A dimensional analysis on chaotic neural networks

We apply a dimensional analysis to a neural network model with chaotic dynamics, or a mathematical model of chaotic neural networks. First, Lyapunov spectra are examined to confirm existence of chaotic network behavior and to calculate the Lyapunov dimension. Then, the correlation dimension is estimated in the following three cases: (1) reconstruction from single variable time series, (2) observation in state subspaces and (3) observation of the macroscopic average activity. The results show good correlation dimension obtained by observation in state subspaces and the characteristics of the correlation dimension estimated by macroscopic observation.

[1]  M. Sano,et al.  Global Spectral Characterization of Chaotic Dynamics , 1986 .

[2]  Walter J. Freeman,et al.  Strange attractors that govern mammalian brain dynamics shown by trajectories of electroencephalographic (EEG) potential , 1988 .

[3]  J. Yorke,et al.  Chaotic behavior of multidimensional difference equations , 1979 .

[4]  K. Aihara,et al.  Chaotic neural networks , 1990 .

[5]  L. Chua,et al.  Chaos: A tutorial for engineers , 1987, Proceedings of the IEEE.

[6]  A. Babloyantz,et al.  Low-dimensional chaos in an instance of epilepsy. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[7]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[8]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[9]  A. Babloyantz,et al.  Evidence of Chaotic Dynamics of Brain Activity During the Sleep Cycle , 1985 .

[10]  I. Dvor̂ak,et al.  On some problems encountered in the estimation of the correlation dimension of the EEG , 1986 .

[11]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

[12]  GOTTFRIED MAYER‐KRESS AND,et al.  Dimensionality of the Human Electroencephalogram , 1987, Annals of the New York Academy of Sciences.

[13]  Theiler,et al.  Spurious dimension from correlation algorithms applied to limited time-series data. , 1986, Physical review. A, General physics.