Line attractors of coupled ring neural networks with block circulant weight matrix

References

• S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet. 27(2) (1977), pp. 77–87. doi: 10.1007/BF00337259
   PubMed Web of Science ®Google Scholar
• R. Ben-Yishai, R.L. Bar-Or, and H. Sompolinsky, Theory of orientation tuning in visual cortex, Proc. Natl. Acad. Sci. USA 92 (1995), pp. 3844–3848. doi: 10.1073/pnas.92.9.3844
   PubMed Web of Science ®Google Scholar
• H.R. Hahnloser, R. Sarpeshkar, M.A. Mahowald, R.J. Douglas, and H.S. Seung, Digital selection and analogue amplication coexist in a cortex-inspired silicon circuit, Nature 405 (2000), pp. 947–951. doi: 10.1038/35016072
   PubMed Web of Science ®Google Scholar
• A. Koulakov, S. Raghavachari, A. Kepecs, and J.E. Lisman, Model for a robust neural integrator, Nat. Neurosci. 5(8) (2002), pp. 775–782. doi: 10.1038/nn893
   PubMed Web of Science ®Google Scholar
• D.D. Lee, B.Y. Reis, H.S. Seung, and D.W. Tank, Nonlinear network models of the oculomotor integrator, in Computational Neuroscience: Trends in Research, James M. Bower, ed., Plenum Press, New York, 1997, 371–377.
   Google Scholar
• C.K. Machens and C.D. Brody, Design of continuous attractor networks with monotonic tuning using a symmetry principle, Neural Comput. 20 (2008), pp. 452–485. doi: 10.1162/neco.2007.07-06-297
   PubMed Web of Science ®Google Scholar
• P. Miller, Analysis of spike statistics in neuronal systems with continuous attractors or multiple, discrete attractor states, Neural Comput. 18 (2006), pp. 1268–1317. doi: 10.1162/neco.2006.18.6.1268
   PubMed Web of Science ®Google Scholar
• T. Nirmaier, C.A. Diez, and J.F. Bille, High-speed CMOS wavefront sensor with resistive-ring networks of winner-take-all circuits, IEEE J. Solid-st Circ. 40(11) (2005), pp. 2315–2322. doi: 10.1109/JSSC.2005.857350
   Web of Science ®Google Scholar
• A. Pouget, P. Dayan, and R. Zemel, Information processing with population codes, Nature Rev. Neurosci. 1 (2000), pp. 125–132. doi: 10.1038/35039062
   PubMed Web of Science ®Google Scholar
• D.A. Robinson, Integrating with neurons, Annu. Rev. Neurosci. 12 (1989), pp. 33–45. doi: 10.1146/annurev.ne.12.030189.000341
   PubMed Web of Science ®Google Scholar
• A. Samsonovich and B.L. McNaughton, Path integration and cognitive mapping in a continuous attractor neural network model, J. Neurosci. 17 (1997), pp. 5900–5920. doi: 10.1523/JNEUROSCI.17-15-05900.1997
   PubMed Web of Science ®Google Scholar
• H.S. Seung, How the brain keeps the eyes still, Proc. Natl. Acad. Sci. USA 93 (1996), pp. 13339–13344. doi: 10.1073/pnas.93.23.13339
   PubMed Web of Science ®Google Scholar
• H.S. Seung, Continouous attractors and oculomotor control, Neural Networks 11 (1998), pp. 1253–1258. doi: 10.1016/S0893-6080(98)00064-1
   PubMed Web of Science ®Google Scholar
• H.S. Seung and D.D. Lee, The manifold ways of perception, Science 290 (2000), pp. 2268–2269. doi: 10.1126/science.290.5500.2268
   PubMed Web of Science ®Google Scholar
• W.E. Skaggs, J.J. Knierim, H.S. Kudrimoti, and B.L. Mcnaughton, A model of the neural basis of the rat's sense of direction, Neural Comput. 7 (7) (1995), pp. 173–180.
