Continuous Attractors of Lotka-Volterra Recurrent Neural Networks

  • Haixian Zhang
  • Jiali Yu
  • Zhang Yi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5768)


Continuous attractor neural network (CANN) models have been studied in conjunction with many diverse brain functions including local cortical processing, working memory, and spatial representation. There is good evidence for continuous stimuli, such as orientation, moving direction, and the spatial location of objects could be encoded as continuous attractors in neural networks. Although their wide applications for the information processing in the brain, representation and stability analysis of continuous attractors in non-linear recurrent neural networks (RNNs) have been reported very little so far. This paper studies the continuous attractors of Lotka-Volterra (LV) recurrent neural networks. Conditions are given to insure the network has continuous attractors. Representation of continuous attractor is obtained under the conditions. Simulations are employed to illustrate the theory.


Continuous attractors Recurrent neural networks Convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amari, S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics 27, 77–87 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben-Yishai, R., Bar-Or, R.L., Sompolinsky, H.: Theory of orientation tuning in visual cortex. Proc. Nat. Acad. Sci. USA 92, 3844–3848 (1995)CrossRefGoogle Scholar
  3. 3.
    Compte, A., Brunel, N., Goldman-Rakic, P.S., Wang, X.J.: Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model. Cereb. Cortex 10, 910–923 (2000)CrossRefGoogle Scholar
  4. 4.
    Yu, J., Zhang, Y., Zhang, L.: Representation of continuous attractors of recurrent neural networks. IEEE Transactions on Neural Networks 20(2), 368–372 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Salinas, E.: Background synaptic activity as a switch between dynamical states in network. Neural computation 15, 1439–1475 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Seung, H.S.: How the brain keeps the eyes still. J. Neurobiology 93, 13339–13344 (1996)Google Scholar
  7. 7.
    Seung, H.S.: Continuous attractors and oculomotor control. Neural Networks 11, 1253–1258 (1998)CrossRefGoogle Scholar
  8. 8.
    Seung, H.S.: Learning continuous attractors in recurrent networks. Adv. Neural Info. Proc. Syst. 10, 654–660 (1998)Google Scholar
  9. 9.
    Seung, H.S., Lee, D.D.: The manifold ways of perception. Science 290, 2268–2269 (2000)CrossRefGoogle Scholar
  10. 10.
    Samsonovich, A., McNaughton, B.L.: Path integration and congnitive mapping in a continuous attractor neural network model. J. Neurosci. 7, 5900 (1997)Google Scholar
  11. 11.
    Stringer, S.M., Trppenberg, T.P., Rolls, E., Aranjo, I.: Self organizing continuous attractor networks and path integration: One-dimensional models of head direction cell. Network: Computation in Neural Systems 13, 217–242 (2002)CrossRefGoogle Scholar
  12. 12.
    Trappenberg, T.P., Standage, D.I.: Self-organising continuous attractor networks with multiple activity packets, and the representation of space. Neural Networks 17, 5–27 (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Tsodyks, M., Sejnowski, T.: Associative memory and hippocampal place cells. International journal of neural systems 6, 81–86 (1995)Google Scholar
  14. 14.
    Wu, S., Amari, S., Nakahara, H.: Population coding and decoding in a neural fields: a computational study. Neural Computation 14, 999–1026 (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Wu, S.: Shun-Ichi Amari, Computing with continuous attractors: Stability and online aspects. Neural Computation 17, 2215–2239 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wu, S., Hamaguchi, K., Amari, S.: Dynamics and computation of continuous attractors. Neural Computation 20, 994–1025 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang, K.C.: Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: A Theory. The journal of neuroscience 16(6), 2110–2126 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Haixian Zhang
    • 1
  • Jiali Yu
    • 1
  • Zhang Yi
    • 2
  1. 1.School of Computer Science and EngineeringUniversity of Electronic Science and Technology of ChinaChengduP.R. China
  2. 2.Machine Intelligence Laboratory, School of Computer ScienceSichuan UniversityChengduP.R. China

Personalised recommendations