Reconstruction of Chaotic Dynamical Systems using MLP by Seongchong Park Department of Physics

1. Reconstruction of Chaotic Dynamical Systems using MLP by Seongchong Park Department of Physics KAIST

2. The Aim of This Study • Inverse problem • How much information on brain dynamics we can extract from EEG? • We hope to reconstruct the underlying dynamics from time series generated from the original systems. • The original systems can be examined by investigating dynamical properties of the simulated signals from reconstructed dynamics. • Time series from real systems • noisy and short data length • non-stationary • Simulated signals from MLP • noise-free • unlimited length • controllable

3. The Characteristics of Chaotic Systems • Governing rules • Nonlinear Differential equations • Nonlinear mapping functions • Deterministic • Unpredictable • sensitive dependence on initial conditions • Fractal Structure • self-similarity

4. Nonlinear measures Correlation Dimension • A metric property of the attractor that estimates the degree of freedom of the time series signal • The number of independent variables which are necessary to describe the dynamics of the time series signal • Systems of deterministic chaos have non-integer dimensions(fractals) : strange attractor. • A measure of complexity of the system

5. Nonlinear measures The First Positive Lyapunov Exponent • Lyapunov exponents estimate the mean exponential divergence of nearby trajectories of the attractor in the phase space • The measure of the sensitive dependence on initial conditions • A dissipative dynamic system, such as brain, possessing at least one positive Lyapunov exponent is chaotic

6. Analysis of Chaotic Dynamic Systems Target System Deterministic Chaotic System Correlation Dimension The first Positive Lyapunov exponents States Time Series Attractor Embedding Procedure 1-dim --> high dimensional dynamics

7. EEG is a deterministic chaotic signal. • The EEG data from the human brain during the sleep cycle had chaotic attractors for sleep stages II and VI (Babloyantz and Salazar, 1985) • The EEG had a finite non-integer correlation dimension and a positive Lyapunov exponent, which means that the EEG is generated by a deterministic chaotic neural process (Babloyantz et al, 1985-1988). • The distinct states of brain activity can also have different chaotic dynamics quantified by nonlinear dynamical measures (Pritchard et al., 1993; Fell et al., 1995; Jeong et al., 1998; Lehnertz and Elger, 1998).

8. Reconstruction method using MLP • Multi-Layer Perceptron • general mapping function • large degrees of freedom • analyticity (differentiability) • well-defined fitting algorithm (BP) • MLP model(Discrete case) • MLP model(Continuous case)

9. Application (Henon map) • Network architecture • MLP(1-1-1) • input node : 2 (+ 1 bias term) • hidden node : 12 • output node : 2 • Henon map • training set

10. Application (Henon map) • Trajectories in the two dimensional phase space

11. Application (Henon map) • Time series generated from Henon map

12. Reconstructed dynamics Henon map • Trajectories in the reconstructed phase space

13. Reconstructed dynamics Henon map • Simulated Henon time series generated from MLP

14. Comparison of Dynamical properties • Dynamical similarity • L1 (the Largest Lyapunov exponent) • Original : 0.60 • Simulated : 0.64 • D2(Correlation dimension) • Original : 1.26 • Simulated : 1.31 • The simulated Henon time series have similar dynamics with the time series obtained from the original Henon map

15. Application • Continuous system (ODE) • EEG(Electroencephalogram)

16. Discussion • BP algorithm (learning steps) • 1st step, learning global structure : fast • 2nd step, learning local structure : very slow • In case of high dimensional chaotic systems, this algorithm may fail to learn local structure. • Future work • We would like to generate the simulated EEG which has similar dynamics with real EEG by using MLP.

17. Summary • The results of our method applied to Henon map showed that MLP can simulate the chaotic dynamics. • BP algorithm is good at learning the global structure of original trajectories, but not local properties.