Some Remarkable Properties of a Hopfield Neural Network with Time Delay
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51881", author = "Kelvin Rozier and Vladimir E. Bondarenko", title = "Some Remarkable Properties of a Hopfield Neural Network with Time Delay", 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", volume = "6", number = "5", pages = "515-6", }