Some Remarkable Properties of a Hopfield Neural Network with Time Delay

Authors: Kelvin Rozier, Vladimir E. Bondarenko

Abstract:

It is known that an analog Hopfield neural network with time delay can generate the outputs which are similar to the human electroencephalogram. To gain deeper insights into the mechanisms of rhythm generation by the Hopfield neural networks and to study the effects of noise on their activities, we investigated the behaviors of the networks with symmetric and asymmetric interneuron connections. The neural network under the study consists of 10 identical neurons. For symmetric (fully connected) networks all interneuron connections aij = +1; the interneuron connections for asymmetric networks form an upper triangular matrix with non-zero entries aij = +1. The behavior of the network is described by 10 differential equations, which are solved numerically. The results of simulations demonstrate some remarkable properties of a Hopfield neural network, such as linear growth of outputs, dependence of synchronization properties on the connection type, huge amplification of oscillation by the external uniform noise, and the capability of the neural network to transform one type of noise to another.

Keywords: Chaos, Hopfield neural network, noise, synchronization

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1057831

References:

[1] C.-J. Cheng, T.-L. Liao, and C.-C. Hwang, "Exponential synchronization of a class of chaotic neural networks," Chaos, Solitons, and Fractals, vol. 24, pp. 197-206, Apr. 2005.
[2] C. M. Marcus and R. M. Westervelt, "Stability of analog neural networks with delay," Phys. Rev. A, vol. 39, pp. 347-359, Jan. 1989.
[3] J. Cao, P. Li, and W. Wang, "Global synchronization in arrays of delayed neural networks with constant and delayed coupling," Phys. Lett. A, vol. 353, pp. 318-325, May 2006.
[4] V. E. Bondarenko, "High-dimensional chaotic neural network under external sinusoidal force," Phys. Lett. A, vol. 236, pp. 513-519, Dec. 1997.
[5] V. E. Bondarenko, "Information processing, memories, and synchronization in chaotic neural network with the time delay," Complexity, vol. 11, pp. 39-52, Nov.-Dec. 2005.
[6] V. E. Bondarenko, "A simple neural network model produces chaos similar to the human EEG," Phys. Lett. A, vol. 196, pp. 195-200, Dec. 1994.
[7] V. E. Bondarenko, "Analog neural network model produces chaos similar to the human EEG," Int. J. Bifurcat. Chaos, vol. 7, pp. 1133- 1140, May 1997.
[8] V. E. Bondarenko, "Self-organization processes in chaotic neural networks under external periodic force," Int. J. Bifurcat. Chaos, vol. 7, pp. 1887-1895, Aug. 1997.
[9] L. M. Hively, V. A. Protopopescu, and P. C. Gailey, "Timely detection of dynamical change in scalp EEG signals," Chaos, vol. 10, pp. 864- 875, Dec. 2000.
[10] V. A. Protopopescu, L. M. Hively, and P. C. Gailey, "Epileptic event forewarning from scalp EEG," J. Clin. Neurophysiol., vol. 18, pp. 223- 245, May 2001.