Abstract
In the design of a neural network, either for biological modeling, cognitive simulation, numerical computation or engineering applications, it is important to describe the dynamics of the network. The success in this area in the early 1980’s was one of the main sources for the resurgence of interest in neural networks, and the current progress towards understanding neural dynamics has been part of exhaustive efforts to lay down a solid theoretical foundation for this fast growing theory and for the applications of neural networks.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Babcock, K.L. and Westevelt, R.M., Dynamics of simple electronic neural networks, Physica D, 28, 1987, 305–316.
Baldi, P. and Attyia, A.F., How delays effect neural dynamics and learning, IEEE Trans. Neural Networks, 5, 1994, 612–621.
Baptistini, M.Z. and Tdboas, P.Z., On the existence and global bifurcation of periodic solutions to planar differential delay equations, J. Differential Equations, 127, 1996, 391–425.
Bélair, J., Campbell, S.A., and van den Driessche, P., Frustration, stability and delay-induced oscillations in a neural network model, SIAM J. Appl. Math., 46, 1996, 245–255.
Cao, J. and Zhou, D., Stability analysis of delayed cellular neural networks, Neural Networks, 11, 1998, 1601–1605.
Chen, Y., Krisztin, T., and Wu, J., Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system, J. Differential Equations, 1999, to appear.
Chen, Y. and Wu, J., Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Advances in Differential Equations, 1998, to appear.
Chen, Y. and Wu, J., Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network, Physics D, 134, 1999a, 185–199.
Chen, Y. and Wu, J., The asymptotic shapes of periodic solutions of a singular delay differential system, J. Differential Equations, 1999b, to appear.
Chen, Y. and Wu, J., Slowly oscillating periodic solutions in a delayed frustrated network of two neurons, 1999c, preprint.
Chua, L.O. and Yang, L., Cellular neural networks: Theory, IEEE Trans. Circuits Syst. I, 35, 1988, 1257–1272.
Cohen, M.A. and Grossberg, S., Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Transactions on Systems, Man, and Cybernetics, 13, 1983, 815–826.
Fiedler, B. and Gedeon, T., A class of convergence neural network dynamics, Physica D, 111, 1998, 288–294.
Gedeon, T., Structure and dynamics of artificial neural networks, in Fields Institute Communications: Differential Equations with Applications to Biology, (S. Ruan, G. Wolkowicz, and J. Wu, eds.), Vol. 21, 1999, 217–224.
Giannakopoulos, F. and Zapp, A., Local and global Hopf bifurcation in a scalar delay differential equation, 1998, preprint.
Gilli, Strange attractors in delayed cellular neural networks, IEEE Trans. Circuits Syst., 40, 1993, 849–853.
Gopalsamy, K. and He, X.-Z., Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76, 1994, 344–358.
Hebb, D.O., The Organization of Behavior, John Wiley & Sons, New York, 1949.
Herz, A.V.M., Salzer, B., Kühn, R., and van Hemmen, J.L., Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets, Biol. Cybem., 60, 1989, 457–467.
Hirsch, M., Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383, 1988, 1–53.
Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci., 81, 1984, 3088–3092.
Huang, L. and Wu, J., Joint effects of the threshold and synaptic delay on dynamics of artificial neural networks with McCulloch-Pitts nonlinearity, 1999a, preprint.
Huang, L. and Wu, J., The role of threshold in preventing delay-induced oscillations of frustrated neural networks with McCulloch-Pitts nonlinearity, 1999b, preprint.
Krisztin, T. and Walther, H.-O., Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynamics and Differential Equations, 1999, to appear.
Krisztin, T., Walther, H.-O., and Wu, J., Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, the Fields Institute Monographs Series 11, Amer. Math. Soc., Rhode Island, 1999.
Levine, D.S., Introduction to Neural and Cognitive Modelling, Lawrence Erlbaum Associate, Inc., New Jersey, 1991.
Marcus, C.M. and Westervelt, R.M., Basins of attraction for electronic neural networks, in Neural Information Processing Systems, (D.Z. Anderson, ed.) American Institute of Physics, New York, 1988, 524–533.
Marcus, C.M. and Westervelt, R.M., Stability of analog neural networks with delay, Physical Review A, 39, 1989, 347–359.
May, R.M. and Leonard, W.J., Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29, 1975, 243–253.
Müller, B. and Reinhardt, J., Neural Networks, An Introduction, Springer-Verlag, Berlin, 1991.
Olien, L. and Bélair, J., Bifurcation’s, stability, and monotonically properties of a delayed neural network model, Physica D, 102, 1997, 349–363.
Pakdaman, K., Grotta-Ragazzo, C.P., Malta, K., and Vibert, J.-F., Delay-induced transient oscillations in a two-neuron network, Resends, 1997, 45–54.
Pakdaman, K., Grotta-Ragazzo, C.P., Malta, C.P., Rain, O., and Vibert, J.-F., Effect of delay on the boundary of the basin of attraction in a system of two neurons, Neural Networks, 11, 1998, 509–519.
Ruan, S. and Wei, J., Periodic solutions of planar systems with two delays, Proc. Royal Soc. Edinburgh (A), 129, 1999, 1017–1032.
Smith, H.L., Monotone semiflows generated by functional differential equations, J. Differential Equations, 66, 1987, 420–442.
Smith, H.L., Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions, Neural Networks, 4, 1991, 41–46.
van den Driessche, P. and Zou, X.F., Global attractivity in delayed Hopfield neural networks models, SIAM J. Appl. Math., 58, 1998, 1878–1890.
Walther, H.-O., The singularities of an attractor of a delay differential equation, Functional Differential Equations, 5, 1998, 513–548.
Wu, J., Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350, 1998, 4799–4838.
Wu, J., Introduction to Neural Dynamics and Signal Delays, 1999, preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Wu, J. (2001). Dynamics of Neural Networks with Delay: Attractors and Content-Addressable Memory. In: Vajravelu, K. (eds) Differential Equations and Nonlinear Mechanics. Mathematics and Its Applications, vol 528. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0277-3_28
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0277-3_28
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-7923-6867-0
Online ISBN: 978-1-4613-0277-3
eBook Packages: Springer Book Archive