Model updating and simulation of Lyapunov exponents

2004

Abstract:-In this paper we consider a nonlinear system with distributed parameters which can be described by a nonlinear lattice model with control functions. The problem of feedback control of Lyapunov exponents by finite dimensional influences is analyzed. We formulate conditions for feedback control in terms of eigenvalues. Using numerical simulations, we examine the controllability of the nonlinear lattice model. Key words: Control, Lyapunov exponents, lattice model, nonlinear dynamics, epileptic seizures.

IEEE transactions on bio-medical engineering, 2003

Epilepsy is a relatively common disease, afflicting 1%-2% of the population, yet many epileptic patients are not sufficiently helped by current pharmacological therapies. Recent reports have suggested that chaos control techniques may be useful for electrically manipulating epileptiform bursting behavior in vitro and could possibly lead to an alternative method for preventing seizures. We implemented chaos control of spontaneous bursting in the rat hippocampal slice using robust control techniques: stable manifold placement (SMP) and an adaptive tracking (AT) algorithm designed to overcome nonstationarity. We examined the effect of several factors, including control radius size and synaptic plasticity, on control efficacy. AT improved control efficacy over basic SMP control, but relatively frequent stimulation was still necessary and very tight control was only achieved for brief stretches. A novel technique was developed for validating period-1 orbit detection in noisy systems by f...

Physical Review Letters, 1998

IEEE Transactions on Biomedical Engineering, 2004

We propose the use of artificial neural networks in an in silico epilepsy model of biological neural networks: 1) to predict the onset of state transitions from higher complexities, possibly chaotic to lower complexity possibly rhythmic activities; and 2) to restore the original higher complexity activity. A coupled nonlinear oscillators model (Bardakjian and Diamant, 1994) was used to represent the spontaneous seizure-like oscillations of CA3 hippocampal neurons (Bardakjian and Aschebrenner-Scheibe, 1995) to illustrate the prediction and control schemes of these state transition onsets. Our prediction scheme consists of a recurrent neural network having Gaussian nonlinearities. When the onset of lower complexity activity is predicted in the in silico model, then our control scheme consists of applying a small perturbation to a system variable (i.e., the transmembrane voltage) when it is sufficiently close to the unstable higher complexity manifold. The system state can be restored back to its higher complexity mode utilizing the forces of the system's vector field.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1999

A coupled ordinary differential equation lattice model for the CA3 region of the hippocampus (a common location of the epileptic focus) is developed. This model consists of a hexagonal lattice of nodes, each describing a subnetwork consisting of a group of prototypical excitatory pyramidal cells and a group of prototypical inhibitory interneurons connected via on/off excitatory and inhibitory synapses. The nodes communicate using simple rules to simulate the diffusion of extracellular potassium. Both the integration time over which a node’s trajectory is integrated before the diffusional event is allowed to occur and the level of inhibition in each node were found to be important parameters. Shorter integration times lead to total synchronization of the lattice (similar to synchronous neural activity occurring during a seizure) whereas longer times cause more random spatiotemporal behavior. Moderately diminished levels of inhibition lead to simple nodal oscillatory behavior. It is p...

Annals of Biomedical Engineering, 2000

Recent reports have suggested that chaos control techniques may be useful for electrically manipulating epileptiform bursting behavior in neuronal ensembles. Because the dynamics of spontaneous in vitro bursting had not been well determined previously, analysis of this behavior in the rat hippocampus was performed. Epileptiform bursting was induced in transverse rat hippocampal slices using three experimental methods. Slices were bathed in artificial cerebrospinal fluid containing: (1) elevated potassium ([K+]o= 10.5 mM), (2) zero magnesium, or (3) the GABAA-receptor antagonists bicuculline (20 microM) and picrotoxin (250 microM). The existence of chaos and determinism was assessed using two different analytical techniques: unstable periodic orbit (UPO) analysis and a new technique for estimating Lyapunov exponents. Significance of these results was assessed by comparing the calculations for each experiment with corresponding randomized surrogate data. UPOs of multiple periods were highly prevalent in experiments from all three epilepsy models: 73% of all experiments contained at least one statistically significant period-1 or period-2 orbit. However, the expansion rate analysis did not provide any evidence of determinism in the data. This suggests that the system may be globally stochastic but contains local pockets of determinism. Thus, manipulation of bursting behavior using chaos control algorithms may yet hold promise for reverting or preventing epileptic seizures.

