this report is in the representation of dynamical and information characteristics in three dimensional vector space in a way that permits practical applications. We discuss an application of this approach to the development novel devices for seizure control though electromagnetic feed-back.