   Google Scholar
• S.M. Stringer, T.P. Trappenberg, E.T. Rolls, and I.E.T. de Araujo, Self-organizing continuous attractor networks and path integration: One-dimensional models of head direction cells, Network Comput. Neural Syst. 13 (2002), pp.217–242. doi: 10.1080/net.13.2.217.242
   PubMed Web of Science ®Google Scholar
• S.M. Stringer, E.T. Rolls, T.P. Trappenberg, and I.E.T. Araujo, Self-organizing continuous attractor networks and motor function, Neural Networks 16 (2003), pp. 161–182; 217–242. doi: 10.1016/S0893-6080(02)00237-X
   PubMed Web of Science ®Google Scholar
• K. Wimmer, D.Q. Nykamp, C. Constantinidis, and A. Compte, Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory, Nat. Neurosci. 17(3) (2014), pp. 431–439. doi: 10.1038/nn.3645
   PubMed Web of Science ®Google Scholar
• S. Wu, K.Y. Wong, C.C. Fung, Y Mi, and W. Zhang, Continuous attractor neural networks: Candidate of a canonical model for neural information representation, F1000research. 5(16) (2016), pp. 209–226.
   Google Scholar
• X. Xie, H.R. Hahnloser, and H.S. Seung, Selectively grouping neurons in recurrent networks of lateral inhibition, Neural Comput. 14 (2002), pp. 2627–2646. doi: 10.1162/089976602760408008
   PubMed Web of Science ®Google Scholar
• Z. Yi, L. Zhang, J. Yu, and K.K. Tan, Permitted and forbidden sets in discrete-time linear threshold recurrent neural networks, IEEE Trans. Neural Netw. 20 (2009), pp. 952–963. doi: 10.1109/TNN.2009.2014373
   PubMed Web of Science ®Google Scholar
• K. Yoon, M.A. Buice, C. Barry, R. Hayman, N. Burgess, and I.R. Fiete, Specific evidence of low-dimensional continuous attractor dynamics in grid cells, Nat. Neurosci. 16 (8) (2013), pp. 1077–1084. doi: 10.1038/nn.3450
   PubMed Web of Science ®Google Scholar
• J. Yu, Z. Yi, and L. Zhang, Representations of continuous attractors of recurrent neural networks, IEEE Trans. Neural Netw. 20 (2009), pp. 368–372. doi: 10.1109/TNN.2008.2010771
   PubMed Web of Science ®Google Scholar
• J. Yu, H. Tang, and H. Li, Dynamics Analysis of A Population Decoding Model, IEEE Trans. Neural Netw. Learn. Syst. 24(3) (2013), pp. 498–503. doi: 10.1109/TNNLS.2012.2236684
   PubMed Web of Science ®Google Scholar
• J. Yu, H. Mao, and Z. Yi, Parameter as a switch between dynamical states of a network in population decoding, IEEE Trans. Neural Netw. Learn. Syst. 28(4) (2017), pp. 911–916. doi: 10.1109/TNNLS.2015.2485263
   PubMed Web of Science ®Google Scholar
• K.C. Zhang, Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: A theory, J. Neurosci. 16 (1996), pp. 2112–2126. doi: 10.1523/JNEUROSCI.16-06-02112.1996
   PubMed Web of Science ®Google Scholar
• W. Zhang and S. Wu, Reciprocally coupled local estimator implement Bayesian information integration distributively, Advances in Neural Information Processing System, 2013, pp. 19–27.
   Google Scholar
• L. Zhao, S. Liao, Y. Wang, Z. Li, J. Tang, V. Pan, and B. Yuan, Theoretical properties for neural networks with weight matrices of low displacement rank, preprint (2017). Available at arXiv: 1703.00144.
   Google Scholar
• L. Zou, H. Tang, K.C. Tan, and W. Zhang, Analysis of continuous attractors for 2-D linear threshold neural networks, IEEE Trans. Neural Netw. 20 (2009), pp. 175–180. doi: 10.1109/TNN.2008.2009535
   PubMed Web of Science ®Google Scholar