2008

The master-slave synchronization problem for chaotic Lur'e systems is studied in this chapter based on time-delayed feedback control. It is assumed that the master system of the synchronization scheme is subject to noise disturbances. Delay-independent and delay-dependent synchronization criteria are presented such that the controlled slave system can robustly track the noise-disturbed master system with guaranteed H∞ performance. It is shown that the design of the time-delayed feedback controller can be accomplished by means of the feasibility of linear matrix inequalities. A simulation example is finally given to demonstrate the effectiveness and performance of the developed approaches.

The Journal of Chemical Physics, 1992

A simple proportional-feedback algorithm for controlling chaos is presented. The scheme is a map-based variation of a method recently proposed by Ott, Grebogi, and Yorke [ Phys. Rev. Lett. 64, 1196 (1990) 1 in which unstable periodic orbits embedded within a strange attractor are stabilized through deliberate perturbations of a system constraint. The simplified method offers advantages for control of systems in which more complicated algorithms might not be feasible due to short time scales or limited computational resources. Applications to chemical and biological models are presented to demonstrate the utility and limitations of the method. Low-dimensional chaos can usually be stabilized through proportional feedback of one parameter; in some cases, however, a linear combination of several parameters must be utilized.

Feature extraction and classification of electro-physiological signals is an important issue in development of disease diagnostic expert system (DDES). In this paper we propose a method based on chaos methodology for EEG signal classification. The nonlinear dynamics of original EEGs are quantified in the form of entropy, largest Lyapunov exponent (LLE), correlation dimension (CD), capacity dimension (CAD) and were considered for discrimination of various categories of EEG signals. After calculating the above mentioned parameters for signals, we found that without going for rigorous time-frequency domain analysis, only chaos based parameters is also suitable to classify various EEG signals.

Journal of Combinatorial Optimization, 2008

Epileptic seizures are manifestations of intermittent spatiotemporal transitions of the human brain from chaos to order. Measures of chaos, namely maximum Lyapunov exponents (STL max), from dynamical analysis of the electroencephalograms (EEGs) at critical sites of the epileptic brain, progressively converge (diverge) before (after) epileptic seizures, a phenomenon that has been called dynamical synchronization (desynchronization). This dynamical synchronization/ desynchronization has already constituted the basis for the design and development of systems for long-term (tens of minutes), on-line, prospective prediction of epileptic seizures. Also, the criterion for the changes in the time constants of the observed synchronization/desynchronization at seizure points has been used to show resetting of the epileptic brain in patients with temporal lobe epilepsy (TLE), a phenomenon that implicates a possible homeostatic role for the seizures themselves to restore normal brain activity. In this paper, we introduce a new criterion to measure this resetting that utilizes changes in the level of observed synchronization/desynchronization. We compare this criterion's sensitivity of resetting with the old one based on the time constants of the observed synchronization/desynchronization. Next, we test the robustness of the resetting phenomena in terms of the utilized measures of EEG dynamics by a comparative study involving STL max , a measure of phase (ϕ max) and a measure of energy (E) using both criteria (i.e. the level and time constants of the observed synchronization/desynchronization). The measures are estimated from intracranial electroencephalographic (iEEG) recordings with subdural and depth electrodes from two patients with focal temporal lobe epilepsy and a total of 43 seizures. Techniques from optimization theory, in particular quadratic bivalent programming, are applied to optimize the performance of the three measures in detecting preictal entrainment. It is shown that using either of the two resetting criteria, and for all three dynamical measures, dynamical resetting at seizures occurs with a significantly higher probability (α = 0.05) than resetting at randomly selected non-seizure points in days of EEG recordings per patient. It is also shown that dynamical resetting at seizures using time constants of STL max synchronization/desynchronization occurs with a higher probability than using the other synchronization measures, whereas dynamical