中国物理B ›› 2009, Vol. 18 ›› Issue (1): 98-105.doi: 10.1088/1674-1056/18/1/017

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Development and transition of spiral wave in the coupled Hindmarsh--Rose neurons in two-dimensional space

刘勇1, 李世荣2, 马军3, 应和平4   

  1. (1)Department of Mathematics, Yancheng Teachers University, Yancheng 224009, China; (2)School of Science, Lanzhou University of Technology, Lanzhou 730050, China; (3)School of Science, Lanzhou University of Technology, Lanzhou 730050, China;Department of Physics, Central China Normal University, Wuhan 430079, China; (4)Zhejiang Institute of Modern Physics in the Zhejiang University, Hangzhou 310027, China
  • 收稿日期:2008-06-12 修回日期:2008-08-30 出版日期:2009-01-20 发布日期:2009-01-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10747005, 10572056 and 10774130), and the Natural Science Foundation of Lanzhou University, China (Grant No Q200706).

Development and transition of spiral wave in the coupled Hindmarsh--Rose neurons in two-dimensional space

Ma Jun(马军)a)b)†, Ying He-Ping(应和平)c), Liu Yong(刘勇)d), and Li Shi-Rong(李世荣)a)   

  1. a School of Science, Lanzhou University of Technology, Lanzhou 730050, China; b Department of Physics, Central China Normal University, Wuhan 430079, China; c Zhejiang Institute of Modern Physics in the Zhejiang University, Hangzhou 310027, China; d Department of Mathematics, Yancheng Teachers University, Yancheng 224009, China
  • Received:2008-06-12 Revised:2008-08-30 Online:2009-01-20 Published:2009-01-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10747005, 10572056 and 10774130), and the Natural Science Foundation of Lanzhou University, China (Grant No Q200706).

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摘要： The dynamics and the transition of spiral waves in the coupled Hindmarsh--Rose (H--R) neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D’, is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons (or sites) and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.

关键词: spiral wave, Hindmarsh--Rose neurons, instability

Abstract: The dynamics and the transition of spiral waves in the coupled Hindmarsh--Rose (H--R) neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D’, is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons (or sites) and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.

Key words: spiral wave, Hindmarsh--Rose neurons, instability

中图分类号:  (Neural networks and synaptic communication)

  • 87.18.Sn
02.30.Oz (Bifurcation theory) 05.45.-a (Nonlinear dynamics and chaos)