Chaotic Behavior of a Class of Neural Network with Discrete Delays

Affiliation(s)

Department of Mathematics, JIS College of Engineering, Kalyani, India.

Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata, India.

Department of Mathematics, JIS College of Engineering, Kalyani, India.

Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata, India.

ABSTRACT

In this paper, the effect of neuronal gain in discrete delayed neural network model is investigated. It is observed that such neural networks become highly chaotic due to the presence of high neuronal gain. On the basis of the largest Lyapunov exponent and largest eigenvalue of Jacobian matrix, chaos analysis has been done. Finally, some numerical simulations are presented to justify our results.

Cite this paper

S. Mandal, D. Jana, A. Roy and N. Majee, "Chaotic Behavior of a Class of Neural Network with Discrete Delays,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 1, 2013, pp. 97-101. doi: 10.4236/ijmnta.2013.21A012.

S. Mandal, D. Jana, A. Roy and N. Majee, "Chaotic Behavior of a Class of Neural Network with Discrete Delays,"

References

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[5] M. Arsiero, C. Lüscher and G. Battaglini, “Gaze-Dependent Visual Neurons in Area V3A of Monkey Prestriate Cortex,” Journal of Neuroscience, Vol. 9, 1989, pp. 1112-1125.

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[11] G. R. Chen, J. Zhou and Z. R. Liu, “Global Synchronization of Coupled Delayed Neural Networks and Applications to Chaotic CNN Models,” International Journal of Bifurcation and Chaos, Vol. 14, No. 7, 2004, pp. 2229-2240. doi:10.1142/S0218127404010655

[12] H. T. Lu, “Chaotic Attractors in Delayed Neural Networks,” Physical Letters A, Vol. 298, No. 2-3, 2002, pp. 109-116. doi:10.1016/S0375-9601(02)00538-8

[13] M. Gilli, “Strange Attractors in Delayed Cellular Neural Networks,” IEEE Transactions on Circuits Systems I Fund Theory Applications, Vol. 40, No. 11, 2006, pp. 849-853. doi:10.1109/81.251826

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[15] X. D. Li and M. Bohner, “Exponential Synchronization of Chaotic Neural Networks with Mixed Delays and Impulsive Effects via Output Coupling with Delay Feedback,” Mathematical and Computer Modelling, Vol. 52, No. 5-6, 2010, pp. 643-653. doi:10.1016/j.mcm.2010.04.011

[16] C. M. Marcus and R. M. Westervelt, “Stability of Analog Neural Networks with Delay,” Physical Review A, Vol. 39, No. 1, 1989, pp. 347-359. doi:10.1103/PhysRevA.39.347

[17] Z. Liu, A. Chen, J. Cao and L. Huang, “Existence and Global Exponential Stability of Periodic Solution for BAM Neural Networks with Periodic Coefficients and Time-Varying Delays,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 50, No. 9, 2003, pp. 1162-1172.

[18] C. Sun and C. Feng, “On Robust Exponential Periodicity of Interval Neural Networks with Delays,” Neural Processing Letters, Vol. 20, No. 1, 2004, pp. 53-61. doi:10.1023/B:NEPL.0000039426.58277.7e

[19] N. C. Majee, “Studies Convergent and Non-Convergent Dynamics of Some Continuous Artificial Neural Network Models with Time-Delay,” Thesis of Doctor of Philosophy, Jadavpur University, Kolkata, 1997.

[20] J. D. Meiss, “Differential Dynamical Systems,” 2007.

[1] S. Mitchell and R. A. Silver, “Shunting Inhibition Modulates Neuronal Gain during Synaptic Excitation,” Neuron, Vol. 38, No. 3, 2003, pp. 433-445. doi:10.1016/S0896-6273(03)00200-9

[2] M. Brozovic, L. F. Abbott and R. A. Andersen, “Mechanism of Gain Modulation at Single Neuron and Network Levels,” Journal of Computing Neuroscience, Vol. 25, No. 1, 2008, pp. 158-168. doi:10.1007/s10827-007-0070-6

[3] R. A. Andersen and V. B. Mountcastle, “The Influence of the Angle of Gaze upon the Excitability of the Light-Sensitive Neurons of the Posterior Parietal Cortex,” Journal of Neuroscience, Vol. 3, 1983, pp. 532-548.

