Stability and Bifurcation in a Hopfield Neuron Model with Delays
- 1 Department of Mathematics, China Agricultural University, Beijing, China
- 2 Department of Mathematics, Bristol University, Bristol, UK
Abstract
The Hopfield system is an artificial neuron model that can be applied to neuron memory and information processing. Like synaptic connections of neurons, the inhibition or excitable feedback often incorporate delay effects, which are either discrete or distributed time delays. With delays varying, Hopf bifurcation of distributed time delay is investigated and stability regime is partitioned by Hopf curves on the parameter plane. Lyapunov-Schmidt reduction skills combined with center manifold theory are applied to discuss the stability of bifurcating periodical solutions arising from Hopf points. In addition, DDE-Biftool software significantly provides the numerical computation of stability analysis of periodical solutions appearing in discrete time delay Hopfield system. The period-doubling bifurcation of periodical solutions, which form P2 circles and P4 circles, respectively, by continuous periodical solutions with varying free parameters, is discussed.
- Hopfield, J.J. (1982) Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proceedings of the National Academy of Sciences , 79, 2554-2558. https://doi.org/10.1073/pnas.79.8.2554
- Hopfield, J.J. (1984) Neurons with Graded Response Have Collective Computational Properties Like Those of Two-State Neurons. Proceedings of the National Academy of Sciences , 81, 3088-3092. https://doi.org/10.1073/pnas.81.10.3088
- Korner, E., Kupper, R., Rahman, M.K.M. and Shkuro, Y. (2007) Neurocomputing Research Developments. Nova Science Publishers.
- Skarda, C.A. and Freeman, W.J. (1987) How Brains Make Chaos in Order to Make Sense of the World. Behavioral and Brain Sciences , 10, 161-173. https://doi.org/10.1017/s0140525x00047336
- Skarda, C.A. and Freeman, W.J. (1990) Chaos and the New Science of the Brain. Neuroscience , 1, 275-285.
- Kundu, A., Das, P. and Roy, A.B. (2013) Complex Dynamics of a Four Neuron Network Model Having a Pair of Short-Cut Connections with Multiple Delays. Nonlinear Dynamics , 72, 643-662. https://doi.org/10.1007/s11071-012-0742-2
- Das, A., Das, P. and Roy, A.B. (2002) Chaos in a Three-Dimensional General Model of Neural Network. International Journal of Bifurcation and Chaos , 12, 2271-2281. https://doi.org/10.1142/s0218127402005820
- Lewis, J.E. and Glass, L. (1991) Steady States, Limit Cycles, and Chaos in Models of Complex Biological Networks. International Journal of Bifurcation and Chaos , 1, 477-483. https://doi.org/10.1142/s0218127491000373
- Huang, Y. and Yang, X. (2006) Hyperchaos and Bifurcation in a New Class of Four-Dimensional Hopfield Neural Networks. Neurocomputing , 69, 1787-1795. https://doi.org/10.1016/j.neucom.2005.11.001
- Yang, X. and Huang, Y. (2007) Chaos and Two-Tori in a New Family of 4-CNNS. International Journal of Bifurcation and Chaos , 17, 953-963. https://doi.org/10.1142/s0218127407017677
- Guckenheimer, J. and Holmes, P. (1997) Nonlinear Oscillation, Dynamical Systems and Bifurcations of Vector Fields. Springer.
- Rech, P.C. (2011) Dynamics of a Neuron Model in Different Two-Dimensional Parameter-Spaces. Physics Letters A , 375, 1461-1464. https://doi.org/10.1016/j.physleta.2011.02.037
- Luonan Chen, and Aihara, K. (1999) Global Searching Ability of Chaotic Neural Networks. IEEE Transactions on Circuits and Systems I : Fundamental Theory and Applications , 46, 974-993. https://doi.org/10.1109/81.780378