%0 Book Section %1 statphys23_0331 %A Metz, F.L. %A Theumann, W.K. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K cycles dynamics network neural statphys23 synchronous topic-9 %T Diversity of period-two cycles in the synchronous dynamics of a recurrent neural network with symmetric interactions %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=331 %X Information processing in synchronous neural network models has been of interest over some time. A replica symmetric analysis of Little's model indicates the presence of period-two cycles in a low-temperature paramagnetic phase for negative values of the self-interaction between units 1. There has also been interest in the dynamics and the stationary states of attractor neural networks that process sequences of patterns. Fixed-point solutions and cycles of period two were found recently for symmetric sequence processing in a feed-forward layered network model 2. We study here the synchronous dynamics of a recurrent neural network with a Hebbian plus symmetric sequential learning rule. The model has been of interest over some time in order to explain experimental recordings in the cortex of monkeys. An exact generating functional approach (GFA) 3 is used here, in the pattern saturation limit, to obtain first the stationary fixed-point solutions that characterize various equilibrium phases. The further behavior of the network is then analyzed through a numerical simulation procedure for the effective single neuron dynamics obtained exactly using the GFA 4. The simulation is first tested on the equilibrium results for a dominant Hebbian term and the procedure is then used to look for cyclic solutions that appear depending on the self-interaction $J_0$. For $J_0>0$, only a fraction of spins flip at each time step, as in Little's model. For $J_0<0$, a variety of period-two cyclic solutions are obtained in this work with macroscopic parameters that change sign at each time step, and characterize a retrieval, a spin-glass or a high-$T$ paramagnetic phase. All spins flip at every time step only for $T 0$ and sufficiently large negative values of $J_0$. For $J_0=0$ and a dominant sequence processing term, the network exhibits a cyclic behavior with oscillating positive overlaps and spin-glass order parameter. The results shown here provide a new insight into the nature of the cyclic states in the parallel dynamics of recurrent neural networks, which could be of interest for information processing. Further features of sequence processing and open questions will be pointed out. 1) J. F. Fontanari, J. Phys. France 49, 13 (1988). \\ 2) F. L. Metz and W. K. Theumann, Phys. Rev. E 75, in press.\\ 3) A. C. C. Coolen, in Handbook of Biological Physics IV: Neuro-Informatics and Neural Modeling, edited by F. Moss and S. Gielen (Elsevier, Amsterdam, 2001), p. 619.\\ 4) H. Eissfeller and M. Opper, Phys. Rev. Lett. 68, 2094 (1992).\\ This work was partially supported by the Brazilian agencies CNPq and FAPERGS.

@incollection{statphys23_0331, abstract = {Information processing in synchronous neural network models has been of interest over some time. A replica symmetric analysis of Little's model indicates the presence of period-two cycles in a low-temperature paramagnetic phase for negative values of the self-interaction between units [1]. There has also been interest in the dynamics and the stationary states of attractor neural networks that process sequences of patterns. Fixed-point solutions and cycles of period two were found recently for symmetric sequence processing in a feed-forward layered network model [2]. We study here the synchronous dynamics of a recurrent neural network with a Hebbian plus symmetric sequential learning rule. The model has been of interest over some time in order to explain experimental recordings in the cortex of monkeys. An exact generating functional approach (GFA) [3] is used here, in the pattern saturation limit, to obtain first the stationary fixed-point solutions that characterize various equilibrium phases. The further behavior of the network is then analyzed through a numerical simulation procedure for the effective single neuron dynamics obtained exactly using the GFA [4]. The simulation is first tested on the equilibrium results for a dominant Hebbian term and the procedure is then used to look for cyclic solutions that appear depending on the self-interaction $J_{0}$. For $J_{0}>0$, only a fraction of spins flip at each time step, as in Little's model. For $J_{0}<0$, a variety of period-two cyclic solutions are obtained in this work with macroscopic parameters that change sign at each time step, and characterize a retrieval, a spin-glass or a high-$T$ paramagnetic phase. All spins flip at every time step only for $T \rightarrow 0$ and sufficiently large negative values of $J_{0}$. For $J_{0}=0$ and a dominant sequence processing term, the network exhibits a cyclic behavior with oscillating positive overlaps and spin-glass order parameter. The results shown here provide a new insight into the nature of the cyclic states in the parallel dynamics of recurrent neural networks, which could be of interest for information processing. Further features of sequence processing and open questions will be pointed out. 1) J. F. Fontanari, J. Phys. France {\bf 49}, 13 (1988). \\ 2) F. L. Metz and W. K. Theumann, Phys. Rev. E {\bf 75}, in press.\\ 3) A. C. C. Coolen, in Handbook of Biological Physics IV: Neuro-Informatics and Neural Modeling, edited by F. Moss and S. Gielen (Elsevier, Amsterdam, 2001), p. 619.\\ 4) H. Eissfeller and M. Opper, Phys. Rev. Lett. {\bf 68}, 2094 (1992).\\ This work was partially supported by the Brazilian agencies CNPq and FAPERGS.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Metz, F.L. and Theumann, W.K.}, biburl = {https://www.bibsonomy.org/bibtex/2dff7a0a489c719f511039e7555d132b0/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {43c92eae97e6a4aafae9c4003d2cdc01}, intrahash = {dff7a0a489c719f511039e7555d132b0}, keywords = {cycles dynamics network neural statphys23 synchronous topic-9}, month = {9-13 July}, timestamp = {2007-06-20T10:16:17.000+0200}, title = {Diversity of period-two cycles in the synchronous dynamics of a recurrent neural network with symmetric interactions}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=331}, year = 2007 }