[4] R. A. Andersen, G. K. Essick and R. M. Siegel, “The Encoding of Spatial Location by Posterior Parietal Neurons,” Science, Vol. 230, No. 4724, 1985, pp. 456-458. doi:10.1126/science.4048942

[5] M. Arsiero, C. Lüscher and G. Battaglini, “Gaze-Dependent Visual Neurons in Area V3A of Monkey Prestriate Cortex,” Journal of Neuroscience, Vol. 9, 1989, pp. 1112-1125.

[6] F. Bremmer, U. J. Ilg, A. Thiele, C. Distler and K. P. Hoffmann,” Eye Position Effects in Monkey Cortex. I. Visual and Pursuit-Related Activity in Extrastriate Areas MT and MST,” Journal of Neurophysiology, Vol. 77, 1997, pp. 944-961.

[7] E. Salinas and T. J. Sejnowski, “Gain Modulation in the Central Nervous System: Where Behavior, Neurophysiology and Computation Meet,” The Neuroscientist, Vol. 7, No. 5, 2001, pp. 430-440. doi:10.1177/107385840100700512

[8] M. H. Higgs, S. J. Slee and W. J. Spain, “Diversity of Gain Modulation by Noise in Neocortical Neurons: Regulation by the Slow after Hyperpolarization Conductance,” The Journal of Neuroscience, Vol. 26, No. 34, 2006, pp. 8787-8799. doi:10.1523/JNEUROSCI.1792-06.2006

[9] K. Aihara, T. Takabe and M. Toyoda, “Chaotic Neural Networks,” Physical Letter A, Vol. 144, No. 6-7, 1990, pp. 333-340. doi:10.1016/0375-9601(90)90136-C

[10] Y. Q. Zhang and Z.-R. He, “A Secure Communication Scheme Based on Cellular Neural Networks,” Proceed- ings of the IEEE International Conference on Intelligent Process Systems, Vol. 1, 1997, pp. 521-524.

[11] G. R. Chen, J. Zhou and Z. R. Liu, “Global Synchronization of Coupled Delayed Neural Networks and Applications to Chaotic CNN Models,” International Journal of Bifurcation and Chaos, Vol. 14, No. 7, 2004, pp. 2229-2240. doi:10.1142/S0218127404010655

[12] H. T. Lu, “Chaotic Attractors in Delayed Neural Networks,” Physical Letters A, Vol. 298, No. 2-3, 2002, pp. 109-116. doi:10.1016/S0375-9601(02)00538-8

[13] M. Gilli, “Strange Attractors in Delayed Cellular Neural Networks,” IEEE Transactions on Circuits Systems I Fund Theory Applications, Vol. 40, No. 11, 2006, pp. 849-853. doi:10.1109/81.251826

[14] Y. Q. Yang and J. D. Cao, “Exponential Lag Synchronization of a Class of Chaotic Delayed Neural Networks with Impulsive Effects,” Physica A, Vol. 386, No. 1, 2007, pp. 492-502. doi:10.1016/j.physa.2007.07.049

[15] X. D. Li and M. Bohner, “Exponential Synchronization of Chaotic Neural Networks with Mixed Delays and Impulsive Effects via Output Coupling with Delay Feedback,” Mathematical and Computer Modelling, Vol. 52, No. 5-6, 2010, pp. 643-653. doi:10.1016/j.mcm.2010.04.011

[16] C. M. Marcus and R. M. Westervelt, “Stability of Analog Neural Networks with Delay,” Physical Review A, Vol. 39, No. 1, 1989, pp. 347-359. doi:10.1103/PhysRevA.39.347

[17] Z. Liu, A. Chen, J. Cao and L. Huang, “Existence and Global Exponential Stability of Periodic Solution for BAM Neural Networks with Periodic Coefficients and Time-Varying Delays,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 50, No. 9, 2003, pp. 1162-1172.

[18] C. Sun and C. Feng, “On Robust Exponential Periodicity of Interval Neural Networks with Delays,” Neural Processing Letters, Vol. 20, No. 1, 2004, pp. 53-61. doi:10.1023/B:NEPL.0000039426.58277.7e

[19] N. C. Majee, “Studies Convergent and Non-Convergent Dynamics of Some Continuous Artificial Neural Network Models with Time-Delay,” Thesis of Doctor of Philosophy, Jadavpur University, Kolkata, 1997.

[20] J. D. Meiss, “Differential Dynamical Systems,” 2007.