Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings
We study a Hopfield-type network that consists of a pair of one-way rings each with three neurons and two-way coupling between the rings. The rings have symmetric group Γ=Z3×Z2, which means the global symmetry Z2 and internal symmetry Z3. We discuss the spatiotemporal patterns of bifurcating periodi...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2014/971520 |
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| author | Baodong Zheng Haidong Yin |
| author_facet | Baodong Zheng Haidong Yin |
| author_sort | Baodong Zheng |
| collection | DOAJ |
| description | We study a Hopfield-type network that consists of a pair of one-way rings each with three neurons and two-way coupling between the rings. The rings have symmetric group Γ=Z3×Z2, which means the global symmetry Z2 and internal symmetry Z3. We discuss the spatiotemporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatiotemporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling; that is, different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase, antiphase, T/3, 2T/3, 4T/3, 5T/6, or 7T/6 periods out of phase depending on the delay. Some numerical simulations support our analysis results. |
| format | Article |
| id | doaj-art-e94d25250d4d4c33a202585f27daa4a6 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-e94d25250d4d4c33a202585f27daa4a62025-02-03T00:59:08ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2014-01-01201410.1155/2014/971520971520Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network RingsBaodong Zheng0Haidong Yin1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaWe study a Hopfield-type network that consists of a pair of one-way rings each with three neurons and two-way coupling between the rings. The rings have symmetric group Γ=Z3×Z2, which means the global symmetry Z2 and internal symmetry Z3. We discuss the spatiotemporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatiotemporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling; that is, different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase, antiphase, T/3, 2T/3, 4T/3, 5T/6, or 7T/6 periods out of phase depending on the delay. Some numerical simulations support our analysis results.http://dx.doi.org/10.1155/2014/971520 |
| spellingShingle | Baodong Zheng Haidong Yin Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings Discrete Dynamics in Nature and Society |
| title | Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings |
| title_full | Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings |
| title_fullStr | Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings |
| title_full_unstemmed | Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings |
| title_short | Z2×Z3 Equivariant Bifurcation in Coupled Two Neural Network Rings |
| title_sort | z2 z3 equivariant bifurcation in coupled two neural network rings |
| url | http://dx.doi.org/10.1155/2014/971520 |
| work_keys_str_mv | AT baodongzheng z2z3equivariantbifurcationincoupledtwoneuralnetworkrings AT haidongyin z2z3equivariantbifurcationincoupledtwoneuralnetworkrings |