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Chaotic itinerancy in the oscillator neural network without Lyapunov functions
1.U. Riedel, R. Kuhn, and J. L. van Hemmen, Phys. Rev. A 38, 1105 (1988).
2.T. Fukai and M. Shiino, Phys. Rev. Lett. 64, 1465 (1990).
3.S. N. Laughton and A. C. C. Coolen, J. Phys. A 27, 8011 (1994).
4.D. Bollé and B. Vinck, Physica A 223, 293 (1996).
5.L. Wang, E. E. Pichler, and J. Ross, Proc. Natl. Acad. Sci. U.S.A. 87, 9467 (1990);
5.L. Wang and J. Ross, Phys. Rev. A 44, 2259 (1991).
6.D. R. C. Dominguez, Phys. Rev. E 54, 4066 (1996).
7.D. Caroppo, M. Mannarelli, G. Nardulli, and S. Stramaglia, Phys. Rev. E 60, 2186 (1999).
8.M. S. Mainieri and R. Erichsen, Jr., Physica A 311, 581 (2002).
9.K. Katayama and T. Horiguchi, J. Phys. Soc. Jpn. 79, 1300 (2001).
10.M. Kawamura, R. Tokunaga, and M. Okada, Europhys. Lett. 62, 657 (2003).
11.H. Sompolinsky, A. Crisanti, and H. J. Sommers, Phys. Rev. Lett. 61, 259 (1988).
12.K. E. Kürten, Phys. Lett. A 129, 157 (1988).
13.H. Gutfreund, J. D. Reger, and A. P. Young, J. Phys. A 21, 2755 (1988).
14.I. Tsuda, Neural Networks 5, 313 (1992);
14.K. Kaneko and I. Tsuda, Complex Systems: Chaos and Beyond (Springer, Berlin, 2001).
15.M. Adachi and K. Aihara, Neural Networks 10, 83 (1997).
16.S. Nara, P. Davis, M. Kawachi, and H. Totsuji, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1205 (1995).
17.K. Kaneko, Physica D 41, 137 (1990).
18.K. Kaneko, Physica D 54, 5 (1991).
19.K. Ikeda, K. Otsuka, and K. Matsumoto, Prog. Theor. Phys. Suppl. 99, 295 (1989).
20.F. T. Arrecchi, Chaos 1, 357 (1991).
21.T. Sameshima, K. Fukushima, and T. Yamada, Physica D 150, 104 (2001).
22.I. Z. Kiss and J. L. Hudson, Chaos 13, 999 (2003).
23.“FOCUS ISSUE: Chaotic Itinerancy,” edited by K. Kaneko and I. Tsuda, Chaos 13, 926 (2003).
24.D. J. Amit, H. Gutfreund, and H. Sompolinsky, Phys. Rev. Lett. 55, 1530 (1985);
24.D. J. Amit, H. Gutfreund, and H. Sompolinsky, Ann. Phys. (N.Y.) 173, 30 (1987).
25.D. J. Amit, Modeling Brain Functions (Cambridge University Press, Cambridge, England, 1989).
26.S. Uchiyama and H. Fujisaka, J. Phys. A 32, 4623 (1999).
27.S. Uchiyama and H. Fujisaka, Phys. Rev. E 65, 061912 (2002).
28.S. Amari and K. Maginu, Neural Networks 1, 63 (1988).
29.M. Okada, Neural Networks 8, 833 (1995).
30.H. Nishimori and I. Opriş, Neural Networks 6, 1061 (1993).
31.E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, England, 1994).
32.T. Aoyagi, Phys. Rev. Lett. 74, 4075 (1995).
33.Note that these two couplings are the same when we consider storing orthogonal random patterns (fully uncorrelated patterns) because A [Eq. (7)] reduces to a unit matrix at that time.
34.The followings are well-known in the case of Hopfield model with the pseudoinverse couplings: (i) The maximum number of storable patterns increases enormously, (ii) the retrieval quality (the relaxed value of at the retrieval–nonretrieval transition point) is improved, and furthermore, (iii) the basin of attraction of the stored patterns get much deeper, while the extent of a basin becomes infinitesimal near the retrieval–nonretrieval transition boundary (Refs. 454647).
35.In contrast to the result of Ref. 27, we did not come across the numerical observation of the sinusoidally oscillating retrieval solutions with the oscillation amplitude greater than 0.01. The bifurcation theory tells us the existence of the sinusoidal solution parameterized between the fixed-point-type retrieval and the chaotic retrieval solutions. However, we stored a number of correlated patterns with the help of the pseudoinverse method. That must negatively work upon the appearance.
36.We embedded the attractor into the 24 dimension space at most, and evaluated its correlation dimension by the Grassberger–Procaccia algorithm, using data points of But it is almost impossible to prepare the sufficient number of data points for the accurate evaluation because the numerical simulations of the system size needs the extremely enormous computation time. Hence, one should regard as a rough estimation.
37.We set α equal to 0.02 because, in our preliminary numerical observations with the case of seems to give the widest parameter range of for the chaotic retrieval. Hence the fine tuning of for CI, which is necessarily blurred by the finite size effect of is expected to be most easily done in the case of If one chooses α slightly larger than 0.02, the corresponding bifurcation diagram would result in more blurred one.
38.We applied different rules for each regions: (a) To characterize the fixed-point-type retrieval solutions most efficiently, the values are sampled from 10 runs and time is restricted to In this region, it is appropriate to pay attentions to the initial condition dependency of the attractor rather than to the long-term behavior; (b) Ten runs and for the best characterization of the chaotic retrieval solutions; (c) One run and for the chaotic itinerancy solutions, in which we need extremely long-term observation to attain its ergodicity. (d) One run and for the nonretrieval solutions. Here its ergodicity is realized in shorter time scale.
39.It is natural to determine the critical values of from the results of the zeroth layer, because the zeroth layer governs the whole behavior of layered dynamics, as mentioned in Sec. II B. The influence of comes only from the zeroth layer. (The other layers have where we have defined as on the layer.)
40.T. Aoyagi and K. Kitano, Neural Comput. 10, 1527 (1998).
41.J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994).
42.I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002).
43.Here the basin size is given by a characteristic size of the attracting domain in which all the -dimensional orbit finally settles down to a concerning attractor. And the size of a ruin means the characteristic size of the destabilized attractor which still contains a dense part of orbit as well as the (pre-destabilized) attractor does contain it. Thus, the basin size is always larger than the size of an attractor ruin.
44.The appearance of the chaotic retrieval solutions has been numerically reported also in the study of the auto-associative oscillator neural networks accompanied with asymmetric distributions of native frequencies (Ref. 48). This system, which is described with coupled differential equations, seems to be mathematically tractable. So future studies on it will attract much attention.
45.I. Kantor and H. Sompolinsky, Phys. Rev. A 35, 380 (1987).
46.I. Kantor, Phys. Rev. A 40, 2611 (1989).
47.B. M. Forrest, J. Phys. A 21, 245 (1988).
48.S. Kawaguchi, Prog. Theor. Phys. 107, 839 (2002).
49.See EPAPS Document No. E-CHAOEH-14-041403 for movies of the retrieval solutions and images of stored patterns. A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information.
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